From 5520c1dc3d919dc221d3f9433b9baedbea384466 Mon Sep 17 00:00:00 2001 From: TGuillerme Date: Sat, 9 May 2020 15:20:45 +0100 Subject: [PATCH] release version --- inst/EcoEvol/CoverLetter.fls | 101 - inst/EcoEvol/CoverLetter.tex | 50 - inst/EcoEvol/shiftingspace_resubmit.tex | 2211 ----------------- inst/EcoEvol/shiftingspace_resubmit_clean.tex | 1870 -------------- inst/EcoEvol/shiftingspace_submit.tex | 1745 ------------- inst/MEE/CoverLetter.tex | 50 - inst/MEE/MEE_sub.md | 31 - inst/MEE/review.Rmd | 350 --- inst/MEE/review.md | 566 ----- inst/defaults | 1075 -------- inst/mee.csl | 94 - inst/methods-in-ecology-and-evolution.csl | 626 ----- inst/shiftingspace.Rmd | 75 +- inst/shiftingspace.tex | 1710 ------------- inst/shiftingspace_bkp.Rmd | 975 -------- inst/shiftingspace_resubmit.tex | 1753 ------------- ...hiftingspace_supplementary_all_results.Rmd | 6 +- inst/shiftingspace_supplementary_sampling.Rmd | 172 -- inst/shiftingspace_text_v.tex | 1622 ------------ 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right=2.5cm, top=1cm, bottom=1cm]{geometry} -\usepackage{hyperref} -\usepackage[osf]{mathpazo} -\signature{Thomas Guillerme \\ (on behalf of my co-authors)} -\address{Department of Animal and Plant Sciences\\The University of Sheffield\\Sheffield S10 2TN, UK\\guillert@tcd.ie} -\longindentation=0pt -\begin{document} - -\begin{letter}{} -\opening{Dear Editors,} - -Using patterns in multidimensional spaces to study biological processes is now a common toolkit in ecology and evolution. -In such analysis, researchers typically use matrices where traits (or transformed traits) are columns (e.g. anatomical measurements, community compositions) and observations are rows (e.g. specimens, field sites, etc.). -These matrices are commonly referred to as morphospaces in evolution or trait-space in ecology. -It is then possible to look at how observations or groups of observations occupy the trait space to understand biological processes. -For example, if plant community A occupies more space trait-space than community B, the former could be more diverse; -or if the morphospace of a group of organisms increases after colonisation of an island, it could be experiencing an adaptive radiation. -However, surprisingly very little work has been done on characterising what should be measured in these multidimensional analysis: what should be measured in a trait-space to accurately capture the pattern of difference between two communities or how trait-space occupancy changes through time? -Furthermore, the parallel between these analysis in ecology and evolution has never been clearly acknowledged (to our knowledge). - -In our research article, entitled ``Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?'', we provide the first interdisciplinary review of 25 space occupancy measurements that uses the broad classification of measurements into size, density and position to capture pattern changes in trait space. -We assess the behaviour of measurements using simulations and a selection of interdisciplinary empirical datasets; these cover a wide range of potential data types and evolutionary/ecological questions. -We also introduce a tool for Measuring Occupancy in Multidimensional Space ([`moms`](https://github.com/TGuillerme/moms)), which is a user-friendly open-source graphical interface tool to allow the tailored testing of metric behaviour for any use case. -This will allow researchers to comprehensively assess the properties of their trait-space and measurements associated with their specific biological question. - -Furthermore, we are are convinced that open data and reproducible papers are the key part of the future of academia. -Therefore, this entire paper and its supplementary is entirely reproducible and based on open access dataset. -In fact, the paper is wrapped in a \texttt{R} package format and can be compile as a vignette through \url{https://github.com/TGuillerme/moms}. -We believe that this extra care put into making the paper easy to reproduce will foster not only a better understanding of multidimensional analysis but also future analysis on how the finding of this paper related beyond the fields of ecology and evolution. - -We look forward to hearing from you soon, - -\closing{Yours sincerely,} - - -% Reviewers: -% Stefano Mammola (ecology) -% Étienne Laliberté (ecology) -% Graeme Lloyd (evolution) -% Melanie Hopkins (evolution) -% Veronica Diaz (ecology) -% Caroline Tucker (ecology) -% Emilie Rayfield (evolution) -% Alexis Mychajliw (evolution) (amychajl@tarpits.org) -% Melissa Kemp (evolution) (mkemp@austin.utexas.edu). - - -\end{letter} -\end{document} diff --git a/inst/EcoEvol/shiftingspace_resubmit.tex b/inst/EcoEvol/shiftingspace_resubmit.tex deleted file mode 100644 index c79f686..0000000 --- a/inst/EcoEvol/shiftingspace_resubmit.tex +++ /dev/null @@ -1,2211 +0,0 @@ -\documentclass[]{article} -\usepackage{xcolor} -\usepackage{lineno} -\usepackage{lmodern} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\usepackage{fixltx2e} % provides \textsubscript -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} -\fi -% use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -% use microtype if available -\IfFileExists{microtype.sty}{% -\usepackage[]{microtype} -\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\PassOptionsToPackage{hyphens}{url} % url is loaded by hyperref -\usepackage[unicode=true]{hyperref} -\hypersetup{ - pdftitle={Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?}, - pdfauthor={Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker}, - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage[margin=1in]{geometry} -\usepackage{longtable,booktabs} -% Fix footnotes in tables (requires footnote package) -\IfFileExists{footnote.sty}{\usepackage{footnote}\makesavenoteenv{long table}}{} -\usepackage{graphicx,grffile} -\makeatletter -\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} -\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} -\makeatother -% Scale images if necessary, so that they will not overflow the page -% margins by default, and it is still possible to overwrite the defaults -% using explicit options in \includegraphics[width, height, ...]{} -\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -% set default figure placement to htbp -\makeatletter -\def\fps@figure{htbp} -\makeatother - - -\title{Shifting spaces: which disparity or dissimilarity measurement best -summarise occupancy in multidimensional spaces?} -\author{Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker} -\date{2020-05-01} - -\linespread{1.6} - -\begin{document} -\maketitle - -\section{Response to reviewers}\label{response-to-reviewers} - -We a really grateful for the reviewers refreshing, positive and -constructive comments and we've addressed them all as detailed below. - -\subsection{Reviewer 1 (Stefano -Mammola)}\label{reviewer-1-stefano-mammola} - -In a conceptually similar study in the context of species distribution -modelling (DOI: 10.1111/2041-210X.12397), Qiao and colleagues termed -this general idea the ``no silver bullet'' paradigm (making a metaphor -with the mythology of lycanthropes). This is certainly a very important -point also in trait-based science. - -\begin{quote} -This is indeed a really similar idea with similar conclusions. We've now -added a reference to niche modeling in the introduction: -\end{quote} - -\textit{This can also be extended to more complex ecological concepts such as -niche modelling (Qiao et al. 2015). l.130-131} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi})} -\tightlist -\item - While reading the text, it was not entirely clear to me what exactly - the ``moms'' tool is. Is this an R package, or just a set of - functions? Would be helpful if you could briefly specify. -\end{enumerate} - -\begin{quote} -We've added an extra section explaining what the \texttt{moms} tool is. -We've also added a sampling (see reviewer 2) and simulation feature -allowing users to replicate the results of this paper with several -clicks or exploring the properties of their own metrics: -\end{quote} - -\textit{Therefore, we propose the -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} shiny app to -allow workers to help them choose their set of space occupancy -measurements (and test the caveats mentioned above). \texttt{moms} is an -online graphical user interface to help analyse multidimensional data. -It allows users to upload their dataset of interest (or simulate one -with specific parameters) and measure space occupancy using a variety of -implemented measures (namely, but not only, the ones used in this -study). Furthermore, the package allows simulation of shifts in trait -space occupancy as also presented in this paper to test whether some -measures capture specific changes in space. However, \texttt{moms} is -not a tool for analysing multidimensional data \emph{per se} but rather -for helping workers to chose the space occupancy measure most -appropriated to their data and question. To run multidimensional -analysis, we suggest using dedicated \texttt{R} packages (such as - but -not limited to: Oksanen et al. (2007), Bonhomme et al. (2014), Cardoso -et al. (2015), Guillerme (2018)). l.374-383} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi})} -\setcounter{enumi}{1} -\tightlist -\item - I suspect that your proposed classification may somewhat overlap with - the general classification scheme of diversity indexes by Pavoine and - Bonsall (2011; Biological review), later expanded by Tucker et al. - (2017; Biological Reviews) in the context of phylogenetic metrics (but - it's actually the same). In a nutshell, they grouped metrics/indexes - in three main category of `Richness', `Divergence' and `Regularity' - components. In their view, these three concepts should capture the - primary mathematical operation inherent to each metric, namely: -\end{enumerate} - -\begin{enumerate} -\def\labelenumi{\roman{enumi})} -\tightlist -\item - the `richness' dimension encompasses indexes representing the sum of - difference among taxa (sum); -\item - the `divergence' dimension encompasses indexes representing functional - (or phylogenetic) dissimilarity, reflecting the average difference - among taxa (mean); and -\item - the `regularity' dimension encompasses indexes representing functional - (or phylogenetic) variance, reflecting how regular the difference - among taxa in an assemblages are (variance). I see some overlap with - your method, but also differences. It would be worth briefly - acknowledging this and perhaps exploring the main conceptual - differences and why your classification is an advance. -\end{enumerate} - -\begin{quote} -We were not aware of these references and we are grateful for the -reviewer pointing them out. There is so much overlapping work on ecology -and evolution but so few visibility between them. Adding these -references and concepts is definitely improving the main idea of this -paper! -\end{quote} - -\textit{Note that this classification bears some similarities with Tucker et al. -(2017) classifying phylogenetic diversity measurements into richness, -divergence and regularity categories. However, while Tucker et al. -(2017) based their classification on the mathematical operation inherent -to each metrics (the sum for richness, the mean for divergence and the -variance for regularity), our three broad classifications are based on -their geometric properties regardless of the formula of each metric -(e.g.~the size of a space can be calculated using a sum, mean or/and -variance). l.113-118} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi})} -\setcounter{enumi}{2} -\tightlist -\item - The approach you choose for simulating changes in the multidimensional - space (Figure 2) is very interesting. While I was thinking about it, I - indulged with an idea (but please ignore if not appropriate): -\end{enumerate} - -I thought it would be useful to try linking the different changes in the -multidimensional space to actual example of biological processes (e.g., -giving few examples in the description in the method or in the -discussion). This, because similar changes in the trait space may occur -in the real world due to different processes and, by providing examples -grounded in the real world you may enhance the appealing of the ms to a -broader audience. For example, from the perspective of a conservation -biologist, you may argue that `limit' and `position' change in the trait -space may occur when there is the destruction of habitats with narrow -environmental conditions that filters for few species possessing -specific traits. I'm primarily a cave biologist: this mental bias made -me think that, for example, if you open a quarry that destroys a cave -(not an unfrequent event! DOI: 10.1093/biosci/biz064), you may end up -removing from the trait space all points from a specific position, -namely those traits clearly selected by the permanent darkness of the -subterranean world. If trait 1 is a gradient of eye size and trait 2 a -gradient of body pigment, you would remove trait contribution of -specialized invertebrates with no eyes and no pigment. The `density' -change may occur, e.g., due to global wildlife trade, when there is the -exploitation of only species with specific traits within the total -functional tree of life (see DOI: 10.1126/science.aav5327). For example, -the fishing industry exploits, within any given species, primarily -fishes of larger size, thus selecting for specific densities in the -size/weight trait space composed of the multiple species. And so on -(these are just random examples to illustrate my point). - -\begin{quote} -We've added this reviewer's example (although not all to also balance -with examples from macroevolution): -\end{quote} - -\textit{This type of change could be due to habitat destruction (e.g. Mammola et -al. 2019) or to mass extinctions (e.g. Wright 2017). l.202-203} - -\textit{This type of change could be due to accelerated rates of evolution -(Close et al. 2015) or to differences in modes of life in macroevolution -(e.g. Healy et al. 2019). l.208-210} - -\textit{This type of change could be due changes in evolutionary trajectories -(Endler et al. 2005) or to differences in ecosystem compositions (e.g. -Jones et al. 2015). l.215-216} - -\subsubsection{Minor Line comments}\label{minor-line-comments} - --L34: ``studied'' instead of ``study'' - -\begin{quote} -We fixed this typo. -\end{quote} - -L35-37: I think merging together these two sentences would put more -emphasis on your idea\ldots{} E.g. ``While different fields use a -different set of terms for such approaches (Table 1), they actually -focus on the same mathematical objects: matrices with columns -representing an original or transformed trait value and rows -representing observations (taxon, field site, etc.; Guillerme 2018). - -\begin{quote} -Merged these sentences. -\end{quote} - --L63: ``are are'' - -\begin{quote} -We fixed this typo. -\end{quote} - --Table 1: `functional space' instead of `function-space'? Morphospace I -think is also widely used in ecology. Please add ``etc.'' at the end of -the text in the cell ``statistic'' intersected ``ecology''. Wouldn't the -``hypervolume'' mentioned in the same cell more fit in the ``Matrix (n x -d)'' row? - -\begin{quote} -We fixed these cells in the table. However, we did not add -``hypervolume'' to the ``Mathematics'' column. We use the term -``statistic'' here as a statistic (i.e.~a measure) which does include -the hypervolume as used in ecology. We've specified this. -\end{quote} - --330--332: True, but there are at least two recent examples of attempt -in this sense, the very recent package TPD (DOI: 10.1002/ecy.2876) and -new functions in BAT (doi: 10.1101/2020.01.25.919373; shameless -self-promotion and still a preprint). Both approaches covers metric in -the Richness, Divergence, and Regularity domains (which should be a -possible equivalent to your Size, Density, and Position view). - -\begin{quote} -We've added these references on lines 396-397. -\end{quote} - --Figure 2: why the dots are in black and white in the first inset? - -\begin{quote} -The black dots are the 50\% randomly removed points. We've specified -this in the caption. -\end{quote} - -\subsection{Reviewer: 2 (Neil -Brocklehurst)}\label{reviewer-2-neil-brocklehurst} - -A general suggestion to start with. The authors state in their -conclusions, very diplomatically, that ``\ldots{}no measure is better -than the next one\ldots{}''. While I agree with the sentiment in the -context of their conclusions (that different metrics show different -things, and you should choose a variety of metrics relevant to the issue -under study), would it not be accurate to say that your results indicate -some metrics are worse than others? Three leapt out at me in table 5: -Sum of Ranges, Minimum spanning tree average distances, and to a lesser -extent average nearest neighbour distances, seem to produce changes in -value following random removals comparable to those seen in not one, but -all three varieties of separation (limit, density and position) . Would -it be fair to say that these metrics are, shall we say, problematic? - -\begin{quote} -We agree with the reviewer and have toned down or ``diplomacy'' level. -We now also state concern about certain types of metrics that are highly -sensitive to outliers or to the number of dimensions: -\end{quote} - -\textit{We insist that although no measure is objectively better than the next -one, some can be more problematic than other in specific contexts. For -example, the results for the Sum of Ranges, Minimum spanning tree -average distances, and to a lesser extent average nearest neighbour -distances produced results in the reduced space often similar to the -randomly reduced spaces (Table 5). This does not make them ``bad'' -measures but rather heavily context dependent. Regardless, we believe -that workers should identify the most appropriate measures based on -their trait space properties as well as their specific biological -question. We believe this could be fostered by following these several -suggestions:} l.385-391 - -Following on from this: its worth testing, or at the very least -acknowledging, that some metrics probably respond to incomplete sampling -worse than others? Or are certain targeted separations (by limit, -density or position) harder to detect with incomplete sampling? A test -of this would be reasonably simple; after carrying out the separations -by limit, density or position, randomly delete different numbers of data -points to see how this affects the results. I realise this would be a -large number of extra analyses to ask for and may be considered beyond -the immediate scope of the paper, so I'll leave the choice of whether or -not to do the actual analyses up to the authors, but I think it should -at least be discussed in the text. - -\begin{quote} -We agree again with this reviewer's point and find this a really -interesting point. We've updated the results in the supplementary -material to highlight what happens when removing 80\% and 20\% of the -elements (rather than 50\%). We've also added a sampling option to the -latest version of the \texttt{moms} app allowing to test the effect of -sampling on the metric of choice and, finally, we've added the following -mentions to sampling issues in the caveat section: -\end{quote} - -\textit{Furthermore, we did not take into account the effect of sampling on -space occupancy measurements (but see additional results with 80\% and -20\% space reduction in the supplementary materials 4). In fact, -sampling has been previously shown to have an effect on measurements -depending on range or volumes (e.g.~the sum of ranges or the hypervolume -Ciampaglio et al. (2001)). This effect is especially expected to be -acerbated in macroevolutionary studies when using the fossil record -(Brocklehurst et al. 2013) but can be tackled using rarefaction and -bootstrapping techniques (Guillerme 2018). l.367-372} - -Some more specific comments: Fig 2, panels E and F, and similar in table -5: the dots and lines representing the changes in occupancy for blue, -orange and random groups. I assume the dot is the median and the solid -and dashed lines are quartiles and ranges, but specify in caption, -please - -\begin{quote} -We already specified that ``the dots represent the median space -occupancy values across all simulations for each scenario of trait space -change (Table 2), the solid and dashed line respectively the 50\% and -95\% confidence intervals'' (Caption Figure 2). We've now also added -this to the Table 5 caption. -\end{quote} - -Table 5 headers are inconsistent with the terms used in the text and -figure 5. In the text and figure 5 you describe Limit and Density -parameters for changes, but the headers talk about size change and -arrangement change. Maybe this will create confusion (particularly the -`arrangement' header, which might not naturally be associated in peoples -minds with density)? - -\begin{quote} -Nice spotting, this comes from a classic version mashup. We've fixed -this in the table header and double checked throughout the manuscript. -\end{quote} - -And finally some typos and wording comments: - -First sentence of the abstrach: ``Multidimensional analysis of traits -are now a common in ecology and evolution\ldots{}''; should be either -``\ldots{}now common in\ldots{}'' or ``\ldots{}now a common method -in\ldots{}'' - -\begin{quote} -We remove ``a''. -\end{quote} - -Abstract, line 8: ``\ldots{}subspace of this space\ldots{}''; maybe say -subset instead of subspace? The two `spaces' next two each other makes -slightly awkward reading - -\begin{quote} -We've changed ``subspace'' to ``subset''. -\end{quote} - -Intro line 48: ``ciampaglio2001''; Space missing between name and year, -name needs capital letter Methods, page 8, line 153: ``hopkins2017''; -Space missing between name and year, name needs capital letter - -\begin{quote} -We fixed the references. -\end{quote} - -Page 20, line 317: ``spacesmight'' missing space between words - -\begin{quote} -We've added a space. -\end{quote} - -\modulolinenumbers[1] % just after the \begin{document} tag -\linenumbers - -\section{Abstract}\label{abstract} - -Multidimensional analysis of traits are now common in ecology and -evolution and are based on trait spaces in which each dimension -summarises the observed trait combination (a morphospace or an -ecospace). Observations of interest will typically occupy a -\textcolor{blue}{subset} of this space, and -researchers will calculate one or more measures to quantify how -organisms inhabit that space. In macroevolution and ecology these -measures \textcolor{blue}{called disparity or -dissimilarity metrics and are} generalised as space occupancy measures. -Researchers use these measures to investigate how space occupancy -changes through time, in relation to other groups of organisms, and in -response to global environmental changes. However, the mathematical and -biological meaning of most space occupancy measures is vague with the -majority of widely-used measures lacking formal description. - -Here we propose a broad classification of space occupancy measures into -three categories that capture changes in size, density, or position. We -study the behaviour of 25 measures to changes in trait space size, -density and position on simulated and empirical datasets. We find that -no measure describes all of trait space aspects but that some are better -at capturing certain aspects. Our results confirm the three broad -categories (size, density and position) and allow us to relate changes -in any of these categories to biological phenomena. - -Because the choice of space occupancy measures is specific to the data -and question, we introduced -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool to -both visualise and capture changes in space occupancy for any -measurement. \href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} -is designed to help workers choose the right space occupancy measures, -given the properties of their trait space and their biological question. -By providing guidelines and common vocabulary for space occupancy -analysis, we hope to help bridging the gap in multidimensional research -between ecology and evolution. - -\section{Introduction}\label{introduction} - -Groups of species and environments share specific, recognisable, -correlated characteristics: guilds or biomes with shared phenotypic, -physiological, phylogenetic or behavioural traits. Organisms or -environments should therefore be studied as a set of traits rather than -some specific traits in isolation (Donohue et al. 2013; Hopkins and -Gerber 2017). Biologists increasingly been using ordination techniques -(see Legendre and Legendre 2012 for a summary) to create -multidimensional trait spaces to either explore properties of data or -test hypotheses (e.g. Oksanen et al. 2007; Blonder 2018; Guillerme -2018). For example, in palaeobiology, Wright (2017) used trait spaces to -study how groups of species' characteristics change through time; in -ecology, Jones et al. (2015) \textcolor{blue}{ studied -} evidence of competition by looking at trait overlap between two -populations. \textcolor{blue}{ While different fields -use a different set of terms for such approaches (Table 1), they -actually focus on the same mathematical objects: matrices with columns -representing an original or transformed trait value and rows -representing observations (taxon, field site, etc.; Guillerme 2018). } - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llll@{}} -\toprule -\begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Mathematics\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Macroevolution\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -This paper\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix (\(n \times d\)) with a structural relation between rows and -columns\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -\textcolor{blue}{ Functional space, morphospace }, -etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Morphospace, traitspace, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -trait space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Rows (\emph{n})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, field sites, environments, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, specimen, populations, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Columns (\emph{d})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -dimensions\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix subset (\(m \times d\); \(m \leq n\))\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Treatments, phylogenetic group (clade), etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Clades, geological stratum, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -group\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Statistic \textcolor{blue}{ (i.e.~a measure) }\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity index or metric, hypervolume, functional diversity, -\textcolor{blue}{ etc. }\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity metric or index\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -space occupancy measure\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Multidimensional analysis\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity analysis, trait analysis, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity analysis, disparity-through-time, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -multidimensional analysis\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 1: \textcolor{blue}{Different terms are used for -equivalent measures in} mathematics, ecology and macroevolution. - -\renewcommand\baselinestretch{1.6}\selectfont - -Ecologists and evolutionary biologists often use trait spaces with -respect to the same fundamental questions: are groups occupying the same -amount of trait space? Do some groups contain more species than others -in the same amount of trait space? Are some specific factors correlated -with different patterns of trait space occupancy? Because of the -multidimensional nature of these trait spaces, it is often not possible -to study them using bi- or tri-variate techniques (Díaz et al. 2016; -Hopkins and Gerber 2017; Mammola 2019). Studying the occupancy of trait -spaces is done using disparity indices in macroevolution (Wills 2001; -Hopkins and Gerber 2017; Guillerme 2018) or comparing hypervolumes in -ecology (Donohue et al. 2013; Díaz et al. 2016; Blonder 2018; Mammola -2019). Despite the commonalities between the measures used in ecology -and evolution (which are often metric but don't necessarily need to be), -surprisingly little work has been published on their behaviour (but see -Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). - -Different occupancy measures capture different aspects of trait space -(Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). -\textcolor{blue}{This} may be widely-known, but to -our knowledge it is infrequently mentioned in peer-reviewed papers. -First, space occupancy measures are often named as the biological aspect -they are describing (``disparity'', ``functional diversity'') rather -than what they are measuring (e.g.~the product of ranges), which -obscures the differences and similarities between studies. Second, in -many studies in ecology and evolution, authors have focused on measuring -the size of the trait space (e.g.~ellipsoid volume Donohue et al. 2013; -hypervolume Díaz et al. 2016; Procrustes variance Marcy et al. 2016; -product of variance Wright 2017). However, the size of the trait space -only represents one aspects of occupancy, disregarding -\textcolor{blue}{other measures} such as the density -(Harmon et al. 2008) or position (Wills 2001; Ciampaglio et al. 2001). -For example, if two groups have the same size, this can support certain -biological conclusions. Yet, an alternative aspect of space occupancy -may indicate that the groups' position are different, leading to a -different biological conclusion (e.g.~the groups are equally diverse but -occupy different niches). Using measures that only capture one aspect of -the trait space may restrain the potential of multidimensional analysis -(Villéger et al. 2008). - -Here we propose a broad classification of space occupancy measures as -used across ecology and evolution and study their power to detect -changes in trait space occupancy in simulated and empirical data. -\textcolor{blue}{Note this does not account whether or -not it is possible for a space to be occupied (e.g., some spaces may -represent biologically impossible shapes); this, however, may be -important in some cases, such as testing whether a region is infinite or -not.} We provide an assessment of each broad type of space occupancy -measures along with a unified terminology to foster communication -between ecology and evolution. Unsurprisingly, we found no one measure -describes all changes \textcolor{blue}{in space} and -that the results from each measures are dependent on the characteristics -of the space and the hypotheses. - -\textcolor{blue}{There can be an infinite number of -measures and that it is thus impossible to propose a comprehensive -analysis for all the measures properties respective to how they measure -changes in trait space. We therefore propose -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool for -researchers to design, experiment and visualise their own space -occupancy measure tailored for their project. The tool will help -researchers understand the ``null'' behaviour of the measures of -interest.} - -\subsection{Space occupancy measures}\label{space-occupancy-measures} - -In this paper, we define trait spaces as any matrix where rows are -observations and columns are traits, where both observations and traits -are structurally related (e.g.~there is a phylogenetic relation between -observations - and traits, etc.). These traits can widely vary in number -and types: they could be coded as discrete (e.g.~presence or absence of -a bone; Beck and Lee 2014; Wright 2017), continuous measurements -(e.g.~leaf area; Díaz et al. 2016) or more sophisticated measures -(Fourier ellipses; Bonhomme et al. 2014; e.g.~landmark position; Marcy -et al. 2016). Traits can also be measured by using relative observations -(e.g.~community compositions; Jones et al. 2015) or distance between -observations (e.g. Close et al. 2015). However, regardless of the -methodology used to build a trait space, three broad occupancy measures -can be used: the \textbf{size} which approximates the amount of space -occupied, the \textbf{density} which approximates the distribution in -space and the \textbf{position} which approximates the location in space -(Fig. 1; Villéger et al. 2008). Of course any combination of these three -aspects is always possible. - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measures_types-1.pdf} -\caption{different type of information captured by space occupancy -measures: (A) size, (B) density and (C) position.} -\end{figure} - -\paragraph{1. Size}\label{size} - -Size captures the spread of a group in the trait space. They can be -interpreted as the amount of the trait space that is occupied by -observations. Typically, larger values for such measures indicate the -presence of more extreme trait combinations. For example, if group A is -bigger than B, the observations in A achieve more extreme trait -combinations than in B. This type of measure is widely used in both -ecology (e.g.~the hypervolume; Blonder 2018) and in evolution (e.g.~the -sum or product of ranges or variances; Wills 2001). - -Although size measures are suitable indicators of a group's trait space -occupancy, they are limited to comparing the range of trait-combinations -between groups. Size measures do not take into account the distribution -of the observations within a group and can often be insensitive to -unoccupied ``holes'' in the trait space (overstimating the size; Blonder -(2018)). They can make it difficult to determine whether all the -observations are on the edge of the group's distribution or whether the -size is simply driven by outliers. - -\paragraph{2. Density}\label{density} - -Density gives an indication of the quantity of observations in the trait -space. They can be interpreted as the distribution of the observations -\emph{within} a group in the trait space. Groups with higher density -contain more observations (i.e.~more observations per approximation of -size) that will tend to be more similar to each other. For example, if -group A is greater is size than group B and both have the same density -(observations are equally distant within each group), similar mechanisms -could be driving both groups' trait space occupancy. Indeed, this -pattern could suggest that A is older and has had more time to achieve -more extreme trait combinations under essentially the same process as -younger, smaller group B (Endler et al. 2005). Note that density based -measures can be sensitive to sampling. Density measures are less common -compared to size measures, but they are still used in both ecology -(e.g.~the minimum spanning tree length; Oksanen et al. 2007) and -evolution (e.g.~the average pairwise distance; Harmon et al. 2008). - -\paragraph{3. Position}\label{position} - -Position captures where a group lies in trait space. They can be -interpreted as where a group lies in the trait space either relative to -the space itself or relative to another group. For example, if group A -has a different position than group B, A will have a different -trait-combination than in B. - -Position measures may be harder to interpret in multidimensional spaces. -In a 2D space, two groups can be equally distant from a fixed point but -in different parts of the space (left, right, up, or down - with the -amount of parts of space increasing with dimensions). However, when -thinking about unidimensional data, this measure is obvious: two groups -A or B could have the same variance (size) with the same number of -observations (density) but could have two different means and thus be in -different positions. These measures are used in ecology to compare the -position of two groups relative to each other (Mammola 2019). - -\textcolor{blue}{ Note that this classification into -size, density and position bears some similarities with Tucker et al. -(2017) classifying phylogenetic diversity measurements into richness, -divergence and regularity categories. However, while Tucker et al. -(2017) based their classification on the mathematical operation inherent -to each metrics (the sum for richness, the mean for divergence and the -variance for regularity), our three broad classifications are based on -their geometric properties regardless of the formula of each metric -(e.g.~the size of a space can be calculated using a sum, mean or/and -variance). } - -\subsection{No measure to rule them all: benefits of considering -multiple -measures}\label{no-measure-to-rule-them-all-benefits-of-considering-multiple-measures} - -The use of multiple measurements to assess trait space occupancy -provides a more detailed characterisation of occupancy changes. If the -question is to look at how space occupancy changes in response to mass -extinction, using a single space occupancy measure can miss part of the -picture: a change in size could be decoupled from a change in position -or density in trait space. For example, the Cretaceous-Paleogene -extinction (66 million years ago) shows an increase in size of the -mammalian trait space (adaptive radiation; Halliday and Goswami 2016) -but more specific questions can be answered by looking at other aspects -of trait space occupancy: does the radiation expand on previously -existing morphologies (elaboration, increase in density; Endler et al. -2005) or does it explore new regions of the trait space (innovation, -change in position; Endler et al. 2005)? Similarly, in ecology, if two -groups have the same trait space size, -\textcolor{blue}{the differences in density within -these two groups is potentially illuminating:} different selection -pressure can lead to different density within equally sized groups. -\textcolor{blue}{This can also be extended to more -complex ecological concepts such as niche modelling (Qiao et al. -2015).} - -Here, we provide the first interdisciplinary review of 25 space -occupancy measures that uses the broad classification of measures into -size, density and position to capture pattern changes in trait space. We -assess the behaviour of measures using simulations and six -interdisciplinary empirical datasets covering a wide range of potential -data types and biological questions. We also introduce a tool for -measuring occupancy in multidimensional space -(\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}), which is -a user-friendly, open-source, graphical interface to allow the tailored -testing of measurement behaviour for any use case. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} will allow -workers to comprehensively assess the properties of their trait space -and the measures associated with their specific biological question. - -\section{Methods}\label{methods} - -We tested how 25 space occupancy measures relate to each other, are -affected by modifications of traits space and affect group comparisons -in empirical data: - - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - We simulated 13 different spaces with different sets of parameters; -\item - We transformed these spaces by removing 50\% of the observations - following four different scenarios corresponding to different - empirical scenarios: randomly, by - \textcolor{blue}{size} (e.g.~expansion or reduction - of niches), by density (e.g.~different degrees of competition within a - guild), and by position (e.g.~ecological niche shift). -\item - We measured occupancy on the resulting transformed spaces using eight - different space occupancy measures; -\item - We applied the same space occupancy measures to six empirical datasets - (covering a range of disciplines and a range of dataset properties). -\end{enumerate} - -Note that the paper contains the results for only eight measures which -were selected as representative of common measures covering the size, -density and position trait space aspects. The results for an additional -17 measures is available in the supplementary material 4. - -\subsection{Generating spaces}\label{generating-spaces} - -We generated trait spaces using the following combinations of size, -distributions, variance and correlation: - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -space name\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.31\columnwidth}\raggedright\strut -distribution(s)\strut -\end{minipage} & \begin{minipage}[b]{0.21\columnwidth}\raggedright\strut -dimensions variance\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -correlation\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform (min = -0.5, max = 0.5)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal (mean = 0, sd = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D with random distributions\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCA-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Multiplicative\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCO-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Additive\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 2: different simulated space distribution. \emph{Name} of the -simulated space; \emph{dimensions} of the matrix (row*columns); -\emph{distribution(s)} of the data on each dimensions (for the `Random', -dimensions are randomly chosen between Normal, Uniform or Lognormal); -\emph{dimension variance}: distribution of the variance between -dimensions (when equal, the dimensions have the same variance); -\emph{correlation} between dimensions. - -\renewcommand\baselinestretch{1.6}\selectfont - - -The differences in trait space sizes (200 elemeents for 3, 15, 50 or 150 -dimensions) reflects the range found in literature (e.g. Hopkins and -Gerber 2017; Mammola 2019). We used a range of distributions (uniform, -normal or a random combination of uniform, normal and lognormal) to test -the effect of observation distributions on the measurements. We used -different levels of variance for each dimensions in the spaces by making -the variance on each dimension either equal -(\(\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}\)) or decreasing -(\(\sigma_{D1} < \sigma_{D2} < \sigma_{Di}\)) with the decreasing factor -being either multiplicative (using the cumulative product of the inverse -of the number of dimensions: \(\prod_i^d(1/d)\)) or additive -(\(\sum_i^d(1/d)\)). Both reductions of variance are used to illustrate -the properties of ordinations where the variance decreases per -dimensions (and normal win Multidimensional Scaling - MDS, PCO or PCoA; -e.g. Close et al. 2015; lognormal in principal components analysis - -PCA; e.g. Marcy et al. 2016; Wright 2017; Healy et al. 2019). Finally, -we added a correlation parameter to illustrate the effect of -co-linearity between traits (especially in non-ordinated trait spaces). -We repeated the simulation of each trait space 20 times (resulting in -260 spaces). - -\subsection{Spatial occupancy -measures}\label{spatial-occupancy-measures} - -We then calculated eight different measures on the resulting transformed -spaces, including a new one, the average displacement, which we expect -to be influenced by changes in trait space position. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Name\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Definition\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Captures\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Source\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Notes\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Laliberté and Legendre (2010)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional dispersion (FDis - without abundance)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\sigma^{2}{k_i}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -common measure used in palaeobiology (Ciampaglio et al. 2001; Wills -2001)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -more sensitive to outliers than the sum of variances\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Donohue et al. (2013)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -less sensitive to outliers than the convex hull hypervolume (Díaz et al. -2016; Blonder 2018)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum(\text{branch length})}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sedgewick (1990)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -similar to the unscaled functional evenness (Villéger et al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Villéger et al. (2008)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional evenness without weighted abundance (FEve; Villéger et -al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sqrt{\sum_{i}^{n}{min({q}_{i}-p_{i})^2}})\times \frac{1}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the density of pairs of observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average displacement\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Position\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -This paper\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the ratio between the observations' position from their centroid and the -centre of the trait space (coordinates: 0, 0, 0, \ldots{}). A value of 1 -indicates that the observations' centroid is the centre of the trait -space\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 3: List of measures with \emph{n} being the number of -observations, \emph{d} the total number of dimensions, \emph{k} any -specific row in the matrix, \emph{Centroid} being their mean and -\(\sigma^{2}\) their variance. \(\Gamma\) is the Gamma distribution and -\(\lambda_{i}\) the eigenvalue of each dimension and \({q}_{i}\) and -\(p_{i}\) are any pairs of coordinates. - -\renewcommand\baselinestretch{1.6}\selectfont - -We selected these eight space occupancy measures to illustrate how they -capture different aspects of space occupancy (not as an expression of -our preference). These measures are specific to Euclidean and isotropic -trait spaces (which is not necessary for all measures). The -supplementary material 4 contains the same analysis as described below, -performed on 17 measures. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -exploration into the effect of many more measures as well as the -customisation of measures by combining them or using user-designed -functions. - -\subsection{Measure comparisons}\label{measure-comparisons} - -We compared the space occupancy measures correlations across all -simulations between each pair of measures to assess their captured -signal (Villéger et al. 2008; Laliberté and Legendre 2010). We used the -measures on the full 13 trait spaces described above. We then scaled the -results and measured the pairwise Pearson correlation to test whether -measures were capturing a similar signals or not using the -\texttt{psych} package (Revelle 2018). - -\subsection{Changing space}\label{changing-spaces} - -To assess how the measures responded to changes within trait spaces, we -removed 50\% of observations each time using the following algorithms: - -\begin{itemize} -\item - \textbf{Randomly:} by randomly removing 50\% of observations (Fig. - 2-A). This reflects a ``null'' biological model of changes in trait - space: the case when observations are removed regardless of their - intrinsic characteristics. For example, if diversity is reduced by - 50\% but the space size remains the same, there is a decoupling - between diversity and space occupancy (Ruta et al. 2013). Our selected - measures are expected to not be affected by this change. -\item - \textbf{\textcolor{blue}{Size}:} by removing - observations within a distance from the centre of the trait space - lower or greater than a radius \(\rho\) (where \(\rho\) is chosen such - that 50\% observations are selected) generating two limit removals: - \emph{maximum} and \emph{minimum} (respectively in orange and blue; - Fig. 2-B). This can reflect a strict selection model where - observations with trait values below or above a threshold are removed - leading to an expansion or a contraction of the trait space. - \textcolor{blue}{This type of change could be due to - habitat destruction (e.g. Mammola et al. 2019) or to mass extinctions - (e.g. Wright 2017).} Size measures are expected to be most affected - by this change. -\end{itemize} - -\begin{itemize} -\item - \textbf{Density:} by removing any pairs of point with a distance \(D\) - from each other where (where \(D\) is chosen such that 50\% - observations are selected) generating two density removals: - \emph{high} and \emph{low} (respectively in orange and blue; Fig. - 2-C). This can reflect changes within groups in the trait space due to - ecological factors (e.g.~niche repulsion resulting in lower density; - Grant and Grant 2006). \textcolor{blue}{This type of - change could be due to accelerated rates of evolution (Close et al. - 2015) or to differences in modes of life in macroevolution (e.g. Healy - et al. 2019).} Density measures are expected to be most affected by - this change. -\item - \textbf{Position:} by removing points similarly as for - \textbf{\textcolor{blue}{Size}} but using the - distance from the furthest point from the centre generating two - position removals: \emph{positive} and \emph{negative} (respectively - in orange and blue; Fig. 2-D). This can reflect global changes in - trait space (e.g.~if an entire group remaining diverse but occupying a - different niche). \textcolor{blue}{This type of - change could be due changes in evolutionary trajectories (Endler et - al. 2005) or to differences in ecosystem compositions (e.g. Jones et - al. 2015).} Position measures are expected to be most affected by - this change. -\end{itemize} - -The algorithm to select \(\rho\) or \(D\) is described in the -Supplementary material 1. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_reduce_space-1.pdf} -\caption{} -\end{figure} - -Figure 2: different type of space reduction. Each panel displays two -groups of 50\% of the data points each. Each group (orange and blue) are -generated using the following algorithm: A - randomly (the removed -elements are displayed in black and the analysed ones in grey); B - by -size (maximum and minimum limit); C - by density (high and low); and D - -by position (positive and negative). Panel E et F represents two typical -display of the reduction results displayed in Table 5: the dots -represent the median space occupancy values across all simulations for -each scenario of trait space change (Table 2), the solid and dashed line -respectively the 50\% and 95\% confidence intervals. Results in grey are -the random 50\% reduction (panel A). Results in blue and orange -represent the opposite scenarios from panels B, C, and D. The displayed -value is the amount of overlap (Bhattacharrya Coefficient) between the -blue or orange distributions and the grey one. Panel E and F shows -respectively the ``ideal'' and ``worst'' results for any type of -measures, where the space occupancy measurement respectively manages or -fails to captures a specific type of reduction (i.e.~size, position or -density; Table 5). - - -\renewcommand\baselinestretch{1.6}\selectfont - -Because occupancy measures are dependent on the space, we scaled and -centred them between -1 and 1 to make them comparable (by subtracting -the observed occupancy without reduction to all the measures of the -reduced spaces and then divided it by the maximum observed occupancy). A -value of 0 indicates no effect of the space reduction and \(>0\) and -\(<0\) respectively indicates an increase or decrease in the measure -value. We then measured the amount of overlap between the non-random -removals (\textcolor{blue}{size}, density and -position) and the random removals using the Bhattacharrya Coefficient -(Bhattacharyya 1943). - -\subsubsection{Measuring the effect of space and -dimensionality}\label{measuring-the-effect-of-space-and-dimensionality} - -Distribution differences and the number of dimensions can have an effect -on the measure results. For example, in a normally distributed space, an -increase in density can often lead to a decrease in size (though this is -not necessarily true if the space is log-normal or uniform). High -dimensional spaces (\textgreater{}10) are subject to the ``curse of -multidimensionality'' (Bellman 1957): data becomes sparser with -increasing number of dimensions. This can have two main consequences: 1) -the probability of overlap between two groups decreases as a product of -the number of dimensions; and 2) the amount of samples needed to -``fill'' the spaces increases exponentially -\href{https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners}{see -this interactive illustration by Toph Tucker}. The ``curse'' can make -the interpretation of high dimensional data counter-intuitive. For -example if a group expands in multiple dimensions (i.e.~increase in -size), the actual hypervolume (\(\prod_{i}^{d} range_{Di}\)) can -decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an -ANOVA (\(occupancy \sim distribution\) and -\(occupancy \sim dimensions\)) by using all spaces with 50 dimensions -and the uniform and normal spaces with equal variance and no correlation -with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing -respectively the effect of distribution and dimensions. The results of -the ANOVAs (F and \emph{p}-values) are reported in Table 5 (full results -in supplementary material 3). - -\subsection{Empirical examples}\label{empirical-examples} - -We analysed the effect of the different space occupancy measures on six -different empirical studies covering a range of fields that employ trait -space analyses. For each of these studies we generated trait spaces from -the data published with the papers. We divided each trait spaces into -two biologically-relevant groups and tested whether the measures -differentiated the groups in different ways. Both the grouping and the -questions were based on a simplified version of the topics of these -papers (with no intention to re-analyse the data and questions). The -procedures to generate the data and the groups varies between studies -and is detailed in the supplementary materials 2. - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}llllllll@{}} -\toprule -\begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -study\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -field\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -taxonomic group\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -traits\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -trait space\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -groups\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -question\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Beck and Lee (2014)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Mammalia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -106*105\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -52 crown vs.~54 stem\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are crown mammals more disparate than stem mammals?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Wright (2017)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Crinoidea\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -42*41\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -16 before vs.~23 after\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Is there a difference in disparity before and after the Ordovician mass -extinction?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Marcy et al. (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Rodentia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -skull 2D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -454*134\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -225 \emph{Megascapheus} vs.~229 \emph{Thomomys}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are two genera of gopher morphologically distinct?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Hopkins and Pearson (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Trilobita\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -3D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -46*46\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -36 adults vs.~10 juveniles\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are juvenile trilobites a subset of adult ones -\textcolor{blue}{in trait space}?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Jones et al. (2015)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Plantae\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Communities species compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Jaccard distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -48*47\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -24 aspens vs.~24 grasslands\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Is there a difference in species composition between aspens and -grasslands?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Healy et al. (2019)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Animalia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Life history traits\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of continuous traits (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -285*6\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -83 ecthotherms vs.~202 endotherms\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Do endotherms have more diversified life history strategies than -ectotherms?\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 4: details of the six empirical trait spaces. - -\renewcommand\baselinestretch{1.6}\selectfont - -For each empirical trait space we bootstrapped each group 500 times -(Guillerme 2018) and applied the eight space occupancy measure to each -pairs of groups. We then compared the means of each groups using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\section{Results}\label{results} - -\subsection{Measure comparisons}\label{measure-comparisons-1} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measure_correlation-1.pdf} -\caption{pairwise correlation between the scaled measures. Numbers on -the upper right corner are the Pearson correlations. The red line are -linear regressions (with the confidence intervals in grey). Av.: -average; dist.: distance; min.: minimum; span.: spanning.} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - -\textcolor{blue}{Most measures of space were -positively correlated (Pearson correlation of 0.99 for the average -Euclidean distance from centroid and sum of variance or 0.97 for the -average nearest neighbour distance and minimum spanning tree average -length; Fig. 3). The remaining measures were either somewhat correlated -or had a negative pairwise distribution } (ranging from 0.66 for the -sum of variances and the ellipsoid volume to -0.09 between the average -displacement and the average Euclidean distance from centroid; Fig. 3). -All measures but the ellipsoid volume were normally (or nearly normally) -distributed (Fig. 3). - -\subsection{Space shifting}\label{space-shifting} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llllll@{}} -\toprule -\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Size change\strut -\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut -Density change\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Position change\strut -\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Distribution effect\strut -\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut -Dimensions effect\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 0.924 ; p = 0.449\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.322 ; p = 0.958\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.285 ; p = 0.274\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.478 ; p = 0.873\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 11.119 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 32.307 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 7.215 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 13.486 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.162 ; p = 0.326\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.998 ; p = 0.435\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 8.152 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 29.358 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.478 ; p = 0.207\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.773 ; p = 0.626\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 10.742 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 26.829 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 5: Results of the effect of space reduction, space dimension -distributions and dimensions number of the different space occupancy -measures. The dots represent the median space occupancy values across -all simulations for each scenario of trait space change (Table 2), the -solid and dashed line respectively the 50\% and 95\% confidence -intervals. See Fig. 2 for details on the interpretation of the figures -distributions and values. F-values for distribution effect and -dimensions effect represents respectively the effect of the ANOVAs space -occupancy \textasciitilde{} distributions and space occupancy -\textasciitilde{} dimension represent the ratio of sum squared -difference within and between groups (the higher, the more the factor -has an effect on the measure) and associated \emph{p}-values (0 `***' -0.001 `**' 0.01 `*' 0.05 `.' 0.1 '' 1). This figure illustrates how -different measures can be influenced by different aspects of changes in -the trait space. E.g. the Average Euclidean distance from centroid (row -1) captures mainly changes in size (column 1), but also captures changes -in density (column 2) but does not capture changes in position (column -3). - -\renewcommand\baselinestretch{1.6}\selectfont - -As expected, some different measures capture different aspects of space -occupancy. However, it can be hard to predict the behaviour of each -measure when 50\% of the observations are removed. We observe a clear -decrease in \textcolor{blue}{the median measure -value} in less than a third of the space reductions (10/36). - -In terms of change in size, only the average Euclidean distance from -centroid and the sum of variances seem to capture a clear change in both -directions. In terms of change in density, only the minimum spanning -tree average distance and the average nearest neighbour distance seem to -capture a clear change in both directions. And in terms of change in -position, only the average displacement metric seems to capture a clear -change in direction (albeit not in both directions). This is not -surprising, since the notion of positions becomes more and more complex -to appreciate as dimensionality increases (i.e.~beyond left/right, -up/down and front/back). - -\subsection{Empirical example}\label{empirical-example} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllllll@{}} -\toprule -\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Beck and Lee 2014\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -Wright 2017\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Marcy et al. 2016\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Hopkins and Pearson 2016\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Jones et al. 2015\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Healy et al. 2019\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Comparisons (orange \emph{vs.} blue)\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -crown \emph{vs.} stem mammals morphologies\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -crinoids morphologies before \emph{vs.} after the end-Ordovician -extinction\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\emph{Megascapheus} \emph{vs.} \emph{Thomomys} skull shapes\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -adults \emph{vs.} juveniles trilobites cephalon shapes\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -aspens \emph{vs.} grasslands communities compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -ecthotherms \emph{vs.} endotherms life history traits\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-25.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-33.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-41.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-26.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-34.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-42.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-27.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-35.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-43.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-28.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-36.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-44.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-29.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-37.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-45.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-30.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-38.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-46.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-31.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-39.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-47.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-32.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-40.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-48.pdf}\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 6: Comparisons of pairs of groups in different empirical trait -spaces. NAs are used for cases where space occupancy could not be -measured due to the curse of multidimensionality. The displayed values -are the amount of overlap between both groups (Bhattacharrya -Coefficient). - -\renewcommand\baselinestretch{1.6}\selectfont - - -\textcolor{blue}{As with the as for the simulations, -there is no measure that summarises all the aspects of distributions for -empirical data.} For all eight measures (except the ellipsoid volume) -we see either one group or the other having a bigger mean than the other -and no consistent case where a group has a bigger mean than the other -for all the measures. For example, in the Beck and Lee (2014)'s dataset, -there is a clear difference in size using the average Euclidean distance -from centroid or the sum of variances (overlaps of respectively 0.175 -and 0.159) but no overlap when measuring the size using the sum of -ranges (0.966). However, for the Hopkins and Pearson (2016)'s dataset, -this pattern is reversed (no clear differences for the average Euclidean -distance from centroid or the sum of variances - 0.701 and 0.865 -respectively - but a clear difference for the sum of ranges (0). For -each dataset, the absolute differences between each groups is not -consistent depending on the measures. For example, in Hopkins and -Pearson (2016)'s dataset, the orange group's mean is clearly higher than -the blue one when measuring the sum of ranges (0) and the inverse is -true when measuring the average displacement (0). - -\section{Discussion}\label{discussion} - -Here we tested 25 measures of trait space occupancy on simulated and -empirical datasets to assess how each measure captures changes in trait -space size, density and position. Our results show that the correlation -between measures can vary both within and between measure categories -(Fig. 3), highlighting the importance of understanding the measure -classification for the interpretation of results. Our simulations show -that different measures capture different types of trait space change -(Table 5), meaning that the use of multiple measures is important for -comprehensive interpretation of trait space occupancy. We also show that -the choice of measure impacts the interpretation of group differences in -empirical datasets (Table 6). - -\paragraph{Measures comparisons}\label{measures-comparisons} - -Measures within the same category of trait space occupancy (size, -density or position) do not have the same level of correlation with each -other. For example, the average Euclidean distance from centroid (size) -is highly correlated to the sum of variances (size - correlation of -0.99) and somewhat correlated with the minimum spanning tree average -distance (density - correlation of 0.66) but poorly with the ellipsoid -volume (size - correlation of 0.17) and the minimum spanning tree -distances evenness (density - correlation of -0.05). Furthermore, the -fact that we have such a range of correlations for normal distributions -suggests that each measure can capture different summaries of space -occupancy ranging from obvious differences (for measures not strongly -correlated) to subtle ones (for measures strongly correlated). - -\paragraph{Space shifting}\label{space-shifting-1} - -Most measures capture no changes in space occupancy for the ``null'' -(random) space reduction (in grey in Table 5). This is a desirable -behaviour for space occupancy measures since it will likely avoid false -positive errors in studies that estimate biological processes from space -occupancy patterns (e.g.~convergence Marcy et al. 2016, life history -traits Healy et al. (2019)). However, the average nearest neighbour -distance and the sum of ranges have a respectively positive and negative -``null'' median. \textcolor{blue}{In itself this is -not necessarily a negative} property but it should be kept in mind that -even random processes can increase or decrease these measures' values. - -For changes in size, the sum of variances and the average Euclidean -distance from centroid are good descriptors (Table 5). However, as -illustrated in the 2D examples in Fig. 2-B only the blue change results -(Table 5) should not result in a direct change in overall size because -the trait space is merely ``hollowed'' out. That said, ``hollowing'' is -harder to conceptualise in many dimensions and the measures can still be -interpreted for comparing groups (orange has a smaller volume than -blue). - -The average nearest neigbhour distance and the minimum spanning tree -average distance consistently detect changes in density with more -precision for low density trait spaces (in blue in Table 5). However, we -can observe some degree of correlation between the changes in density -and the changes in size for most measure picking either signal. This -could be due to the use of normally distributed spaces where a change in -density often leads to a change in size. This is not necessarily the -case with empirical data. - -Regarding the changes in position, only the average displacement measure -seems able to distinguish between a random change and a displacement of -the trait space (Table 5). However, the average displacement measure -does not distinguish between positive or negative displacement: this -might be due to the inherent complexity of \emph{position} in a -multidimensional trait space. - -\paragraph{Empirical examples}\label{empirical-examples-1} - -Although most differences are fairly consistent within each dataset with -one group having a higher space occupancy score than the other for -multiple measures, this difference can be more or less pronounced within -each dataset (ranging from no to nearly full overlap - BC -\(\in(0;0.995)\)) and sometimes even reversed. This indicates that -opposite conclusions can be drawn from a dataset depending on which -space occupancy measure is considered. For example, in Wright (2017), -crinoids after the Ordovician mass extinction have a higher median -measure value for all measures but for the average displacement. These -differences depending on the measures are also more pronounced in the -empirical datasets where the observations per group are unequal (Hopkins -and Pearson 2016; Healy et al. 2019). - -\subsubsection{Caveats}\label{caveats} - -While our simulations are useful to illustrate the behaviour of diverse -space occupancy measures, they have several caveats. First, the -simulated observations in the trait spaces are independent. This is not -the case in biology where observations can be spatially (Jones et al. -2015) or phylogenetically correlated (e.g. Beck and Lee 2014). Second, -the algorithm used to reduce the trait -\textcolor{blue}{ spaces might } not always -accurately reflect changes. This might favour some specific measures -over others, in particular for the changes in density that modify the -nearest neighbour density rather than changing the global density. This -algorithmic choice was made in order to not confound changes in density -along with changes in size. However, the results presented here probably -capture the general behaviour of each measure since results are -consistent between the simulated and empirical analysis. - -\textcolor{blue}{ Furthermore, we did not take into -account the effect of sampling on space occupancy measurements (but see -additional results with 80\% and 20\% space reduction in the -supplementary materials 4). In fact, sampling has been previously shown -to have an effect on measurements depending on range or volumes -(e.g.~the sum of ranges or the hypervolume Ciampaglio et al. (2001)). -This effect is especially expected to be acerbated in macroevolutionary -studies when using the fossil record (Brocklehurst et al. 2013) but can -be tackled using rarefaction and bootstrapping techniques (Guillerme -2018). } - -\subsubsection{\texorpdfstring{Using \texttt{moms} to choose the -appropriate -measurements}{Using moms to choose the appropriate measurements}}\label{using-moms-to-choose-the-appropriate-measurements} - -\textcolor{blue}{Therefore, we propose the -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} shiny app to -allow workers to help them choose their set of space occupancy -measurements (and test the caveats mentioned above). \texttt{moms} is an -online graphical user interface to help analyse multidimensional data. -It allows users to upload their dataset of interest (or simulate one -with specific parameters) and measure space occupancy using a variety of -implemented measures (namely, but not only, the ones used in this -study). Furthermore, the package allows simulation of shifts in trait -space occupancy as also presented in this paper to test whether some -measures capture specific changes in space. However, \texttt{moms} is -not a tool for analysing multidimensional data \emph{per se} but rather -for helping workers to chose the space occupancy measure most -appropriated to their data and question. To run multidimensional -analysis, we suggest using dedicated \texttt{R} packages (such as - but -not limited to: Oksanen et al. (2007), Bonhomme et al. (2014), Cardoso -et al. (2015), Guillerme (2018)). } - -\subsubsection{Conclusions}\label{conclusions} - -\textcolor{blue}{We insist that although no measure is -objectively better than the next one, some can be more problematic than -other in specific contexts. For example, the results for the Sum of -Ranges, Minimum spanning tree average distances, and to a lesser extent -average nearest neighbour distances produced results in the reduced -space often similar to the randomly reduced spaces (Table 5). This does -not make them ``bad'' measures but rather heavily context dependent. -Regardless, we believe that workers should identify the most appropriate -measures based on their trait space properties as well as their specific -biological question. We believe this could be fostered by following -these several suggestions: } - -First, we suggest using multiple measures to tackle different aspects of -the trait space. This follows the same logical thinking that the mean -might not be sufficient to describe a distribution (e.g.~the variance -might be a good additional descriptor). Although using multiple measures -is not uncommon in macroevolutionary studies (e.g. Halliday and Goswami -2016) or in ecology (Mammola 2019), they often do no cover more than one -of the three categories of trait space measures -\textcolor{blue}{(but see the recent work of Carmona -et al. (2019) and Mammola and Cardoso (2020)).} - -Second, we suggest selecting the measures that best address the -biological question at hand. If one studies an adaptive radiation in a -group of organisms, it is worth thinking what would be the expected null -model: would the group's size increase (radiation in all directions), -would it increase in density (niche specialisation) or would it shift in -position (radiation into a new set of niches)? - -Third, we suggest not naming measures after the biological aspect they -describe which can be vague (e.g. ``disparity'' or ``functional -dispersion'') but rather after what they are measuring and why (e.g. -``we used sum of ranges to measure the space size''). We believe this -will support both a clearer understanding of what \emph{is} measured as -well as better communication between ecology and evolution research -where measures can be similar but have different names. - -Multidimensional analyses have been acknowledged as essential tools in -modern biology but they can often be counter-intuitive (Bellman 1957). -It is thus crucial to accurately describe patterns in multidimensional -trait spaces to be able to link them to biological processes. When -summarising trait spaces, it is important to remember that a pattern -captured by a specific space occupancy measure is often dependent on the -properties of the space and of the particular biological question of -interest. We believe that having a clearer understanding of both the -properties of the trait space and the associated space occupancy -measures (e.g.~using -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}) as well as -using novel space occupancy measures to answer specific questions will -be of great use to study biological processes in a multidimensional -world. - -\section{Acknowledgements}\label{acknowledgements} - -We thank Natalie Jones and Kevin Healy for helping with the empirical -datasets and Stefano Mammola and Neil Brocklehurst for their positive -and encouraging reviews. We acknowledge funding from the Australian -Research Council DP170103227 and FT180100634 awarded to VW. - -\section{Authors contributions}\label{authors-contributions} - -TG, MNP, AEM and VW designed the project. TG and AEM collected the -empirical dataset. TG ran the analyses and designed the software. TG, -MNP, AEM and VW wrote the manuscript. - -\section{Data Availability, repeatability and -reproducibility}\label{data-availability-repeatability-and-reproducibility} - -The raw empirical data is available from the original papers (Beck and -Lee 2014; Jones et al. 2015, Marcy et al. (2016); Hopkins and Pearson -2016; Wright 2017; Healy et al. 2019). The subsets of the empirical data -used in this analysis are available on figshare -\href{https://doi.org/10.6084/m9.figshare.9943181.v1}{DOI: -10.6084/m9.figshare.9943181.v1}. The modified empirical data are -available in the package accompanying this manuscript -(\texttt{data(moms::demo\_data)}). This manuscript (including the -figures, tables and supplementary material) is repeatable and -reproducible by compiling the vignette of the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. - -\section*{References}\label{references} -\addcontentsline{toc}{section}{References} - -\hypertarget{refs}{} -\hypertarget{ref-beck2014}{} -Beck R.M.D., Lee M.S.Y. 2014. Ancient dates or accelerated rates? -Morphological clocks and the antiquity of placental mammals. Proceedings -of the Royal Society B: Biological Sciences. 281:20141278. - -\hypertarget{ref-cursedimensionality}{} -Bellman R.E. 1957. Dynamic programming. Princeton University Press. - -\hypertarget{ref-bhattacharyya1943}{} -Bhattacharyya A. 1943. On a measure of divergence between two -statistical populations defined by their probability distributions. -Bulletin of the Calcutta Mathematical Society. 35:99--109. - -\hypertarget{ref-blonder2018}{} -Blonder B. 2018. Hypervolume concepts in niche-and trait-based ecology. -Ecography. 41:1441--1455. - -\hypertarget{ref-momocs}{} -Bonhomme V., Picq S., Gaucherel C., Claude J. 2014. Momocs: Outline -analysis using R. Journal of Statistical Software. 56:1--24. - -\hypertarget{ref-brocklehurst2013}{} -Brocklehurst N., Kammerer C.F., Fröbisch J. 2013. The early evolution of -synapsids, and the influence of sampling on their fossil record. -Paleobiology. 39:470--490. - -\hypertarget{ref-bat2015}{} -Cardoso P., Rigal F., Carvalho J.C. 2015. BAT -- biodiversity assessment -tools, an r package for the measurement and estimation of alpha and beta -taxon, phylogenetic and functional diversity. Methods in Ecology and -Evolution. 6:232--236. - -\hypertarget{ref-carmona2019}{} -Carmona C.P., Bello F. de, Mason N.W.H., Lepš J. 2019. Trait probability -density (tpd): Measuring functional diversity across scales based on tpd -with r. Ecology. 100:e02876. - -\hypertarget{ref-ciampaglio2001}{} -Ciampaglio C.N., Kemp M., McShea D.W. 2001. Detecting changes in -morphospace occupation patterns in the fossil record: Characterization -and analysis of measures of disparity. Paleobiology. 71:695--715. - -\hypertarget{ref-close2015}{} -Close R.A., Friedman M., Lloyd G.T., Benson R.B. 2015. Evidence for a -mid-Jurassic adaptive radiation in mammals. Current Biology. - -\hypertarget{ref-diaz2016}{} -Díaz S., Kattge J., Cornelissen J.H., Wright I.J., Lavorel S., Dray S., -Reu B., Kleyer M., Wirth C., Prentice I.C., others. 2016. The global -spectrum of plant form and function. Nature. 529:167. - -\hypertarget{ref-donohue2013}{} -Donohue I., Petchey O.L., Montoya J.M., Jackson A.L., McNally L., Viana -M., Healy K., Lurgi M., O'Connor N.E., Emmerson M.C. 2013. On the -dimensionality of ecological stability. Ecology Letters. 16:421--429. - -\hypertarget{ref-endler2005}{} -Endler J.A., Westcott D.A., Madden J.R., Robson T. 2005. Animal visual -systems and the evolution of color patterns: Sensory processing -illuminates signal evolution. Evolution. 59:1795--1818. - -\hypertarget{ref-foote1992}{} -Foote M. 1992. Rarefaction analysis of morphological and taxonomic -diversity. Paleobiology. 18:1--16. - -\hypertarget{ref-grant2006}{} -Grant P.R., Grant B.R. 2006. Evolution of character displacement in -darwins finches. Science. 313:224--226. - -\hypertarget{ref-disprity}{} -Guillerme T. 2018. dispRity: A modular R package for measuring -disparity. Methods in Ecology and Evolution. 9:1755--1763. - -\hypertarget{ref-halliday2015}{} -Halliday T.J.D., Goswami A. 2016. Eutherian morphological disparity -across the end-cretaceous mass extinction. Biological Journal of the -Linnean Society. 118:152--168. - -\hypertarget{ref-geiger2008}{} -Harmon L.J., Weir J.T., Brock C.D., Glor R.E., Challenger W. 2008. -GEIGER: Investigating evolutionary radiations. Bioinformatics. -24:129--131. - -\hypertarget{ref-healy2019}{} -Healy K., Ezard T.H.G., Jones O.R., Salguero-G'omez R., Buckley Y.M. -2019. Animal life history is shaped by the pace of life and the -distribution of age-specific mortality and reproduction. Nature Ecology -\& Evolution. 2397-334X. - -\hypertarget{ref-hopkins2016}{} -Hopkins M., Pearson K. 2016. Non-linear ontogenetic shape change in -cryptolithus tesselatus (trilobita) using three-dimensional geometric -morphometrics. Palaeontologia Electronica. 19:1--54. - -\hypertarget{ref-hopkins2017}{} -Hopkins M.J., Gerber S. 2017. Morphological disparity. In: Nuno de la -Rosa L., Müller G., editors. Evolutionary developmental biology: A -reference guide. Cham: Springer International Publishing. p. 1--12. - -\hypertarget{ref-jones2015}{} -Jones N.T., Germain R.M., Grainger T.N., Hall A.M., Baldwin L., Gilbert -B. 2015. Dispersal mode mediates the effect of patch size and patch -connectivity on metacommunity diversity. Journal of Ecology. -103:935--944. - -\hypertarget{ref-lalibertuxe92010}{} -Laliberté É., Legendre P. 2010. A distance-based framework for measuring -functional diversity from multiple traits. Ecology. 91:299--305. - -\hypertarget{ref-legendre2012}{} -Legendre P., Legendre L.F. 2012. Numerical ecology. Elsevier. - -\hypertarget{ref-mammola2019}{} -Mammola S. 2019. Assessing similarity of n-dimensional hypervolumes: -Which metric to use? Journal of Biogeography. 0. - -\hypertarget{ref-mammola2020}{} -Mammola S., Cardoso P. 2020. Functional diversity metrics using kernel -density n-dimensional hypervolumes. bioRxiv. - -\hypertarget{ref-mammola2019b}{} -Mammola S., Cardoso P., Culver D.C., Deharveng L., Ferreira R.L., Fišer -C., Galassi D.M.P., Griebler C., Halse S., Humphreys W.F., Isaia M., -Malard F., Martinez A., Moldovan O.T., Niemiller M.L., Pavlek M., -Reboleira A.S.P.S., Souza-Silva M., Teeling E.C., Wynne J.J., Zagmajster -M. 2019. Scientists' Warning on the Conservation of Subterranean -Ecosystems. BioScience. 69:641--650. - -\hypertarget{ref-marcy2016}{} -Marcy A.E., Hadly E.A., Sherratt E., Garland K., Weisbecker V. 2016. -Getting a head in hard soils: Convergent skull evolution and divergent -allometric patterns explain shape variation in a highly diverse genus of -pocket gophers (thomomys). BMC evolutionary biology. 16:207. - -\hypertarget{ref-oksanen2007vegan}{} -Oksanen J., Kindt R., Legendre P., O'Hara B., Stevens M.H.H., Oksanen -M.J., Suggests M. 2007. The vegan package. Community ecology package. -10:631--637. - -\hypertarget{ref-qiao2015}{} -Qiao H., Soberón J., Peterson A.T. 2015. No silver bullets in -correlative ecological niche modelling: Insights from testing among many -potential algorithms for niche estimation. Methods in Ecology and -Evolution. 6:1126--1136. - -\hypertarget{ref-psych}{} -Revelle W. 2018. Psych: Procedures for psychological, psychometric, and -personality research. Evanston, Illinois: Northwestern University. - -\hypertarget{ref-ruta2013}{} -Ruta M., Angielczyk K.D., Fröbisch J., Benton M.J. 2013. Decoupling of -morphological disparity and taxic diversity during the adaptive -radiation of anomodont therapsids. Proceedings of the Royal Society of -London B: Biological Sciences. 280. - -\hypertarget{ref-sedgewick1990}{} -Sedgewick R. 1990. Algorithms in c. Addison-Wesley, Reading. - -\hypertarget{ref-tucker2017}{} -Tucker C.M., Cadotte M.W., Carvalho S.B., Davies T.J., Ferrier S., Fritz -S.A., Grenyer R., Helmus M.R., Jin L.S., Mooers A.O., Pavoine S., -Purschke O., Redding D.W., Rosauer D.F., Winter M., Mazel F. 2017. A -guide to phylogenetic metrics for conservation, community ecology and -macroecology. Biological Reviews. 92:698--715. - -\hypertarget{ref-villuxe9ger2008}{} -Villéger S., Mason N.W.H., Mouillot D. 2008. New multidimensional -functional diversity indices for a multifaceted framework in functional -ecology. Ecology. 89:2290--2301. - -\hypertarget{ref-wills2001}{} -Wills M.A. 2001. Morphological disparity: A primer. In: Adrain J.M., -Edgecombe G.D., Lieberman B.S., editors. Fossils, phylogeny, and form. -Springer US. p. 55--144. - -\hypertarget{ref-wright2017}{} -Wright D.F. 2017. Phenotypic innovation and adaptive constraints in the -evolutionary radiation of palaeozoic crinoids. Scientific Reports. -7:13745. - -\end{document} diff --git a/inst/EcoEvol/shiftingspace_resubmit_clean.tex b/inst/EcoEvol/shiftingspace_resubmit_clean.tex deleted file mode 100644 index 599e3f9..0000000 --- a/inst/EcoEvol/shiftingspace_resubmit_clean.tex +++ /dev/null @@ -1,1870 +0,0 @@ -\documentclass[]{article} -\usepackage{xcolor} -\usepackage{lineno} -\usepackage{lmodern} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\usepackage{fixltx2e} % provides \textsubscript -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} -\fi -% use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -% use microtype if available -\IfFileExists{microtype.sty}{% -\usepackage[]{microtype} -\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\PassOptionsToPackage{hyphens}{url} % url is loaded by hyperref -\usepackage[unicode=true]{hyperref} -\hypersetup{ - pdftitle={Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?}, - pdfauthor={Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker}, - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage[margin=1in]{geometry} -\usepackage{longtable,booktabs} -% Fix footnotes in tables (requires footnote package) -\IfFileExists{footnote.sty}{\usepackage{footnote}\makesavenoteenv{long table}}{} -\usepackage{graphicx,grffile} -\makeatletter -\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} -\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} -\makeatother -% Scale images if necessary, so that they will not overflow the page -% margins by default, and it is still possible to overwrite the defaults -% using explicit options in \includegraphics[width, height, ...]{} -\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -% set default figure placement to htbp -\makeatletter -\def\fps@figure{htbp} -\makeatother - - -\title{Shifting spaces: which disparity or dissimilarity measurement best -summarise occupancy in multidimensional spaces?} -\author{Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker} -\date{2020-05-01} - -\linespread{1.6} - -\begin{document} -\maketitle - -\modulolinenumbers[1] % just after the \begin{document} tag -\linenumbers - -\section{Abstract}\label{abstract} - -Multidimensional analysis of traits are now common in ecology and -evolution and are based on trait spaces in which each dimension -summarises the observed trait combination (a morphospace or an -ecospace). Observations of interest will typically occupy a -\textcolor{black}{subset} of this space, and -researchers will calculate one or more measures to quantify how -organisms inhabit that space. In macroevolution and ecology these -measures \textcolor{black}{called disparity or -dissimilarity metrics and are} generalised as space occupancy measures. -Researchers use these measures to investigate how space occupancy -changes through time, in relation to other groups of organisms, and in -response to global environmental changes. However, the mathematical and -biological meaning of most space occupancy measures is vague with the -majority of widely-used measures lacking formal description. - -Here we propose a broad classification of space occupancy measures into -three categories that capture changes in size, density, or position. We -study the behaviour of 25 measures to changes in trait space size, -density and position on simulated and empirical datasets. We find that -no measure describes all of trait space aspects but that some are better -at capturing certain aspects. Our results confirm the three broad -categories (size, density and position) and allow us to relate changes -in any of these categories to biological phenomena. - -Because the choice of space occupancy measures is specific to the data -and question, we introduced -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool to -both visualise and capture changes in space occupancy for any -measurement. \href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} -is designed to help workers choose the right space occupancy measures, -given the properties of their trait space and their biological question. -By providing guidelines and common vocabulary for space occupancy -analysis, we hope to help bridging the gap in multidimensional research -between ecology and evolution. - -\section{Introduction}\label{introduction} - -Groups of species and environments share specific, recognisable, -correlated characteristics: guilds or biomes with shared phenotypic, -physiological, phylogenetic or behavioural traits. Organisms or -environments should therefore be studied as a set of traits rather than -some specific traits in isolation (Donohue et al. 2013; Hopkins and -Gerber 2017). Biologists increasingly been using ordination techniques -(see Legendre and Legendre 2012 for a summary) to create -multidimensional trait spaces to either explore properties of data or -test hypotheses (e.g. Oksanen et al. 2007; Blonder 2018; Guillerme -2018). For example, in palaeobiology, Wright (2017) used trait spaces to -study how groups of species' characteristics change through time; in -ecology, Jones et al. (2015) \textcolor{black}{ studied -} evidence of competition by looking at trait overlap between two -populations. \textcolor{black}{ While different fields -use a different set of terms for such approaches (Table 1), they -actually focus on the same mathematical objects: matrices with columns -representing an original or transformed trait value and rows -representing observations (taxon, field site, etc.; Guillerme 2018). } - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llll@{}} -\toprule -\begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Mathematics\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Macroevolution\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -This paper\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix (\(n \times d\)) with a structural relation between rows and -columns\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -\textcolor{black}{ Functional space, morphospace }, -etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Morphospace, traitspace, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -trait space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Rows (\emph{n})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, field sites, environments, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, specimen, populations, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Columns (\emph{d})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -dimensions\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix subset (\(m \times d\); \(m \leq n\))\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Treatments, phylogenetic group (clade), etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Clades, geological stratum, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -group\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Statistic \textcolor{black}{ (i.e.~a measure) }\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity index or metric, hypervolume, functional diversity, -\textcolor{black}{ etc. }\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity metric or index\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -space occupancy measure\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Multidimensional analysis\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity analysis, trait analysis, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity analysis, disparity-through-time, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -multidimensional analysis\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 1: \textcolor{black}{Different terms are used for -equivalent measures in} mathematics, ecology and macroevolution. - -\renewcommand\baselinestretch{1.6}\selectfont - -Ecologists and evolutionary biologists often use trait spaces with -respect to the same fundamental questions: are groups occupying the same -amount of trait space? Do some groups contain more species than others -in the same amount of trait space? Are some specific factors correlated -with different patterns of trait space occupancy? Because of the -multidimensional nature of these trait spaces, it is often not possible -to study them using bi- or tri-variate techniques (Díaz et al. 2016; -Hopkins and Gerber 2017; Mammola 2019). Studying the occupancy of trait -spaces is done using disparity indices in macroevolution (Wills 2001; -Hopkins and Gerber 2017; Guillerme 2018) or comparing hypervolumes in -ecology (Donohue et al. 2013; Díaz et al. 2016; Blonder 2018; Mammola -2019). Despite the commonalities between the measures used in ecology -and evolution (which are often metric but don't necessarily need to be), -surprisingly little work has been published on their behaviour (but see -Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). - -Different occupancy measures capture different aspects of trait space -(Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). -\textcolor{black}{This} may be widely-known, but to -our knowledge it is infrequently mentioned in peer-reviewed papers. -First, space occupancy measures are often named as the biological aspect -they are describing (``disparity'', ``functional diversity'') rather -than what they are measuring (e.g.~the product of ranges), which -obscures the differences and similarities between studies. Second, in -many studies in ecology and evolution, authors have focused on measuring -the size of the trait space (e.g.~ellipsoid volume Donohue et al. 2013; -hypervolume Díaz et al. 2016; Procrustes variance Marcy et al. 2016; -product of variance Wright 2017). However, the size of the trait space -only represents one aspects of occupancy, disregarding -\textcolor{black}{other measures} such as the density -(Harmon et al. 2008) or position (Wills 2001; Ciampaglio et al. 2001). -For example, if two groups have the same size, this can support certain -biological conclusions. Yet, an alternative aspect of space occupancy -may indicate that the groups' position are different, leading to a -different biological conclusion (e.g.~the groups are equally diverse but -occupy different niches). Using measures that only capture one aspect of -the trait space may restrain the potential of multidimensional analysis -(Villéger et al. 2008). - -Here we propose a broad classification of space occupancy measures as -used across ecology and evolution and study their power to detect -changes in trait space occupancy in simulated and empirical data. -\textcolor{black}{Note this does not account whether or -not it is possible for a space to be occupied (e.g., some spaces may -represent biologically impossible shapes); this, however, may be -important in some cases, such as testing whether a region is infinite or -not.} We provide an assessment of each broad type of space occupancy -measures along with a unified terminology to foster communication -between ecology and evolution. Unsurprisingly, we found no one measure -describes all changes \textcolor{black}{in space} and -that the results from each measures are dependent on the characteristics -of the space and the hypotheses. - -\textcolor{black}{There can be an infinite number of -measures and that it is thus impossible to propose a comprehensive -analysis for all the measures properties respective to how they measure -changes in trait space. We therefore propose -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool for -researchers to design, experiment and visualise their own space -occupancy measure tailored for their project. The tool will help -researchers understand the ``null'' behaviour of the measures of -interest.} - -\subsection{Space occupancy measures}\label{space-occupancy-measures} - -In this paper, we define trait spaces as any matrix where rows are -observations and columns are traits, where both observations and traits -are structurally related (e.g.~there is a phylogenetic relation between -observations - and traits, etc.). These traits can widely vary in number -and types: they could be coded as discrete (e.g.~presence or absence of -a bone; Beck and Lee 2014; Wright 2017), continuous measurements -(e.g.~leaf area; Díaz et al. 2016) or more sophisticated measures -(Fourier ellipses; Bonhomme et al. 2014; e.g.~landmark position; Marcy -et al. 2016). Traits can also be measured by using relative observations -(e.g.~community compositions; Jones et al. 2015) or distance between -observations (e.g. Close et al. 2015). However, regardless of the -methodology used to build a trait space, three broad occupancy measures -can be used: the \textbf{size} which approximates the amount of space -occupied, the \textbf{density} which approximates the distribution in -space and the \textbf{position} which approximates the location in space -(Fig. 1; Villéger et al. 2008). Of course any combination of these three -aspects is always possible. - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measures_types-1.pdf} -\caption{different type of information captured by space occupancy -measures: (A) size, (B) density and (C) position.} -\end{figure} - -\paragraph{1. Size}\label{size} - -Size captures the spread of a group in the trait space. They can be -interpreted as the amount of the trait space that is occupied by -observations. Typically, larger values for such measures indicate the -presence of more extreme trait combinations. For example, if group A is -bigger than B, the observations in A achieve more extreme trait -combinations than in B. This type of measure is widely used in both -ecology (e.g.~the hypervolume; Blonder 2018) and in evolution (e.g.~the -sum or product of ranges or variances; Wills 2001). - -Although size measures are suitable indicators of a group's trait space -occupancy, they are limited to comparing the range of trait-combinations -between groups. Size measures do not take into account the distribution -of the observations within a group and can often be insensitive to -unoccupied ``holes'' in the trait space (overstimating the size; Blonder -(2018)). They can make it difficult to determine whether all the -observations are on the edge of the group's distribution or whether the -size is simply driven by outliers. - -\paragraph{2. Density}\label{density} - -Density gives an indication of the quantity of observations in the trait -space. They can be interpreted as the distribution of the observations -\emph{within} a group in the trait space. Groups with higher density -contain more observations (i.e.~more observations per approximation of -size) that will tend to be more similar to each other. For example, if -group A is greater is size than group B and both have the same density -(observations are equally distant within each group), similar mechanisms -could be driving both groups' trait space occupancy. Indeed, this -pattern could suggest that A is older and has had more time to achieve -more extreme trait combinations under essentially the same process as -younger, smaller group B (Endler et al. 2005). Note that density based -measures can be sensitive to sampling. Density measures are less common -compared to size measures, but they are still used in both ecology -(e.g.~the minimum spanning tree length; Oksanen et al. 2007) and -evolution (e.g.~the average pairwise distance; Harmon et al. 2008). - -\paragraph{3. Position}\label{position} - -Position captures where a group lies in trait space. They can be -interpreted as where a group lies in the trait space either relative to -the space itself or relative to another group. For example, if group A -has a different position than group B, A will have a different -trait-combination than in B. - -Position measures may be harder to interpret in multidimensional spaces. -In a 2D space, two groups can be equally distant from a fixed point but -in different parts of the space (left, right, up, or down - with the -amount of parts of space increasing with dimensions). However, when -thinking about unidimensional data, this measure is obvious: two groups -A or B could have the same variance (size) with the same number of -observations (density) but could have two different means and thus be in -different positions. These measures are used in ecology to compare the -position of two groups relative to each other (Mammola 2019). - -\textcolor{black}{ Note that this classification into -size, density and position bears some similarities with Tucker et al. -(2017) classifying phylogenetic diversity measurements into richness, -divergence and regularity categories. However, while Tucker et al. -(2017) based their classification on the mathematical operation inherent -to each metrics (the sum for richness, the mean for divergence and the -variance for regularity), our three broad classifications are based on -their geometric properties regardless of the formula of each metric -(e.g.~the size of a space can be calculated using a sum, mean or/and -variance). } - -\subsection{No measure to rule them all: benefits of considering -multiple -measures}\label{no-measure-to-rule-them-all-benefits-of-considering-multiple-measures} - -The use of multiple measurements to assess trait space occupancy -provides a more detailed characterisation of occupancy changes. If the -question is to look at how space occupancy changes in response to mass -extinction, using a single space occupancy measure can miss part of the -picture: a change in size could be decoupled from a change in position -or density in trait space. For example, the Cretaceous-Paleogene -extinction (66 million years ago) shows an increase in size of the -mammalian trait space (adaptive radiation; Halliday and Goswami 2016) -but more specific questions can be answered by looking at other aspects -of trait space occupancy: does the radiation expand on previously -existing morphologies (elaboration, increase in density; Endler et al. -2005) or does it explore new regions of the trait space (innovation, -change in position; Endler et al. 2005)? Similarly, in ecology, if two -groups have the same trait space size, -\textcolor{black}{the differences in density within -these two groups is potentially illuminating:} different selection -pressure can lead to different density within equally sized groups. -\textcolor{black}{This can also be extended to more -complex ecological concepts such as niche modelling (Qiao et al. -2015).} - -Here, we provide the first interdisciplinary review of 25 space -occupancy measures that uses the broad classification of measures into -size, density and position to capture pattern changes in trait space. We -assess the behaviour of measures using simulations and six -interdisciplinary empirical datasets covering a wide range of potential -data types and biological questions. We also introduce a tool for -measuring occupancy in multidimensional space -(\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}), which is -a user-friendly, open-source, graphical interface to allow the tailored -testing of measurement behaviour for any use case. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} will allow -workers to comprehensively assess the properties of their trait space -and the measures associated with their specific biological question. - -\section{Methods}\label{methods} - -We tested how 25 space occupancy measures relate to each other, are -affected by modifications of traits space and affect group comparisons -in empirical data: - - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - We simulated 13 different spaces with different sets of parameters; -\item - We transformed these spaces by removing 50\% of the observations - following four different scenarios corresponding to different - empirical scenarios: randomly, by - \textcolor{black}{size} (e.g.~expansion or reduction - of niches), by density (e.g.~different degrees of competition within a - guild), and by position (e.g.~ecological niche shift). -\item - We measured occupancy on the resulting transformed spaces using eight - different space occupancy measures; -\item - We applied the same space occupancy measures to six empirical datasets - (covering a range of disciplines and a range of dataset properties). -\end{enumerate} - -Note that the paper contains the results for only eight measures which -were selected as representative of common measures covering the size, -density and position trait space aspects. The results for an additional -17 measures is available in the supplementary material 4. - -\subsection{Generating spaces}\label{generating-spaces} - -We generated trait spaces using the following combinations of size, -distributions, variance and correlation: - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -space name\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.31\columnwidth}\raggedright\strut -distribution(s)\strut -\end{minipage} & \begin{minipage}[b]{0.21\columnwidth}\raggedright\strut -dimensions variance\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -correlation\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform (min = -0.5, max = 0.5)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal (mean = 0, sd = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D with random distributions\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCA-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Multiplicative\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCO-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Additive\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 2: different simulated space distribution. \emph{Name} of the -simulated space; \emph{dimensions} of the matrix (row*columns); -\emph{distribution(s)} of the data on each dimensions (for the `Random', -dimensions are randomly chosen between Normal, Uniform or Lognormal); -\emph{dimension variance}: distribution of the variance between -dimensions (when equal, the dimensions have the same variance); -\emph{correlation} between dimensions. - -\renewcommand\baselinestretch{1.6}\selectfont - - -The differences in trait space sizes (200 elemeents for 3, 15, 50 or 150 -dimensions) reflects the range found in literature (e.g. Hopkins and -Gerber 2017; Mammola 2019). We used a range of distributions (uniform, -normal or a random combination of uniform, normal and lognormal) to test -the effect of observation distributions on the measurements. We used -different levels of variance for each dimensions in the spaces by making -the variance on each dimension either equal -(\(\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}\)) or decreasing -(\(\sigma_{D1} < \sigma_{D2} < \sigma_{Di}\)) with the decreasing factor -being either multiplicative (using the cumulative product of the inverse -of the number of dimensions: \(\prod_i^d(1/d)\)) or additive -(\(\sum_i^d(1/d)\)). Both reductions of variance are used to illustrate -the properties of ordinations where the variance decreases per -dimensions (and normal win Multidimensional Scaling - MDS, PCO or PCoA; -e.g. Close et al. 2015; lognormal in principal components analysis - -PCA; e.g. Marcy et al. 2016; Wright 2017; Healy et al. 2019). Finally, -we added a correlation parameter to illustrate the effect of -co-linearity between traits (especially in non-ordinated trait spaces). -We repeated the simulation of each trait space 20 times (resulting in -260 spaces). - -\subsection{Spatial occupancy -measures}\label{spatial-occupancy-measures} - -We then calculated eight different measures on the resulting transformed -spaces, including a new one, the average displacement, which we expect -to be influenced by changes in trait space position. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Name\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Definition\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Captures\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Source\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Notes\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Laliberté and Legendre (2010)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional dispersion (FDis - without abundance)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\sigma^{2}{k_i}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -common measure used in palaeobiology (Ciampaglio et al. 2001; Wills -2001)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -more sensitive to outliers than the sum of variances\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Donohue et al. (2013)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -less sensitive to outliers than the convex hull hypervolume (Díaz et al. -2016; Blonder 2018)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum(\text{branch length})}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sedgewick (1990)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -similar to the unscaled functional evenness (Villéger et al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Villéger et al. (2008)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional evenness without weighted abundance (FEve; Villéger et -al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sqrt{\sum_{i}^{n}{min({q}_{i}-p_{i})^2}})\times \frac{1}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the density of pairs of observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average displacement\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Position\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -This paper\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the ratio between the observations' position from their centroid and the -centre of the trait space (coordinates: 0, 0, 0, \ldots{}). A value of 1 -indicates that the observations' centroid is the centre of the trait -space\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 3: List of measures with \emph{n} being the number of -observations, \emph{d} the total number of dimensions, \emph{k} any -specific row in the matrix, \emph{Centroid} being their mean and -\(\sigma^{2}\) their variance. \(\Gamma\) is the Gamma distribution and -\(\lambda_{i}\) the eigenvalue of each dimension and \({q}_{i}\) and -\(p_{i}\) are any pairs of coordinates. - -\renewcommand\baselinestretch{1.6}\selectfont - -We selected these eight space occupancy measures to illustrate how they -capture different aspects of space occupancy (not as an expression of -our preference). These measures are specific to Euclidean and isotropic -trait spaces (which is not necessary for all measures). The -supplementary material 4 contains the same analysis as described below, -performed on 17 measures. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -exploration into the effect of many more measures as well as the -customisation of measures by combining them or using user-designed -functions. - -\subsection{Measure comparisons}\label{measure-comparisons} - -We compared the space occupancy measures correlations across all -simulations between each pair of measures to assess their captured -signal (Villéger et al. 2008; Laliberté and Legendre 2010). We used the -measures on the full 13 trait spaces described above. We then scaled the -results and measured the pairwise Pearson correlation to test whether -measures were capturing a similar signals or not using the -\texttt{psych} package (Revelle 2018). - -\subsection{Changing space}\label{changing-spaces} - -To assess how the measures responded to changes within trait spaces, we -removed 50\% of observations each time using the following algorithms: - -\begin{itemize} -\item - \textbf{Randomly:} by randomly removing 50\% of observations (Fig. - 2-A). This reflects a ``null'' biological model of changes in trait - space: the case when observations are removed regardless of their - intrinsic characteristics. For example, if diversity is reduced by - 50\% but the space size remains the same, there is a decoupling - between diversity and space occupancy (Ruta et al. 2013). Our selected - measures are expected to not be affected by this change. -\item - \textbf{\textcolor{black}{Size}:} by removing - observations within a distance from the centre of the trait space - lower or greater than a radius \(\rho\) (where \(\rho\) is chosen such - that 50\% observations are selected) generating two limit removals: - \emph{maximum} and \emph{minimum} (respectively in orange and blue; - Fig. 2-B). This can reflect a strict selection model where - observations with trait values below or above a threshold are removed - leading to an expansion or a contraction of the trait space. - \textcolor{black}{This type of change could be due to - habitat destruction (e.g. Mammola et al. 2019) or to mass extinctions - (e.g. Wright 2017).} Size measures are expected to be most affected - by this change. -\end{itemize} - -\begin{itemize} -\item - \textbf{Density:} by removing any pairs of point with a distance \(D\) - from each other where (where \(D\) is chosen such that 50\% - observations are selected) generating two density removals: - \emph{high} and \emph{low} (respectively in orange and blue; Fig. - 2-C). This can reflect changes within groups in the trait space due to - ecological factors (e.g.~niche repulsion resulting in lower density; - Grant and Grant 2006). \textcolor{black}{This type of - change could be due to accelerated rates of evolution (Close et al. - 2015) or to differences in modes of life in macroevolution (e.g. Healy - et al. 2019).} Density measures are expected to be most affected by - this change. -\item - \textbf{Position:} by removing points similarly as for - \textbf{\textcolor{black}{Size}} but using the - distance from the furthest point from the centre generating two - position removals: \emph{positive} and \emph{negative} (respectively - in orange and blue; Fig. 2-D). This can reflect global changes in - trait space (e.g.~if an entire group remaining diverse but occupying a - different niche). \textcolor{black}{This type of - change could be due changes in evolutionary trajectories (Endler et - al. 2005) or to differences in ecosystem compositions (e.g. Jones et - al. 2015).} Position measures are expected to be most affected by - this change. -\end{itemize} - -The algorithm to select \(\rho\) or \(D\) is described in the -Supplementary material 1. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_reduce_space-1.pdf} -\caption{} -\end{figure} - -Figure 2: different type of space reduction. Each panel displays two -groups of 50\% of the data points each. Each group (orange and blue) are -generated using the following algorithm: A - randomly (the removed -elements are displayed in black and the analysed ones in grey); B - by -size (maximum and minimum limit); C - by density (high and low); and D - -by position (positive and negative). Panel E et F represents two typical -display of the reduction results displayed in Table 5: the dots -represent the median space occupancy values across all simulations for -each scenario of trait space change (Table 2), the solid and dashed line -respectively the 50\% and 95\% confidence intervals. Results in grey are -the random 50\% reduction (panel A). Results in blue and orange -represent the opposite scenarios from panels B, C, and D. The displayed -value is the amount of overlap (Bhattacharrya Coefficient) between the -blue or orange distributions and the grey one. Panel E and F shows -respectively the ``ideal'' and ``worst'' results for any type of -measures, where the space occupancy measurement respectively manages or -fails to captures a specific type of reduction (i.e.~size, position or -density; Table 5). - - -\renewcommand\baselinestretch{1.6}\selectfont - -Because occupancy measures are dependent on the space, we scaled and -centred them between -1 and 1 to make them comparable (by subtracting -the observed occupancy without reduction to all the measures of the -reduced spaces and then divided it by the maximum observed occupancy). A -value of 0 indicates no effect of the space reduction and \(>0\) and -\(<0\) respectively indicates an increase or decrease in the measure -value. We then measured the amount of overlap between the non-random -removals (\textcolor{black}{size}, density and -position) and the random removals using the Bhattacharrya Coefficient -(Bhattacharyya 1943). - -\subsubsection{Measuring the effect of space and -dimensionality}\label{measuring-the-effect-of-space-and-dimensionality} - -Distribution differences and the number of dimensions can have an effect -on the measure results. For example, in a normally distributed space, an -increase in density can often lead to a decrease in size (though this is -not necessarily true if the space is log-normal or uniform). High -dimensional spaces (\textgreater{}10) are subject to the ``curse of -multidimensionality'' (Bellman 1957): data becomes sparser with -increasing number of dimensions. This can have two main consequences: 1) -the probability of overlap between two groups decreases as a product of -the number of dimensions; and 2) the amount of samples needed to -``fill'' the spaces increases exponentially -\href{https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners}{see -this interactive illustration by Toph Tucker}. The ``curse'' can make -the interpretation of high dimensional data counter-intuitive. For -example if a group expands in multiple dimensions (i.e.~increase in -size), the actual hypervolume (\(\prod_{i}^{d} range_{Di}\)) can -decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an -ANOVA (\(occupancy \sim distribution\) and -\(occupancy \sim dimensions\)) by using all spaces with 50 dimensions -and the uniform and normal spaces with equal variance and no correlation -with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing -respectively the effect of distribution and dimensions. The results of -the ANOVAs (F and \emph{p}-values) are reported in Table 5 (full results -in supplementary material 3). - -\subsection{Empirical examples}\label{empirical-examples} - -We analysed the effect of the different space occupancy measures on six -different empirical studies covering a range of fields that employ trait -space analyses. For each of these studies we generated trait spaces from -the data published with the papers. We divided each trait spaces into -two biologically-relevant groups and tested whether the measures -differentiated the groups in different ways. Both the grouping and the -questions were based on a simplified version of the topics of these -papers (with no intention to re-analyse the data and questions). The -procedures to generate the data and the groups varies between studies -and is detailed in the supplementary materials 2. - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}llllllll@{}} -\toprule -\begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -study\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -field\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -taxonomic group\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -traits\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -trait space\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -groups\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -question\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Beck and Lee (2014)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Mammalia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -106*105\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -52 crown vs.~54 stem\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are crown mammals more disparate than stem mammals?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Wright (2017)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Crinoidea\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -42*41\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -16 before vs.~23 after\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Is there a difference in disparity before and after the Ordovician mass -extinction?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Marcy et al. (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Rodentia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -skull 2D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -454*134\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -225 \emph{Megascapheus} vs.~229 \emph{Thomomys}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are two genera of gopher morphologically distinct?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Hopkins and Pearson (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Trilobita\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -3D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -46*46\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -36 adults vs.~10 juveniles\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are juvenile trilobites a subset of adult ones -\textcolor{black}{in trait space}?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Jones et al. (2015)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Plantae\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Communities species compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Jaccard distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -48*47\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -24 aspens vs.~24 grasslands\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Is there a difference in species composition between aspens and -grasslands?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Healy et al. (2019)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Animalia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Life history traits\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of continuous traits (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -285*6\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -83 ecthotherms vs.~202 endotherms\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Do endotherms have more diversified life history strategies than -ectotherms?\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 4: details of the six empirical trait spaces. - -\renewcommand\baselinestretch{1.6}\selectfont - -For each empirical trait space we bootstrapped each group 500 times -(Guillerme 2018) and applied the eight space occupancy measure to each -pairs of groups. We then compared the means of each groups using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\section{Results}\label{results} - -\subsection{Measure comparisons}\label{measure-comparisons-1} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measure_correlation-1.pdf} -\caption{pairwise correlation between the scaled measures. Numbers on -the upper right corner are the Pearson correlations. The red line are -linear regressions (with the confidence intervals in grey). Av.: -average; dist.: distance; min.: minimum; span.: spanning.} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - -\textcolor{black}{Most measures of space were -positively correlated (Pearson correlation of 0.99 for the average -Euclidean distance from centroid and sum of variance or 0.97 for the -average nearest neighbour distance and minimum spanning tree average -length; Fig. 3). The remaining measures were either somewhat correlated -or had a negative pairwise distribution } (ranging from 0.66 for the -sum of variances and the ellipsoid volume to -0.09 between the average -displacement and the average Euclidean distance from centroid; Fig. 3). -All measures but the ellipsoid volume were normally (or nearly normally) -distributed (Fig. 3). - -\subsection{Space shifting}\label{space-shifting} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llllll@{}} -\toprule -\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Size change\strut -\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut -Density change\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Position change\strut -\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Distribution effect\strut -\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut -Dimensions effect\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 0.924 ; p = 0.449\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.322 ; p = 0.958\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.285 ; p = 0.274\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.478 ; p = 0.873\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 11.119 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 32.307 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 7.215 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 13.486 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.162 ; p = 0.326\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.998 ; p = 0.435\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 8.152 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 29.358 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.478 ; p = 0.207\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.773 ; p = 0.626\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 10.742 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 26.829 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 5: Results of the effect of space reduction, space dimension -distributions and dimensions number of the different space occupancy -measures. The dots represent the median space occupancy values across -all simulations for each scenario of trait space change (Table 2), the -solid and dashed line respectively the 50\% and 95\% confidence -intervals. See Fig. 2 for details on the interpretation of the figures -distributions and values. F-values for distribution effect and -dimensions effect represents respectively the effect of the ANOVAs space -occupancy \textasciitilde{} distributions and space occupancy -\textasciitilde{} dimension represent the ratio of sum squared -difference within and between groups (the higher, the more the factor -has an effect on the measure) and associated \emph{p}-values (0 `***' -0.001 `**' 0.01 `*' 0.05 `.' 0.1 '' 1). This figure illustrates how -different measures can be influenced by different aspects of changes in -the trait space. E.g. the Average Euclidean distance from centroid (row -1) captures mainly changes in size (column 1), but also captures changes -in density (column 2) but does not capture changes in position (column -3). - -\renewcommand\baselinestretch{1.6}\selectfont - -As expected, some different measures capture different aspects of space -occupancy. However, it can be hard to predict the behaviour of each -measure when 50\% of the observations are removed. We observe a clear -decrease in \textcolor{black}{the median measure -value} in less than a third of the space reductions (10/36). - -In terms of change in size, only the average Euclidean distance from -centroid and the sum of variances seem to capture a clear change in both -directions. In terms of change in density, only the minimum spanning -tree average distance and the average nearest neighbour distance seem to -capture a clear change in both directions. And in terms of change in -position, only the average displacement metric seems to capture a clear -change in direction (albeit not in both directions). This is not -surprising, since the notion of positions becomes more and more complex -to appreciate as dimensionality increases (i.e.~beyond left/right, -up/down and front/back). - -\subsection{Empirical example}\label{empirical-example} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllllll@{}} -\toprule -\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Beck and Lee 2014\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -Wright 2017\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Marcy et al. 2016\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Hopkins and Pearson 2016\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Jones et al. 2015\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Healy et al. 2019\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Comparisons (orange \emph{vs.} blue)\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -crown \emph{vs.} stem mammals morphologies\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -crinoids morphologies before \emph{vs.} after the end-Ordovician -extinction\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\emph{Megascapheus} \emph{vs.} \emph{Thomomys} skull shapes\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -adults \emph{vs.} juveniles trilobites cephalon shapes\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -aspens \emph{vs.} grasslands communities compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -ecthotherms \emph{vs.} endotherms life history traits\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-25.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-33.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-41.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-26.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-34.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-42.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-27.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-35.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-43.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-28.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-36.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-44.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-29.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-37.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-45.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-30.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-38.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-46.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-31.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-39.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-47.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-32.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-40.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-48.pdf}\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 6: Comparisons of pairs of groups in different empirical trait -spaces. NAs are used for cases where space occupancy could not be -measured due to the curse of multidimensionality. The displayed values -are the amount of overlap between both groups (Bhattacharrya -Coefficient). - -\renewcommand\baselinestretch{1.6}\selectfont - - -\textcolor{black}{As with the as for the simulations, -there is no measure that summarises all the aspects of distributions for -empirical data.} For all eight measures (except the ellipsoid volume) -we see either one group or the other having a bigger mean than the other -and no consistent case where a group has a bigger mean than the other -for all the measures. For example, in the Beck and Lee (2014)'s dataset, -there is a clear difference in size using the average Euclidean distance -from centroid or the sum of variances (overlaps of respectively 0.175 -and 0.159) but no overlap when measuring the size using the sum of -ranges (0.966). However, for the Hopkins and Pearson (2016)'s dataset, -this pattern is reversed (no clear differences for the average Euclidean -distance from centroid or the sum of variances - 0.701 and 0.865 -respectively - but a clear difference for the sum of ranges (0). For -each dataset, the absolute differences between each groups is not -consistent depending on the measures. For example, in Hopkins and -Pearson (2016)'s dataset, the orange group's mean is clearly higher than -the blue one when measuring the sum of ranges (0) and the inverse is -true when measuring the average displacement (0). - -\section{Discussion}\label{discussion} - -Here we tested 25 measures of trait space occupancy on simulated and -empirical datasets to assess how each measure captures changes in trait -space size, density and position. Our results show that the correlation -between measures can vary both within and between measure categories -(Fig. 3), highlighting the importance of understanding the measure -classification for the interpretation of results. Our simulations show -that different measures capture different types of trait space change -(Table 5), meaning that the use of multiple measures is important for -comprehensive interpretation of trait space occupancy. We also show that -the choice of measure impacts the interpretation of group differences in -empirical datasets (Table 6). - -\paragraph{Measures comparisons}\label{measures-comparisons} - -Measures within the same category of trait space occupancy (size, -density or position) do not have the same level of correlation with each -other. For example, the average Euclidean distance from centroid (size) -is highly correlated to the sum of variances (size - correlation of -0.99) and somewhat correlated with the minimum spanning tree average -distance (density - correlation of 0.66) but poorly with the ellipsoid -volume (size - correlation of 0.17) and the minimum spanning tree -distances evenness (density - correlation of -0.05). Furthermore, the -fact that we have such a range of correlations for normal distributions -suggests that each measure can capture different summaries of space -occupancy ranging from obvious differences (for measures not strongly -correlated) to subtle ones (for measures strongly correlated). - -\paragraph{Space shifting}\label{space-shifting-1} - -Most measures capture no changes in space occupancy for the ``null'' -(random) space reduction (in grey in Table 5). This is a desirable -behaviour for space occupancy measures since it will likely avoid false -positive errors in studies that estimate biological processes from space -occupancy patterns (e.g.~convergence Marcy et al. 2016, life history -traits Healy et al. (2019)). However, the average nearest neighbour -distance and the sum of ranges have a respectively positive and negative -``null'' median. \textcolor{black}{In itself this is -not necessarily a negative} property but it should be kept in mind that -even random processes can increase or decrease these measures' values. - -For changes in size, the sum of variances and the average Euclidean -distance from centroid are good descriptors (Table 5). However, as -illustrated in the 2D examples in Fig. 2-B only the blue change results -(Table 5) should not result in a direct change in overall size because -the trait space is merely ``hollowed'' out. That said, ``hollowing'' is -harder to conceptualise in many dimensions and the measures can still be -interpreted for comparing groups (orange has a smaller volume than -blue). - -The average nearest neigbhour distance and the minimum spanning tree -average distance consistently detect changes in density with more -precision for low density trait spaces (in blue in Table 5). However, we -can observe some degree of correlation between the changes in density -and the changes in size for most measure picking either signal. This -could be due to the use of normally distributed spaces where a change in -density often leads to a change in size. This is not necessarily the -case with empirical data. - -Regarding the changes in position, only the average displacement measure -seems able to distinguish between a random change and a displacement of -the trait space (Table 5). However, the average displacement measure -does not distinguish between positive or negative displacement: this -might be due to the inherent complexity of \emph{position} in a -multidimensional trait space. - -\paragraph{Empirical examples}\label{empirical-examples-1} - -Although most differences are fairly consistent within each dataset with -one group having a higher space occupancy score than the other for -multiple measures, this difference can be more or less pronounced within -each dataset (ranging from no to nearly full overlap - BC -\(\in(0;0.995)\)) and sometimes even reversed. This indicates that -opposite conclusions can be drawn from a dataset depending on which -space occupancy measure is considered. For example, in Wright (2017), -crinoids after the Ordovician mass extinction have a higher median -measure value for all measures but for the average displacement. These -differences depending on the measures are also more pronounced in the -empirical datasets where the observations per group are unequal (Hopkins -and Pearson 2016; Healy et al. 2019). - -\subsubsection{Caveats}\label{caveats} - -While our simulations are useful to illustrate the behaviour of diverse -space occupancy measures, they have several caveats. First, the -simulated observations in the trait spaces are independent. This is not -the case in biology where observations can be spatially (Jones et al. -2015) or phylogenetically correlated (e.g. Beck and Lee 2014). Second, -the algorithm used to reduce the trait -\textcolor{black}{ spaces might } not always -accurately reflect changes. This might favour some specific measures -over others, in particular for the changes in density that modify the -nearest neighbour density rather than changing the global density. This -algorithmic choice was made in order to not confound changes in density -along with changes in size. However, the results presented here probably -capture the general behaviour of each measure since results are -consistent between the simulated and empirical analysis. - -\textcolor{black}{ Furthermore, we did not take into -account the effect of sampling on space occupancy measurements (but see -additional results with 80\% and 20\% space reduction in the -supplementary materials 4). In fact, sampling has been previously shown -to have an effect on measurements depending on range or volumes -(e.g.~the sum of ranges or the hypervolume Ciampaglio et al. (2001)). -This effect is especially expected to be acerbated in macroevolutionary -studies when using the fossil record (Brocklehurst et al. 2013) but can -be tackled using rarefaction and bootstrapping techniques (Guillerme -2018). } - -\subsubsection{\texorpdfstring{Using \texttt{moms} to choose the -appropriate -measurements}{Using moms to choose the appropriate measurements}}\label{using-moms-to-choose-the-appropriate-measurements} - -\textcolor{black}{Therefore, we propose the -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} shiny app to -allow workers to help them choose their set of space occupancy -measurements (and test the caveats mentioned above). \texttt{moms} is an -online graphical user interface to help analyse multidimensional data. -It allows users to upload their dataset of interest (or simulate one -with specific parameters) and measure space occupancy using a variety of -implemented measures (namely, but not only, the ones used in this -study). Furthermore, the package allows simulation of shifts in trait -space occupancy as also presented in this paper to test whether some -measures capture specific changes in space. However, \texttt{moms} is -not a tool for analysing multidimensional data \emph{per se} but rather -for helping workers to chose the space occupancy measure most -appropriated to their data and question. To run multidimensional -analysis, we suggest using dedicated \texttt{R} packages (such as - but -not limited to: Oksanen et al. (2007), Bonhomme et al. (2014), Cardoso -et al. (2015), Guillerme (2018)). } - -\subsubsection{Conclusions}\label{conclusions} - -\textcolor{black}{We insist that although no measure is -objectively better than the next one, some can be more problematic than -other in specific contexts. For example, the results for the Sum of -Ranges, Minimum spanning tree average distances, and to a lesser extent -average nearest neighbour distances produced results in the reduced -space often similar to the randomly reduced spaces (Table 5). This does -not make them ``bad'' measures but rather heavily context dependent. -Regardless, we believe that workers should identify the most appropriate -measures based on their trait space properties as well as their specific -biological question. We believe this could be fostered by following -these several suggestions: } - -First, we suggest using multiple measures to tackle different aspects of -the trait space. This follows the same logical thinking that the mean -might not be sufficient to describe a distribution (e.g.~the variance -might be a good additional descriptor). Although using multiple measures -is not uncommon in macroevolutionary studies (e.g. Halliday and Goswami -2016) or in ecology (Mammola 2019), they often do no cover more than one -of the three categories of trait space measures -\textcolor{black}{(but see the recent work of Carmona -et al. (2019) and Mammola and Cardoso (2020)).} - -Second, we suggest selecting the measures that best address the -biological question at hand. If one studies an adaptive radiation in a -group of organisms, it is worth thinking what would be the expected null -model: would the group's size increase (radiation in all directions), -would it increase in density (niche specialisation) or would it shift in -position (radiation into a new set of niches)? - -Third, we suggest not naming measures after the biological aspect they -describe which can be vague (e.g. ``disparity'' or ``functional -dispersion'') but rather after what they are measuring and why (e.g. -``we used sum of ranges to measure the space size''). We believe this -will support both a clearer understanding of what \emph{is} measured as -well as better communication between ecology and evolution research -where measures can be similar but have different names. - -Multidimensional analyses have been acknowledged as essential tools in -modern biology but they can often be counter-intuitive (Bellman 1957). -It is thus crucial to accurately describe patterns in multidimensional -trait spaces to be able to link them to biological processes. When -summarising trait spaces, it is important to remember that a pattern -captured by a specific space occupancy measure is often dependent on the -properties of the space and of the particular biological question of -interest. We believe that having a clearer understanding of both the -properties of the trait space and the associated space occupancy -measures (e.g.~using -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}) as well as -using novel space occupancy measures to answer specific questions will -be of great use to study biological processes in a multidimensional -world. - -\section{Acknowledgements}\label{acknowledgements} - -We thank Natalie Jones and Kevin Healy for helping with the empirical -datasets and Stefano Mammola and Neil Brocklehurst for their positive -and encouraging reviews. We acknowledge funding from the Australian -Research Council DP170103227 and FT180100634 awarded to VW. - -\section{Authors contributions}\label{authors-contributions} - -TG, MNP, AEM and VW designed the project. TG and AEM collected the -empirical dataset. TG ran the analyses and designed the software. TG, -MNP, AEM and VW wrote the manuscript. - -\section{Data Availability, repeatability and -reproducibility}\label{data-availability-repeatability-and-reproducibility} - -The raw empirical data is available from the original papers (Beck and -Lee 2014; Jones et al. 2015, Marcy et al. (2016); Hopkins and Pearson -2016; Wright 2017; Healy et al. 2019). The subsets of the empirical data -used in this analysis are available on figshare -\href{https://doi.org/10.6084/m9.figshare.9943181.v1}{DOI: -10.6084/m9.figshare.9943181.v1}. The modified empirical data are -available in the package accompanying this manuscript -(\texttt{data(moms::demo\_data)}). This manuscript (including the -figures, tables and supplementary material) is repeatable and -reproducible by compiling the vignette of the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. - -\section*{References}\label{references} -\addcontentsline{toc}{section}{References} - -\hypertarget{refs}{} -\hypertarget{ref-beck2014}{} -Beck R.M.D., Lee M.S.Y. 2014. Ancient dates or accelerated rates? -Morphological clocks and the antiquity of placental mammals. Proceedings -of the Royal Society B: Biological Sciences. 281:20141278. - -\hypertarget{ref-cursedimensionality}{} -Bellman R.E. 1957. Dynamic programming. Princeton University Press. - -\hypertarget{ref-bhattacharyya1943}{} -Bhattacharyya A. 1943. On a measure of divergence between two -statistical populations defined by their probability distributions. -Bulletin of the Calcutta Mathematical Society. 35:99--109. - -\hypertarget{ref-blonder2018}{} -Blonder B. 2018. Hypervolume concepts in niche-and trait-based ecology. -Ecography. 41:1441--1455. - -\hypertarget{ref-momocs}{} -Bonhomme V., Picq S., Gaucherel C., Claude J. 2014. Momocs: Outline -analysis using R. Journal of Statistical Software. 56:1--24. - -\hypertarget{ref-brocklehurst2013}{} -Brocklehurst N., Kammerer C.F., Fröbisch J. 2013. The early evolution of -synapsids, and the influence of sampling on their fossil record. -Paleobiology. 39:470--490. - -\hypertarget{ref-bat2015}{} -Cardoso P., Rigal F., Carvalho J.C. 2015. BAT -- biodiversity assessment -tools, an r package for the measurement and estimation of alpha and beta -taxon, phylogenetic and functional diversity. Methods in Ecology and -Evolution. 6:232--236. - -\hypertarget{ref-carmona2019}{} -Carmona C.P., Bello F. de, Mason N.W.H., Lepš J. 2019. Trait probability -density (tpd): Measuring functional diversity across scales based on tpd -with r. Ecology. 100:e02876. - -\hypertarget{ref-ciampaglio2001}{} -Ciampaglio C.N., Kemp M., McShea D.W. 2001. Detecting changes in -morphospace occupation patterns in the fossil record: Characterization -and analysis of measures of disparity. Paleobiology. 71:695--715. - -\hypertarget{ref-close2015}{} -Close R.A., Friedman M., Lloyd G.T., Benson R.B. 2015. Evidence for a -mid-Jurassic adaptive radiation in mammals. Current Biology. - -\hypertarget{ref-diaz2016}{} -Díaz S., Kattge J., Cornelissen J.H., Wright I.J., Lavorel S., Dray S., -Reu B., Kleyer M., Wirth C., Prentice I.C., others. 2016. The global -spectrum of plant form and function. Nature. 529:167. - -\hypertarget{ref-donohue2013}{} -Donohue I., Petchey O.L., Montoya J.M., Jackson A.L., McNally L., Viana -M., Healy K., Lurgi M., O'Connor N.E., Emmerson M.C. 2013. On the -dimensionality of ecological stability. Ecology Letters. 16:421--429. - -\hypertarget{ref-endler2005}{} -Endler J.A., Westcott D.A., Madden J.R., Robson T. 2005. Animal visual -systems and the evolution of color patterns: Sensory processing -illuminates signal evolution. Evolution. 59:1795--1818. - -\hypertarget{ref-foote1992}{} -Foote M. 1992. Rarefaction analysis of morphological and taxonomic -diversity. Paleobiology. 18:1--16. - -\hypertarget{ref-grant2006}{} -Grant P.R., Grant B.R. 2006. Evolution of character displacement in -darwins finches. Science. 313:224--226. - -\hypertarget{ref-disprity}{} -Guillerme T. 2018. dispRity: A modular R package for measuring -disparity. Methods in Ecology and Evolution. 9:1755--1763. - -\hypertarget{ref-halliday2015}{} -Halliday T.J.D., Goswami A. 2016. Eutherian morphological disparity -across the end-cretaceous mass extinction. Biological Journal of the -Linnean Society. 118:152--168. - -\hypertarget{ref-geiger2008}{} -Harmon L.J., Weir J.T., Brock C.D., Glor R.E., Challenger W. 2008. -GEIGER: Investigating evolutionary radiations. Bioinformatics. -24:129--131. - -\hypertarget{ref-healy2019}{} -Healy K., Ezard T.H.G., Jones O.R., Salguero-G'omez R., Buckley Y.M. -2019. Animal life history is shaped by the pace of life and the -distribution of age-specific mortality and reproduction. Nature Ecology -\& Evolution. 2397-334X. - -\hypertarget{ref-hopkins2016}{} -Hopkins M., Pearson K. 2016. Non-linear ontogenetic shape change in -cryptolithus tesselatus (trilobita) using three-dimensional geometric -morphometrics. Palaeontologia Electronica. 19:1--54. - -\hypertarget{ref-hopkins2017}{} -Hopkins M.J., Gerber S. 2017. Morphological disparity. In: Nuno de la -Rosa L., Müller G., editors. Evolutionary developmental biology: A -reference guide. Cham: Springer International Publishing. p. 1--12. - -\hypertarget{ref-jones2015}{} -Jones N.T., Germain R.M., Grainger T.N., Hall A.M., Baldwin L., Gilbert -B. 2015. Dispersal mode mediates the effect of patch size and patch -connectivity on metacommunity diversity. Journal of Ecology. -103:935--944. - -\hypertarget{ref-lalibertuxe92010}{} -Laliberté É., Legendre P. 2010. A distance-based framework for measuring -functional diversity from multiple traits. Ecology. 91:299--305. - -\hypertarget{ref-legendre2012}{} -Legendre P., Legendre L.F. 2012. Numerical ecology. Elsevier. - -\hypertarget{ref-mammola2019}{} -Mammola S. 2019. Assessing similarity of n-dimensional hypervolumes: -Which metric to use? Journal of Biogeography. 0. - -\hypertarget{ref-mammola2020}{} -Mammola S., Cardoso P. 2020. Functional diversity metrics using kernel -density n-dimensional hypervolumes. bioRxiv. - -\hypertarget{ref-mammola2019b}{} -Mammola S., Cardoso P., Culver D.C., Deharveng L., Ferreira R.L., Fišer -C., Galassi D.M.P., Griebler C., Halse S., Humphreys W.F., Isaia M., -Malard F., Martinez A., Moldovan O.T., Niemiller M.L., Pavlek M., -Reboleira A.S.P.S., Souza-Silva M., Teeling E.C., Wynne J.J., Zagmajster -M. 2019. Scientists' Warning on the Conservation of Subterranean -Ecosystems. BioScience. 69:641--650. - -\hypertarget{ref-marcy2016}{} -Marcy A.E., Hadly E.A., Sherratt E., Garland K., Weisbecker V. 2016. -Getting a head in hard soils: Convergent skull evolution and divergent -allometric patterns explain shape variation in a highly diverse genus of -pocket gophers (thomomys). BMC evolutionary biology. 16:207. - -\hypertarget{ref-oksanen2007vegan}{} -Oksanen J., Kindt R., Legendre P., O'Hara B., Stevens M.H.H., Oksanen -M.J., Suggests M. 2007. The vegan package. Community ecology package. -10:631--637. - -\hypertarget{ref-qiao2015}{} -Qiao H., Soberón J., Peterson A.T. 2015. No silver bullets in -correlative ecological niche modelling: Insights from testing among many -potential algorithms for niche estimation. Methods in Ecology and -Evolution. 6:1126--1136. - -\hypertarget{ref-psych}{} -Revelle W. 2018. Psych: Procedures for psychological, psychometric, and -personality research. Evanston, Illinois: Northwestern University. - -\hypertarget{ref-ruta2013}{} -Ruta M., Angielczyk K.D., Fröbisch J., Benton M.J. 2013. Decoupling of -morphological disparity and taxic diversity during the adaptive -radiation of anomodont therapsids. Proceedings of the Royal Society of -London B: Biological Sciences. 280. - -\hypertarget{ref-sedgewick1990}{} -Sedgewick R. 1990. Algorithms in c. Addison-Wesley, Reading. - -\hypertarget{ref-tucker2017}{} -Tucker C.M., Cadotte M.W., Carvalho S.B., Davies T.J., Ferrier S., Fritz -S.A., Grenyer R., Helmus M.R., Jin L.S., Mooers A.O., Pavoine S., -Purschke O., Redding D.W., Rosauer D.F., Winter M., Mazel F. 2017. A -guide to phylogenetic metrics for conservation, community ecology and -macroecology. Biological Reviews. 92:698--715. - -\hypertarget{ref-villuxe9ger2008}{} -Villéger S., Mason N.W.H., Mouillot D. 2008. New multidimensional -functional diversity indices for a multifaceted framework in functional -ecology. Ecology. 89:2290--2301. - -\hypertarget{ref-wills2001}{} -Wills M.A. 2001. Morphological disparity: A primer. In: Adrain J.M., -Edgecombe G.D., Lieberman B.S., editors. Fossils, phylogeny, and form. -Springer US. p. 55--144. - -\hypertarget{ref-wright2017}{} -Wright D.F. 2017. Phenotypic innovation and adaptive constraints in the -evolutionary radiation of palaeozoic crinoids. Scientific Reports. -7:13745. - -\end{document} diff --git a/inst/EcoEvol/shiftingspace_submit.tex b/inst/EcoEvol/shiftingspace_submit.tex deleted file mode 100644 index 91b701a..0000000 --- a/inst/EcoEvol/shiftingspace_submit.tex +++ /dev/null @@ -1,1745 +0,0 @@ -\documentclass[]{article} -\usepackage{lineno} -\usepackage{lmodern} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\usepackage{fixltx2e} % provides \textsubscript -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} -\fi -% use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -% use microtype if available -\IfFileExists{microtype.sty}{% -\usepackage{microtype} -\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\usepackage[margin=1in]{geometry} -\usepackage{hyperref} -\hypersetup{unicode=true, - pdftitle={Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?}, - pdfauthor={Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker}, - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage{longtable,booktabs} -\usepackage{graphicx,grffile} -\makeatletter -\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} -\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} -\makeatother -% Scale images if necessary, so that they will not overflow the page -% margins by default, and it is still possible to overwrite the defaults -% using explicit options in \includegraphics[width, height, ...]{} -\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -%%% Use protect on footnotes to avoid problems with footnotes in titles -\let\rmarkdownfootnote\footnote% -\def\footnote{\protect\rmarkdownfootnote} - -%%% Change title format to be more compact -\usepackage{titling} - -% Create subtitle command for use in maketitle -\providecommand{\subtitle}[1]{ - \posttitle{ - \begin{center}\large#1\end{center} - } -} - -\setlength{\droptitle}{-2em} - - \title{Shifting spaces: which disparity or dissimilarity measurement best -summarise occupancy in multidimensional spaces?} - \pretitle{\vspace{\droptitle}\centering\huge} - \posttitle{\par} - \author{Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker} - \preauthor{\centering\large\emph} - \postauthor{\par} - \predate{\centering\large\emph} - \postdate{\par} - \date{2020-03-27} - -\linespread{1.6} - -\begin{document} -\modulolinenumbers[1] -\linenumbers - -\maketitle - -\section{Abstract}\label{abstract} - -Multidimensional analysis of traits are now a common in ecology and -evolution and are based on trait spaces in which each dimension -summarises the observed trait combination (a morphospace or an -ecospace). Observations of interest will typically occupy a subspace of -this space, and researchers will calculate one or more measures to -quantify how organisms ``inhabit'' that space. In macroevolution and -ecology these measures are referred to as disparity or dissimilarity -metrics and can be generalised as space occupancy measures. Researchers -use these measures to investigate how space occupancy changes through -time, in relation to other groups of organisms, and in response to -global environmental changes. However, the mathematical and biological -meaning of most space occupancy measures is vague with the majority of -widely-used measures lacking formal description. - -Here we propose a broad classification of space occupancy measures into -three categories that capture changes in size, density, or position. We -study the behaviour of 25 measures to changes in trait space size, -density and position on simulated and empirical datasets. We find that -no measure describes all of trait space aspects but that some are better -at capturing certain aspects. Our results confirm the three broad -categories (size, density and position) and allow us to relate changes -in any of these categories to biological phenomena. - -Because the choice of space occupancy measures is specific to the data -and question, we introduced -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool -allowing users to both visualise and capture changes in space occupancy -for any measurement. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} is designed -to help workers choose the right space occupancy measures, given the -properties of their trait space and their biological question. By -providing guidelines and common vocabulary for space occupancy analysis, -we hope to help bridging the gap in multidimensional research between -ecology and evolution. - -\section{Introduction}\label{introduction} - -Groups of species and environments share specific, recognisable, -correlated characteristics: guilds or biomes with shared phenotypic, -physiological, phylogenetic or behavioural traits. Organisms or -environments should therefore be studied as a set of traits rather than -some specific traits in isolation (Donohue et al. 2013; Hopkins and -Gerber 2017). Biologists have increasingly been using ordination -techniques (see Legendre and Legendre 2012 for a summary) to create -multidimensional trait spaces to either explore properties of the data -or test hypotheses (e.g. Oksanen et al. 2007; Blonder 2018; Guillerme -2018). For example, in palaeobiology, Wright (2017) used trait spaces to -study how groups of species' characteristics change through time; in -ecology, Jones et al. (2015) study evidence of competition by looking at -trait overlap between two populations. However, different fields use a -different set of terms for such approaches (Table 1). Nonetheless, they -use the same mathematical objects: matrices with columns representing an -original or transformed trait value and rows representing observations -(taxon, field site, etc.; Guillerme 2018). - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llll@{}} -\toprule -\begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Mathematics\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Macroevolution\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -This paper\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix (\(n \times d\)) with a structural relation between rows and -columns\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Function-space, Eco-space, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Morphospace, traitspace, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -trait space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Rows (\emph{n})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, field sites, environments, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, specimen, populations, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Columns (\emph{d})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -dimensions\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix subset (\(m \times d\); \(m \leq n\))\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Treatments, phylogenetic group (clade), etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Clades, geological stratum, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -group\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Statistic\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity index or metric, hypervolume, functional diversity\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity metric or index\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -space occupancy measure\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Multidimensional analysis\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity analysis, trait analysis, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity analysis, disparity-through-time, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -multidimensional analysis\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{terms and equivalence between mathematics, ecology and -macroevolution.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -Ecologists and evolutionary biologists often use trait spaces with -respect to the same fundamental questions: are groups occupying the same -amount of trait space? Do some groups contain more species than others -in the same amount of trait space? Are some specific factors correlated -with different patterns of trait space occupancy? Because of the -multidimensional nature of these trait spaces, it is often not possible -to study them using bi- or tri-variate techniques (Díaz et al. 2016; -Hopkins and Gerber 2017; Mammola 2019). Studying the occupancy of trait -spaces is done using disparity indices in macroevolution (Wills 2001; -Hopkins and Gerber 2017; Guillerme 2018) or comparing hypervolumes in -ecology (Donohue et al. 2013; Díaz et al. 2016; Blonder 2018; Mammola -2019). Despite the commonalities between the measures used in ecology -and evolution (which are often metric but don't necessarily need to be), -surprisingly little work has been published on their behaviour (but see -Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). - -Different occupancy measures capture different aspects of trait space -(ciampaglio2001; Villéger et al. 2008; Mammola 2019). It may be -widely-known, but to our knowledge is infrequently mentioned in -peer-reviewed papers. First, space occupancy measures are often named as -the biological aspect they are describing (``disparity'', ``functional -diversity'') rather than what they are measuring (e.g.~the product of -ranges), which obscures the differences and similarities between -studies. Second, in many studies in ecology and evolution, authors have -focused on measuring the size of the trait space (e.g.~ellipsoid volume -Donohue et al. 2013; hypervolume Díaz et al. 2016; Procrustes variance -Marcy et al. 2016; product of variance Wright 2017). However, the size -of the trait space only represents one aspects of occupancy, -disregarding others such as the density (Harmon et al. 2008) or position -(Wills 2001; Ciampaglio et al. 2001). For example, if two groups have -the same size, this can support certain biological conclusions. Yet, an -alternative aspect of space occupancy may indicate that the groups' -position are different, leading to a different biological conclusion -(e.g.~the groups are equally diverse but occupy different niches). Using -measures that only capture one aspect of the trait space may restrain -the potential of multidimensional analysis (Villéger et al. 2008). - -Here we propose a broad classification of space occupancy measures as -used across ecology and evolution and study their power to detect -changes in trait space occupancy in simulated and empirical data -(regardless of whether spaces are are truly ``occupiable'' which might -be important in some cases - e.g.~if the space is infinite or if some -regions inapplicable). We provide an assessment of each broad type of -space occupancy measures along with a unified terminology to foster -communication between ecology and evolution. Unsurprisingly, we found no -one measure describes all changes and that the results from each -measures are dependent on the characteristics of the space and the -hypotheses. Furthermore, because there can be an infinite number of -measures, it would be impossible to propose clear generalities to space -occupancy measures behaviour. Therefore, we propose -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool -allowing researchers to design, experiment and visualise their own space -occupancy measure tailored for their specific project and helping them -understanding the ``null'' behaviour of the measures of interest. - -\subsection{Space occupancy measures}\label{space-occupancy-measures} - -In this paper, we define trait spaces as any matrix where rows are -observations and columns are traits, where both observations and traits -are structurally related (e.g.~there is a phylogenetic relation between -observations - and traits, etc.). These traits can widely vary in number -and types: they could be coded as discrete (e.g.~presence or absence of -a bone; Beck and Lee 2014; Wright 2017), continuous measurements -(e.g.~leaf area; Díaz et al. 2016) or more sophisticated measures -(Fourier ellipses; Bonhomme et al. 2014; e.g.~landmark position; Marcy -et al. 2016). Traits can also be measured by using relative observations -(e.g.~community compositions; Jones et al. 2015) or distance between -observations (e.g. Close et al. 2015). However, regardless of the -methodology used to build a trait space, three broad occupancy measures -can be used: the size which approximates the amount of space occupied, -the density which approximates the distribution in space and the -position which approximates the location in space (Fig. 1; Villéger et -al. 2008). Of course any combination of these three aspects is always -possible. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measures_types-1.pdf} -\caption{different type of information captured by space occupancy -measures: (A) size, (B) density and (C) position.} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - -\paragraph{1. Size}\label{size} - -Size captures the spread of a group in the trait space. They can be -interpreted as the amount of the trait space that is occupied by -observations. Typically, larger values for such measures indicate the -presence of more extreme trait combinations. For example, if group A is -bigger than B, the observations in A achieve more extreme trait -combinations than in B. This type of measure is widely used in both -ecology (e.g.~the hypervolume; Blonder 2018) and in evolution (e.g.~the -sum or product of ranges or variances; Wills 2001). - -Although size measures are suitable indicators of a group's trait space -occupancy, they are limited to comparing the range of trait-combinations -between groups. Size measures do not take into account the distribution -of the observations within a group and can often be insensitive to -unoccupied ``holes'' in the trait space (overstimating the size; Blonder -(2018)). They can make it difficult to determine whether all the -observations are on the edge of the group's distribution or whether the -size is simply driven by outliers. - -\paragraph{2. Density}\label{density} - -Density gives an indication of the quantity of observations in the trait -space. They can be interpreted as the distribution of the observations -\emph{within} a group in the trait space. Groups with higher density -contain more observations (i.e.~more observations per approximation of -size) that will tend to be more similar to each other. For example, if -group A is greater is size than group B and both have the same density -(observations are equally distant within each group), similar mechanisms -could be driving both groups' trait space occupancy. Indeed, this -pattern could suggest that A is older and has had more time to achieve -more extreme trait combinations under essentially the same process as -younger, smaller group B (Endler et al. 2005). Note that density based -measures can be sensitive to sampling. Density measures are less common -compared to size measures, but they are still used in both ecology -(e.g.~the minimum spanning tree length; Oksanen et al. 2007) and -evolution (e.g.~the average pairwise distance; Harmon et al. 2008). - -\paragraph{3. Position}\label{position} - -Position captures where a group lies in trait space. They can be -interpreted as where a group lies in the trait space either relative to -the space itself or relative to another group. For example, if group A -has a different position than group B, A will have a different -trait-combination than in B. - -Position measures may be harder to interpret in multidimensional spaces. -In a 2D space, two groups can be equally distant from a fixed point but -in different parts of the space (left, right, up, or down - with the -amount of parts of space increasing with dimensions). However, when -thinking about unidimensional data, this measure is obvious: two groups -A or B could have the same variance (size) with the same number of -observations (density) but could have two different means and thus be in -different positions. These measures are used in ecology to compare the -position of two groups relative to each other (Mammola 2019). - -\subsection{No measure to rule them all: benefits of considering -multiple -measures}\label{no-measure-to-rule-them-all-benefits-of-considering-multiple-measures} - -The use of multiple measurements to assess trait space occupancy has the -benefit of providing a more detailed characterisation of occupancy -changes. If the question is to look at how space occupancy changes in -response to mass extinction, using a single space occupancy measure can -miss part of the picture: a change in size could be decoupled from a -change in position or density in trait space. For example, the -Cretaceous-Paleogene extinction (66 million years ago) shows an increase -in size of the mammalian trait space (adaptive radiation; Halliday and -Goswami 2016) but more specific questions can be answered by looking at -other aspects of trait space occupancy: does the radiation expand on -previously existing morphologies (elaboration, increase in density; -Endler et al. 2005) or does it explore new regions of the trait space -(innovation, change in position; Endler et al. 2005)? Similarly, in -ecology, if two groups have the same trait space size, it can be -interesting to look at differences in density within these two groups: -different selection pressure can lead to different density within -equally sized groups. - -Here, we provide the first interdisciplinary review of 25 space -occupancy measures that uses the broad classification of measures into -size, density and position to capture pattern changes in trait space. We -assess the behaviour of measures using simulations and six -interdisciplinary empirical datasets covering a wide range of potential -data types and biological questions. We also introduce a tool for -measuring occupancy in multidimensional space -(\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}), which is -a user-friendly, open-source, graphical interface to allow the tailored -testing of measurement behaviour for any use case. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} will allow -workers to comprehensively assess the properties of their trait space -and the measures associated with their specific biological question. - -\section{Methods}\label{methods} - -We tested how 25 space occupancy measures relate to each other, are -affected by modifications of traits space and affect group comparisons -in empirical data: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - We simulated 13 different spaces with different sets of parameters; -\item - We transformed these spaces by removing 50\% of the observations - following four different scenarios corresponding to different - empirical scenarios: randomly, by limit (e.g.~expansion or reduction - of niches), by density (e.g.~different degrees of competition within a - guild) and by position (e.g.~ecological niche shift). -\item - We measured occupancy on the resulting transformed spaces using eight - different space occupancy measures; -\item - We applied the same space occupancy measures to six empirical datasets - (covering a range of disciplines and a range of dataset properties). -\end{enumerate} - -Note that the paper contains the results for only eight measures which -were selected as representative of common measures covering the size, -density and position trait space aspects. The results for an additional -17 measures is available in the supplementary material 4. - -\subsection{Generating spaces}\label{generating-spaces} - -We generated trait spaces using the following combinations of size, -distributions, variance and correlation: - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -space name\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.31\columnwidth}\raggedright\strut -distribution(s)\strut -\end{minipage} & \begin{minipage}[b]{0.21\columnwidth}\raggedright\strut -dimensions variance\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -correlation\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform (min = -0.5, max = 0.5)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal (mean = 0, sd = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D with random distributions\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCA-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Multiplicative\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCO-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Additive\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{different simulated space distribution. \emph{Name} of the -simulated space; \emph{dimensions} of the matrix (row*columns); -\emph{distribution(s)} of the data on each dimensions (for the `Random', -dimensions are randomly chosen between Normal, Uniform or Lognormal); -\emph{dimension variance}: distribution of the variance between -dimensions (when equal, the dimensions have the same variance); -\emph{correlation} between dimensions.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -The differences in trait space sizes (200 elemeents for 3, 15, 50 or 150 -dimensions) reflects the range found in literature (e.g.~hopkins2017; -Mammola 2019). We used a range of distributions (uniform, normal or a -random combination of uniform, normal and lognormal) to test the effect -of observation distributions on the measurements. We used different -levels of variance for each dimensions in the spaces by making the -variance on each dimension either equal -(\(\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}\)) or decreasing -(\(\sigma_{D1} < \sigma_{D2} < \sigma_{Di}\)) with the decreasing factor -being either multiplicative (using the cumulative product of the inverse -of the number of dimensions: \(\prod_i^d(1/d)\)) or additive -(\(\sum_i^d(1/d)\)). Both reductions of variance are used to illustrate -the properties of ordinations where the variance decreases per -dimensions (and normal win Multidimensional Scaling - MDS, PCO or PCoA; -e.g. Close et al. 2015; lognormal in principal components analysis - -PCA; e.g. Marcy et al. 2016; Wright 2017; Healy et al. 2019). Finally, -we added a correlation parameter to illustrate the effect of -co-linearity between traits (especially in non-ordinated trait spaces). -We repeated the simulation of each trait space 20 times (resulting in -260 spaces). - -\subsection{Spatial occupancy -measures}\label{spatial-occupancy-measures} - -We then calculated eight different measures on the resulting transformed -spaces, including a new one, the average displacement, which we expect -to be influenced by changes in trait space position. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Name\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Definition\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Captures\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Source\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Notes\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Laliberté and Legendre (2010)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional dispersion (FDis - without abundance)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\sigma^{2}{k_i}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -common measure used in palaeobiology (Ciampaglio et al. 2001; Wills -2001)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -more sensitive to outliers than the sum of variances\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Donohue et al. (2013)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -less sensitive to outliers than the convex hull hypervolume (Díaz et al. -2016; Blonder 2018)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum(\text{branch length})}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sedgewick (1990)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -similar to the unscaled functional evenness (Villéger et al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Villéger et al. (2008)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional evenness without weighted abundance (FEve; Villéger et -al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sqrt{\sum_{i}^{n}{min({q}_{i}-p_{i})^2}})\times \frac{1}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the density of pairs of observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average displacement\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Position\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -This paper\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the ratio between the observations' position from their centroid and the -centre of the trait space (coordinates: 0, 0, 0, \ldots{}). A value of 1 -indicates that the observations' centroid is the centre of the trait -space\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{list of measures with \emph{n} being the number of -observations, \emph{d} the total number of dimensions, \emph{k} any -specific row in the matrix, \emph{Centroid} being their mean and -\(\sigma^{2}\) their variance. \(\Gamma\) is the Gamma distribution and -\(\lambda_{i}\) the eigenvalue of each dimension and \({q}_{i}\) and -\(p_{i}\) are any pairs of coordinates.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -We selected these eight space occupancy measures to illustrate how they -capture different aspects of space occupancy (not as an expression of -our preference). These measures are specific to Euclidean and isotropic -trait spaces (which is not necessary for all measures). The -supplementary material 4 contains the same analysis as described below, -performed on 17 measures. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -exploration into the effect of many more measures as well as the -customisation of measures by combining them or using user-designed -functions. - -\subsection{Measure comparisons}\label{measure-comparisons} - -We compared the space occupancy measures correlations across all -simulations between each pair of measures to assess their captured -signal (Villéger et al. 2008; Laliberté and Legendre 2010). We used the -measures on the full 13 trait spaces described above. We then scaled the -results and measured the pairwise Pearson correlation to test whether -measures were capturing a similar signals or not using the -\texttt{psych} package (Revelle 2018). - -\subsection{Changing space}\label{changing-spaces} - -To assess how the measures responded to changes within trait spaces, we -removed 50\% of observations each time using the following algorithms: - -\begin{itemize} -\item - \textbf{Randomly:} by randomly removing 50\% of observations (Fig. - 2-A). This reflects a ``null'' biological model of changes in trait - space: the case when observations are removed regardless of their - intrinsic characteristics. For example, if diversity is reduced by - 50\% but the space size remains the same, there is a decoupling - between diversity and space occupancy (Ruta et al. 2013). Our selected - measures are expected to not be affected by this change. -\item - \textbf{Limit:} by removing observations within a distance from the - centre of the trait space lower or greater than a radius \(\rho\) - (where \(\rho\) is chosen such that 50\% observations are selected) - generating two limit removals: \emph{maximum} and \emph{minimum} - (respectively in orange and blue; Fig. 2-B). This can reflect a strict - selection model where observations with trait values below or above a - threshold are removed leading to an expansion or a contraction of the - trait space. Size measures are expected to be most affected by this - change. -\item - \textbf{Density:} by removing any pairs of point with a distance \(D\) - from each other where (where \(D\) is chosen such that 50\% - observations are selected) generating two density removals: - \emph{high} and \emph{low} (respectively in orange and blue; Fig. - 2-C). This can reflect changes within groups in the trait space due to - ecological factors (e.g.~niche repulsion resulting in lower density; - Grant and Grant 2006). Density measures are expected to be most - affected by this change. -\item - \textbf{Position:} by removing points similarly as for \textbf{Limit} - but using the distance from the furthest point from the centre - generating two position removals: \emph{positive} and \emph{negative} - (respectively in orange and blue; Fig. 2-D). This can reflect global - changes in trait space (e.g.~if an entire group remaining diverse but - occupying a different niche). Position measures are expected to be - most affected by this change. -\end{itemize} - -The algorithm to select \(\rho\) or \(D\) is described in the -Supplementary material 1. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics[width=0.8\textwidth]{shiftingspace_files/figure-latex/fig_reduce_space-1.pdf} -\caption{\small{different type of space reduction. Each panel displays two -groups of 50\% of the data points each. Each group (orange and blue) are -generated using the following algorithm: A - randomly; B - by limit -(maximum and minimum limit); C - by density (high and low); and D - by -position (positive and negative). Panel E et F represents two typical -display of the reduction results displayed in Table 5: the dots -represent the median space occupancy values across all simulations for -each scenario of trait space change (Table 2), the solid and dashed line -respectively the 50\% and 95\% confidence intervals. Results in grey are -the random 50\% reduction (panel A). Results in blue and orange -represent the opposite scenarios from panels B, C, and D. The displayed -value is the amount of overlap (Bhattacharrya Coefficient) between the -blue or orange distributions and the grey one. Panel E and F shows -respectively the ``ideal'' and ``worst'' results for any type of -measures, where the space occupancy measurement respectively manages or -fails to captures a specific type of reduction (i.e.~size, position or -density; Table 5).}} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - -Because occupancy measures are dependent on the space, we scaled and -centred them between -1 and 1 to make them comparable (by subtracting -the observed occupancy without reduction to all the measures of the -reduced spaces and then divided it by the maximum observed occupancy). A -value of 0 indicates no effect of the space reduction and \(>0\) and -\(<0\) respectively indicates an increase or decrease in the measure -value. We then measured the amount of overlap between the non-random -removals (limit, density and position) and the random removals using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\subsubsection{Measuring the effect of space and -dimensionality}\label{measuring-the-effect-of-space-and-dimensionality} - -Distribution differences and the number of dimensions can have an effect -on the measure results. For example, in a normally distributed space, an -increase in density can often lead to a decrease in size (though this is -not necessarily true if the space is log-normal or uniform). High -dimensional spaces (\textgreater{}10) are subject to the ``curse of -multidimensionality'' (Bellman 1957): data becomes sparser with -increasing number of dimensions. This can have two main consequences: 1) -the probability of overlap between two groups decreases as a product of -the number of dimensions; and 2) the amount of samples needed to -``fill'' the spaces increases exponentially -\href{https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners}{see -this interactive illustration by Toph Tucker}. The ``curse'' can make -the interpretation of high dimensional data counter-intuitive. For -example if a group expands in multiple dimensions (i.e.~increase in -size), the actual hypervolume (\(\prod_{i}^{d} range_{Di}\)) can -decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an -ANOVA (\(occupancy \sim distribution\) and -\(occupancy \sim dimensions\)) by using all spaces with 50 dimensions -and the uniform and normal spaces with equal variance and no correlation -with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing -respectively the effect of distribution and dimensions. The results of -the ANOVAs (F and \emph{p}-values) are reported in Table 5 (full results -in supplementary material 3). - -\subsection{Empirical examples}\label{empirical-examples} - -We analysed the effect of the different space occupancy measures on six -different empirical studies covering a range of fields that employ trait -space analyses. For each of these studies we generated trait spaces from -the data published with the papers. We divided each trait spaces into -two biologically-relevant groups and tested whether the measures -differentiated the groups in different ways. Both the grouping and the -questions were based on a simplified version of the topics of these -papers (with no intention to re-analyse the data and questions). The -procedures to generate the data and the groups varies between studies -and is detailed in the supplementary materials 2. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llllllll@{}} -\toprule -\begin{minipage}[b]{0.06\columnwidth}\raggedright\strut -study\strut -\end{minipage} & \begin{minipage}[b]{0.06\columnwidth}\raggedright\strut -field\strut -\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut -taxonomic group\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -traits\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -trait space\strut -\end{minipage} & \begin{minipage}[b]{0.06\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut -groups\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -question\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Beck and Lee (2014)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -Mammalia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -106*105\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -52 crown vs.~54 stem\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are crown mammals more disparate than stem mammals?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Wright (2017)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -Crinoidea\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -42*41\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -16 before vs.~23 after\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Is there a difference in disparity before and after the Ordovician mass -extinction?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Marcy et al. (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -Rodentia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -skull 2D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -454*134\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -225 \emph{Megascapheus} vs.~229 \emph{Thomomys}\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are two genera of gopher morphologically distinct?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Hopkins and Pearson (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -Trilobita\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -3D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -46*46\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -36 adults vs.~10 juveniles\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are juvenile trilobites a subset of adult ones?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Jones et al. (2015)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -Plantae\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Communities species compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Jaccard distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -48*47\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -24 aspens vs.~24 grasslands\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Is there a difference in species composition between aspens and -grasslands?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Healy et al. (2019)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -Animalia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Life history traits\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of continuous traits (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -285*6\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -83 ecthotherms vs.~202 endotherms\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Do endotherms have more diversified life history strategies than -ectotherms?\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{details of the six empirical trait spaces.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -For each empirical trait space we bootstrapped each group 500 times -(Guillerme 2018) and applied the eight space occupancy measure to each -pairs of groups. We then compared the means of each groups using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\section{Results}\label{results} - -\subsection{Measure comparisons}\label{measure-comparisons-1} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measure_correlation-1.pdf} -\caption{pairwise correlation between the scaled measures. Numbers on -the upper right corner are the Pearson correlations. The red line are -linear regressions (with the confidence intervals in grey). Av.: -average; dist.: distance; min.: minimum; span.: spanning.} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - -All the measures were either positively correlated (Pearson correlation -of 0.99 for the average Euclidean distance from centroid and sum of -variance or 0.97 for the average nearest neighbour distance and minimum -spanning tree average length; Fig. 3) or somewhat correlated (ranging -from 0.66 for the sum of variances and the ellipsoid volume to -0.09 -between the average displacement and the average Euclidean distance from -centroid; Fig. 3). All measures but the ellipsoid volume were normally -(or nearly normally) distributed (Fig. 3). - -\subsection{Space shifting}\label{space-shifting} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llllll@{}} -\toprule -\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Size change\strut -\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut -Arrangement change\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Position change\strut -\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Distribution effect\strut -\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut -Dimensions effect\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 0.924 ; p = 0.449\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.322 ; p = 0.958\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.285 ; p = 0.274\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.478 ; p = 0.873\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 11.119 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 32.307 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 7.215 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 13.486 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.162 ; p = 0.326\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.998 ; p = 0.435\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 8.152 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 29.358 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.478 ; p = 0.207\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.773 ; p = 0.626\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 10.742 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 26.829 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{results of the effect of space reduction, space dimension -distributions and dimensions number of the different space occupancy -measures. See Fig. 2 for interpretation of the figures distributions and -values. F-values for distribution effect and dimensions effect -represents respectively the effect of the ANOVAs space occupancy -\textasciitilde{} distributions and space occupancy \textasciitilde{} -dimension represent the ratio of sum squared difference within and -between groups (the higher, the more the factor has an effect on the -measure) and associated \emph{p}-values (0 `***' 0.001 `**' 0.01 `*' -0.05 `.' 0.1 '' 1). This figure illustrates how different measures can -be influenced by different aspects of changes in the trait space. E.g. -the Average Euclidean distance from centroid (row 1) captures mainly -changes in size (column 1), but also captures changes in density (column -2) but does not capture changes in position (column 3).} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -As expected, some different measures capture different aspects of space -occupancy. However, it can be hard to predict the behaviour of each -measure when 50\% of the observations are removed. We observe a clear -decrease in median metric in less than a third of the space reductions -(10/36). - -In terms of change in size, only the average Euclidean distance from -centroid and the sum of variances seem to capture a clear change in both -directions. In terms of change in density, only the minimum spanning -tree average distance and the average nearest neighbour distance seem to -capture a clear change in both directions. And in terms of change in -position, only the average displacement metric seems to capture a clear -change in direction (albeit not in both directions). This is not -surprising, since the notion of positions becomes more and more complex -to appreciate as dimensionality increases (i.e.~beyond left/right, -up/down and front/back). - -\subsection{Empirical example}\label{empirical-example} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllllll@{}} -\toprule -\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Beck and Lee 2014\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -Wright 2017\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Marcy et al. 2016\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Hopkins and Pearson 2016\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Jones et al. 2015\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Healy et al. 2019\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Comparisons (orange \emph{vs.} blue)\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -crown \emph{vs.} stem mammals morphologies\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -crinoids morphologies before \emph{vs.} after the end-Ordovician -extinction\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\emph{Megascapheus} \emph{vs.} \emph{Thomomys} skull shapes\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -adults \emph{vs.} juveniles trilobites cephalon shapes\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -aspens \emph{vs.} grasslands communities compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -ecthotherms \emph{vs.} endotherms life history traits\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-25.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-33.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-41.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-26.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-34.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-42.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-27.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-35.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-43.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-28.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-36.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-44.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-29.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-37.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-45.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-30.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-38.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-46.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-31.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-39.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-47.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-32.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-40.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-48.pdf}\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{comparisons of pairs of groups in different empirical trait -spaces. NAs are used for cases where space occupancy could not be -measured due to the curse of multidimensionality. The displayed values -are the amount of overlap between both groups (Bhattacharrya -Coefficient).} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -Similarly as for the simulated results, the empirical ones indicate that -there can be no perfect one-size-fit all measurement. For all eight -measures (except the ellipsoid volume) we see either one group or the -other having a bigger mean than the other and no consistent case where a -group has a bigger mean than the other for all the measures. For -example, in the Beck and Lee (2014)'s dataset, there is a clear -difference in size using the average Euclidean distance from centroid or -the sum of variances (overlaps of respectively 0.175 and 0.159) but no -overlap when measuring the size using the sum of ranges (0.966). -However, for the Hopkins and Pearson (2016)'s dataset, this pattern is -reversed (no clear differences for the average Euclidean distance from -centroid or the sum of variances - 0.701 and 0.865 respectively - but a -clear difference for the sum of ranges (0). For each dataset, the -absolute differences between each groups is not consistent depending on -the measures. For example, in Hopkins and Pearson (2016)'s dataset, the -orange group's mean is clearly higher than the blue one when measuring -the sum of ranges (0) and the inverse is true when measuring the average -displacement (0). - -\section{Discussion}\label{discussion} - -Here we tested 25 measures of trait space occupancy on simulated and -empirical datasets to assess how each measure captures changes in trait -space size, density and position. Our results show that the correlation -between measures can vary both within and between measure categories -(Fig. 3), highlighting the importance of understanding the measure -classification for the interpretation of results. Our simulations show -that different measures capture different types of trait space change -(Table 5), meaning that the use of multiple measures is important for -comprehensive interpretation of trait space occupancy. We also show that -the choice of measure impacts the interpretation of group differences in -empirical datasets (Table 6). - -\paragraph{Measures comparisons}\label{measures-comparisons} - -Measures within the same category of trait space occupancy (size, -density or position) do not have the same level of correlation with each -other. For example, the average Euclidean distance from centroid (size) -is highly correlated to the sum of variances (size - correlation of -0.99) and somewhat correlated with the minimum spanning tree average -distance (density - correlation of 0.66) but poorly with the ellipsoid -volume (size - correlation of 0.17) and the minimum spanning tree -distances evenness (density - correlation of -0.05). Furthermore, the -fact that we have such a range of correlations for normal distributions -suggests that each measure can capture different summaries of space -occupancy ranging from obvious differences (for measures not strongly -correlated) to subtle ones (for measures strongly correlated). - -\paragraph{Space shifting}\label{space-shifting-1} - -Most measures capture no changes in space occupancy for the ``null'' -(random) space reduction (in grey in Table 5). This is a desirable -behaviour for space occupancy measures since it will likely avoid false -positive errors in studies that estimate biological processes from space -occupancy patterns (e.g.~convergence Marcy et al. 2016, life history -traits Healy et al. (2019)). However, the average nearest neighbour -distance and the sum of ranges have a respectively positive and negative -``null'' median. This is not especially a bad property but it should be -kept in mind that even random processes can increase or decrease these -measures' values. - -For changes in size, the sum of variances and the average Euclidean -distance from centroid are good descriptors (Table 5). However, as -illustrated in the 2D examples in Fig. 2-B only the blue change results -(Table 5) should not result in a direct change in overall size because -the trait space is merely ``hollowed'' out. That said, ``hollowing'' is -harder to conceptualise in many dimensions and the measures can still be -interpreted for comparing groups (orange has a smaller volume than -blue). - -The average nearest neigbhour distance and the minimum spanning tree -average distance consistently detect changes in density with more -precision for low density trait spaces (in blue in Table 5). However, we -can observe some degree of correlation between the changes in density -and the changes in size for most measure picking either signal. This -could be due to the use of normally distributed spaces where a change in -density often leads to a change in size. This is not necessarily the -case with empirical data. - -Regarding the changes in position, only the average displacement measure -seems able to distinguish between a random change and a displacement of -the trait space (Table 5). However, the average displacement measure -does not distinguish between positive or negative displacement: this -might be due to the inherent complexity of \emph{position} in a -multidimensional trait space. - -\paragraph{Empirical examples}\label{empirical-examples-1} - -Although most differences are fairly consistent within each dataset with -one group having a higher space occupancy score than the other for -multiple measures, this difference can be more or less pronounced within -each dataset (ranging from no to nearly full overlap - BC -\(\in(0;0.995)\)) and sometimes even reversed. This indicates that -opposite conclusions can be drawn from a dataset depending on which -space occupancy measure is considered. For example, in Wright (2017), -crinoids after the Ordovician mass extinction have a higher median -measure value for all measures but for the average displacement. - -These differences depending on the measures are also more pronounced in -the empirical datasets where the observations per group are unequal -(Hopkins and Pearson 2016; Healy et al. 2019). - -\subsubsection{Caveats}\label{caveats} - -While our simulations are useful to illustrate the behaviour of diverse -space occupancy measures, they have several caveats. First, the -simulated observations in the trait spaces are independent. This is not -the case in biology where observations can be spatially (Jones et al. -2015) or phylogenetically correlated (e.g. Beck and Lee 2014). Second, -the algorithm used to reduce the trait spacesmight not always accurately -reflect changes. This might favour some specific measures over others, -in particular for the changes in density that modify the nearest -neighbour density rather than changing the global density. This -algorithmic choice was made in order to not confound changes in density -along with changes in size. However, the results presented here probably -capture the general behaviour of each measure since results are -consistent between the simulated and empirical analysis. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -workers to test the caveats mentioned above by uploading empirical trait -spaces. - -\subsubsection{Conclusions}\label{conclusions} - -We insist that no measure is better than the next one and that workers -should identify the most appropriate measures based on their trait space -properties as well as their specific biological question. However, -following the findings of this study we make several suggestions: - -First, we suggest using multiple measures to tackle different aspects of -the trait space. This follows the same logical thinking that the mean -might not be sufficient to describe a distribution (e.g.~the variance -might be a good additional descriptor). Although using multiple measures -is not uncommon in macroevolutionary studies (e.g. Halliday and Goswami -2016) or in ecology (Mammola 2019), they often do no cover more than one -of the three categories of trait space measures. - -Second, we suggest selecting the measures that best address the -biological question at hand. If one studies an adaptive radiation in a -group of organisms, it is worth thinking what would be the expected null -model: would the group's size increase (radiation in all directions), -would it increase in density (niche specialisation) or would it shift in -position (radiation into a new set of niches)? - -Third, we suggest not naming measures after the biological aspect they -describe which can be vague (e.g. ``disparity'' or ``functional -dispersion'') but rather after what they are measuring and why (e.g. -``we used sum of ranges to measure the space size''). We believe this -will support both a clearer understanding of what \emph{is} measured as -well as better communication between ecology and evolution research -where measures can be similar but have different names. - -Multidimensional analyses have been acknowledged as essential tools in -modern biology but they can often be counter-intuitive (Bellman 1957). -It is thus crucial to accurately describe patterns in multidimensional -trait spaces to be able to link them to biological processes. When -summarising trait spaces, it is important to remember that a pattern -captured by a specific space occupancy measure is often dependent on the -properties of the space and of the particular biological question of -interest. We believe that having a clearer understanding of both the -properties of the trait space and the associated space occupancy -measures (e.g.~using -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}) as well as -using novel space occupancy measures to answer specific questions will -be of great use to study biological processes in a multidimensional -world. - -\section{Acknowledgements}\label{acknowledgements} - -We thank Natalie Jones and Kevin Healy for helping with the empirical -datasets and two anonymous reviewer for their comments. We acknowledge -funding from the Australian Research Council DP170103227 and FT180100634 -awarded to VW. - -\section{Authors contributions}\label{authors-contributions} - -TG, MNP, AEM and VW designed the project. TG and AEM collected the -empirical dataset. TG ran the analyses and designed the software. TG, -MNP, AEM and VW wrote the manuscript. - -\section{Data Availability, repeatability and -reproducibility}\label{data-availability-repeatability-and-reproducibility} - -The raw empirical data is available from the original papers (Beck and -Lee 2014; Jones et al. 2015, Marcy et al. (2016); Hopkins and Pearson -2016; Wright 2017; Healy et al. 2019). The subsets of the empirical data -used in this analysis are available on figshare -\href{https://doi.org/10.6084/m9.figshare.9943181.v1}{DOI: -10.6084/m9.figshare.9943181.v1}. The modified empirical data are -available in the package accompanying this manuscript -(\texttt{data(moms::demo\_data)}). This manuscript (including the -figures, tables and supplementary material) is repeatable and -reproducible by compiling the vignette of the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. - -\section*{References}\label{references} -\addcontentsline{toc}{section}{References} - -\hypertarget{refs}{} -\hypertarget{ref-beck2014}{} -Beck R.M.D., Lee M.S.Y. 2014. Ancient dates or accelerated rates? -Morphological clocks and the antiquity of placental mammals. Proceedings -of the Royal Society B: Biological Sciences. 281:20141278. - -\hypertarget{ref-cursedimensionality}{} -Bellman R.E. 1957. Dynamic programming. Princeton University Press. - -\hypertarget{ref-bhattacharyya1943}{} -Bhattacharyya A. 1943. On a measure of divergence between two -statistical populations defined by their probability distributions. -Bulletin of the Calcutta Mathematical Society. 35:99--109. - -\hypertarget{ref-blonder2018}{} -Blonder B. 2018. Hypervolume concepts in niche-and trait-based ecology. -Ecography. 41:1441--1455. - -\hypertarget{ref-momocs}{} -Bonhomme V., Picq S., Gaucherel C., Claude J. 2014. Momocs: Outline -analysis using R. Journal of Statistical Software. 56:1--24. - -\hypertarget{ref-ciampaglio2001}{} -Ciampaglio C.N., Kemp M., McShea D.W. 2001. Detecting changes in -morphospace occupation patterns in the fossil record: Characterization -and analysis of measures of disparity. Paleobiology. 71:695--715. - -\hypertarget{ref-close2015}{} -Close R.A., Friedman M., Lloyd G.T., Benson R.B. 2015. Evidence for a -mid-Jurassic adaptive radiation in mammals. Current Biology. - -\hypertarget{ref-diaz2016}{} -Díaz S., Kattge J., Cornelissen J.H., Wright I.J., Lavorel S., Dray S., -Reu B., Kleyer M., Wirth C., Prentice I.C., others. 2016. The global -spectrum of plant form and function. Nature. 529:167. - -\hypertarget{ref-donohue2013}{} -Donohue I., Petchey O.L., Montoya J.M., Jackson A.L., McNally L., Viana -M., Healy K., Lurgi M., O'Connor N.E., Emmerson M.C. 2013. On the -dimensionality of ecological stability. Ecology Letters. 16:421--429. - -\hypertarget{ref-endler2005}{} -Endler J.A., Westcott D.A., Madden J.R., Robson T. 2005. Animal visual -systems and the evolution of color patterns: Sensory processing -illuminates signal evolution. Evolution. 59:1795--1818. - -\hypertarget{ref-foote1992}{} -Foote M. 1992. Rarefaction analysis of morphological and taxonomic -diversity. Paleobiology. 18:1--16. - -\hypertarget{ref-grant2006}{} -Grant P.R., Grant B.R. 2006. Evolution of character displacement in -darwins finches. Science. 313:224--226. - -\hypertarget{ref-disprity}{} -Guillerme T. 2018. dispRity: A modular R package for measuring -disparity. Methods in Ecology and Evolution. 9:1755--1763. - -\hypertarget{ref-halliday2015}{} -Halliday T.J.D., Goswami A. 2016. Eutherian morphological disparity -across the end-cretaceous mass extinction. Biological Journal of the -Linnean Society. 118:152--168. - -\hypertarget{ref-geiger2008}{} -Harmon L.J., Weir J.T., Brock C.D., Glor R.E., Challenger W. 2008. -GEIGER: Investigating evolutionary radiations. Bioinformatics. -24:129--131. - -\hypertarget{ref-healy2019}{} -Healy K., Ezard T.H.G., Jones O.R., Salguero-G'omez R., Buckley Y.M. -2019. Animal life history is shaped by the pace of life and the -distribution of age-specific mortality and reproduction. Nature Ecology -\& Evolution. 2397-334X. - -\hypertarget{ref-hopkins2016}{} -Hopkins M., Pearson K. 2016. Non-linear ontogenetic shape change in -cryptolithus tesselatus (trilobita) using three-dimensional geometric -morphometrics. Palaeontologia Electronica. 19:1--54. - -\hypertarget{ref-hopkins2017}{} -Hopkins M.J., Gerber S. 2017. Morphological disparity. In: Nuno de la -Rosa L., Müller G., editors. Evolutionary developmental biology: A -reference guide. Cham: Springer International Publishing. p. 1--12. - -\hypertarget{ref-jones2015}{} -Jones N.T., Germain R.M., Grainger T.N., Hall A.M., Baldwin L., Gilbert -B. 2015. Dispersal mode mediates the effect of patch size and patch -connectivity on metacommunity diversity. Journal of Ecology. -103:935--944. - -\hypertarget{ref-lalibertuxe92010}{} -Laliberté É., Legendre P. 2010. A distance-based framework for measuring -functional diversity from multiple traits. Ecology. 91:299--305. - -\hypertarget{ref-legendre2012}{} -Legendre P., Legendre L.F. 2012. Numerical ecology. Elsevier. - -\hypertarget{ref-mammola2019}{} -Mammola S. 2019. Assessing similarity of n-dimensional hypervolumes: -Which metric to use? Journal of Biogeography. 0. - -\hypertarget{ref-marcy2016}{} -Marcy A.E., Hadly E.A., Sherratt E., Garland K., Weisbecker V. 2016. -Getting a head in hard soils: Convergent skull evolution and divergent -allometric patterns explain shape variation in a highly diverse genus of -pocket gophers (thomomys). BMC evolutionary biology. 16:207. - -\hypertarget{ref-oksanen2007vegan}{} -Oksanen J., Kindt R., Legendre P., O'Hara B., Stevens M.H.H., Oksanen -M.J., Suggests M. 2007. The vegan package. Community ecology package. -10:631--637. - -\hypertarget{ref-psych}{} -Revelle W. 2018. Psych: Procedures for psychological, psychometric, and -personality research. Evanston, Illinois: Northwestern University. - -\hypertarget{ref-ruta2013}{} -Ruta M., Angielczyk K.D., Fröbisch J., Benton M.J. 2013. Decoupling of -morphological disparity and taxic diversity during the adaptive -radiation of anomodont therapsids. Proceedings of the Royal Society of -London B: Biological Sciences. 280. - -\hypertarget{ref-sedgewick1990}{} -Sedgewick R. 1990. Algorithms in c. Addison-Wesley, Reading. - -\hypertarget{ref-villuxe9ger2008}{} -Villéger S., Mason N.W.H., Mouillot D. 2008. New multidimensional -functional diversity indices for a multifaceted framework in functional -ecology. Ecology. 89:2290--2301. - -\hypertarget{ref-wills2001}{} -Wills M.A. 2001. Morphological disparity: A primer. In: Adrain J.M., -Edgecombe G.D., Lieberman B.S., editors. Fossils, phylogeny, and form. -Springer US. p. 55--144. - -\hypertarget{ref-wright2017}{} -Wright D.F. 2017. Phenotypic innovation and adaptive constraints in the -evolutionary radiation of palaeozoic crinoids. Scientific Reports. -7:13745. - - -\end{document} diff --git a/inst/MEE/CoverLetter.tex b/inst/MEE/CoverLetter.tex deleted file mode 100755 index e091472..0000000 --- a/inst/MEE/CoverLetter.tex +++ /dev/null @@ -1,50 +0,0 @@ -\documentclass[11pt]{letter} -\usepackage[a4paper,left=2.5cm, right=2.5cm, top=1cm, bottom=1cm]{geometry} -\usepackage{hyperref} -\usepackage[osf]{mathpazo} -\signature{Thomas Guillerme \\ (on behalf of my co-authors)} -\address{The University of Queensland \\School of Biological Sciences \\St Lucia QLD 4067, Australia \\guillert@tcd.ie} -\longindentation=0pt -\begin{document} - -\begin{letter}{} -\opening{Dear Editors,} - -Using patterns in multidimensional spaces to study biological processes is now a common toolkit in ecology and evolution. -In such analysis, researchers typically use matrices where traits (or transformed traits) are columns (e.g. anatomical measurements, community compositions) and observations are rows (e.g. specimens, field sites, etc.). -These matrices are commonly referred to as morphospaces in evolution or trait-space in ecology. -It is then possible to look at how observations or groups of observations occupy the trait space to understand biological processes. -For example, if plant community A occupies more space trait-space than community B, the former could be more diversed; -or if the morphospace of a group of organisms increases after colonisation of an island, it could be experiencing an adaptive radiation. -However, surprisingly very little work has been done on characterising what should be measured in these multidimensional analysis: what should be measured in a trait-space to accurately capture the pattern of difference between two communities or how trait-space occupancy changes through time? -Furthermore, the parallel between these analysis in ecology and evolution has never been clearly acknowledged (to our knowledge). - -In our research article, entitled ``Shifting spaces: how to summarise multidimensional spaces occupancy?'', we provide the first interdisciplinary review of 25 space occupancy metrics that uses the broad classification of metrics into volume, density and position to capture pattern changes in trait space. -We assess the behaviour of metrics using simulations and a selection of interdisciplinary empirical datasets; these cover a wide range of potential data types and evolutionary/ecological questions. -We also introduce a tool for Measuring Occupancy in Multidimensional Space ([`moms`](@@@)), which is a user-friendly open-source graphical interface tool to allow the tailored testing of metric behaviour for any use case. -This will allow researchers to comprehensively assess the properties of their trait-space and metrics associated with their specific biological question. - -Furthermore, we are are convinced that open data and reproducible papers are the key part of the future of academia. -Therefore, this entire paper and its supplementary is easily reproducible and based on open access dataset. -In fact, the paper is wrapped in a \texttt{R} package format and can be compile as a vignette through \url{https://github.com/TGuillerme/moms}. -We believe that this extra care put into making the paper easy to reproduce will foster not only a better understanding of multidimensional analysis but also future analysis on how the finding of this paper related beyond the fields of ecology and evolution. - -We look forward to hearing from you soon, - -\closing{Yours sincerely,} - - -% Reviewers: -% Stefano Mammola (ecology) -% Étienne Laliberté (ecology) -% Graeme Lloyd (evolution) -% Melanie Hopkins (evolution) -% Veronica Diaz (ecology) -% Caroline Tucker (ecology) -% Emilie Rayfield (evolution) -% Alexis Mychajliw (evolution) (amychajl@tarpits.org) -% Melissa Kemp (evolution) (mkemp@austin.utexas.edu). - - -\end{letter} -\end{document} diff --git a/inst/MEE/MEE_sub.md b/inst/MEE/MEE_sub.md deleted file mode 100644 index 859f2ab..0000000 --- a/inst/MEE/MEE_sub.md +++ /dev/null @@ -1,31 +0,0 @@ -# MEE submission points - * Single column, double line spaced - * Within the word count (6000-7000 words for Standard Articles, 3000 words for Applications) - * Continuous line and page numbering throughout - * Clearly defined manuscript structure as standard: Author details, Abstract (must be numbered according to Manuscript Specifications), Keywords, Introduction, Materials and Methods, Results, Discussion, Figures and Tables with captions - * Figures and Tables can be embedded within the text where referenced to facilitate reviewing - * Statement of where you intend to archive your data - -Title Page should include: - - * A concise and informative title. Do not include the authorities for taxonomic names. - * A list of all authors' names with names and addresses of Institutions. - * The name, address and e-mail address of the correspondence author. - * A running headline of not more than 45 characters. - > Multidimensional Spaces Occupancy - - -Abstract - -The Abstract must not exceed 350 words and should list the main results and conclusions, using simple, factual, numbered statements: - -Point 1: set the context for and purpose of the work; - -Point 2: indicate the approach and methods; - -Point 3: outline the main results; - -Point 4: identify the conclusions and the wider implications. - - - \ No newline at end of file diff --git a/inst/MEE/review.Rmd b/inst/MEE/review.Rmd deleted file mode 100644 index 8db12c6..0000000 --- a/inst/MEE/review.Rmd +++ /dev/null @@ -1,350 +0,0 @@ ---- -title: "MEE-19-10-743 Shifting spaces: which disparity or dissimilarity metrics best summarise occupancy in multidimensional spaces?" -author: "Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker" -date: "`r Sys.Date()`" -output: - pdf_document: - fig_width: 8 - fig_height: 8 - self_contained: false ---- - -Dear Editor, - - -Please find below our detailed response to the reviewers comments. For clarity, the reviewer's comments are in \textcolor{blue}{blue} and our response in black with the modified text in the manuscript in *italic*. -We've provided a version of the manuscript where the changes are highlighted in \textcolor{blue}{blue}. - -We believe that some of the parts that the manuscript that the reviewers found unclear were due to the differences of usage of the term "disparity" (which seems to have a disparate meanings throughout the literature depending on the author). -This is why we have deliberately tried to not refer to this term in the manuscript (appart for several specific instances) in order to distance ourselves from the "disparity" meaning*s*, and to try to bridge the gap between evolution and ecology, the latter of which rarely uses "disparity". - -# Reviewer: 1 - -\textcolor{blue}{In terms of potential modifications to the paper I would actually urge the authors to remove some material rather than add more. Specifically the paper entertains the notion (e.g., L281) that there might be a "magic bullet" metric that captures all three types of measure and that any metric could be applied to capture any of the three types of distribution. This seems a priori to be foolish. For example, sum of ranges could not reasonably be expected to capture density as it does not consider values between the edges (ranges) on any given axis. Similarly, if all points in a distribution were shifted exactly ten units in a specific direction then sum of ranges would remain identical. Thus this is clearly only ever going to be reasonably applied as a volume metric. One could also say something similar for nearest neighbor measures, which logically capture density, but would make no sense as a volume or position metric.} - -> We are sorry that the reviewer missed the point and we totally agree with them: it is foolish to have one-size-fit-all metric. This is why we mentioned it throughout the manuscript and have now highlighted it more to be sure there is no confusion. On line 281, we did actually insist that there is "that there is no perfect one-size-fit all", we have now changed it to "there **can be** no perfect one-size-fit all metric" to highlight that we, like the reviewer agree that finding a "magic bullet" is foolish. We have now highlighted that throughout the manuscript, for example: - -*no one measure describes all changes through a trait space and the results from each measurement are dependent on the characteristics of the space and the hypotheses* lines 65-66. - -*Similarly as for the simulated results, the empirical ones indicate that there can be no perfect one-size-fit all measurement.* lines 251-252 - -*We suggest using multiple measures to tackle different aspects of the trait space.* line 318. - -*We insist that no measure is better than the next one and that researchers should identify the most appropriate measures based on their trait space properties as well as their specific biological question.* lines 315-316. - - -\textcolor{blue}{This is especially true of the graphing (i.e., Figure 2E) where the "ideal" scenario will vary depending on the type of measure intended. Currently this makes Table 5 harder to understand than is necessary. Indeed, I would wager that the authors came up with their novel position metric purely because they realised none of the other metrics are logically intended to do this job. } - -> We've added a new panel to Figure 2 and improved the caption describing two "ideal" and "worst" scenarios for a space occupancy measure that actually captures the desired change in space size, density or position. The panels E and D on figure 2 are not describing a measure that would capture all three aspects (as we already highlighted above, we also believe this is foolish). In fact, we've now made clear in the figure caption that this figure is a key for interpreting Table 5: - -*Figure 2: this figure illustrates the different type of space reduction and how this could affect the measures for the simulation results displayed in table 5.* lines 221-221. - -> Whereas for the point regarding the novel position metric, yes, this was indeed the objective: not only did we aim to highlight the state-of-the-art in space occupancy measurements (or the lack thereof) but we also aim to move the field forward with novel propositions that will be hopefully be the basis of novel approaches in the field. - -\textcolor{blue}{I thus suggest a priori defining each metric as a volume, density or position metric and showing only the tests for each in the main text (alongside the ideal plot for a metric that captures that feature well).} - -> We are giving a "definition" of what each measure captures (or rather a loose expectancy on what they should capture) in table 3 column 3 ("Captures"). Of course this is to be contrasted with the results in table 5 where some measures can capture different aspects of space occupancy (columns 2, 3 and 4). - - -\textcolor{blue}{In other words, sum of ranges would only appear in the volume tests and nearest neighbour only in the density tests. The full results can be moved to the SI, but I think Figure 3 already captures the fact that these metrics measure very different things for the most part (very few show strong positive correlation), as we would logically expect.} - -> We clarified Figure 2 to show the "ideal" scenario for any type of metric with two panels showing the best and worst scenarios for any metrics. We did not, however separate the Table 5 for each different aspect (size, position, density) since we wanted this table to illustrate how different metrics can capture different aspects at different degrees. For example, the Average distance from centroid (row 1 Table 5) captures changes in size fairly well (column 1) but can also be influenced by changes in density (column 2). Thus changes in the Average distance from centroid are likely to correspond to changes in the size of the trait space but could also be due to changes in density of the trait space (though not likely due to a change in position of the trait space). We added the following to the Table 5 caption to make it clearer: - -*This figure illustrates how different metrics can be influenced by different aspects of changes in the trait space. For example, the Average distance from centroid (row 1) captures mainly changes in size (column 1), but also captures changes in density (column 2) but does not capture changes in position (column 3).* lines 221-222. - -\textcolor{blue}{Aside from this, the only potentially fatal flaw in the paper is that it seems to assume features about the spaces that may not be true. For example, that they are Euclidean (distances in the space reflect true distances between points) and isotropic (it is as easy to traverse the space in any direction). E.g., if the latter is not true then distance based measures (nearest neighbor and minimum spanning trees) are confounded. Similarly, if spaces are anisotropic then position based measures may be confounded. } - -> We've added a disclaimer specifying that the metrics presented here work for Euclidean isotropic spaces: - -*These measures are specific to Euclidean and isotropic trait spaces (which is not necessary for all measures).* lines 165-166. - -\textcolor{blue}{Additionally, the authors use the term "occupancy" many times, but I do not think they ever really consider occupancy in the truest sense, as this would require knowing how much of the space is truly /occupiable/. For example, where are the true limits of a space (i.e., the "edges")? And are all points inside those limits realistically occupiable? Note that these are simply hard problems, and the authors do already consider other aspects of empirical spaces such as total number of dimensions and distributions of variance across axes. However, I think they could go further in making it clear to readers that these issues must be considered. A nice example of how this could be done was shown by Hillis et al. (2005; Systematic Biology, 54, 471–482). Their Figure 10 shows how ordinations can drastically distort distances. Otherwise, I simply think the authors need to state these issues plainly to help guide readers in their metric choice.} - ->We thank the reviewer for point out this really interesting aspect and we have added the following disclaimer around the term "occupancy": - -*regardless of whether spaces are are truly "occupiable" which might be important in some cases - e.g. if the space is infinite or if some regions inapplicable.* lines 61-63. - -\textcolor{blue}{(I would also note that the authors currently post several questions that their metrics do not really answer, i.e., L47-48 "are groups overlapping in the trait-space? Are some regions of the trait-space not occupied? How do specific factors influence the occupancy of the trait-space?" These are good questions, but are not what the volume, density and position metrics actually capture. I would suggest rewording, or more explicitly state how these questions can be answered.)} - -> We reworded the generic questions to correspond to what the size, density and position can actually capture: - -*are groups occupying the same amount of trait space? Do some groups contain more species than others in the same amount of trait space? Are some specific factors correlated with different patterns of trait space occupancy?* lines 38-40. - -\textcolor{blue}{Some other thoughts the authors might want to consider (e.g., as additional discussion topics):} - -> We agree with the reviewers' three points below but due to the restricted amount of words allowed for this manuscript we could unfortunately only add short caveats. Note that these exact three points are discussed in a review paper (still in prep.) where they are discussed in more details. - -1. \textcolor{blue}{At no point do the authors really discuss visualisation of spaces. E.g., why are these metrics important? Why can't workers simply make bivariate plots and interpet those? (There are good reasons, of course, but I think it would help to make these explicit to emphasise the need for this paper.)} - -> We added a note saying that trait space occupancy cannot be studied beyond two or three dimensions using bi/tri-variate techniques: - -*Because of the multidimensional nature of these trait spaces, it is often not possible to study them using bi- or tri-variate techniques (Díaz et al. 2016; Hopkins and Gerber 2017; Mammola 2019).* lines 39-41. - -2. \textcolor{blue}{Volume measures generally assume convexity (hypercubes, hyperellipsoids, ranges), but what if the point distributions are concave or have voids (inoccupiable "holes")? This seems like another consideration of what a good metric is that is currently not discussed. Concavities can also be artefactual features of some ordinations (e.g., the "horseshoeing" seen with some spaces).} - -> We've added the following caveat for the size metrics: - -*Size measures do not take into account the distribution of the observations within a group and can often be insensitive to unoccupied "holes" in the trait space (overstimating the size; Blonder 2018).* lines 88-90. - -3. \textcolor{blue}{What about sampling considerations? I would imagine both density and range would be correlated with sample size, albeit negatively and positively respectively. I don't think the authors are obliged to solve this issue, but it is another consideration of what makes a "good" metric that could be discussed.} - -> We've added the following caveats for the density metrics: - -*Note that density based measures can be sensitive to sampling (e.g. if only living species are present).* lines 98-99. - -## Other minor corrections: - -* \textcolor{blue}{L125 - "Paleogene" (spelling). NB: This is a name and hence there is not a formal UK- (palaeo-) versus US-spelling (paleo-) option here.} - -> We fixed this typo. - -* \textcolor{blue}{L127 - "expand" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L152 - I applaud this approach over bombarding readers with information, but a brief note on why would be helpful (and what helped the ones in the main text make "the cut"; are these the most commonly used metrics?).} - -> We chose these metrics because of their contrasting results and because they were the ones commonly used to represent size and position in the trait space (expect for the average displacement). We added a justification in the paper: - -*Note that the paper contains the results for only eight measures which were selected as representative of common measures covering the size, density and position trait space aspects.* lines 144-145 - -* \textcolor{blue}{L155-156 - Both the table and contents require more information to understand them. Currently I have to go into the main text to get even a basic idea of what size means, for example.} - -> We expanded the table caption to: - -*Table 2: different simulated space distribution.* Name *of the simulated space;* dimensions *of the matrix (row\*columns);* distribution(s) *of the data on each dimensions (for the 'Random' space, each dimension is randomly chosen to be Normal, Uniform or Lognormal);* dimension variance *:diistribution of the variance between dimensions (when equal, the dimensions have the same variance);* correlation *between dimensions.* lines 148-149 - -* \textcolor{blue}{L165 - Missing \cite{} in TeX for healy?} - -> We fixed this typo. - -* \textcolor{blue}{L167 - Isn't a correlation (i.e., co-linearity) between traits exactly what many ordination techniques use to reduce the dimensionality of the space? I.e., should they be correlated?} - -> Ordination techniques (e.g. PCA) do indeed produce spaces with uncorrelated dimensions. We added this extra simulation parameter for the cases when trait spaces where not ordinated and could thus contain some degree of co-linearity. We've specified this in the main text: - -*Finally, we added a correlation parameter to illustrate the effect of co-linearity between traits (especially in non-ordinated trait spaces).* lines 158-159 - -* \textcolor{blue}{L173 - I suspect many of the attributions for these metrics are inaccurate, referring instead to later synthetic works. E.g., sum of variances or ranges certainly goes back to Foote 1992. I think the original authors should get their due here!} - -> We've updated the attributions of the Sum of ranges and sum of variances to Foote 1992 as well as the Minimum Spanning tree distance to Sedgewick 1990. - -* \textcolor{blue}{L176 - "their" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L214 - Grammar.} - -> We fixed this typo. - -* \textcolor{blue}{L215 - Surely you mean degree of overlap? It seems like you can say absolutely if they do overlap (p=1). Clarification would help.} - -> We've changed "probability of overlap" to "amount of overlap" throughout the text. - -* \textcolor{blue}{L216 - This could do with being a bit more informative for what each panel means. Specifically, panel E is probably the most important as it will need to be understood for the results to come to make sense. I.e., what does an ideal plot look like? A worst case scenario?} - -> We've modified figure 2 by showing a "worst" and "best" scenario (see above). - -* \textcolor{blue}{L227 - "a decrease in density can often lead to an increase in volume" Surely this depends on the metric. E.g., how could range increase if data are removed? It seems like this is also a consideration for whether a metric is doing a good job. } - -> We apologise for this typo and meant that a **increase** in density (in a normal space) can lead to a **decrease** of size (not the opposite): when increasing density of point in a normal space, the points further away from the space (which occur at a lower denisty) could be removed which would thus decrease the size. More simply: tightening a space will reduce it's density (again, if the space is normally distributed). We've correct this error. - -* \textcolor{blue}{L229 - Again, I think the correct citation for this is an older work.} - -> We changed the reference to Bellman 1957. - -* \textcolor{blue}{L234 - This seems like a failure of the metric not a hyperdimensionality issue. I.e., shouldn't a good metric capture an expansion regardless of the number of axes involved? Perhaps I am missing something here, but if I specifically think of sum of ranges in a fixed dimension space then increasing the range on any single axis will increase the overall sum.} - -> The reviewer is totally correct here for the metric "sum of ranges", we here meant that the metric "hypervolume" (i.e. product of ranges) would decrease. We've clarified this in the manuscript: - -*if a group expands in multiple dimensions (i.e. increase in size), the actual hypervolume <($\prod_{i}^{d} range_{Di}$) can decrease.* lines 215-216. - -* \textcolor{blue}{L245 - "were" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L262 - This Table/Figure is not as easy to interpret as the authors seem to think. I think it would help to show the ideal result for each case as a comparator.} - -> We improved the table readability by editing Figure 2 as detailed in comment about L216 above. - -* \textcolor{blue}{L274 - "a change a clear change" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L276 - It should be much easier to identify what the groups are here.} - -> We've added the groups names for each dataset in the table caption. - -* \textcolor{blue}{L282 - "except" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L318 - "values" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L342 - "\#\#\# Caveats" A Markdown subheader without a line break?} - -> We fixed this typo. - -* \textcolor{blue}{L346 - "accurately" Missing word? "reflect"?} - -> We fixed this typo. - -* \textcolor{blue}{L358 - Missing "a"?} - -> We fixed this typo. - -* \textcolor{blue}{L362 - "at" (grammar).} - -> We fixed this typo. - -* \textcolor{blue}{L366 - I would actually argue the opposite here. "Disparity" is simply poorly defined, but density and volume have clear meanings. However, specific tests make sense too. In other words, a good use would be something like "We used sum of ranges to capture volume of space occupied".} - -> We've changed the sentence to reflect the reviewer's opinion that we wholeheartedly agree with: - -*Third, we suggest not naming measures after the biological aspect they describe which can be vague (e.g. "disparity" or "functional dispersion") but rather after what they are measuring and why (e.g. "we used sum of ranges to measure the space size").* lines 316-328. - -* \textcolor{blue}{SI4, P3 - Plot seems to get cropped by bottom of page.} - ->We've fixed this display bug. - - - - - - - - - - - - - - - - - - - - - - - - - - -# Reviewer: 2 - -> We apologise for the imprecision in the terminology highlight by this reviewer and thanks them for pointing them out. The terms used in the previous version (and their vagueness) are unfortunately very common in the palaeontological literature (somehow slightly less in ecology) and the purpose of this paper was to strike a balance between mathematical precision and using a language that both fields can easily understand. We hope the updated version manages to keep this balance (see our detailed responses below): - -1. \textcolor{blue}{Many statistical and mathematical terms are used imprecisely or incorrectly. For instance: } - - * \textcolor{blue}{ The term "metric" has a particular meaning in statistics and geometry, but the term is used more vaguely in the sense of "measure" here. } - - > We changed mentions of metric to measure to make it clearer we do not mean metric in a geometric sense. - - * \textcolor{blue}{ "Volume" is a clear geometric concept, but it is used here for all kinds of formulas. The product of lengths (ranges, standard deviations, inter-quartile distances) in orthogonal directions can indeed be interpreted as a volume, but Procrustes variance (l 61) cannot (this is a sum, not a product of variances). Also the example in Figure 1A is wrong: a sum of ranges is not a volume. Also in Table 3, the first two formulas, "average distance from centroid" (which should read "square root of summed squared distances from centroid") and "sum of variances", are not volumes (and, by the way, they are the same except for the root and the division by n; hence their correlation in the example).} - - > We have chosen to change our broad category "volume" to "size" to avoid confusion with the mathematical definition of volume. - We have renamed the "average distance from centroid" to "average Euclidean distance from centroid". - Also, the similarity between the average Euclidean distance from centroid is indeed expected but we thought we leave both in the main text of the manuscript since they are commonly used. - - * \textcolor{blue}{ Likewise, the term "density" is used in a way that I fail to understand. In statistics, "probability density" is a standard concept that also applies to multivariate data spaces. Sometimes, "density" is used here to describe a variance, e.g. in Fig. 1B, but it is not clear why. Also I don't understand the description in l 103-110.} - - > Throughout the manuscript we use density in a more colloquial (or physical) way as suggest by the reviewer where density is an indication of the quantity of observations per units of size (volume, area, etc.): some trait spaces can have the same size but contain different number of observations. - In a sense, this can be linked to probability density since a high value (density) in a probability density function of a distribution corresponds to the parts of the distribution with a high number of observations. - We have tried to make this clearer in our definition of the category of measurement: - - *Density gives an indication of the quantity of observations in the trait space. They can be interpreted as the distribution of the observations* within *a group in the trait space. Groups with higher density contain more observations that will tend to be more similar to each other. For example, if group A is greater is size than group B and both have the same density, similar mechanisms could be driving both groups’ trait space occupancy. This pattern could suggest that A is older and has had more time to achieve extreme trait combinations under essentially the same process as younger, smaller group B (Endler et al. 2005). Note that density based measures can be sensitive to sampling (e.g. if only living species are present). Density measures are less common compared to size measures, but they are still used in both ecology (e.g. the minimum spanning tree length; Oksanen et al. 2007) and evolution (e.g. the average pairwise distance; Harmon et al. 2008).* lines 93-102. - - * \textcolor{blue}{ The concept of a "position" of a group in trait space is not clear throughout the manuscript. E.g., in Fig 1C, the median distance from the centroid is considered a "position", but it is just another measure of disparity/variance. And why should "position metrics be harder to interpret in multidimensional spaces" (l 115)? } - - > We've changed the position measure in Fig. 1C from "average distance from centroid" to "distances to centre". In this specific case the centre (0,0) of the space and the centroid (-0.23, -0.46) actually differ (though this is not always the case, especially when looking at all the data in a PCA for example). Although we agree that the "distances from centroid" gives an approximation of size, the "distances from centre" gives an approximation of position (i.e. for observations that are further away from the centre). This can be generalised to any measurements like "distances from a fixed point" which will give an idea of the position of a group in trait space. This also links to our comment on line 115 about the difficulty of interpreting position measures in trait space: if two groups have the same position in trait space (say using the average displacement measurement), it just indicates that are equally away from a fixed point (say the centre) but not whether they are in the same position in the trait space (e.g. in a 2D space, groups can be equally distant from the centre left, right up or down - these possible positions will of course increases with each additional dimensions). We've specified this in the main text: - - *In a 2D space, two groups can be equally distant from a fixed point but in different parts of the space (left, right, up, or down - with the amount of parts of space increasing with dimensions).* lines 107-109. - - * \textcolor{blue}{ A data matrix is not yet a "space" (l 80). In addition to a set of elements, a mathematical space requires some relations between the elements, such as a metric or nearness relationship (for assessable discussions of mathematical spaces in biology see, e.g., Stadler et al. 2002 J Theor Biol, Mitteroecker \& Huttegger 2009 Biol Theory).} - - Here we followed the general mathematical definition of space as a set with some mathematical structure. The exact definition of a mathematical set and structure is way beyond our mathematical comfort zone (and towards logics and philosophy - we specifically followed the [simplification explained here](https://math.stackexchange.com/questions/177937/difference-between-space-and-mathematical-structure)). - The definition in Mitteroecker \& Huttegger 2009 is specific to "morphospaces" whereas the Stadler et al 2002 one is more close to trait spaces as vaguely defined in our manuscript which we have now updated to: - - *Here we define trait spaces as any matrix where rows are observations and columns are their related traits..* line 72. - - > as well as in Table 1: - - *Matrix (n $\times$ d) with a structural relation between rows and columns* lines 36-37 - - * \textcolor{blue}{ What is a "random" distribution, as opposed to a normal or uniform distribution (e.g., l 160)?} - - >By random we meant a random selection between normal, lognormal and uniform. We've fixed it in the main text: - - *We used a range of distributions (uniform, normal or a random combination of uniform, normal and lognormal) to test the effect of observation distributions on the measurements* lines 150-152. - -2. \textcolor{blue}{I find the statement/conclusion "no one metric describes all of trait-space" somewhat trivial. Also, it is pretty clear that a measure of variance/disparity does not inform about the mean or central tendency; one does not need a simulation for this insight. If one cares about a single measure that captures group differences in both mean and variance (e.g., to assess group overlap), one should use appropriate statistics, such as Mahalanobis distance, Fisher information metric, Bhattacharrya coefficient (which the authors cite in a different context), or one of the many others derived from them.} - -> We do wholeheartedly agree with this reviewer that the conclusion (and the topic) of this paper is trivial. -This was actually also highlighted by reviewer 1 ("Much of what the authors discuss here could be considered common sense"). -However, as reviewer 1 also highlights: "sadly much of the existing literature has failed to take much of this on board". -We of course share the same opinion and this observation was at the origin of this whole paper: providing a basis for "helping workers consider what may be best to use for their own data" (reviewer 1). -Furthermore, we are not only proposing a "trivial" conclusion but we also put some effort into providing a novel tool (`moms`) allowing researchers to make their own decision on which metric to use in their specific case (rather than "blindly" following our suggestions) and propose some specific terms (space occupancy, trait spaces, size, density and position measures, etc...) that aim to bridge the gap between ecology and evolution that both use multidimensional methods to study similar essential biological questions but with different approaches and a different jargon. -Through this paper we thus propose some basis for both fields to learn from each other and hopefully allowing future fruitful collaborations. - -3. \textcolor{blue}{The brief discussion of the "curse of dimensionality" is interesting and could be extended, as this is relevant for many modern datasets. However, the statement about the "probability of two points A and B overlapping in n dimensions" is inappropriate, as this probability is zero for continuous variables, regardless of dimension.} - -> We've removed the bit about the probability of overlap (we should have mentioned "groups" rather than "points") and replaced it by a more general point on the "curse": - -*This can have two main consequences: 1) the probability of overlap between two groups decreases as a product of the number of dimensions; and 2) the amount of samples needed to "fill" the spaces increases exponentially [see this interactive illustration by Toph Tucker](https://observablehq.com/@tophtucker/theres-plenty-of214room-in-the-corners).* lines 211-213. - -4. \textcolor{blue}{I miss a discussion of how the measurement scales of the variables can influence the described statistics. It seems that all the variables are assumed to have an interval scale, for some statistics also a ratio scale. Is this realistic? Also nothing is mentioned about the topology of the multivariate data spaces: Do the spaces need to be metric or even Euclidean? Do the units and scales of the variables all need to be the same?} - -> See our response to reviewer 1's comment on the trait space properties. Furthermore, we specified that we scale the measurements as to make them comparable since they are relative to each trait space: - -*Because occupancy measures are dependent on the space, we scaled and centred them between -1 and 1 to make them comparable (by subtracting the observed occupancy without reduction to all the measures of the reduced spaces and then divided it by the maximum observed occupancy).* lines 200-202. - -> We did not mention, however, whether the units in the trait space need to be Euclidean, scaled, homoscedastic, etc. since this can vary between studies and measurements. In our case most our measures are assuming only a Euclidean isotropic space but not necessarily scaled. The scaling issue of the traits (variables) has more importance in representing the data (e.g. plotting a projection of the trait space) but less in measuring occupancy. This is illustrated in the exaggerated example below: - -```{r, eval = FALSE} -## A small trait space with a huge scale and distribution difference per axis -space <- cbind(rnorm(10, sd = 1e3), rlnorm(10, sdlog = 1e-3)) -## The ellipsoid volume can be calculated using both axis: -dispRity::ellipse.volume(space) -## Operations discussed in this paper on the space are valid -## However, representing without scaling the axis is missleading -par(mfrow = c(1,2)) -plot(space, main = "scaled axis") -## Since the real space looks like this -plot(space, xlim = c(-1.5e3, 1.5e3), ylim = c(-1.5e3, 1.5e3), - main = "unscaled axis") -``` - -5. \textcolor{blue}{The different statistics were evaluated and compared across datasets with different dimensionalities and distributions. While this seems feasible, the computation of p-values by ANOVA is not. This is not a question of type I error as you are not testing against sampling variation. In other words, you treat each dataset as a case drawn independently from the same distribution of datasets. Clearly, this is not what you are doing.} - -> We somewhat disagree with this reviewer's comment suggesting that the space occupancy measurements are not drawn independently from the same distribution (*sensu* the statistical population) and that we can therefore not estimate the p-value correctly from a simple ANOVA. In fact, because of the centering and scaling of all space occupancy metrics, the metrics are effectively distributed near normally (figure 3) and thus form a coherent population. For example, any sum of variance measured in any simulated space from any parameter forms the "sum of variance population" - the ANOVA then tests whether there is a difference within groups (i.e. the sum of variance measured in a specific simulated space from specific parameters) and between groups (the sum of variance measured in any simulated space). However, we understand the reviewers' concern that these data could be considered as independent but we do not know of any way to report the within/between variation and interpreting the results without using the F-distribution. Maybe the reviewer could help us with this one? Meanwhile, we reported the F-statistic AND the p-value in the table for more clarity. - -### Minor comments - -* \textcolor{blue}{The term "trait space" does not need to be hyphenated.} - -> We removed the hyphen throughout the manuscript. - -* \textcolor{blue}{line 11: "a subset of this trait-space" Either a "subset of this set" or a "subspace of this space" } - -> We changed it to: - -*a subspace of this space* line 4. - -* \textcolor{blue}{l 39: typo "used"} - -> We fixed this typo. - -* \textcolor{blue}{ 175: "eigenvalue" is a single word} - -> We fixed this typo. - -* \textcolor{blue}{Table 3: the formula for "average nearest neighbourhood distance" does not seem correct to me; last line, right column: what is a "centre of trait space"? } - -> We fixed the average nearest neighbour distance and specified the coordinates of the trait space as the point with coordinates (0, 0, ...). - -* \textcolor{blue}{Table 5: A p-value cannot be 0.} - -> We changed our p-value rounding to display p-values rounded to 0 as <1e-3. diff --git a/inst/MEE/review.md b/inst/MEE/review.md deleted file mode 100644 index 2cae28a..0000000 --- a/inst/MEE/review.md +++ /dev/null @@ -1,566 +0,0 @@ ---- -title: "MEE-19-10-743 Shifting spaces: which disparity or dissimilarity metrics best summarise occupancy in multidimensional spaces?" -author: "Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker" -date: "2020-02-12" -output: - html_document: - fig_width: 8 - fig_height: 8 ---- - -Dear Editor, - -> Mark, Ariel and Vera: this is still a drafty version! Ignore the colours business and the blocks with three @ (@@@) are still not written properly, so if you want to comment them, please don't worry about the tone and the grammar. (the places with a single @ are just the line tags that I'll have to update before resubmission) - -@@@please find below our detailed response to the reviewers comments. For clarity, the reviewer's comments are in blue and our response in black with the modified text in the manuscript in *italic*. -We've provided a version of the manuscript where the changes are highlighted in blue. - - -We believe that some of the parts that the manuscript that the reviewers found unclear were due to the differences of usage of the term "disparity" (which seems to have a disparate meanings throughout the literature depending on the author). -This is why we have deliberately tried to not refer to this term in the manuscript (appart for several specific instances) in order to distance ourselves from the "disparity" meaning*s*, and to try to bridge the gap between evolution and ecology where "disparity" is actually rarely used. - - -@@@TODO: also update (compile) the supplementaries -@@@Add reference to Bazzi et al Curr. Biol. -@@@Cite Mammalo new paper https://www.biorxiv.org/content/10.1101/2020.01.25.919373v1.full.pdf - - -# Associate Editor Comments to Author: - -*

[...]The main problem pointed out by the second reviewer is that the manuscript suffers from a lack of accuracy and clarity in the use of mathematical terminology. I share this concern. I fact I found the manuscript difficult to follow at any level of detail. After reading I was left with little added understanding of what the different measures are supposed to measure or how they perform in these goals. For example, I did not understand the results in the seemingly central Table 5, and like the reviewers I found many mathematical statements to be odd or nonsensical. [...] The topic is certainly appropriate, but a revised manuscript must be substantially improved in mathematical clarity and rigor. The motivation and goals of the measures must be clearly laid out and evaluated. [...]

- -# Reviewer: 1 - -

In terms of potential modifications to the paper I would actually urge the authors to remove some material rather than add more. Specifically the paper entertains the notion (e.g., L281) that there might be a "magic bullet" metric that captures all three types of measure and that any metric could be applied to capture any of the three types of distribution. This seems a priori to be foolish. For example, sum of ranges could not reasonably be expected to capture density as it does not consider values between the edges (ranges) on any given axis. Similarly, if all points in a distribution were shifted exactly ten units in a specific direction then sum of ranges would remain identical. Thus this is clearly only ever going to be reasonably applied as a volume metric. One could also say something similar for nearest neighbor measures, which logically capture density, but would make no sense as a volume or position metric.

- -> On line 281, we did actually insist that there is "that there is no perfect one-size-fit all", we have now changed it to "there **can be** no perfect one-size-fit all metric" to highlight that we, like the reviewer agree that finding a "magic bullet" is foolish. - -

This is especially true of the graphing (i.e., Figure 2E) where the "ideal" scenario will vary depending on the type of measure intended. Currently this makes Table 5 harder to understand than is necessary. Indeed, I would wager that the authors came up with their novel position metric purely because they realised none of the other metrics are logically intended to do this job. I thus suggest a priori defining each metric as a volume, density or position metric and showing only the tests for each in the main text (alongside the ideal plot for a metric that captures that feature well). In other words, sum of ranges would only appear in the volume tests and nearest neighbour only in the density tests. The full results can be moved to the SI, but I think Figure 3 already captures the fact that these metrics measure very different things for the most part (very few show strong positive correlation), as we would logically expect.

- -> We clarified the Figure 2 to show the "ideal" scenario for any type of metric with two panels showing the best and worst scenarios for any metrics. We did not however separated the Table 5 for each different aspect (size, position, density) since we wanted this table to illustrate how different metrics can capture different aspects at different degrees. For example, the Average distance from centroid (row 1 Table 5) captures changes in size fairly well (column 1) but can also be influenced by changes in density (column 2). Thus changes in the Average distance from centroid are likely to correspond to changes in the size of the trait space but could also be due to changes in density of the trait space (though not likely due to a change in position of the trait space). We added the following to the Table 5 caption to make it clearer: - -*This figure illustrates how different metrics can be influenced by different aspects of changes in the trait space. For example, the Average distance from centroid (row 1) captures mainly changes in size (column 1), but also captures changes in density (column 2) but does not capture changes in position (column 3).* lines@ 298-301. - - -

Aside from this, the only potentially fatal flaw in the paper is that it seems to assume features about the spaces that may not be true. For example, that they are Euclidean (distances in the space reflect true distances between points) and isotropic (it is as easy to traverse the space in any direction). E.g., if the latter is not true then distance based measures (nearest neighbor and minimum spanning trees) are confounded. Similarly, if spaces are anisotropic then position based measures may be confounded.

- -> We've added a disclaimer specifying that the metrics presented here work for Euclidean isotropic spaces: - -*Note that these metrics are specific to trait spaces that are Euclidean (distances in the space reflect true distances between points) and isotropic (it is as easy to traverse the space in any direction).* lines@ 195-197. - -

Additionally, the authors use the term "occupancy" many times, but I do not think they ever really consider occupancy in the truest sense, as this would require knowing how much of the space is truly /occupiable/. For example, where are the true limits of a space (i.e., the "edges")? And are all points inside those limits realistically occupiable? Note that these are simply hard problems, and the authors do already consider other aspects of empirical spaces such as total number of dimensions and distributions of variance across axes. However, I think they could go further in making it clear to readers that these issues must be considered. A nice example of how this could be done was shown by Hillis et al. (2005; Systematic Biology, 54, 471–482). Their Figure 10 shows how ordinations can drastically distort distances. Otherwise, I simply think the authors need to state these issues plainly to help guide readers in their metric choice.

- ->We thank the reviewer for point out this really interesting aspect and we have added the following disclaimer around the term "occupancy": - -*Note that we refer to occupancy here as the general term for where observations are in a trait space. -This definition excludes whether spaces are are truly "occupiable" (e.g. are there limits to the space, are some regions of the space inapplicable, etc.) which might be of importance in particulare spaces (e.g. in tree spaces; @Hillis2005).* lines@ 70-74 - - -

(I would also note that the authors currently post several questions that their metrics do not really answer, i.e., L47-48 "are groups overlapping in the trait-space? Are some regions of the trait-space not occupied? How do specific factors influence the occupancy of the trait-space?" These are good questions, but are not what the volume, density and position metrics actually capture. I would suggest rewording, or more explicitly state how these questions can be answered.)

- -> We reworded the generic questions to correspond to what the size, density and position can actually capture: - -*are groups occupying the same amount of trait space? -Do some groups have contain more species than others in the same amount of trait space? -Are some specific factors correlated to different patterns of occupancy of the trait space (e.g. are some factors correlated with more occupancy)?* lines@ 45-47. - -

Some other thoughts the authors might want to consider (e.g., as additional discussion topics):

- -> We agree with the reviewers' three points below but due to the restricted amount of words allowed for this manuscript we could unfortunately only add short caveats. Note that these exact three points are discussed in a review paper (still in prep.) where they are discussed in more details. - -1.

At no point do the authors really discuss visualisation of spaces. E.g., why are these metrics important? Why can't workers simply make bivariate plots and interpet those? (There are good reasons, of course, but I think it would help to make these explicit to emphasise the need for this paper.)

- -> We've added a note saying that trait space occupancy cannot be studied beyond 2 or 3 dimensions using bi/tri-variate techniques: - -*Because of the multidimensional nature of these trait spaces, it is often not possible to study them use bi- or tri-variate techniques.* lines@ 47-49 - -2.

Volume measures generally assume convexity (hypercubes, hyperellipsoids, ranges), but what if the point distributions are concave or have voids (inoccupiable "holes")? This seems like another consideration of what a good metric is that is currently not discussed. Concavities can also be artefactual features of some ordinations (e.g., the "horseshoeing" seen with some spaces).

- -> We've added the following caveat for the size metrics: - -*Size measures do not take into account the distribution of the observations within a group and can be sensible to innocupied "holes" in it (Blonder et al. 2018).* lines@ 104-105 - -3.

What about sampling considerations? I would imagine both density and range would be correlated with sample size, albeit negatively and positively respectively. I don't think the authors are obliged to solve this issue, but it is another consideration of what makes a "good" metric that could be discussed.

- -> We've added the following caveats for the density metrics: - -*Note that density based metrics can be sensitive to sampling*. lines@ 115-116 - -## Other minor corrections: - -*

L125 - "Paleogene" (spelling). NB: This is a name and hence there is not a formal UK- (palaeo-) versus US-spelling (paleo-) option here.

- -> We fixed this typo. - -*

L127 - "expand" (grammar).

- -> We fixed this typo. - -*

L152 - I applaud this approach over bombarding readers with information, but a brief note on why would be helpful (and what helped the ones in the main text make "the cut"; are these the most commonly used metrics?).

- -> We chose these metrics because of their contrasted results and because the where the ones commonly used to represent size and position in the trait-space (expect for the average displacement). We've added a justification in the paper: - -*Note that the paper contains the results for only eight metrics which were selected as representative of common metrics covering the size, density and position trait space aspects. However, the results for the additional 17 metrics is available in the supplementary material 4.* lines@ 162-164 - -*

L155-156 - Both the table and contents require more information to understand them. Currently I have to go into the main text to get even a basic idea of what size means, for example.

- -> We expanded the table caption to: - -*Table 2: different simulated space distribution.* Name *of the simulated space in this paper;* dimensions *of the matrix (row\*columns);* distribution(s) *of the data on each dimensions (for the 'Random' space, each dimension is randomly chosen to be Normal, Uniform or Lognormal);* dimension variance *:distribution of the variance between dimensions (when equal, the dimensions have the same variance, otherwise, the variance per dimensions is decreasing either lognormally (multiplicative) or normally (additive);* correlation *between dimensions.* lines 167-172 - -*

L165 - Missing \cite{} in TeX for healy?

- -> We fixed this typo. - -*

L167 - Isn't a correlation (i.e., co-linearity) between traits exactly what many ordination techniques use to reduce the dimensionality of the space? I.e., should they be correlated?

- -> Ordination techniques (e.g. PCA) do indeed produce spaces with uncorrelated dimensions. We added this extra simulation parameters for the cases when trait-spaces where not ordinated and could thus contain some degree of co-linearity. We've specified this in the main text: - -*Finally, we added a correlation parameter to take into account the potential correlation between different traits to illustrate the effect of co-linearity between traits (especially in non-ordinated trait spaces).* lines@ 183-185 - -*

L173 - I suspect many of the attributions for these metrics are inaccurate, referring instead to later synthetic works. E.g., sum of variances or ranges certainly goes back to Foote 1992. I think the original authors should get their due here!

- -> We've updated the attributions of the Sum of ranges and sum of variances to Foote 1992 as well as the Minimum Spanning tree distance to Sedgewick 1990. - -*

L176 - "their" (grammar).

- -> We fixed this typo. - -*

L214 - Grammar.

- -> We fixed this typo. - -*

L215 - Surely you mean degree of overlap? It seems like you can say absolutely if they do overlap (p=1). Clarification would help.

- -> We've changed "probability of overlap" to "amount of overlap" throughout the text. - -*

L216 - This could do with being a bit more informative for what each panel means. Specifically, panel E is probably the most important as it will need to be understood for the results to come to make sense. I.e., what does an ideal plot look like? A worst case scenario?

- -> We've modified figure 2 by showing a "worst" and "best" scenario (see above). - -*

L227 - "a decrease in density can often lead to an increase in volume" Surely this depends on the metric. E.g., how could range increase if data are removed? It seems like this is also a consideration for whether a metric is doing a good job.

- -> We apologise for this typo and meant that a **increase** in density (in a normal space) can lead to a **decrease** of size (not the opposite): when increasing density of point in a normal space, the points further away from the space (which occur at a lower denisty) could be removed which would thus decrease the size. More simply: tightening a space will reduce it's density (again, if the space is normally distributed). We've correct this error. - -*

L229 - Again, I think the correct citation for this is an older work.

- ->We change the reference to Bellman 1957. - -*

L234 - This seems like a failure of the metric not a hyperdimensionality issue. I.e., shouldn't a good metric capture an expansion regardless of the number of axes involved? Perhaps I am missing something here, but if I specifically think of sum of ranges in a fixed dimension space then increasing the range on any single axis will increase the overall sum.

- -> The reviewer is totally correct here for the metric "sum of ranges", we here meant that the metric "hypervolume" (i.e. product of ranges) would decrease. We've clarified this in the manuscript: - -*if a group expands in multiple dimensions (i.e. increase in size), the actual hypervolume <($\prod_{i}^{d} range_{Di}$) can decrease.* lines@ 262-263 - -*

L245 - "were" (grammar).

- -> We fixed this typo. - -*

L262 - This Table/Figure is not as easy to interpret as the authors seem to think. I think it would help to show the ideal result for each case as a comparator.

- -> We improved the table readability by editing Figure 2 as detailed in comment about L216 above. - -*

L274 - "a change a clear change" (grammar).

- -> We fixed this typo. - -*

L276 - It should be much easier to identify what the groups are here.

- -> We've added the groups names for each dataset in the table caption. - -*

L282 - "except" (grammar).

- -> We fixed this typo. - -*

L318 - "values" (grammar).

- -> We fixed this typo. - -*

L342 - "### Caveats" A Markdown subheader without a line break?

- -> We fixed this typo. - -*

L346 - "accurately" Missing word? "reflect"?

- -> We fixed this typo. - -*

L358 - Missing "a"?

- -> We fixed this typo. - -*

L362 - "at" (grammar).

- -> We fixed this typo. - -*

L366 - I would actually argue the opposite here. "Disparity" is simply poorly defined, but density and volume have clear meanings. However, specific tests make sense too. In other words, a good use would be something like "We used sum of ranges to capture volume of space occupied".

- -> We've changed the sentence to reflect the reviewer's opinion that we wholeheartedly agree with: - -*Third, we suggest to not name measures as the biological aspect they are describing which can be vague (e.g. "disparity" or "functional dispersion") but rather what they are measuring and why (e.g. "we used sum of ranges to capture the trait space size").* lines@ 405-497 - -*

SI4, P3 - Plot seems to get cropped by bottom of page.

- ->We've fixed this display bug. - - - - - - - - - - - - - - - - - - - - - - - - - - -# Reviewer: 2 - -1.

Many statistical and mathematical terms are used imprecisely or incorrectly. For instance:

- - *

The term "metric" has a particular meaning in statistics and geometry, but the term is used more vaguely in the sense of "measure" here.

- - >We changed mentions of metric to measurements to make it clearer we do not mean metric in a geometric sense. - - *

"Volume" is a clear geometric concept, but it is used here for all kinds of formulas. The product of lengths (ranges, standard deviations, inter-quartile distances) in orthogonal directions can indeed be interpreted as a volume, but Procrustes variance (l 61) cannot (this is a sum, not a product of variances). Also the example in Figure 1A is wrong: a sum of ranges is not a volume. Also in Table 3, the first two formulas, "average distance from centroid" (which should read "square root of summed squared distances from centroid") and "sum of variances", are not volumes (and, by the way, they are the same except for the root and the division by n; hence their correlation in the example).

- - >We have chosen to change our broad category "volume" to "size" to avoid confusion with the mathematical definition of volume. - We have renamed the "average distance from centroid" to "average Euclidean distance from centroid". - Also, the similarity between the average Euclidean distance from centroid is indeed expected but we thought we leave both in the main text of the manuscript since they are commonly used. - - *

Likewise, the term "density" is used in a way that I fail to understand. In statistics, "probability density" is a standard concept that also applies to multivariate data spaces. Sometimes, "density" is used here to describe a variance, e.g. in Fig. 1B, but it is not clear why. Also I don't understand the description in l 103-110.

- - >Throughout the manuscript we use density in a more colloquial (or physical) way as suggest by the reviewer where density is an indication of the quantity of observations per units of size (volume, area, etc.): some trait-spaces can have the same size but contain different number of observations. - In a sense, this can be linked to probability density since a high value (density) in a probability density function of a distribution corresponds to the parts of the distribution with a high number of observations. - We have tried to make this clearer in our definition of the category of measurement: - - *Density measures an indication of the quantity of observations in the trait space. They can be interpreted as the distribution of the observations within a group in the trait space. Groups with higher density have more observations within it (i.e. more observations per approximation of size) that will tend to be more similar to each other. For example, if group A is greater than group B and both have the same density (observations are equally distant within each group), similar mechanisms could be driving both groups’ trait space occupancy. However, this might suggest that A is older and had more time to achieve more extreme trait combinations under essentially the same process (Endler et al. 2005). Note that density based measures can be sensitive to sampling. Density is less commonly measured compared to size, but it is still used in both ecology (e.g. the minimum spanning tree length; Oksanen et al. 2007) and evolution (e.g. the average pairwise distance; Harmon et al. 2008).* lines@ 109-118 - - *

The concept of a "position" of a group in trait space is not clear throughout the manuscript. E.g., in Fig 1C, the median distance from the centroid is considered a "position", but it is just another measure of disparity/variance. And why should "position metrics be harder to interpret in multidimensional spaces" (l 115)?

- - > We've changed the position metric in Fig. 1C from "average distance from centroid" to "distances to centre". In this specific case the centre (0,0) of the space and the centroid (-0.23, -0.46) actually differ (though this is not always the case, especially when looking at all the data in a PCA for example). Although we agree that the "distances from centroid" gives an approximation of size, the "distances from centre" gives an approximation of position (i.e. for observations that are further away from the centre). This can be generalised to any measurements like "distances from a fixed point" which will give an idea of the position of a group in trait space. This also links to our comment on line 115 about the difficulty of interpreting position measures in trait space: if two groups have the same position in trait space (say using the average displacement measurement), it just indicates that are equally away from a fixed point (say the centre) but not whether they are in the same position in the trait space (e.g. in a 2D space, groups can be equally distant from the centre left, right up or down - these possible positions will of course increases with each additional dimensions). We've specified this in the main text: - - *For example in a 2D space, two groups can be equally distant from a fixed point but in different parts of the space left, right up or down (with the position possibilities increasing with the number of dimensions).* lines@ 123-125 - - *

A data matrix is not yet a "space" (l 80). In addition to a set of elements, a mathematical space requires some relations between the elements, such as a metric or nearness relationship (for assessable discussions of mathematical spaces in biology see, e.g., Stadler et al. 2002 J Theor Biol, Mitteroecker & Huttegger 2009 Biol Theory).

- - Here we followed the general mathematical definition of space as a set with some mathematical structure. The exact definition of a mathematical set and structure is way beyond our mathematical comfort zone (and towards logics and philosophy - we specifically followed the [simplification explained here](https://math.stackexchange.com/questions/177937/difference-between-space-and-mathematical-structure)). - The definition in Mitteroecker & Huttegger 2009 is specific to "morphospaces" whereas the Stadler et al 2002 one is more close to trait spaces as vaguely defined in our manuscript which we have now updated to: - - *In this paper, we define trait spaces as any matrix where rows are observations and columns are traits, where both observations and traits are structurally related (e.g. there is a phylogenetic relation between observations - and traits, etc.).* lines@ 83-85 - - *

What is a "random" distribution, as opposed to a normal or uniform distribution (e.g., l 160)?

- - >By random we meant a random selection between normal, lognormal and uniform. We've fixed it in the main text: - - *We used a range of distributions (uniform, normal or a random combination of uniform, normal and lognormal) to test the effect of observation distributions on the measurements.* lines@ 175-177 - -2.

I find the statement/conclusion "no one metric describes all of trait-space" somewhat trivial. Also, it is pretty clear that a measure of variance/disparity does not inform about the mean or central tendency; one does not need a simulation for this insight. If one cares about a single measure that captures group differences in both mean and variance (e.g., to assess group overlap), one should use appropriate statistics, such as Mahalanobis distance, Fisher information metric, Bhattacharrya coefficient (which the authors cite in a different context), or one of the many others derived from them.

- -> We do wholeheartedly agree with this reviewer that the conclusion (and the topic) of this paper is trivial. -This was actually also highlighted by reviewer 1 ("Much of what the authors discuss here could be considered common sense"). -However, as reviewer 1 also highlights: "sadly much of the existing literature has failed to take much of this on board". -We of course share the same opinion and this observation was at the origin of this whole paper: providing a basis for "helping workers consider what may be best to use for their own data" (reviewer 1). -Furthermore, we are not only proposing a "trivial" conclusion but we also put some effort into providing a novel tool allowing researchers to make their own decision on which metric to use in their specific case (rather than "blindly" following our suggestions) and propose some specific terms (space occupancy, trait spaces, size, density and position measurements, etc...) that aim to bridge the gap between ecology and evolution that both use multidimensional methods to study similar essential biological questions but with different approaches and a different jargon. -Through this paper we thus propose some basis for both fields to learn from each other and hopefully allowing future fruitful collaborations. - - -3.

The brief discussion of the "curse of dimensionality" is interesting and could be extended, as this is relevant for many modern datasets. However, the statement about the "probability of two points A and B overlapping in n dimensions" is inappropriate, as this probability is zero for continuous variables, regardless of dimension.

- -> We've removed the bit about the probability of overlap (we should have mentioned "groups" rather than "points") and replaced it by a more general point on the "curse": - -*This can have two main unforeseen mathematical consequence: 1) the probability of overlap between two groups decreases as a product of the number of dimensions; and 2) the amount of samples needed to "fill" the spaces increases exponentially [see this interactive illustration by Toph Tucker](https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners). -Furthermore computational time can be increased exponentially rather than linearly with the number of dimensions.* lines@ 257-261 - -4.

I miss a discussion of how the measurement scales of the variables can influence the described statistics. I seems that all the variables are assumed to have an interval scale, for some statistics also a ratio scale. Is this realistic? Also nothing is mentioned about the topology of the multivariate data spaces: Do the spaces need to be metric or even Euclidean? Do the units and scales of the variables all need to be the same?

- -> See our response to reviewer 1 comment on the trait space properties (lines @@@). Furthermore, we specified that we scale the measurements as to make them comparable since they are relative to each trait space: - -*Since the occupancy measures are dependent on their trait space, we scaled and centred them between -1 and 1 to make them comparable.* lines@ 246-247 - -> We did not mention however whether the units in the trait space need to be Euclidean, scaled, homoscedastic, etc. since this can vary between studies and measurements. In our case most our measures are assuming only a Euclidean isotropic space but not necessarily scaled. The scaling issue of the traits (variables) has more importance in representing the data (e.g. plotting a projection of the trait space) but less in measuring occupancy. This is illustrated in the exaggerated example below: - - -```r -## A small trait space with a huge scale and distribution difference per axis -space <- cbind(rnorm(10, sd = 1e3), rlnorm(10, sdlog = 1e-3)) -## The ellipsoid volume can be calculated using both axis: -dispRity::ellipse.volume(space) -## Operations discussed in this paper on the space are valid -## However, representing without scaling the axis is missleading -par(mfrow = c(1,2)) -plot(space, main = "scaled axis") -## Since the real space looks like this -plot(space, xlim = c(-1.5e3, 1.5e3), ylim = c(-1.5e3, 1.5e3), - main = "unscaled axis") -``` - -5.

The different statistics were evaluated and compared across datasets with different dimensionalities and distributions. While this seems feasible, the computation of p-values by ANOVA is not. This is not a question of type I error as you are not testing against sampling variation. In other words, you treat each dataset as a case drawn independently from the same distribution of datasets. Clearly, this is not what you are doing.

- -> @@@We somewhat disagree with this reviewer's comment suggesting that the space occupancy measurements are not drawn independently from the same distribution (*sensu* the statistical population) and that we can therefore not estimate the p-value correctly from a simple ANOVA. In fact, because of the centering and scaling of all space occupancy metrics, the metrics are effectively distributed near normally (figure 3) and thus form a coherent population. For example, any sum of variance measured in any simulated space from any parameter forms the "sum of variance population" - the ANOVA then tests whether there is a difference within groups (i.e. the sum of variance measured in a specific simulated space from specific parameters) and between groups (the sum of variance measured in any simulated space). However, we understand the reviewers' concern that these data could be considered as independent but we do not know of any way to report the within/between variation and interpreting the results without using the F-distribution. Maybe the reviewer could help us with this one? Meanwhile, we reported the F-statistic AND the p-value in the table for more clarity. - - - - - - -```r -## The data -data <- matrix(c(6, 8, 13, 8, 12, 9, 4, 9, 11, 5, 11, 8, 3, 6, 7, 4, 8, 12 ), ncol = 3, nrow = 6, byrow = TRUE) - -## Step 1, calculate the treatments means -means <- apply(data, 2, mean) - -## Step 2, calculate the overall mean -overall_mean <- mean(data) - -## Step 3, between groups sum of squared differences -between_group_df <- ncol(data) - 1 -between_group_ss <- (sum(nrow(data) * (means - overall_mean)^2))/between_group_df - -## Step 4, within groups sum of squared differences -within_group_df <- ncol(data) * (nrow(data) - 1) -within_group_ss <- sum(t((t(data)-means)^2))/within_group_df - -## Step 5, calculate the F-ratio -F_ratio <- between_group_ss/within_group_ss - -# F-ratio thresholf at 0.05 for both degrees of freedom F(2, 15) = 3.68 -# Since F_ratio > F(2, 15), the hypothesis can be rejected (all groups are the same) - -# The fuck is wrong with that... -``` - -### Minor comments - -*

The term "trait space" does not need to be hyphenated.

- -> We removed the hyphen throughout the manuscript. - -*

line 11: "a subset of this trait-space" Either a "subset of this set" or a "subspace of this space"

- -> We changed it to: - -*a subspace of this space* line@ 9 - -*

l 39: typo "used"

- -> We fixed this typo. - -*

175: "eigenvalue" is a single word

- -> We fixed this typo. - -*

Table 3: the formula for "average nearest neighbourhood distance" does not seem correct to me; last line, right column: what is a "centre of trait space"?

- -> We fixed the average nearest neighbour distance and specified the coordinates of the trait space as the point with coordinates (0, 0, ...). - -*

Table 5: A p-value cannot be 0.

- -> We changed our p-value rounding to display p-values rounded to 0 as <1e-3. - - - - - - - - - - - - - - - - - diff --git a/inst/defaults b/inst/defaults deleted file mode 100644 index 8a2acef..0000000 --- a/inst/defaults +++ /dev/null @@ -1,1075 +0,0 @@ ---- -title: "Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?" -author: "Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker" -bibliography: references.bib -csl: mee.csl -date: "`r Sys.Date()`" -output: - pdf_document: - fig_width: 8 - fig_height: 8 - keep_tex: true - self_contained: false ---- - - - - - - - - - - -# Response to reviewers - -We a really grateful for the reviewers refreshing, positive and constructive comments and we've addressed them all as detailed below. - -## Reviewer 1 (Stefano Mammola) - -In a conceptually similar study in the context of species distribution modelling (DOI: 10.1111/2041-210X.12397), Qiao and colleagues termed this general idea the “no silver bullet” paradigm (making a metaphor with the mythology of lycanthropes). This is certainly a very important point also in trait-based science. - -> This is indeed a really similar idea with similar conclusions. We've now added a reference to niche modeling in the introduction: - -"This can also be extended to more complex ecological concepts such as niche modelling [@qiao2015]." l.@@@ - -1) While reading the text, it was not entirely clear to me what exactly the “moms” tool is. Is this an R package, or just a set of functions? Would be helpful if you could briefly specify. - -> We've added an extra section explaining what the `moms` tool is. We've also added a sampling (see reviewer 2) and simulation feature allowing users to replicate the results of this paper with several clicks or exploring the properties of their own metrics: - -"Therefore, we propose the [`moms`](https://tguillerme.shinyapps.io/moms/) shiny app to allow workers to help them choose their set of space occupancy measurements (and test the caveats mentioned above). -`moms` is an online graphical user interface to help analyse multidimensional data. -It allows users to upload their dataset of interest (or simulate one with specific parameters) and measure space occupancy using a variety of implemented measures (namely, but not only, the ones used in this study). -Furthermore, the package allows simulation of shifts in trait space occupancy as also presented in this paper to test whether some measures capture specific changes in space. -However, `moms` is not a tool for analysing multidimensional data _per se_ but rather for helping workers to chose the space occupancy measure most appropriated to their data and question. -To run multidimensional analysis, we suggest using dedicated `R` packages (such as - but not limited to: @oksanen2007vegan, @momocs, @bat2015, @disprity)." l.@@@ - -2) I suspect that your proposed classification may somewhat overlap with the general classification scheme of diversity indexes by Pavoine and Bonsall (2011; Biological review), later expanded by Tucker et al. (2017; Biological Reviews) in the context of phylogenetic metrics (but it’s actually the same). In a nutshell, they grouped metrics/indexes in three main category of ‘Richness’, ‘Divergence’ and ‘Regularity’ components. In their view, these three concepts should capture the primary mathematical operation inherent to each metric, namely: -i) the ‘richness’ dimension encompasses indexes representing the sum of difference among taxa (sum); -ii) the ‘divergence’ dimension encompasses indexes representing functional (or phylogenetic) dissimilarity, reflecting the average difference among taxa (mean); and -iii) the ‘regularity’ dimension encompasses indexes representing functional (or phylogenetic) variance, reflecting how regular the difference among taxa in an assemblages are (variance). -I see some overlap with your method, but also differences. It would be worth briefly acknowledging this and perhaps exploring the main conceptual differences and why your classification is an advance. - ->We were not aware of these references and we are grateful for the reviewer pointing them out. There is so much overlapping work on ecology and evolution but so few visibility between them. Adding these references and concepts is definitely improving the main idea of this paper! - -Note that this classification bears some similarities with @tucker2017 classifying phylogenetic diversity measurements into richness, divergence and regularity categories. -However, while @tucker2017 based their classification on the mathematical operation inherent to each metrics (the sum for richness, the mean for divergence and the variance for regularity), our three broad classifications are based on their geometric properties regardless of the formula of each metric (e.g. the size of a space can be calculated using a sum, mean or/and variance). l.@@@ - - -3) The approach you choose for simulating changes in the multidimensional space (Figure 2) is very interesting. While I was thinking about it, I indulged with an idea (but please ignore if not appropriate): - -I thought it would be useful to try linking the different changes in the multidimensional space to actual example of biological processes (e.g., giving few examples in the description in the method or in the discussion). This, because similar changes in the trait space may occur in the real world due to different processes and, by providing examples grounded in the real world you may enhance the appealing of the ms to a broader audience. -For example, from the perspective of a conservation biologist, you may argue that ‘limit’ and ‘position’ change in the trait space may occur when there is the destruction of habitats with narrow environmental conditions that filters for few species possessing specific traits. I’m primarily a cave biologist: this mental bias made me think that, for example, if you open a quarry that destroys a cave (not an unfrequent event! DOI: 10.1093/biosci/biz064), you may end up removing from the trait space all points from a specific position, namely those traits clearly selected by the permanent darkness of the subterranean world. If trait 1 is a gradient of eye size and trait 2 a gradient of body pigment, you would remove trait contribution of specialized invertebrates with no eyes and no pigment. The ‘density’ change may occur, e.g., due to global wildlife trade, when there is the exploitation of only species with specific traits within the total functional tree of life (see DOI: 10.1126/science.aav5327). For example, the fishing industry exploits, within any given species, primarily fishes of larger size, thus selecting for specific densities in the size/weight trait space composed of the multiple species. And so on (these are just random examples to illustrate my point). - -> We've added this reviewer's example (although not all to also balance with examples from macroevolution): - -This type of change could be due to habitat destruction [e.g. @mammola2019b] or to mass extinctions [e.g. @wright2017]. l.@@@ - -This type of change could be due to accelerated rates of evolution [@close2015] or to differences in modes of life in macroevolution [e.g. @healy2019]. l.@@@ - -This type of change could be due changes in evolutionary trajectories [@endler2005] or to differences in ecosystem compositions [e.g. @jones2015]. l.@@@ - - -### Minor Line comments - --L34: “studied” instead of “study” - -> We fixed this typo. - -L35-37: I think merging together these two sentences would put more emphasis on your idea... E.g. “While different fields use a different set of terms for such approaches (Table 1), they actually focus on the same mathematical objects: matrices with columns representing an original or transformed trait value and rows representing observations (taxon, field site, etc.; Guillerme 2018). - -> Merged these sentences. - --L63: “are are” - -> We fixed this typo. - --Table 1: -‘functional space’ instead of ‘function-space’? -Morphospace I think is also widely used in ecology. -Please add “etc.” at the end of the text in the cell “statistic” intersected “ecology”. Wouldn’t the “hypervolume” mentioned in the same cell more fit in the “Matrix (n x d)” row? - -> We fixed these cells in the table. However, we did not add "hypervolume" to the "Mathematics" column. We use the term "statistic" here as a statistic (i.e. a measure) which does include the hypervolume as used in ecology. We've specified this. - --330–332: True, but there are at least two recent examples of attempt in this sense, the very recent package TPD (DOI: 10.1002/ecy.2876) and new functions in BAT (doi: 10.1101/2020.01.25.919373; shameless self-promotion and still a preprint). Both approaches covers metric in the Richness, Divergence, and Regularity domains (which should be a possible equivalent to your Size, Density, and Position view). - -> We've added these references on line @@@. - --Figure 2: why the dots are in black and white in the first inset? - -> The black dots are the 50% randomly removed points. We've specified this in the caption. - - -## Reviewer: 2 (Neil Brocklehurst) - -A general suggestion to start with. The authors state in their conclusions, very diplomatically, that "...no measure is better than the next one...". While I agree with the sentiment in the context of their conclusions (that different metrics show different things, and you should choose a variety of metrics relevant to the issue under study), would it not be accurate to say that your results indicate some metrics are worse than others? Three leapt out at me in table 5: Sum of Ranges, Minimum spanning tree average distances, and to a lesser extent average nearest neighbour distances, seem to produce changes in value following random removals comparable to those seen in not one, but all three varieties of separation (limit, density and position) . Would it be fair to say that these metrics are, shall we say, problematic? - -> We agree with the reviewer and have toned down or "diplomacy" level. We now also state concern about certain types of metrics that are highly sensitive to outliers or to the number of dimensions: - -We insist that although no measure is objectively better than the next one, some can be more problematic than other in specific contexts. -For example, the results for the Sum of Ranges, Minimum spanning tree average distances, and to a lesser extent average nearest neighbour distances produced results in the reduced space often similar to the randomly reduced spaces (Table 5). -This does not make them "bad" measures but rather heavily context dependent. -Regardless, we believe that workers should identify the most appropriate measures based on their trait space properties as well as their specific biological question. -We believe this could be fostered by following these several suggestions: l.@@@ - - -Following on from this: its worth testing, or at the very least acknowledging, that some metrics probably respond to incomplete sampling worse than others? Or are certain targeted separations (by limit, density or position) harder to detect with incomplete sampling? A test of this would be reasonably simple; after carrying out the separations by limit, density or position, randomly delete different numbers of data points to see how this affects the results. I realise this would be a large number of extra analyses to ask for and may be considered beyond the immediate scope of the paper, so I’ll leave the choice of whether or not to do the actual analyses up to the authors, but I think it should at least be discussed in the text. - -> We agree again with this reviewer's point and find this a really interesting point. We've updated the results in the supplementary material to highlight what happens when removing 80% and 20% of the elements (rather than 50%). -We've also added a sampling option to the latest version of the `moms` app allowing to test the effect of sampling on the metric of choice and, finally, we've added the following mentions to sampling issues in the caveat section: - -Furthermore, we did not take into account the effect of sampling on space occupancy measurements (but see additional results with 80% and 20% space reduction in the supplementary materials 4). -In fact, sampling has been previously shown to have an effect on measurements depending on range or volumes (e.g. the sum of ranges or the hypervolume @ciampaglio). -This effect is especially expected to be acerbated in macroevolutionary studies when using the fossil record [@brocklehurst2013] but can be tackled using rarefaction and bootstrapping techniques [@disprity]. l.@@@ - -Some more specific comments: -Fig 2, panels E and F, and similar in table 5: the dots and lines representing the changes in occupancy for blue, orange and random groups. I assume the dot is the median and the solid and dashed lines are quartiles and ranges, but specify in caption, please - -> We already specified that "the dots represent the median space occupancy values across all simulations for each scenario of trait space change (Table 2), the solid and dashed line respectively the 50% and 95% confidence intervals" (Caption Figure 2). We've now also added this to the Table 5 caption. - -Table 5 headers are inconsistent with the terms used in the text and figure 5. In the text and figure 5 you describe Limit and Density parameters for changes, but the headers talk about size change and arrangement change. Maybe this will create confusion (particularly the ‘arrangement’ header, which might not naturally be associated in peoples minds with density)? - -> Nice spotting, this comes from a classic version mashup. We've fixed this in the table header and double checked throughout the manuscript. - -And finally some typos and wording comments: - -First sentence of the abstrach: “Multidimensional analysis of traits are now a common in ecology and evolution…”; should be either “…now common in…” or “…now a common method in…” - -> We remove "a". - -Abstract, line 8: “…subspace of this space…”; maybe say subset instead of subspace? The two ‘spaces’ next two each other makes slightly awkward reading - -> We've changed "subspace" to "subset". - -Intro line 48: “ciampaglio2001”; Space missing between name and year, name needs capital letter -Methods, page 8, line 153: “hopkins2017”; Space missing between name and year, name needs capital letter - -> We fixed the references. - -Page 20, line 317: “spacesmight” missing space between words - -> We've added a space. - - - -```{r header, echo = FALSE, results = 'hide', message = FALSE, warning = FALSE} -## Repeatability note: -## This whole paper is entirely reproducible and compiles as a single document. The data, figures -## and tables are all generated through the code snippets in this file. Note that the first time -## that this paper is compiled, it will generate the data which will take substantial time -## as the shiftingspace_supplementary.Rmd file completion takes ~10 minutes. Subsequent compilations -## will be much faster! - -## Loading the packages -if(!require(devtools)) install.packages("devtools") -if(!require(knitr)) install.packages("knitr"); library(knitr) -if(!require(rmarkdown)) install.packages("rmarkdown"); library(rmarkdown) -if(!require(ape)) install.packages("ape"); library(ape) -if(!require(dispRity)) install.packages("dispRity"); library(dispRity) -if(packageVersion("dispRity") < "1.2.4") { - ## dispRity must be above v1.2.3 - devtools::install_github("TGuillerme/dispRity"); library(dispRity) -} -if(!require(moms)) devtools::install_github("TGuillerme/moms"); library(moms) - -## Setting the datapath -DATA_PATH <- "../data/processed/" - -## Create data directory -if(!dir.exists(DATA_PATH)) { - dir.create(path = DATA_PATH, showWarnings = FALSE) -} - -## Setting the default parameters for the space plots -defaults <- list(pch = 20, - xlim = c(-3, 3), - ylim = c(-3, 3), - col1 = "grey", - col2 = "black", - xlab = "Trait", - ylab = "Trait", - cex = 1) -## Generating the default palette -default.palette <- function(n) { - hues <- seq(15, 375, length = n + 1) - grDevices::hcl(h <- hues, l = 65, c = 100)[1:n] -} - -## Checking whether the data exists -if(!all(c("remove_05.Rda") %in% list.files(DATA_PATH))) { - ## Run The supplementary material (simulation) ~ 30 minutes - rmarkdown::render("shiftingspace_supplementary_simulation.Rmd", "html_document") -} -if(!all(c("empirical_results.Rda") %in% list.files(DATA_PATH))) { - ## Run The supplementary material ~ 15 minutes - rmarkdown::render("shiftingspace_supplementary_empirical.Rmd", "html_document") -} -``` - -```{r compilation_html, echo = FALSE, eval = FALSE} -## Changing defaults -body(plot.id)[[2]] <- substitute(type <- ".png") -``` - -# Abstract - - -Multidimensional analysis of traits are now common in ecology and evolution and are based on trait spaces in which each dimension summarises the observed trait combination (a morphospace or an ecospace). -Observations of interest will typically occupy a subset of this space, and researchers will calculate one or more measures to quantify how organisms inhabit that space. -In macroevolution and ecology these measures called disparity or dissimilarity metrics and are generalised as space occupancy measures. -Researchers use these measures to investigate how space occupancy changes through time, in relation to other groups of organisms, and in response to global environmental changes. -However, the mathematical and biological meaning of most space occupancy measures is vague with the majority of widely-used measures lacking formal description. - -Here we propose a broad classification of space occupancy measures into three categories that capture changes in size, density, or position. -We study the behaviour of 25 measures to changes in trait space size, density and position on simulated and empirical datasets. -We find that no measure describes all of trait space aspects but that some are better at capturing certain aspects. -Our results confirm the three broad categories (size, density and position) and allow us to relate changes in any of these categories to biological phenomena. - -Because the choice of space occupancy measures is specific to the data and question, we introduced [`moms`](https://tguillerme.shinyapps.io/moms/), a tool to both visualise and capture changes in space occupancy for any measurement. -[`moms`](https://tguillerme.shinyapps.io/moms/) is designed to help workers choose the right space occupancy measures, given the properties of their trait space and their biological question. -By providing guidelines and common vocabulary for space occupancy analysis, we hope to help bridging the gap in multidimensional research between ecology and evolution. - -# Introduction - -Groups of species and environments share specific, recognisable, correlated characteristics: guilds or biomes with shared phenotypic, physiological, phylogenetic or behavioural traits. -Organisms or environments should therefore be studied as a set of traits rather than some specific traits in isolation [@donohue2013; @hopkins2017]. -Biologists increasingly use ordination techniques [see @legendre2012 for a summary] to create multidimensional trait spaces to either explore properties of data or test hypotheses [e.g. @oksanen2007vegan; @blonder2018; @disprity]. -For example, in palaeobiology, @wright2017 used trait spaces to study how groups of species' characteristics change through time; in ecology, @jones2015 studied evidence of competition by looking at trait overlap between two populations. - While different fields use a different set of terms for such approaches (Table 1), they actually focus on the same mathematical objects: matrices with columns representing an original or transformed trait value and rows representing observations [taxon, field site, etc.; @disprity]. - -Mathematics | Ecology | Macroevolution | This paper ---------------------|--------------------|--------------------|------------ -Matrix ($n \times d$) with a structural relation between rows and columns | Functional space, morphospace , etc. | Morphospace, traitspace, etc. | trait space -Rows (*n*) | Taxa, field sites, environments, etc. | Taxa, specimen, populations, etc. | observations -Columns (*d*) | Traits, Ordination scores, distances, etc. | Traits, ordination scores, distances, etc. | dimensions -Matrix subset ($m \times d$; $m \leq n$) | Treatments, phylogenetic group (clade), etc. | Clades, geological stratum, etc. | group -Statistic (i.e. a measure) | Dissimilarity index or metric, hypervolume, functional diversity, etc. | Disparity metric or index | space occupancy measure -Multidimensional analysis | Dissimilarity analysis, trait analysis, etc. | Disparity analysis, disparity-through-time, etc. | multidimensional analysis -Table 1: Different terms are used for equivalent measures in mathematics, ecology and macroevolution. - -Ecologists and evolutionary biologists often use trait spaces with respect to the same fundamental questions: -are groups occupying the same amount of trait space? -Do some groups contain more species than others in the same amount of trait space? -Are some specific factors correlated with different patterns of trait space occupancy? -Because of the multidimensional nature of these trait spaces, it is often not possible to study them using bi- or tri-variate techniques [@diaz2016; @hopkins2017; @mammola2019]. -Studying the occupancy of trait spaces is done using disparity indices in macroevolution [@wills2001; @hopkins2017; @disprity] or comparing hypervolumes in ecology [@donohue2013; @diaz2016; @blonder2018; @mammola2019]. -Despite the commonalities between the measures used in ecology and evolution (which are often metric but don't necessarily need to be) [[NOTE - INSERT TABLE REF HERE?]], surprisingly little work has been published on their behaviour [but see @ciampaglio2001; @villéger2008; @mammola2019]. - -Different occupancy measures capture different aspects of trait space [@ciampaglio2001; @villéger2008; @mammola2019]. -This may be widely-known, but to our knowledge it is infrequently mentioned in peer-reviewed papers. -First, space occupancy measures are often named as the biological aspect they are describing ("disparity", "functional diversity") rather than what they are measuring (e.g. the product of ranges), which obscures the differences and similarities between studies. -Second, in many studies in ecology and evolution, authors have focused on measuring the size of the trait space [e.g. ellipsoid volume @donohue2013; hypervolume @diaz2016; product of variance @wright2017; Procrustes variance @marcy2016]. -However, the size of the trait space only represents one aspects of occupancy, disregarding other measures such as the density [@geiger2008] or position [@wills2001; @ciampaglio2001]. -For example, if two groups have the same size, this can support certain biological conclusions. -Yet, an alternative aspect of space occupancy may indicate that the groups' position are different, leading to a different biological conclusion (e.g. the groups are equally diverse but occupy different niches). -Using measures that only capture one aspect of the trait space may restrain the potential of multidimensional analysis [@villéger2008]. - -Here we propose a broad classification of space occupancy measures as used across ecology and evolution and study their power to detect changes in trait space occupancy in simulated and empirical data. Note this does not account whether or not it is possible for a space to be occupied (e.g., some spaces may represent biologically impossible shapes); this, however, may be important in some cases, such as testing whether a region is infinite or not. We provide an assessment of each broad type of space occupancy measures along with a unified terminology to foster communication between ecology and evolution. -Unsurprisingly, we found that no one measure describes all changes in space and that the results from each measures are dependent on the characteristics of the space and the hypotheses. -Furthermore, because there can be an infinite number of measures, it would be impossible to propose clear generalities to space occupancy measures behaviour. [[NOTE -- I DO QUITE UNDERSTAND THIS SENTENCE, PLEASE CLARIFY]]. -Therefore, we propose [`moms`](https://tguillerme.shinyapps.io/moms/), a tool for researchers to design, experiment and visualise their own space occupancy measure tailored for their project. The tool will help researchers understand the "null" behaviour of the measures of interest. - - -## Space occupancy measures - -In this paper, we define trait spaces as any matrix where rows are observations and columns are traits, where both observations and traits are structurally related (e.g. there is a phylogenetic relation between observations - and traits, etc.). -These traits can widely vary in number and types: they could be coded as discrete [e.g. presence or absence of a bone; @beck2014; @wright2017], continuous measurements [e.g. leaf area; @diaz2016] or more sophisticated measures [e.g. landmark position; @marcy2016; Fourier ellipses; @momocs]. -Traits can also be measured by using relative observations [e.g. community compositions; @jones2015] or distance between observations [e.g. @close2015]. -However, regardless of the methodology used to build a trait space, three broad occupancy measures can be used: the size [[NOTE - maybe put this in bold]] which approximates the amount of space occupied, the density which approximates the distribution in space, and the position which approximates the location in space [Fig. 1; @villéger2008]. -Of course any combination of these three aspects is always possible. - -```{r fig_measures_types, echo = FALSE, fig.height = 3, fig.width = 9, results = 'hide', fig.cap = paste("different type of information captured by space occupancy measures: (A) size, (B) density and (C) position.") } - -set.seed(11) -## Plot space function (utility shortcut) -## The elements -elements <- 10 - -## Trait space -trait_space <- space.maker(elements, 2, distribution = rnorm) - -## Graphical parameters -op <- par(mfrow = c(1,3), bty = "n") - -## The size -plot(trait_space, pch = defaults$pch, main = "A - Size\n(e.g. Sum of ranges)", xlab = "Trait 1", ylab = "Trait 2") -## The range lines -lines(x = range(trait_space[,1]), y = rep(mean(trait_space[,2]), 2), col = defaults$col1) -lines(x = rep(mean(trait_space[,1]), 2), y = range(trait_space[,2]), col = defaults$col1) - -## The density -plot(trait_space, pch = defaults$pch, main = "B - Density\n(e.g. pairwise distances)", xlab = "Trait 1", ylab = "") -## Plotting the pairwise lines -pair.line <- function(line, trait_space, defaults) { - lines(x = trait_space[line,1], y = trait_space[line,2], col = defaults$col1) -} -apply(combn(1:elements, 2), 2, pair.line, trait_space, defaults) -points(trait_space, pch = defaults$pch) - -## The position -plot(trait_space, pch = defaults$pch, main = "C - Position\n(e.g. distance from centre)", xlab = "Trait 1", ylab = "") -## Plotting the center of the space -arrows(x0 = 0, y0 = 0, x1 = trait_space[,1], y1 = trait_space[,2], code = 0, col = defaults$col1) -points(trait_space, pch = defaults$pch) -points(0,0, pch = 13, cex = 2) - -par(op) -``` - -#### 1. Size - -Size captures the spread of a group in the trait space. -They can be interpreted as the amount of the trait space that is occupied by observations. -Typically, larger values for such measures indicate the presence of more extreme trait combinations. -For example, if group A is bigger than B, the observations in A achieve more extreme trait combinations than in B. -This type of measure is widely used in both ecology [e.g. the hypervolume; @blonder2018] and in evolution [e.g. the sum or product of ranges or variances; @wills2001]. - -Although size measures are suitable indicators of a group's trait space occupancy, they are limited to comparing the range of trait-combinations between groups. -Size measures do not take into account the distribution of the observations within a group and can often be insensitive to unoccupied "holes" in the trait space (overstimating the size; @blonder2018). -They can make it difficult to determine whether all the observations are on the edge of the group's distribution or whether the size is simply driven by outliers. - - -#### 2. Density - -Density gives an indication of the quantity of observations in the trait space. -They can be interpreted as the distribution of the observations *within* a group in the trait space. -Groups with higher density contain more observations (i.e. more observations per approximation of size) that will tend to be more similar to each other. -For example, if group A is greater is size than group B and both have the same density (observations are equally distant within each group), similar mechanisms could be driving both groups' trait space occupancy. -Indeed, this pattern could suggest that A is older and has had more time to achieve more extreme trait combinations under essentially the same process as younger, smaller group B [@endler2005]. -Note that density based measures can be sensitive to sampling. -Density measures are less common compared to size measures, but they are still used in both ecology [e.g. the minimum spanning tree length; @oksanen2007vegan] and evolution [e.g. the average pairwise distance; @geiger2008]. - -#### 3. Position - -Position captures where a group lies in trait space. -They can be interpreted as where a group lies in the trait space either relative to the space itself or relative to another group. -For example, if group A has a different position than group B, A will have a different trait-combination than in B. - -Position measures may be harder to interpret in multidimensional spaces. -In a 2D space, two groups can be equally distant from a fixed point but in different parts of the space (left, right, up, or down - with the amount of parts of space increasing with dimensions). -However, when thinking about unidimensional data, this measure is obvious: two groups A or B could have the same variance (size) with the same number of observations (density) but could have two different means and thus be in different positions. -These measures are used in ecology to compare the position of two groups relative to each other [@mammola2019]. - - -Note that this classification into size, density and position bears some similarities with @tucker2017 classifying phylogenetic diversity measurements into richness, divergence and regularity categories. -However, while @tucker2017 based their classification on the mathematical operation inherent to each metrics (the sum for richness, the mean for divergence and the variance for regularity), our three broad classifications are based on their geometric properties regardless of the formula of each metric (e.g. the size of a space can be calculated using a sum, mean or/and variance). - - -## No measure to rule them all: benefits of considering multiple measures - -The use of multiple measurements to assess trait space occupancy provides a more detailed characterisation of occupancy changes. -If the question is to look at how space occupancy changes in response to mass extinction, using a single space occupancy measure can miss part of the picture: a change in size could be decoupled from a change in position or density in trait space. -For example, the Cretaceous-Paleogene extinction (66 million years ago) shows an increase in size of the mammalian trait space [adaptive radiation; @halliday2015] but more specific questions can be answered by looking at other aspects of trait space occupancy: does the radiation expand on previously existing morphologies [elaboration, increase in density; @endler2005] or does it explore new regions of the trait space [innovation, change in position; @endler2005]? -Similarly, in ecology, if two groups have the same trait space size, the differences in density within these two groups is potentially illuminating: different selection pressure can lead to different density within equally sized groups. -This can also be extended to more complex ecological concepts such as niche modelling [@qiao2015]. - - - - -```{r fig_measure_captures, echo = FALSE, fig.height = 3, fig.width = 9, results = 'hide', fig.cap = paste("Illustration how three different size (size. - product of ranges), density (den. - average nearest neighbour distance) and position (pos. - average displacement) measurements capture different occupancy aspects in this simplified trait space. This illustrates how using only one specific space occupancy measure (e.g. the size) will fail to capture changes in B (no change in size), or partial changes in C (also change in density and position)."), eval = FALSE} - -## Making the volume/density/position space -vdp_space <- moms::vdp.make() - -## Measuring disparity -vdp_disp <- moms::vdp.dispRity(vdp_space, - volume = c(prod, ranges), - density = c(mean, neighbours), - position = c(mean, displacements)) - -## Checking the table of volume changes -silent <- moms::vdp.check.table(vdp_disp, vdp_space) - -## Plotting the results -moms::vdp.plot(vdp_space, plots = c(1, 4, 8), disparity = vdp_disp, mfrow = c(1, 3), - plot.names = c("A - Base (no changes)", #A - "B - Change in position", #D - "C - Change in vol. den. & pos.")) #I -``` - - -Here, we provide the first interdisciplinary review of 25 space occupancy measures that uses the broad classification of measures into size, density and position to capture pattern changes in trait space. -We assess the behaviour of measures using simulations and six interdisciplinary empirical datasets covering a wide range of potential data types and biological questions. -We also introduce a tool for measuring occupancy in multidimensional space ([`moms`](https://tguillerme.shinyapps.io/moms/)), which is a user-friendly, open-source, graphical interface to allow the tailored testing of measurement behaviour for any use case. -[`moms`](https://tguillerme.shinyapps.io/moms/) will allow workers to comprehensively assess the properties of their trait space and the measures associated with their specific biological question. - - - - - - - - - - - - - - - - - - - - - -# Methods - -```{r metrics_list, echo = FALSE} -## Loading the list of measures -source("list.of.metrics.R") -``` - -We tested how 25 space occupancy measures relate to each other, are affected by modifications of traits space and affect group comparisons in empirical data: - -1. We simulated 13 different spaces with different sets of parameters; -2. We transformed these spaces by removing 50% of the observations following four different scenarios corresponding to different empirical scenarios: randomly, by limit (e.g. expansion or reduction of niches), by density (e.g. different degrees of competition within a guild), and by position (e.g. ecological niche shift). -3. We measured occupancy on the resulting transformed spaces using `r metric.names()` different space occupancy measures; -4. We applied the same space occupancy measures to six empirical datasets (covering a range of disciplines and a range of dataset properties). - -Note that the paper contains the results for only `r metric.names()` measures which were selected as representative of common measures covering the size, density and position trait space aspects. -The results for an additional 17 measures is available in the supplementary material 4. - -## Generating spaces - -We generated trait spaces using the following combinations of size, distributions, variance and correlation: - -| space name | size | distribution(s) | dimensions variance | correlation | -|------------|-------|--------------------------------|---------------------|-------------| -| 3D uniform |200*3 | Uniform (min = -0.5, max = 0.5)| Equal | None | -| 15D uniform |200*15 | Uniform | Equal | None | -| 50D uniform |200*50 | Uniform | Equal | None | -| 150D uniform |200*150| Uniform | Equal | None | -| 50D uniform correlated |200*50 | Uniform | Equal | Random (between 0.1 and 0.9) | -| 3D normal |200*3 | Normal (mean = 0, sd = 1) | Equal | None | -| 15D normal |200*15 | Normal | Equal | None | -| 50D normal |200*50 | Normal | Equal | None | -| 150D normal |200*150| Normal | Equal | None | -| 50D normal correlated |200*50 | Normal | Equal | Random (between 0.1 and 0.9) | -| 50D with random distributions |200*50 | Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)| Equal | None | -| 50D PCA-like |200*50 | Normal | Multiplicative | None | -| 50D PCO-like |200*50 | Normal | Additive | None | -Table 2: different simulated space distribution. - *Name* of the simulated space; *dimensions* of the matrix (row\*columns); *distribution(s)* of the data on each dimensions (for the 'Random', dimensions are randomly chosen between Normal, Uniform or Lognormal); *dimension variance*: distribution of the variance between dimensions (when equal, the dimensions have the same variance); *correlation* between dimensions. - -The differences in trait space sizes (200 elemeents for 3, 15, 50 or 150 dimensions) reflects the range found in literature [e.g. @hopkins2017; @mammola2019]. -We used a range of distributions (uniform, normal or a random combination of uniform, normal and lognormal) to test the effect of observation distributions on the measurements. -We used different levels of variance for each dimensions in the spaces by making the variance on each dimension either equal ($\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}$) or decreasing ($\sigma_{D1} < \sigma_{D2} < \sigma_{Di}$) with the decreasing factor being either multiplicative (using the cumulative product of the inverse of the number of dimensions: $\prod_i^d(1/d)$) or additive ($\sum_i^d(1/d)$). -Both reductions of variance are used to illustrate the properties of ordinations where the variance decreases per dimensions [lognormal in principal components analysis - PCA; e.g. @marcy2016; @healy2019; and normal win Multidimensional Scaling - MDS, PCO or PCoA; e.g. @close2015; @wright2017]. -Finally, we added a correlation parameter to illustrate the effect of co-linearity between traits (especially in non-ordinated trait spaces). -We repeated the simulation of each trait space 20 times (resulting in 260 spaces). - -## Spatial occupancy measures - -We then calculated `r metric.names()` different measures on the resulting transformed spaces, including a new one, the average displacement, which we expect to be influenced by changes in trait space position. - -| Name | Definition | Captures | Source | Notes | -|--------------------|------------------------------|----------|----------|------------------------------| -| Average Euclidean distance from centroid | $\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}$ | Size | @laliberté2010 | the functional dispersion (FDis - without abundance) | -| Sum of variances | $\sum_{i}^{d}{\sigma^{2}{k_i}}$ | Size | @foote1992 | common measure used in palaeobiology [@ciampaglio2001; @wills2001] | -| Sum of ranges | $\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}$ | Size | @foote1992 | more sensitive to outliers than the sum of variances | -| Ellipsoid volume | $\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})$ | Size | @donohue2013 | less sensitive to outliers than the convex hull hypervolume [@diaz2016; @blonder2018] | -| Minimum spanning tree average distance | $\frac{\sum(\text{branch length})}{n}$ | Density | @sedgewick1990 | similar to the unscaled functional evenness [@villéger2008] | -| Minimum spanning tree distances evenness | $\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}$ | Density | @villéger2008 | the functional evenness without weighted abundance [FEve; @villéger2008] | -| Average nearest neighbour distance | $\sqrt{\sum_{i}^{n}{min({q}_{i}-p_{i})^2}})\times \frac{1}{n}$ | Density | @foote1992 | the density of pairs of observations | -| Average displacement | $\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}$ | Position | This paper | the ratio between the observations' position from their centroid and the centre of the trait space (coordinates: 0, 0, 0, ...). A value of 1 indicates that the observations' centroid is the centre of the trait space | -Table 3: List of measures with *n* being the number of observations, *d* the total number of dimensions, *k* any specific row in the matrix, *Centroid* being their mean and $\sigma^{2}$ their variance. $\Gamma$ is the Gamma distribution and $\lambda_{i}$ the eigenvalue of each dimension and ${q}_{i}$ and $p_{i}$ are any pairs of coordinates. - -We selected these `r metric.names()` space occupancy measures to illustrate how they capture different aspects of space occupancy (not as an expression of our preference). -These measures are specific to Euclidean and isotropic trait spaces (which is not necessary for all measures). -The supplementary material 4 contains the same analysis as described below, performed on 17 measures. -Furthermore, [`moms`](https://tguillerme.shinyapps.io/moms/) allows exploration into the effect of many more measures as well as the customisation of measures by combining them or using user-designed functions. - - - - - - - - - - -## Measure comparisons - -We compared the space occupancy measures correlations across all simulations between each pair of measures to assess their captured signal [@villéger2008; @laliberté2010]. -We used the measures on the full 13 trait spaces described above. -We then scaled the results and measured the pairwise Pearson correlation to test whether measures were capturing a similar signals or not using the `psych` package [@psych]. - - - - - - - - - -## Changing space {#changing-spaces} - -To assess how the measures responded to changes within trait spaces, we removed 50% of observations each time using the following algorithms: - -* **Randomly:** by randomly removing 50% of observations (Fig. 2-A). -This reflects a "null" biological model of changes in trait space: the case when observations are removed regardless of their intrinsic characteristics. -For example, if diversity is reduced by 50% but the space size remains the same, there is a decoupling between diversity and space occupancy [@ruta2013]. -Our selected measures are expected to not be affected by this change. - -* **Limit:** by removing observations within a distance from the centre of the trait space lower or greater than a radius $\rho$ (where $\rho$ is chosen such that 50% observations are selected) generating two limit removals: *maximum* and *minimum* (respectively in orange and blue; Fig. 2-B). -This can reflect a strict selection model where observations with trait values below or above a threshold are removed leading to an expansion or a contraction of the trait space. -This type of change could be due to habitat destruction [e.g. @mammola2019b] or to mass extinctions [e.g. @wright2017]. -Size measures are expected to be most affected by this change. - - -[[NOTE - I think the explanation for density is slightly unclear. I think it may also be slightly confusing to have density as the name of a measure and a simulation procedure (same for 'position').]] - -* **Density:** by removing any pairs of point with a distance $D$ from each other where (where $D$ is chosen such that 50% observations are selected) generating two density removals: *high* and *low* (respectively in orange and blue; Fig. 2-C). -This can reflect changes within groups in the trait space due to ecological factors [e.g. niche repulsion resulting in lower density; @grant2006]. -This type of change could be due to accelerated rates of evolution [@close2015] or to differences in modes of life in macroevolution [e.g. @healy2019]. -Density measures are expected to be most affected by this change. - -* **Position:** by removing points similarly as for **Limit** but using the distance from the furthest point from the centre generating two position removals: *positive* and *negative* (respectively in orange and blue; Fig. 2-D). -This can reflect global changes in trait space (e.g. if an entire group remaining diverse but occupying a different niche). -This type of change could be due changes in evolutionary trajectories [@endler2005] or to differences in ecosystem compositions [e.g. @jones2015]. -Position measures are expected to be most affected by this change. - -The algorithm to select $\rho$ or $D$ is described in the Supplementary material 1. - - -```{r fig_reduce_space, echo = FALSE, fig.height = 12, fig.width = 8, results = 'hide', fig.cap = paste("different type of space reduction. Each panel displays two groups of 50% of the data points each. Each group (orange and blue) are generated using the following algorithm: A - randomly (the removed elements are displayed in black and the analysed ones in grey); B - by limit (maximum and minimum limit); C - by density (high and low); and D - by position (positive and negative). Panel E et F represents two typical display of the reduction results displayed in Table 5: the dots represent the median space occupancy values across all simulations for each scenario of trait space change (Table 2), the solid and dashed line respectively the 50% and 95% confidence intervals. Results in grey are the random 50% reduction (panel A). Results in blue and orange represent the opposite scenarios from panels B, C, and D. The displayed value is the amount of overlap (Bhattacharrya Coefficient) between the blue or orange distributions and the grey one. Panel E and F shows respectively the \"ideal\" and \"worst\" results for any type of measures, where the space occupancy measurement respectively manages or fails to captures a specific type of reduction (i.e. size, position or density; Table 5).")} - -set.seed(42) - -## Change colors -defaults$col1 <- "blue" -defaults$col2 <- "orange" - -op <- par(mfrow = c(3,2), bty = "n") -## The elements -elements <- 300 -## The amount to remove -remove <- 0.5 - -## Trait space -trait_space <- space.maker(elements, 2, distribution = rnorm) - -## Random removal colours -defaults$col1 <- "black" -defaults$col2 <- "grey" - -## The randomly removed space -random_rm <- reduce.space(trait_space, type = "random", remove = 0.5) -## Plotting the reduction -plot.space(trait_space, random_rm, main = "A - Random removal", defaults) - -## Back to normal colours -defaults$col1 <- "blue" -defaults$col2 <- "orange" - -## The limit removal -limit_rm <- reduce.space(trait_space, type = "limit", remove = 0.5) -## Plotting the reduction -plot.space(trait_space, limit_rm, main = "B - Limit", defaults) - -## The density removal -density_rm <- reduce.space(trait_space, type = "density", remove = 0.5) -## Plotting the reduction -plot.space(trait_space, density_rm, main = "C - Density", defaults) - -## The displacement removal -displace_rm <- reduce.space(trait_space, type = "displacement", remove = 0.5) -## Moving the space to the upper right corner (positive) -plot.space(trait_space, displace_rm, main = "D - Position", defaults) - - -## Test example (positive) -set.seed(42) -colours <- c("grey", "orange", "blue") -par(bty = "n") -## Plot size -plot(NULL, ylim = c(0.8, 3.2), pch = 19, xlim = c(-1,1), xlab = "Centred and scaled space occupancy", ylab = "", xaxt = "n", yaxt = "n", main = "E - Space reduction \"ideal\" results example (Table 5)") -## Adding lines -abline(v = 0, lty = 2, col = colours, lwd = 2) -## Adding the x axis -axis(1, at = c(-1, -0.5, 0, 0.5, 1), labels = TRUE, tick = TRUE, col.ticks = "black", col = "black", lwd = 2) -## Adding the values -values <- cbind(rnorm(30, mean = 0, sd = 0.15), - rnorm(30, mean = -0.75, sd = 0.15), - rnorm(30, mean = 0.75, sd = 0.2)) -quantile_vals <- apply(values, 2, quantile, probs = c(0.025, 0.250, 0.750, 0.975)) -centtend_vals <- apply(values, 2, median) - -## Loop through the lines -for(column in 1:3) { - ## Get the x y values - line_x_vals <- quantile_vals[, column] - line_y_vals <- rep(column, 2) - - ## Add the lines - n_cis <- 4 - for(ci in 1:(n_cis/2)) { - lines(x = line_x_vals[c(ci, n_cis-(ci-1))], y = line_y_vals, lty = (n_cis/2 - ci + 1), lwd = ci * 1.5 * 2, col = colours[column]) - } -} -## Add the points -points(x = centtend_vals, y = 1:ncol(values), pch = 19, col = colours, cex = 1.5 + 2) - -bc_1 <- dispRity::bhatt.coeff(values[,2], values[,1]) -bc_2 <- dispRity::bhatt.coeff(values[,3], values[,1]) - -text(x = 0.2, y = 3, labels = round(bc_2, 3), cex = 1.4) -text(x = 0.2, y = 2.9, labels = paste0("\n(Probability of overlap between\nthe blue and the grey distributions)"), cex = 0.8) -text(x = -0.2, y = 2, labels = round(bc_1, 3), cex = 1.4) -text(x = - 0.2, y = 1.9, labels = paste0("\n(Probability of overlap between\nthe orange and the grey distributions)"), cex = 0.8) - -## Test example (negative) -set.seed(42) -colours <- c("grey", "orange", "blue") -par(bty = "n") -## Plot size -plot(NULL, ylim = c(0.8, 3.2), pch = 19, xlim = c(-1,1), xlab = "Centred and scaled space occupancy", ylab = "", xaxt = "n", yaxt = "n", main = "F -Space reduction \"worst\" results example (Table 5)") -## Adding lines -abline(v = 0, lty = 2, col = colours, lwd = 2) -## Adding the x axis -axis(1, at = c(-1, -0.5, 0, 0.5, 1), labels = TRUE, tick = TRUE, col.ticks = "black", col = "black", lwd = 2) -## Adding the values -values <- cbind(rnorm(30, mean = 0, sd = 0.15), - rnorm(30, mean = 0, sd = 0.17), - rnorm(30, mean = 0, sd = 0.2)) -quantile_vals <- apply(values, 2, quantile, probs = c(0.025, 0.250, 0.750, 0.975)) -centtend_vals <- apply(values, 2, median) - -## Loop through the lines -for(column in 1:3) { - ## Get the x y values - line_x_vals <- quantile_vals[, column] - line_y_vals <- rep(column, 2) - - ## Add the lines - n_cis <- 4 - for(ci in 1:(n_cis/2)) { - lines(x = line_x_vals[c(ci, n_cis-(ci-1))], y = line_y_vals, lty = (n_cis/2 - ci + 1), lwd = ci * 1.5 * 2, col = colours[column]) - } -} -## Add the points -points(x = centtend_vals, y = 1:ncol(values), pch = 19, col = colours, cex = 1.5 + 2) - -bc_1 <- dispRity::bhatt.coeff(values[,2], values[,1]) -bc_2 <- dispRity::bhatt.coeff(values[,3], values[,1]) - -text(x = 0.5, y = 3, labels = round(bc_2, 3), cex = 1.4) -text(x = -0.6, y = 2, labels = round(bc_1, 3), cex = 1.4) - -``` - -Because occupancy measures are dependent on the space, we scaled and centred them between -1 and 1 to make them comparable (by subtracting the observed occupancy without reduction to all the measures of the reduced spaces and then divided it by the maximum observed occupancy). -A value of 0 indicates no effect of the space reduction and $>0$ and $<0$ respectively indicates an increase or decrease in the measure value. -We then measured the amount of overlap between the non-random removals (limit, density and position) and the random removals using the Bhattacharrya Coefficient [@bhattacharyya1943]. - -### Measuring the effect of space and dimensionality - -Distribution differences and the number of dimensions can have an effect on the measure results. -For example, in a normally distributed space, an increase in density can often lead to a decrease in size (though this is not necessarily true if the space is log-normal or uniform). -High dimensional spaces (>10) are subject to the "curse of multidimensionality" [@cursedimensionality]: data becomes sparser with increasing number of dimensions. -This can have two main consequences: 1) the probability of overlap between two groups decreases as a product of the number of dimensions; and 2) the amount of samples needed to "fill" the spaces increases exponentially [see this interactive illustration by Toph Tucker](https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners). -The "curse" can make the interpretation of high dimensional data counter-intuitive. -For example if a group expands in multiple dimensions (i.e. increase in size), the actual hypervolume ($\prod_{i}^{d} range_{Di}$) can decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an ANOVA ($occupancy \sim distribution$ and $occupancy \sim dimensions$) by using all spaces with 50 dimensions and the uniform and normal spaces with equal variance and no correlation with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing respectively the effect of distribution and dimensions. -The results of the ANOVAs (F and *p*-values) are reported in Table 5 (full results in supplementary material 3). - - - - - - - - - - - -## Empirical examples - -We analysed the effect of the different space occupancy measures on six different empirical studies covering a range of fields that employ trait space analyses. -For each of these studies we generated trait spaces from the data published with the papers. -We divided each trait spaces into two biologically-relevant groups and tested whether the measures differentiated the groups in different ways. -Both the grouping and the questions were based on a simplified version of the topics of these papers (with no intention to re-analyse the data and questions). -The procedures to generate the data and the groups varies between studies and is detailed in the supplementary materials 2. - - -study | field | taxonomic group | traits | trait space | size | groups | question | -----------|-----------|-----------|--------------|--------------|----------|--------------|---------------- -@beck2014 | Palaeontology | Mammalia | discrete morphological phylogenetic data | Ordination of a distance matrix (PCO) | 106*105 | 52 crown vs. 54 stem | Are crown mammals more disparate than stem mammals?| -@wright2017 | Palaeontology | Crinoidea | discrete morphological phylogenetic data | Ordination of a distance matrix (PCO) | 42*41 | 16 before vs. 23 after | Is there a difference in disparity before and after the Ordovician mass extinction?| -@marcy2016 | Evolution | Rodentia | skull 2D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 454*134 | 225 *Megascapheus* vs. 229 *Thomomys* | Are two genera of gopher morphologically distinct? | -@hopkins2016 | Evolution | Trilobita | 3D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 46*46 | 36 adults vs. 10 juveniles | Are juvenile trilobites a subset of adult ones in trait space? | -@jones2015 | Ecology | Plantae | Communities species compositions | Ordination of a Jaccard distance matrix (PCO) | 48*47 | 24 aspens vs. 24 grasslands | Is there a difference in species composition between aspens and grasslands? | -@healy2019 | Ecology | Animalia | Life history traits | Ordination of continuous traits (PCA) | 285*6 | 83 ecthotherms vs. 202 endotherms | Do endotherms have more diversified life history strategies than ectotherms? | -Table 4: details of the six empirical trait spaces. - -For each empirical trait space we bootstrapped each group 500 times [@disprity] and applied the `r metric.names()` space occupancy measure to each pairs of groups. -We then compared the means of each groups using the Bhattacharrya Coefficient [@bhattacharyya1943]. - - - - - - - - - - - - - - - - -# Results - -```{r loading_results, echo = FALSE} -## Loading the results -remove_05 <- load.results("remove_05") -``` - -```{r running_tests, echo = FALSE} -## Anova function -anova.fun <- function(data) {return(aov(glm(disparity ~ factor, data = data)))} - -## Running the tests -all_test <- test.simulation(remove_05, test = anova.fun, scale = TRUE) -all_dim_test <- test.simulation(remove_05, test = anova.fun, scale = TRUE, - factors = c("uniform3", "uniform15", "uniform50", "uniform100", "uniform150", - "normal3", "normal15", "normal50", "normal100", "normal150")) -space_test <- test.simulation(remove_05, test = anova.fun, scale = TRUE, - factors = c("uniform50", "uniform50c", "normal50", "normal50c", - "random50", "pca_like", "pco_like")) -``` - -## Measure comparisons - - -```{r fig_measure_correlation, echo = FALSE, fig.height = 8, fig.width = 8, results = 'hide', fig.cap = paste("pairwise correlation between the scaled measures. Numbers on the upper right corner are the Pearson correlations. The red line are linear regressions (with the confidence intervals in grey). Av.: average; dist.: distance; min.: minimum; span.: spanning.")} - -## Adding proper names -results_pairwise <- remove_05 -names(results_pairwise[[1]]) <- metric_names -## Shortening the names for plotting -names(results_pairwise[[1]])[1] <- "Av. dist.\nfrom centroid" -names(results_pairwise[[1]])[5] <- "Min. span. tree\nav. dist." -names(results_pairwise[[1]])[6] <- "Min. span. tree\ndist. evenness" -names(results_pairwise[[1]])[7] <- "Av. nearest\nneighbour dist." -names(results_pairwise[[1]])[8] <- "Av. displacements" - -## Plotting the pairwise results -pairwise.plot(results_pairwise, scale = TRUE, type = "base", plot = "cor") -``` - -Most measures of space were positively correlated (Pearson correlation of 0.99 for the average Euclidean distance from centroid and sum of variance or 0.97 for the average nearest neighbour distance and minimum spanning tree average length; Fig. 3). The remaining measures were either somewhat correlated or had a negative pairwise distribution (ranging from 0.66 for the sum of variances and the ellipsoid volume to -0.09 between the average displacement and the average Euclidean distance from centroid; Fig. 3). -All measures but the ellipsoid volume were normally (or nearly normally) distributed (Fig. 3). - -## Space shifting - -```{r fable_results, fig.show='hide', echo=FALSE, fig.height=3, fig.width=3} -## The measures names (shortened vector) -name <- metric_names - -## Making a list of parameters for each mini plot -plot.param <- list(scaler = 3, - bg.col = "black", - col = c("grey", "orange", "blue"), - quantiles = c(95, 50), - cent.tend = median, - pch = 19, - metric.max = length(metrics_list), - cex = 2) - -## Looping through each mini plot for every measure -for(metric in 1:length(metrics_list)) { - generate.fable.plot(data = remove_05, metric = metric, what = "limits", plot.param = plot.param, overlap = TRUE) - generate.fable.plot(data = remove_05, metric = metric, what = "densit", plot.param = plot.param, overlap = TRUE) - generate.fable.plot(data = remove_05, metric = metric, what = "displa", plot.param = plot.param, overlap = TRUE) -} - -``` - -```{r, echo = FALSE, eval = FALSE} -## Function for printing the table in the R console -print.fable <- function(n_measures, byrow, ncol, test) { - ## Make the plot.id table - ids <- matrix(1:(n_measures*ncol), ncol = ncol, byrow = byrow) - - for(one_measure in 1:n_measures) { - ## Print all the rows one by one - text <- c(paste0("`r name[", one_measure, "]`"), paste0("`r plot.id(", ids[one_measure, ], ")`")) - ## Add some tests? - if(test) { - text <- c(text, paste0("`r s.test(", one_measure, ", \"s\")`"), paste0("`r s.test(", one_measure, ", \"r\")`")) - } - ## Print the line - cat(paste(text, collapse = " | ")) - cat("|\n") - } -} -## Getting the fable to copy paste under the table header below. -print.fable(length(metrics_list), byrow = TRUE, ncol = 3, test = TRUE) -``` - -Measure | Size change | Density change | Position change | Distribution effect | Dimensions effect | -:-----------|----------------|-----------------|----------------|---------------------|-------------------| -`r name[1]` | `r plot.id(1)` | `r plot.id(2)` | `r plot.id(3)` | `r s.test(1, "s")` | `r s.test(1, "r")`| -`r name[2]` | `r plot.id(4)` | `r plot.id(5)` | `r plot.id(6)` | `r s.test(2, "s")` | `r s.test(2, "r")`| -`r name[3]` | `r plot.id(7)` | `r plot.id(8)` | `r plot.id(9)` | `r s.test(3, "s")` | `r s.test(3, "r")`| -`r name[4]` | `r plot.id(10)` | `r plot.id(11)` | `r plot.id(12)` | `r s.test(4, "s")` | `r s.test(4, "r")`| -`r name[5]` | `r plot.id(13)` | `r plot.id(14)` | `r plot.id(15)` | `r s.test(5, "s")` | `r s.test(5, "r")`| -`r name[6]` | `r plot.id(16)` | `r plot.id(17)` | `r plot.id(18)` | `r s.test(6, "s")` | `r s.test(6, "r")`| -`r name[7]` | `r plot.id(19)` | `r plot.id(20)` | `r plot.id(21)` | `r s.test(7, "s")` | `r s.test(7, "r")`| -`r name[8]` | `r plot.id(22)` | `r plot.id(23)` | `r plot.id(24)` | `r s.test(8, "s")` | `r s.test(8, "r")`| -Table 5: Results of the effect of space reduction, space dimension distributions and dimensions number of the different space occupancy measures. The dots represent the median space occupancy values across all simulations for each scenario of trait space change (Table 2), the solid and dashed line respectively the 50% and 95% confidence intervals. See Fig. 2 for details on the interpretation of the figures distributions and values. F-values for distribution effect and dimensions effect represents respectively the effect of the ANOVAs space occupancy ~ distributions and space occupancy ~ dimension represent the ratio of sum squared difference within and between groups (the higher, the more the factor has an effect on the measure) and associated _p_-values (0 '\*\*\*' 0.001 '\*\*' 0.01 '\*' 0.05 '.' 0.1 '' 1). This figure illustrates how different measures can be influenced by different aspects of changes in the trait space. E.g. the Average Euclidean distance from centroid (row 1) captures mainly changes in size (column 1), but also captures changes in density (column 2) but does not capture changes in position (column 3). - -As expected, some different measures capture different aspects of space occupancy. -However, it can be hard to predict the behaviour of each measure when 50% of the observations are removed. -We observe a clear decrease in the median metric value in less than a third of the space reductions (10/36). - -In terms of change in size, only the average Euclidean distance from centroid and the sum of variances seem to capture a clear change in both directions. - -In terms of change in density, only the minimum spanning tree average distance and the average nearest neighbour distance seem to capture a clear change in both directions. -And in terms of change in position, only the average displacement metric seems to capture a clear change in direction (albeit not in both directions). -This is not surprising, since the notion of positions becomes more and more complex to appreciate as dimensionality increases (i.e. beyond left/right, up/down and front/back). - -## Empirical example - -```{r test_empirical, echo = FALSE} -## Loading the results -empirical_results <- load.results("empirical_results") - -## Testing the differences for each distributions -bhatt.coeff.safe <- function(x, y, tol = 1e-16, ...) { - if(all(x < tol) | all(y < tol)) { - bhatt.coeff(1, 0) - } else { - bhatt.coeff(x, y, ...) - } -} -disparity_test <- lapply(empirical_results, lapply, test.dispRity, test = bhatt.coeff.safe) -``` - -```{r fable_results_empirical, fig.show='hide', echo=FALSE, fig.height=3, fig.width=3} -## Plotting parameters -plot.param <- list(cex = 2, - col = c("#F7B27E", "#BFE4E3"), - border = c("#F65205", "#3E9CBA"), - na.cex = 3, - scaler = 3) - -## Looping through each mini plot for every measure -data_names <- c("Beck and Lee 2014", "Wright 2017", "Marcy et al. 2016", - "Hopkins and Pearson 2016", "Jones et al. 2015", "Healy et al. 2019") - -for(dataset in 1:length(data_names)){ - for(measure in 1:length(metrics_list)){ - ## Plotting the results - generate.fable.empirical(data = empirical_results[[dataset]][[measure]], - test = disparity_test[[dataset]][[measure]], - precision = 1e-5, plot.param, dataset = dataset) - } -} -``` - -```{r change_plotid_chain, echo = FALSE, eval = TRUE} -## Changing defaults -body(plot.id)[[3]] <- substitute(chain <- "fable_results_empirical") -``` - -```{r, echo = FALSE, eval = FALSE} -print.fable(length(metrics_list), byrow = FALSE, ncol = 6, test = FALSE) -``` - -Measure | Beck and Lee 2014 | Wright 2017 | Marcy et al. 2016 | Hopkins and Pearson 2016 | Jones et al. 2015 | Healy et al. 2019 | -:-----------|----------------|-----------------|------------------|----------------|------------------|----------------| -Comparisons (orange *vs.* blue) | crown *vs.* stem mammals morphologies | crinoids morphologies before *vs.* after the end-Ordovician extinction | *Megascapheus* *vs.* *Thomomys* skull shapes | adults *vs.* juveniles trilobites cephalon shapes | aspens *vs.* grasslands communities compositions | ecthotherms *vs.* endotherms life history traits | -`r name[1]` | `r plot.id(1)` | `r plot.id(9)` | `r plot.id(17)` | `r plot.id(25)` | `r plot.id(33)` | `r plot.id(41)`| -`r name[2]` | `r plot.id(2)` | `r plot.id(10)` | `r plot.id(18)` | `r plot.id(26)` | `r plot.id(34)` | `r plot.id(42)`| -`r name[3]` | `r plot.id(3)` | `r plot.id(11)` | `r plot.id(19)` | `r plot.id(27)` | `r plot.id(35)` | `r plot.id(43)`| -`r name[4]` | `r plot.id(4)` | `r plot.id(12)` | `r plot.id(20)` | `r plot.id(28)` | `r plot.id(36)` | `r plot.id(44)`| -`r name[5]` | `r plot.id(5)` | `r plot.id(13)` | `r plot.id(21)` | `r plot.id(29)` | `r plot.id(37)` | `r plot.id(45)`| -`r name[6]` | `r plot.id(6)` | `r plot.id(14)` | `r plot.id(22)` | `r plot.id(30)` | `r plot.id(38)` | `r plot.id(46)`| -`r name[7]` | `r plot.id(7)` | `r plot.id(15)` | `r plot.id(23)` | `r plot.id(31)` | `r plot.id(39)` | `r plot.id(47)`| -`r name[8]` | `r plot.id(8)` | `r plot.id(16)` | `r plot.id(24)` | `r plot.id(32)` | `r plot.id(40)` | `r plot.id(48)`| -Table 6: Comparisons of pairs of groups in different empirical trait spaces. NAs are used for cases where space occupancy could not be measured due to the curse of multidimensionality. The displayed values are the amount of overlap between both groups (Bhattacharrya Coefficient). - -As with the as for the simulations, there is no measure that summarises all the aspects of distributions for empirical data. -For all `r metric.names()` measures (except the ellipsoid volume) we see either one group or the other having a bigger mean than the other and no consistent case where a group has a bigger mean than the other for all the measures. -For example, in the @beck2014's dataset, there is a clear difference in size using the average Euclidean distance from centroid or the sum of variances (overlaps of respectively 0.175 and 0.159) but no overlap when measuring the size using the sum of ranges (0.966). -However, for the @hopkins2016's dataset, this pattern is reversed (no clear differences for the average Euclidean distance from centroid or the sum of variances - 0.701 and 0.865 respectively - but a clear difference for the sum of ranges (0). -For each dataset, the absolute differences between each groups is not consistent depending on the measures. -For example, in @hopkins2016's dataset, the orange group's mean is clearly higher than the blue one when measuring the sum of ranges (0) and the inverse is true when measuring the average displacement (0). - - - - - - - - - - - - - - - - - -# Discussion - -Here we tested 25 measures of trait space occupancy on simulated and empirical datasets to assess how each measure captures changes in trait space size, density and position. -Our results show that the correlation between measures can vary both within and between measure categories (Fig. 3), highlighting the importance of understanding the measure classification for the interpretation of results. -Our simulations show that different measures capture different types of trait space change (Table 5), meaning that the use of multiple measures is important for comprehensive interpretation of trait space occupancy. -We also show that the choice of measure impacts the interpretation of group differences in empirical datasets (Table 6). - - -#### Measures comparisons - -Measures within the same category of trait space occupancy (size, density or position) do not have the same level of correlation with each other. -For example, the average Euclidean distance from centroid (size) is highly correlated to the sum of variances (size - correlation of 0.99) and somewhat correlated with the minimum spanning tree average distance (density - correlation of 0.66) but poorly with the ellipsoid volume (size - correlation of 0.17) and the minimum spanning tree distances evenness (density - correlation of -0.05). -Furthermore, the fact that we have such a range of correlations for normal distributions suggests that each measure can capture different summaries of space occupancy ranging from obvious differences (for measures not strongly correlated) to subtle ones (for measures strongly correlated). - - -#### Space shifting - -Most measures capture no changes in space occupancy for the "null" (random) space reduction (in grey in Table 5). -This is a desirable behaviour for space occupancy measures since it will likely avoid false positive errors in studies that estimate biological processes from space occupancy patterns [e.g. convergence @marcy2016, life history traits @healy2019]. -However, the average nearest neighbour distance and the sum of ranges have a respectively positive and negative "null" median. -In itself this is not necessarily a negative property but it should be kept in mind that even random processes can increase or decrease these measures' values. - -For changes in size, the sum of variances and the average Euclidean distance from centroid are good descriptors (Table 5). -However, as illustrated in the 2D examples in Fig. 2-B only the blue change results (Table 5) should not result in a direct change in overall size because the trait space is merely "hollowed" out. -That said, "hollowing" is harder to conceptualise in many dimensions and the measures can still be interpreted for comparing groups (orange has a smaller volume than blue). - -The average nearest neigbhour distance and the minimum spanning tree average distance consistently detect changes in density with more precision for low density trait spaces (in blue in Table 5). -However, we can observe some degree of correlation between the changes in density and the changes in size for most measure picking either signal. -This could be due to the use of normally distributed spaces where a change in density often leads to a change in size. -This is not necessarily the case with empirical data. - -Regarding the changes in position, only the average displacement measure seems able to distinguish between a random change and a displacement of the trait space (Table 5). -However, the average displacement measure does not distinguish between positive or negative displacement: this might be due to the inherent complexity of *position* in a multidimensional trait space. - -#### Empirical examples - -Although most differences are fairly consistent within each dataset with one group having a higher space occupancy score than the other for multiple measures, this difference can be more or less pronounced within each dataset (ranging from no to nearly full overlap - BC $\in(0;0.995)$) and sometimes even reversed. -This indicates that opposite conclusions can be drawn from a dataset depending on which space occupancy measure is considered. -For example, in @wright2017, crinoids after the Ordovician mass extinction have a higher median measure value for all measures but for the average displacement. -These differences depending on the measures are also more pronounced in the empirical datasets where the observations per group are unequal [@hopkins2016; @healy2019]. - -### Caveats - -While our simulations are useful to illustrate the behaviour of diverse space occupancy measures, they have several caveats. -First, the simulated observations in the trait spaces are independent. -This is not the case in biology where observations can be spatially [@jones2015] or phylogenetically correlated [e.g. @beck2014]. -Second, the algorithm used to reduce the trait spaces might not always accurately reflect changes. -This might favour some specific measures over others, in particular for the changes in density that modify the nearest neighbour density rather than changing the global density. -This algorithmic choice was made in order to not confound changes in density along with changes in size. -However, the results presented here probably capture the general behaviour of each measure since results are consistent between the simulated and empirical analysis. - - -Furthermore, we did not take into account the effect of sampling on space occupancy measurements (but see additional results with 80% and 20% space reduction in the supplementary materials 4). -In fact, sampling has been previously shown to have an effect on measurements depending on range or volumes (e.g. the sum of ranges or the hypervolume @ciampaglio2001). -This effect is especially expected to be acerbated in macroevolutionary studies when using the fossil record [@brocklehurst2013] but can be tackled using rarefaction and bootstrapping techniques [@disprity]. - - -### Using `moms` to choose the appropriate measurements - - -Therefore, we propose the [`moms`](https://tguillerme.shinyapps.io/moms/) shiny app to allow workers to help them choose their set of space occupancy measurements (and test the caveats mentioned above). -`moms` is an online graphical user interface to help analyse multidimensional data. -It allows users to upload their dataset of interest (or simulate one with specific parameters) and measure space occupancy using a variety of implemented measures (namely, but not only, the ones used in this study). -Furthermore, the package allows simulation of shifts in trait space occupancy as also presented in this paper to test whether some measures capture specific changes in space. -However, `moms` is not a tool for analysing multidimensional data _per se_ but rather for helping workers to chose the space occupancy measure most appropriated to their data and question. -To run multidimensional analysis, we suggest using dedicated `R` packages (such as - but not limited to: @oksanen2007vegan, @momocs, @bat2015, @disprity). - - - -### Conclusions - - -We insist that although no measure is objectively better than the next one, some can be more problematic than other in specific contexts. -For example, the results for the Sum of Ranges, Minimum spanning tree average distances, and to a lesser extent average nearest neighbour distances produced results in the reduced space often similar to the randomly reduced spaces (Table 5). -This does not make them "bad" measures but rather heavily context dependent. -Regardless, we believe that workers should identify the most appropriate measures based on their trait space properties as well as their specific biological question. -We believe this could be fostered by following these several suggestions: - - -First, we suggest using multiple measures to tackle different aspects of the trait space. -This follows the same logical thinking that the mean might not be sufficient to describe a distribution (e.g. the variance might be a good additional descriptor). -Although using multiple measures is not uncommon in macroevolutionary studies [e.g. @halliday2015] or in ecology [@mammola2019], they often do no cover more than one of the three categories of trait space measures (but see the recent work of @carmona2019 and @mammola2020). - -Second, we suggest selecting the measures that best address the biological question at hand. -If one studies an adaptive radiation in a group of organisms, it is worth thinking what would be the expected null model: would the group's size increase (radiation in all directions), would it increase in density (niche specialisation) or would it shift in position (radiation into a new set of niches)? - -Third, we suggest not naming measures after the biological aspect they describe which can be vague (e.g. "disparity" or "functional dispersion") but rather after what they are measuring and why (e.g. "we used sum of ranges to measure the space size"). -We believe this will support both a clearer understanding of what *is* measured as well as better communication between ecology and evolution research where measures can be similar but have different names. - -Multidimensional analyses have been acknowledged as essential tools in modern biology but they can often be counter-intuitive [@cursedimensionality]. -It is thus crucial to accurately describe patterns in multidimensional trait spaces to be able to link them to biological processes. -When summarising trait spaces, it is important to remember that a pattern captured by a specific space occupancy measure is often dependent on the properties of the space and of the particular biological question of interest. -We believe that having a clearer understanding of both the properties of the trait space and the associated space occupancy measures (e.g. using [`moms`](https://tguillerme.shinyapps.io/moms/)) as well as using novel space occupancy measures to answer specific questions will be of great use to study biological processes in a multidimensional world. - - -# Acknowledgements - -We thank Natalie Jones and Kevin Healy for helping with the empirical datasets and Stefano Mammola and Neil Brocklehurst for their positive and encouraging reviews. -We acknowledge funding from the Australian Research Council DP170103227 and FT180100634 awarded to VW. - -# Authors contributions - -TG, MNP, AEM and VW designed the project. -TG and AEM collected the empirical dataset. -TG ran the analyses and designed the software. -TG, MNP, AEM and VW wrote the manuscript. - -# Data Availability, repeatability and reproducibility - -The raw empirical data is available from the original papers [@beck2014; @jones2015, @marcy2016; @hopkins2016; @wright2017; @healy2019]. -The subsets of the empirical data used in this analysis are available on figshare -[DOI: 10.6084/m9.figshare.9943181.v1](https://doi.org/10.6084/m9.figshare.9943181.v1). -The modified empirical data are available in the package accompanying this manuscript (`data(moms::demo_data)`). -This manuscript (including the figures, tables and supplementary material) is repeatable and reproducible by compiling the vignette of the [GitHub `moms R` package](https://github/TGuillerme/moms). - -# References diff --git a/inst/mee.csl b/inst/mee.csl deleted file mode 100644 index d7b426d..0000000 --- a/inst/mee.csl +++ /dev/null @@ -1,94 +0,0 @@ - - diff --git a/inst/methods-in-ecology-and-evolution.csl b/inst/methods-in-ecology-and-evolution.csl deleted file mode 100644 index 5e0a631..0000000 --- a/inst/methods-in-ecology-and-evolution.csl +++ /dev/null @@ -1,626 +0,0 @@ - - diff --git a/inst/shiftingspace.Rmd b/inst/shiftingspace.Rmd index 8dee144..f8b5c70 100644 --- a/inst/shiftingspace.Rmd +++ b/inst/shiftingspace.Rmd @@ -9,7 +9,7 @@ output: fig_width: 8 fig_height: 8 keep_tex: true - self_contained: false + self_contained: true --- @@ -36,7 +36,7 @@ output: Switch the text highlighting by replacing -<@@@font style="color:blue"> +<@@@font style="color:black"> to: \@@@textcolor{blue}{ @@ -58,7 +58,7 @@ and edit the save .tex file to add the following \modulolinenumbers[1] % just after the \begin{document} tag \linenumbers --> - + ```{r header, echo = FALSE, results = 'hide', message = FALSE, warning = FALSE} @@ -253,8 +253,8 @@ body(plot.id)[[2]] <- substitute(type <- ".png") Multidimensional analysis of traits are now common in ecology and evolution and are based on trait spaces in which each dimension summarises the observed trait combination (a morphospace or an ecospace). -Observations of interest will typically occupy a subset of this space, and researchers will calculate one or more measures to quantify how organisms inhabit that space. -In macroevolution and ecology these measures called disparity or dissimilarity metrics and are generalised as space occupancy measures. +Observations of interest will typically occupy a subset of this space, and researchers will calculate one or more measures to quantify how organisms inhabit that space. +In macroevolution and ecology these measures called disparity or dissimilarity metrics and are generalised as space occupancy measures. Researchers use these measures to investigate how space occupancy changes through time, in relation to other groups of organisms, and in response to global environmental changes. However, the mathematical and biological meaning of most space occupancy measures is vague with the majority of widely-used measures lacking formal description. @@ -272,18 +272,18 @@ By providing guidelines and common vocabulary for space occupancy analysis, we h Groups of species and environments share specific, recognisable, correlated characteristics: guilds or biomes with shared phenotypic, physiological, phylogenetic or behavioural traits. Organisms or environments should therefore be studied as a set of traits rather than some specific traits in isolation [@donohue2013; @hopkins2017]. Biologists increasingly been using ordination techniques [see @legendre2012 for a summary] to create multidimensional trait spaces to either explore properties of data or test hypotheses [e.g. @oksanen2007vegan; @blonder2018; @disprity]. -For example, in palaeobiology, @wright2017 used trait spaces to study how groups of species' characteristics change through time; in ecology, @jones2015 studied evidence of competition by looking at trait overlap between two populations. - While different fields use a different set of terms for such approaches (Table 1), they actually focus on the same mathematical objects: matrices with columns representing an original or transformed trait value and rows representing observations [taxon, field site, etc.; @disprity]. +For example, in palaeobiology, @wright2017 used trait spaces to study how groups of species' characteristics change through time; in ecology, @jones2015 studied evidence of competition by looking at trait overlap between two populations. + While different fields use a different set of terms for such approaches (Table 1), they actually focus on the same mathematical objects: matrices with columns representing an original or transformed trait value and rows representing observations [taxon, field site, etc.; @disprity]. Mathematics | Ecology | Macroevolution | This paper --------------------|--------------------|--------------------|------------ -Matrix ($n \times d$) with a structural relation between rows and columns | Functional space, morphospace , etc. | Morphospace, traitspace, etc. | trait space +Matrix ($n \times d$) with a structural relation between rows and columns | Functional space, morphospace , etc. | Morphospace, traitspace, etc. | trait space Rows (*n*) | Taxa, field sites, environments, etc. | Taxa, specimen, populations, etc. | observations Columns (*d*) | Traits, Ordination scores, distances, etc. | Traits, ordination scores, distances, etc. | dimensions Matrix subset ($m \times d$; $m \leq n$) | Treatments, phylogenetic group (clade), etc. | Clades, geological stratum, etc. | group -Statistic (i.e. a measure) | Dissimilarity index or metric, hypervolume, functional diversity, etc. | Disparity metric or index | space occupancy measure +Statistic (i.e. a measure) | Dissimilarity index or metric, hypervolume, functional diversity, etc. | Disparity metric or index | space occupancy measure Multidimensional analysis | Dissimilarity analysis, trait analysis, etc. | Disparity analysis, disparity-through-time, etc. | multidimensional analysis -Table 1: Different terms are used for equivalent measures in mathematics, ecology and macroevolution. +Table 1: Different terms are used for equivalent measures in mathematics, ecology and macroevolution. Ecologists and evolutionary biologists often use trait spaces with respect to the same fundamental questions: are groups occupying the same amount of trait space? @@ -294,20 +294,20 @@ Studying the occupancy of trait spaces is done using disparity indices in macroe Despite the commonalities between the measures used in ecology and evolution (which are often metric but don't necessarily need to be), surprisingly little work has been published on their behaviour [but see @ciampaglio2001; @villéger2008; @mammola2019]. Different occupancy measures capture different aspects of trait space [@ciampaglio2001; @villéger2008; @mammola2019]. -This may be widely-known, but to our knowledge it is infrequently mentioned in peer-reviewed papers. +This may be widely-known, but to our knowledge it is infrequently mentioned in peer-reviewed papers. First, space occupancy measures are often named as the biological aspect they are describing ("disparity", "functional diversity") rather than what they are measuring (e.g. the product of ranges), which obscures the differences and similarities between studies. Second, in many studies in ecology and evolution, authors have focused on measuring the size of the trait space [e.g. ellipsoid volume @donohue2013; hypervolume @diaz2016; product of variance @wright2017; Procrustes variance @marcy2016]. -However, the size of the trait space only represents one aspects of occupancy, disregarding other measures such as the density [@geiger2008] or position [@wills2001; @ciampaglio2001]. +However, the size of the trait space only represents one aspects of occupancy, disregarding other measures such as the density [@geiger2008] or position [@wills2001; @ciampaglio2001]. For example, if two groups have the same size, this can support certain biological conclusions. Yet, an alternative aspect of space occupancy may indicate that the groups' position are different, leading to a different biological conclusion (e.g. the groups are equally diverse but occupy different niches). Using measures that only capture one aspect of the trait space may restrain the potential of multidimensional analysis [@villéger2008]. Here we propose a broad classification of space occupancy measures as used across ecology and evolution and study their power to detect changes in trait space occupancy in simulated and empirical data. -Note this does not account whether or not it is possible for a space to be occupied (e.g., some spaces may represent biologically impossible shapes); this, however, may be important in some cases, such as testing whether a region is infinite or not. +Note this does not account whether or not it is possible for a space to be occupied (e.g., some spaces may represent biologically impossible shapes); this, however, may be important in some cases, such as testing whether a region is infinite or not. We provide an assessment of each broad type of space occupancy measures along with a unified terminology to foster communication between ecology and evolution. -Unsurprisingly, we found no one measure describes all changes in space and that the results from each measures are dependent on the characteristics of the space and the hypotheses. +Unsurprisingly, we found no one measure describes all changes in space and that the results from each measures are dependent on the characteristics of the space and the hypotheses. -There can be an infinite number of measures and that it is thus impossible to propose a comprehensive analysis for all the measures properties respective to how they measure changes in trait space. +There can be an infinite number of measures and that it is thus impossible to propose a comprehensive analysis for all the measures properties respective to how they measure changes in trait space. We therefore propose [`moms`](https://tguillerme.shinyapps.io/moms/), a tool for researchers to design, experiment and visualise their own space occupancy measure tailored for their project. The tool will help researchers understand the "null" behaviour of the measures of interest. @@ -391,7 +391,7 @@ In a 2D space, two groups can be equally distant from a fixed point but in diffe However, when thinking about unidimensional data, this measure is obvious: two groups A or B could have the same variance (size) with the same number of observations (density) but could have two different means and thus be in different positions. These measures are used in ecology to compare the position of two groups relative to each other [@mammola2019]. - Note that this classification into size, density and position bears some similarities with @tucker2017 classifying phylogenetic diversity measurements into richness, divergence and regularity categories. + Note that this classification into size, density and position bears some similarities with @tucker2017 classifying phylogenetic diversity measurements into richness, divergence and regularity categories. However, while @tucker2017 based their classification on the mathematical operation inherent to each metrics (the sum for richness, the mean for divergence and the variance for regularity), our three broad classifications are based on their geometric properties regardless of the formula of each metric (e.g. the size of a space can be calculated using a sum, mean or/and variance). ## No measure to rule them all: benefits of considering multiple measures @@ -399,8 +399,8 @@ However, while @tucker2017 based their classification on the mathematical operat The use of multiple measurements to assess trait space occupancy provides a more detailed characterisation of occupancy changes. If the question is to look at how space occupancy changes in response to mass extinction, using a single space occupancy measure can miss part of the picture: a change in size could be decoupled from a change in position or density in trait space. For example, the Cretaceous-Paleogene extinction (66 million years ago) shows an increase in size of the mammalian trait space [adaptive radiation; @halliday2015] but more specific questions can be answered by looking at other aspects of trait space occupancy: does the radiation expand on previously existing morphologies [elaboration, increase in density; @endler2005] or does it explore new regions of the trait space [innovation, change in position; @endler2005]? -Similarly, in ecology, if two groups have the same trait space size, the differences in density within these two groups is potentially illuminating: different selection pressure can lead to different density within equally sized groups. -This can also be extended to more complex ecological concepts such as niche modelling [@qiao2015]. +Similarly, in ecology, if two groups have the same trait space size, the differences in density within these two groups is potentially illuminating: different selection pressure can lead to different density within equally sized groups. +This can also be extended to more complex ecological concepts such as niche modelling [@qiao2015]. Here, we provide the first interdisciplinary review of 25 space occupancy measures that uses the broad classification of measures into size, density and position to capture pattern changes in trait space. We assess the behaviour of measures using simulations and six interdisciplinary empirical datasets covering a wide range of potential data types and biological questions. @@ -437,7 +437,7 @@ source("list.of.metrics.R") We tested how 25 space occupancy measures relate to each other, are affected by modifications of traits space and affect group comparisons in empirical data: 1. We simulated 13 different spaces with different sets of parameters; -2. We transformed these spaces by removing 50% of the observations following four different scenarios corresponding to different empirical scenarios: randomly, by size (e.g. expansion or reduction of niches), by density (e.g. different degrees of competition within a guild), and by position (e.g. ecological niche shift). +2. We transformed these spaces by removing 50% of the observations following four different scenarios corresponding to different empirical scenarios: randomly, by size (e.g. expansion or reduction of niches), by density (e.g. different degrees of competition within a guild), and by position (e.g. ecological niche shift). 3. We measured occupancy on the resulting transformed spaces using `r metric.names()` different space occupancy measures; 4. We applied the same space occupancy measures to six empirical datasets (covering a range of disciplines and a range of dataset properties). @@ -526,21 +526,21 @@ This reflects a "null" biological model of changes in trait space: the case when For example, if diversity is reduced by 50% but the space size remains the same, there is a decoupling between diversity and space occupancy [@ruta2013]. Our selected measures are expected to not be affected by this change. -* **Size:** by removing observations within a distance from the centre of the trait space lower or greater than a radius $\rho$ (where $\rho$ is chosen such that 50% observations are selected) generating two limit removals: *maximum* and *minimum* (respectively in orange and blue; Fig. 2-B). +* **Size:** by removing observations within a distance from the centre of the trait space lower or greater than a radius $\rho$ (where $\rho$ is chosen such that 50% observations are selected) generating two limit removals: *maximum* and *minimum* (respectively in orange and blue; Fig. 2-B). This can reflect a strict selection model where observations with trait values below or above a threshold are removed leading to an expansion or a contraction of the trait space. -This type of change could be due to habitat destruction [e.g. @mammola2019b] or to mass extinctions [e.g. @wright2017]. +This type of change could be due to habitat destruction [e.g. @mammola2019b] or to mass extinctions [e.g. @wright2017]. Size measures are expected to be most affected by this change. * **Density:** by removing any pairs of point with a distance $D$ from each other where (where $D$ is chosen such that 50% observations are selected) generating two density removals: *high* and *low* (respectively in orange and blue; Fig. 2-C). This can reflect changes within groups in the trait space due to ecological factors [e.g. niche repulsion resulting in lower density; @grant2006]. -This type of change could be due to accelerated rates of evolution [@close2015] or to differences in modes of life in macroevolution [e.g. @healy2019]. +This type of change could be due to accelerated rates of evolution [@close2015] or to differences in modes of life in macroevolution [e.g. @healy2019]. Density measures are expected to be most affected by this change. -* **Position:** by removing points similarly as for **Size** but using the distance from the furthest point from the centre generating two position removals: *positive* and *negative* (respectively in orange and blue; Fig. 2-D). +* **Position:** by removing points similarly as for **Size** but using the distance from the furthest point from the centre generating two position removals: *positive* and *negative* (respectively in orange and blue; Fig. 2-D). This can reflect global changes in trait space (e.g. if an entire group remaining diverse but occupying a different niche). -This type of change could be due changes in evolutionary trajectories [@endler2005] or to differences in ecosystem compositions [e.g. @jones2015]. +This type of change could be due changes in evolutionary trajectories [@endler2005] or to differences in ecosystem compositions [e.g. @jones2015]. Position measures are expected to be most affected by this change. The algorithm to select $\rho$ or $D$ is described in the Supplementary material 1. @@ -674,7 +674,7 @@ text(x = -0.6, y = 2, labels = round(bc_1, 3), cex = 1.4) Because occupancy measures are dependent on the space, we scaled and centred them between -1 and 1 to make them comparable (by subtracting the observed occupancy without reduction to all the measures of the reduced spaces and then divided it by the maximum observed occupancy). A value of 0 indicates no effect of the space reduction and $>0$ and $<0$ respectively indicates an increase or decrease in the measure value. -We then measured the amount of overlap between the non-random removals (size, density and position) and the random removals using the Bhattacharrya Coefficient [@bhattacharyya1943]. +We then measured the amount of overlap between the non-random removals (size, density and position) and the random removals using the Bhattacharrya Coefficient [@bhattacharyya1943]. ### Measuring the effect of space and dimensionality @@ -712,7 +712,7 @@ study | field | taxonomic group | traits | trait space | size | groups | questio @beck2014 | Palaeontology | Mammalia | discrete morphological phylogenetic data | Ordination of a distance matrix (PCO) | 106*105 | 52 crown vs. 54 stem | Are crown mammals more disparate than stem mammals?| @wright2017 | Palaeontology | Crinoidea | discrete morphological phylogenetic data | Ordination of a distance matrix (PCO) | 42*41 | 16 before vs. 23 after | Is there a difference in disparity before and after the Ordovician mass extinction?| @marcy2016 | Evolution | Rodentia | skull 2D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 454*134 | 225 *Megascapheus* vs. 229 *Thomomys* | Are two genera of gopher morphologically distinct? | -@hopkins2016 | Evolution | Trilobita | 3D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 46*46 | 36 adults vs. 10 juveniles | Are juvenile trilobites a subset of adult ones in trait space? | +@hopkins2016 | Evolution | Trilobita | 3D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 46*46 | 36 adults vs. 10 juveniles | Are juvenile trilobites a subset of adult ones in trait space? | @jones2015 | Ecology | Plantae | Communities species compositions | Ordination of a Jaccard distance matrix (PCO) | 48*47 | 24 aspens vs. 24 grasslands | Is there a difference in species composition between aspens and grasslands? | @healy2019 | Ecology | Animalia | Life history traits | Ordination of continuous traits (PCA) | 285*6 | 83 ecthotherms vs. 202 endotherms | Do endotherms have more diversified life history strategies than ectotherms? | Table 4: details of the six empirical trait spaces. @@ -775,7 +775,7 @@ names(results_pairwise[[1]])[8] <- "Av. displacements" pairwise.plot(results_pairwise, scale = TRUE, type = "base", plot = "cor") ``` -Most measures of space were positively correlated (Pearson correlation of 0.99 for the average Euclidean distance from centroid and sum of variance or 0.97 for the average nearest neighbour distance and minimum spanning tree average length; Fig. 3). +Most measures of space were positively correlated (Pearson correlation of 0.99 for the average Euclidean distance from centroid and sum of variance or 0.97 for the average nearest neighbour distance and minimum spanning tree average length; Fig. 3). The remaining measures were either somewhat correlated or had a negative pairwise distribution (ranging from 0.66 for the sum of variances and the ellipsoid volume to -0.09 between the average displacement and the average Euclidean distance from centroid; Fig. 3). All measures but the ellipsoid volume were normally (or nearly normally) distributed (Fig. 3). @@ -840,7 +840,7 @@ Table 5: Results of the effect of space reduction, space dimension distributions As expected, some different measures capture different aspects of space occupancy. However, it can be hard to predict the behaviour of each measure when 50% of the observations are removed. -We observe a clear decrease in the median measure value in less than a third of the space reductions (10/36). +We observe a clear decrease in the median measure value in less than a third of the space reductions (10/36). In terms of change in size, only the average Euclidean distance from centroid and the sum of variances seem to capture a clear change in both directions. @@ -909,7 +909,7 @@ Comparisons (orange *vs.* blue) | crown *vs.* stem mammals morphologies | crinoi `r name[8]` | `r plot.id(8)` | `r plot.id(16)` | `r plot.id(24)` | `r plot.id(32)` | `r plot.id(40)` | `r plot.id(48)`| Table 6: Comparisons of pairs of groups in different empirical trait spaces. NAs are used for cases where space occupancy could not be measured due to the curse of multidimensionality. The displayed values are the amount of overlap between both groups (Bhattacharrya Coefficient). -As with the as for the simulations, there is no measure that summarises all the aspects of distributions for empirical data. +As with the as for the simulations, there is no measure that summarises all the aspects of distributions for empirical data. For all `r metric.names()` measures (except the ellipsoid volume) we see either one group or the other having a bigger mean than the other and no consistent case where a group has a bigger mean than the other for all the measures. For example, in the @beck2014's dataset, there is a clear difference in size using the average Euclidean distance from centroid or the sum of variances (overlaps of respectively 0.175 and 0.159) but no overlap when measuring the size using the sum of ranges (0.966). However, for the @hopkins2016's dataset, this pattern is reversed (no clear differences for the average Euclidean distance from centroid or the sum of variances - 0.701 and 0.865 respectively - but a clear difference for the sum of ranges (0). @@ -952,7 +952,7 @@ Furthermore, the fact that we have such a range of correlations for normal distr Most measures capture no changes in space occupancy for the "null" (random) space reduction (in grey in Table 5). This is a desirable behaviour for space occupancy measures since it will likely avoid false positive errors in studies that estimate biological processes from space occupancy patterns [e.g. convergence @marcy2016, life history traits @healy2019]. However, the average nearest neighbour distance and the sum of ranges have a respectively positive and negative "null" median. -In itself this is not necessarily a negative property but it should be kept in mind that even random processes can increase or decrease these measures' values. +In itself this is not necessarily a negative property but it should be kept in mind that even random processes can increase or decrease these measures' values. For changes in size, the sum of variances and the average Euclidean distance from centroid are good descriptors (Table 5). However, as illustrated in the 2D examples in Fig. 2-B only the blue change results (Table 5) should not result in a direct change in overall size because the trait space is merely "hollowed" out. @@ -978,18 +978,18 @@ These differences depending on the measures are also more pronounced in the empi While our simulations are useful to illustrate the behaviour of diverse space occupancy measures, they have several caveats. First, the simulated observations in the trait spaces are independent. This is not the case in biology where observations can be spatially [@jones2015] or phylogenetically correlated [e.g. @beck2014]. -Second, the algorithm used to reduce the trait spaces might not always accurately reflect changes. +Second, the algorithm used to reduce the trait spaces might not always accurately reflect changes. This might favour some specific measures over others, in particular for the changes in density that modify the nearest neighbour density rather than changing the global density. This algorithmic choice was made in order to not confound changes in density along with changes in size. However, the results presented here probably capture the general behaviour of each measure since results are consistent between the simulated and empirical analysis. - Furthermore, we did not take into account the effect of sampling on space occupancy measurements (but see additional results with 80% and 20% space reduction in the supplementary materials 4). + Furthermore, we did not take into account the effect of sampling on space occupancy measurements (but see additional results with 80% and 20% space reduction in the supplementary materials 4). In fact, sampling has been previously shown to have an effect on measurements depending on range or volumes (e.g. the sum of ranges or the hypervolume @ciampaglio2001). This effect is especially expected to be acerbated in macroevolutionary studies when using the fossil record [@brocklehurst2013] but can be tackled using rarefaction and bootstrapping techniques [@disprity]. ### Using `moms` to choose the appropriate measurements -Therefore, we propose the [`moms`](https://tguillerme.shinyapps.io/moms/) shiny app to allow workers to help them choose their set of space occupancy measurements (and test the caveats mentioned above). +Therefore, we propose the [`moms`](https://tguillerme.shinyapps.io/moms/) shiny app to allow workers to help them choose their set of space occupancy measurements (and test the caveats mentioned above). `moms` is an online graphical user interface to help analyse multidimensional data. It allows users to upload their dataset of interest (or simulate one with specific parameters) and measure space occupancy using a variety of implemented measures (namely, but not only, the ones used in this study). Furthermore, the package allows simulation of shifts in trait space occupancy as also presented in this paper to test whether some measures capture specific changes in space. @@ -999,7 +999,7 @@ To run multidimensional analysis, we suggest using dedicated `R` packages (such ### Conclusions -We insist that although no measure is objectively better than the next one, some can be more problematic than other in specific contexts. +We insist that although no measure is objectively better than the next one, some can be more problematic than other in specific contexts. For example, the results for the Sum of Ranges, Minimum spanning tree average distances, and to a lesser extent average nearest neighbour distances produced results in the reduced space often similar to the randomly reduced spaces (Table 5). This does not make them "bad" measures but rather heavily context dependent. Regardless, we believe that workers should identify the most appropriate measures based on their trait space properties as well as their specific biological question. @@ -1007,7 +1007,7 @@ We believe this could be fostered by following these several suggestions: (but see the recent work of @carmona2019 and @mammola2020). +Although using multiple measures is not uncommon in macroevolutionary studies [e.g. @halliday2015] or in ecology [@mammola2019], they often do no cover more than one of the three categories of trait space measures (but see the recent work of @carmona2019 and @mammola2020). Second, we suggest selecting the measures that best address the biological question at hand. If one studies an adaptive radiation in a group of organisms, it is worth thinking what would be the expected null model: would the group's size increase (radiation in all directions), would it increase in density (niche specialisation) or would it shift in position (radiation into a new set of niches)? @@ -1041,4 +1041,7 @@ The subsets of the empirical data used in this analysis are available on figshar The modified empirical data are available in the package accompanying this manuscript (`data(moms::demo_data)`). This manuscript (including the figures, tables and supplementary material) is repeatable and reproducible by compiling the vignette of the [GitHub `moms R` package](https://github/TGuillerme/moms). +# Conflict of interest +None declared. + # References diff --git a/inst/shiftingspace.tex b/inst/shiftingspace.tex deleted file mode 100644 index 2077ef9..0000000 --- a/inst/shiftingspace.tex +++ /dev/null @@ -1,1710 +0,0 @@ -\documentclass[]{article} -\usepackage{lmodern} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\usepackage{fixltx2e} % provides \textsubscript -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} -\fi -% use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -% use microtype if available -\IfFileExists{microtype.sty}{% -\usepackage{microtype} -\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\usepackage[margin=1in]{geometry} -\usepackage{hyperref} -\hypersetup{unicode=true, - pdftitle={Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?}, - pdfauthor={Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker}, - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage{longtable,booktabs} -\usepackage{graphicx,grffile} -\makeatletter -\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} -\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} -\makeatother -% Scale images if necessary, so that they will not overflow the page -% margins by default, and it is still possible to overwrite the defaults -% using explicit options in \includegraphics[width, height, ...]{} -\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -%%% Use protect on footnotes to avoid problems with footnotes in titles -\let\rmarkdownfootnote\footnote% -\def\footnote{\protect\rmarkdownfootnote} - -%%% Change title format to be more compact -\usepackage{titling} - -% Create subtitle command for use in maketitle -\providecommand{\subtitle}[1]{ - \posttitle{ - \begin{center}\large#1\end{center} - } -} - -\setlength{\droptitle}{-2em} - - \title{Shifting spaces: which disparity or dissimilarity measurement best -summarise occupancy in multidimensional spaces?} - \pretitle{\vspace{\droptitle}\centering\huge} - \posttitle{\par} - \author{Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker} - \preauthor{\centering\large\emph} - \postauthor{\par} - \predate{\centering\large\emph} - \postdate{\par} - \date{2020-03-27} - - -\begin{document} -\maketitle - -\section{Abstract}\label{abstract} - -Multidimensional analysis of traits are now a common in ecology and -evolution and are based on trait spaces in which each dimension -summarises the observed trait combination (a morphospace or an -ecospace). Observations of interest will typically occupy a subspace of -this space, and researchers will calculate one or more measures to -quantify how organisms ``inhabit'' that space. In macroevolution and -ecology these measures are referred to as disparity or dissimilarity -metrics and can be generalised as space occupancy measures. Researchers -use these measures to investigate how space occupancy changes through -time, in relation to other groups of organisms, and in response to -global environmental changes. However, the mathematical and biological -meaning of most space occupancy measures is vague with the majority of -widely-used measures lacking formal description. - -Here we propose a broad classification of space occupancy measures into -three categories that capture changes in size, density, or position. We -study the behaviour of 25 measures to changes in trait space size, -density and position on simulated and empirical datasets. We find that -no measure describes all of trait space aspects but that some are better -at capturing certain aspects. Our results confirm the three broad -categories (size, density and position) and allow us to relate changes -in any of these categories to biological phenomena. - -Because the choice of space occupancy measures is specific to the data -and question, we introduced -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool -allowing users to both visualise and capture changes in space occupancy -for any measurement. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} is designed -to help workers choose the right space occupancy measures, given the -properties of their trait space and their biological question. By -providing guidelines and common vocabulary for space occupancy analysis, -we hope to help bridging the gap in multidimensional research between -ecology and evolution. - -\section{Introduction}\label{introduction} - -Groups of species and environments share specific, recognisable, -correlated characteristics: guilds or biomes with shared phenotypic, -physiological, phylogenetic or behavioural traits. Organisms or -environments should therefore be studied as a set of traits rather than -some specific traits in isolation (Donohue et al. 2013; Hopkins and -Gerber 2017). Biologists have increasingly been using ordination -techniques (see Legendre and Legendre 2012 for a summary) to create -multidimensional trait spaces to either explore properties of the data -or test hypotheses (e.g. Oksanen et al. 2007; Blonder 2018; Guillerme -2018). For example, in palaeobiology, Wright (2017) used trait spaces to -study how groups of species' characteristics change through time; in -ecology, Jones et al. (2015) study evidence of competition by looking at -trait overlap between two populations. However, different fields use a -different set of terms for such approaches (Table 1). Nonetheless, they -use the same mathematical objects: matrices with columns representing an -original or transformed trait value and rows representing observations -(taxon, field site, etc.; Guillerme 2018). - -\begin{longtable}[]{@{}llll@{}} -\toprule -\begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Mathematics\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[b]{0.24\columnwidth}\raggedright\strut -Macroevolution\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -This paper\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix (\(n \times d\)) with a structural relation between rows and -columns\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Function-space, Eco-space, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Morphospace, traitspace, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -trait space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Rows (\emph{n})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, field sites, environments, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Taxa, specimen, populations, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Columns (\emph{d})\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -dimensions\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Matrix subset (\(m \times d\); \(m \leq n\))\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Treatments, phylogenetic group (clade), etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Clades, geological stratum, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -group\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Statistic\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity index or metric, hypervolume, functional diversity\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity metric or index\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -space occupancy measure\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Multidimensional analysis\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Dissimilarity analysis, trait analysis, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut -Disparity analysis, disparity-through-time, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -multidimensional analysis\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 1: terms and equivalence between mathematics, ecology and -macroevolution. - -Ecologists and evolutionary biologists often use trait spaces with -respect to the same fundamental questions: are groups occupying the same -amount of trait space? Do some groups contain more species than others -in the same amount of trait space? Are some specific factors correlated -with different patterns of trait space occupancy? Because of the -multidimensional nature of these trait spaces, it is often not possible -to study them using bi- or tri-variate techniques (Díaz et al. 2016; -Hopkins and Gerber 2017; Mammola 2019). Studying the occupancy of trait -spaces is done using disparity indices in macroevolution (Wills 2001; -Hopkins and Gerber 2017; Guillerme 2018) or comparing hypervolumes in -ecology (Donohue et al. 2013; Díaz et al. 2016; Blonder 2018; Mammola -2019). Despite the commonalities between the measures used in ecology -and evolution (which are often metric but don't necessarily need to be), -surprisingly little work has been published on their behaviour (but see -Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). - -Different occupancy measures capture different aspects of trait space -(ciampaglio2001; Villéger et al. 2008; Mammola 2019). It may be -widely-known, but to our knowledge is infrequently mentioned in -peer-reviewed papers. First, space occupancy measures are often named as -the biological aspect they are describing (``disparity'', ``functional -diversity'') rather than what they are measuring (e.g.~the product of -ranges), which obscures the differences and similarities between -studies. Second, in many studies in ecology and evolution, authors have -focused on measuring the size of the trait space (e.g.~ellipsoid volume -Donohue et al. 2013; hypervolume Díaz et al. 2016; Procrustes variance -Marcy et al. 2016; product of variance Wright 2017). However, the size -of the trait space only represents one aspects of occupancy, -disregarding others such as the density (Harmon et al. 2008) or position -(Wills 2001; Ciampaglio et al. 2001). For example, if two groups have -the same size, this can support certain biological conclusions. Yet, an -alternative aspect of space occupancy may indicate that the groups' -position are different, leading to a different biological conclusion -(e.g.~the groups are equally diverse but occupy different niches). Using -measures that only capture one aspect of the trait space may restrain -the potential of multidimensional analysis (Villéger et al. 2008). - -Here we propose a broad classification of space occupancy measures as -used across ecology and evolution and study their power to detect -changes in trait space occupancy in simulated and empirical data -(regardless of whether spaces are are truly ``occupiable'' which might -be important in some cases - e.g.~if the space is infinite or if some -regions inapplicable). We provide an assessment of each broad type of -space occupancy measures along with a unified terminology to foster -communication between ecology and evolution. Unsurprisingly, we found no -one measure describes all changes and that the results from each -measures are dependent on the characteristics of the space and the -hypotheses. Furthermore, because there can be an infinite number of -measures, it would be impossible to propose clear generalities to space -occupancy measures behaviour. Therefore, we propose -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool -allowing researchers to design, experiment and visualise their own space -occupancy measure tailored for their specific project and helping them -understanding the ``null'' behaviour of the measures of interest. - -\subsection{Space occupancy measures}\label{space-occupancy-measures} - -In this paper, we define trait spaces as any matrix where rows are -observations and columns are traits, where both observations and traits -are structurally related (e.g.~there is a phylogenetic relation between -observations - and traits, etc.). These traits can widely vary in number -and types: they could be coded as discrete (e.g.~presence or absence of -a bone; Beck and Lee 2014; Wright 2017), continuous measurements -(e.g.~leaf area; Díaz et al. 2016) or more sophisticated measures -(Fourier ellipses; Bonhomme et al. 2014; e.g.~landmark position; Marcy -et al. 2016). Traits can also be measured by using relative observations -(e.g.~community compositions; Jones et al. 2015) or distance between -observations (e.g. Close et al. 2015). However, regardless of the -methodology used to build a trait space, three broad occupancy measures -can be used: the size which approximates the amount of space occupied, -the density which approximates the distribution in space and the -position which approximates the location in space (Fig. 1; Villéger et -al. 2008). Of course any combination of these three aspects is always -possible. - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measures_types-1.pdf} -\caption{different type of information captured by space occupancy -measures: (A) size, (B) density and (C) position.} -\end{figure} - -\paragraph{1. Size}\label{size} - -Size captures the spread of a group in the trait space. They can be -interpreted as the amount of the trait space that is occupied by -observations. Typically, larger values for such measures indicate the -presence of more extreme trait combinations. For example, if group A is -bigger than B, the observations in A achieve more extreme trait -combinations than in B. This type of measure is widely used in both -ecology (e.g.~the hypervolume; Blonder 2018) and in evolution (e.g.~the -sum or product of ranges or variances; Wills 2001). - -Although size measures are suitable indicators of a group's trait space -occupancy, they are limited to comparing the range of trait-combinations -between groups. Size measures do not take into account the distribution -of the observations within a group and can often be insensitive to -unoccupied ``holes'' in the trait space (overstimating the size; Blonder -(2018)). They can make it difficult to determine whether all the -observations are on the edge of the group's distribution or whether the -size is simply driven by outliers. - -\paragraph{2. Density}\label{density} - -Density gives an indication of the quantity of observations in the trait -space. They can be interpreted as the distribution of the observations -\emph{within} a group in the trait space. Groups with higher density -contain more observations (i.e.~more observations per approximation of -size) that will tend to be more similar to each other. For example, if -group A is greater is size than group B and both have the same density -(observations are equally distant within each group), similar mechanisms -could be driving both groups' trait space occupancy. Indeed, this -pattern could suggest that A is older and has had more time to achieve -more extreme trait combinations under essentially the same process as -younger, smaller group B (Endler et al. 2005). Note that density based -measures can be sensitive to sampling. Density measures are less common -compared to size measures, but they are still used in both ecology -(e.g.~the minimum spanning tree length; Oksanen et al. 2007) and -evolution (e.g.~the average pairwise distance; Harmon et al. 2008). - -\paragraph{3. Position}\label{position} - -Position captures where a group lies in trait space. They can be -interpreted as where a group lies in the trait space either relative to -the space itself or relative to another group. For example, if group A -has a different position than group B, A will have a different -trait-combination than in B. - -Position measures may be harder to interpret in multidimensional spaces. -In a 2D space, two groups can be equally distant from a fixed point but -in different parts of the space (left, right, up, or down - with the -amount of parts of space increasing with dimensions). However, when -thinking about unidimensional data, this measure is obvious: two groups -A or B could have the same variance (size) with the same number of -observations (density) but could have two different means and thus be in -different positions. These measures are used in ecology to compare the -position of two groups relative to each other (Mammola 2019). - -\subsection{No measure to rule them all: benefits of considering -multiple -measures}\label{no-measure-to-rule-them-all-benefits-of-considering-multiple-measures} - -The use of multiple measurements to assess trait space occupancy has the -benefit of providing a more detailed characterisation of occupancy -changes. If the question is to look at how space occupancy changes in -response to mass extinction, using a single space occupancy measure can -miss part of the picture: a change in size could be decoupled from a -change in position or density in trait space. For example, the -Cretaceous-Paleogene extinction (66 million years ago) shows an increase -in size of the mammalian trait space (adaptive radiation; Halliday and -Goswami 2016) but more specific questions can be answered by looking at -other aspects of trait space occupancy: does the radiation expand on -previously existing morphologies (elaboration, increase in density; -Endler et al. 2005) or does it explore new regions of the trait space -(innovation, change in position; Endler et al. 2005)? Similarly, in -ecology, if two groups have the same trait space size, it can be -interesting to look at differences in density within these two groups: -different selection pressure can lead to different density within -equally sized groups. - -Here, we provide the first interdisciplinary review of 25 space -occupancy measures that uses the broad classification of measures into -size, density and position to capture pattern changes in trait space. We -assess the behaviour of measures using simulations and six -interdisciplinary empirical datasets covering a wide range of potential -data types and biological questions. We also introduce a tool for -measuring occupancy in multidimensional space -(\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}), which is -a user-friendly, open-source, graphical interface to allow the tailored -testing of measurement behaviour for any use case. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} will allow -workers to comprehensively assess the properties of their trait space -and the measures associated with their specific biological question. - -\section{Methods}\label{methods} - -We tested how 25 space occupancy measures relate to each other, are -affected by modifications of traits space and affect group comparisons -in empirical data: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - We simulated 13 different spaces with different sets of parameters; -\item - We transformed these spaces by removing 50\% of the observations - following four different scenarios corresponding to different - empirical scenarios: randomly, by limit (e.g.~expansion or reduction - of niches), by density (e.g.~different degrees of competition within a - guild) and by position (e.g.~ecological niche shift). -\item - We measured occupancy on the resulting transformed spaces using eight - different space occupancy measures; -\item - We applied the same space occupancy measures to six empirical datasets - (covering a range of disciplines and a range of dataset properties). -\end{enumerate} - -Note that the paper contains the results for only eight measures which -were selected as representative of common measures covering the size, -density and position trait space aspects. The results for an additional -17 measures is available in the supplementary material 4. - -\subsection{Generating spaces}\label{generating-spaces} - -We generated trait spaces using the following combinations of size, -distributions, variance and correlation: - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -space name\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.31\columnwidth}\raggedright\strut -distribution(s)\strut -\end{minipage} & \begin{minipage}[b]{0.21\columnwidth}\raggedright\strut -dimensions variance\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -correlation\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform (min = -0.5, max = 0.5)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal (mean = 0, sd = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D normal\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal correlated\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D with random distributions\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCA-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Multiplicative\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCO-like\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Additive\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 2: different simulated space distribution. \emph{Name} of the -simulated space; \emph{dimensions} of the matrix (row*columns); -\emph{distribution(s)} of the data on each dimensions (for the `Random', -dimensions are randomly chosen between Normal, Uniform or Lognormal); -\emph{dimension variance}: distribution of the variance between -dimensions (when equal, the dimensions have the same variance); -\emph{correlation} between dimensions. - -The differences in trait space sizes (200 elemeents for 3, 15, 50 or 150 -dimensions) reflects the range found in literature (e.g.~hopkins2017; -Mammola 2019). We used a range of distributions (uniform, normal or a -random combination of uniform, normal and lognormal) to test the effect -of observation distributions on the measurements. We used different -levels of variance for each dimensions in the spaces by making the -variance on each dimension either equal -(\(\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}\)) or decreasing -(\(\sigma_{D1} < \sigma_{D2} < \sigma_{Di}\)) with the decreasing factor -being either multiplicative (using the cumulative product of the inverse -of the number of dimensions: \(\prod_i^d(1/d)\)) or additive -(\(\sum_i^d(1/d)\)). Both reductions of variance are used to illustrate -the properties of ordinations where the variance decreases per -dimensions (and normal win Multidimensional Scaling - MDS, PCO or PCoA; -e.g. Close et al. 2015; lognormal in principal components analysis - -PCA; e.g. Marcy et al. 2016; Wright 2017; Healy et al. 2019). Finally, -we added a correlation parameter to illustrate the effect of -co-linearity between traits (especially in non-ordinated trait spaces). -We repeated the simulation of each trait space 20 times (resulting in -260 spaces). - -\subsection{Spatial occupancy -measures}\label{spatial-occupancy-measures} - -We then calculated eight different measures on the resulting transformed -spaces, including a new one, the average displacement, which we expect -to be influenced by changes in trait space position. - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Name\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Definition\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Captures\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Source\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Notes\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Laliberté and Legendre (2010)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional dispersion (FDis - without abundance)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\sigma^{2}{k_i}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -common measure used in palaeobiology (Ciampaglio et al. 2001; Wills -2001)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -more sensitive to outliers than the sum of variances\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Donohue et al. (2013)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -less sensitive to outliers than the convex hull hypervolume (Díaz et al. -2016; Blonder 2018)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum(\text{branch length})}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sedgewick (1990)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -similar to the unscaled functional evenness (Villéger et al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Villéger et al. (2008)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the functional evenness without weighted abundance (FEve; Villéger et -al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sqrt{\sum_{i}^{n}{min({q}_{i}-p_{i})^2}})\times \frac{1}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the density of pairs of observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -Average displacement\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Position\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -This paper\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -the ratio between the observations' position from their centroid and the -centre of the trait space (coordinates: 0, 0, 0, \ldots{}). A value of 1 -indicates that the observations' centroid is the centre of the trait -space\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 3: List of measures with \emph{n} being the number of -observations, \emph{d} the total number of dimensions, \emph{k} any -specific row in the matrix, \emph{Centroid} being their mean and -\(\sigma^{2}\) their variance. \(\Gamma\) is the Gamma distribution and -\(\lambda_{i}\) the eigenvalue of each dimension and \({q}_{i}\) and -\(p_{i}\) are any pairs of coordinates. - -We selected these eight space occupancy measures to illustrate how they -capture different aspects of space occupancy (not as an expression of -our preference). These measures are specific to Euclidean and isotropic -trait spaces (which is not necessary for all measures). The -supplementary material 4 contains the same analysis as described below, -performed on 17 measures. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -exploration into the effect of many more measures as well as the -customisation of measures by combining them or using user-designed -functions. - -\subsection{Measure comparisons}\label{measure-comparisons} - -We compared the space occupancy measures correlations across all -simulations between each pair of measures to assess their captured -signal (Villéger et al. 2008; Laliberté and Legendre 2010). We used the -measures on the full 13 trait spaces described above. We then scaled the -results and measured the pairwise Pearson correlation to test whether -measures were capturing a similar signals or not using the -\texttt{psych} package (Revelle 2018). - -\subsection{Changing space}\label{changing-spaces} - -To assess how the measures responded to changes within trait spaces, we -removed 50\% of observations each time using the following algorithms: - -\begin{itemize} -\item - \textbf{Randomly:} by randomly removing 50\% of observations (Fig. - 2-A). This reflects a ``null'' biological model of changes in trait - space: the case when observations are removed regardless of their - intrinsic characteristics. For example, if diversity is reduced by - 50\% but the space size remains the same, there is a decoupling - between diversity and space occupancy (Ruta et al. 2013). Our selected - measures are expected to not be affected by this change. -\item - \textbf{Limit:} by removing observations within a distance from the - centre of the trait space lower or greater than a radius \(\rho\) - (where \(\rho\) is chosen such that 50\% observations are selected) - generating two limit removals: \emph{maximum} and \emph{minimum} - (respectively in orange and blue; Fig. 2-B). This can reflect a strict - selection model where observations with trait values below or above a - threshold are removed leading to an expansion or a contraction of the - trait space. Size measures are expected to be most affected by this - change. -\item - \textbf{Density:} by removing any pairs of point with a distance \(D\) - from each other where (where \(D\) is chosen such that 50\% - observations are selected) generating two density removals: - \emph{high} and \emph{low} (respectively in orange and blue; Fig. - 2-C). This can reflect changes within groups in the trait space due to - ecological factors (e.g.~niche repulsion resulting in lower density; - Grant and Grant 2006). Density measures are expected to be most - affected by this change. -\item - \textbf{Position:} by removing points similarly as for \textbf{Limit} - but using the distance from the furthest point from the centre - generating two position removals: \emph{positive} and \emph{negative} - (respectively in orange and blue; Fig. 2-D). This can reflect global - changes in trait space (e.g.~if an entire group remaining diverse but - occupying a different niche). Position measures are expected to be - most affected by this change. -\end{itemize} - -The algorithm to select \(\rho\) or \(D\) is described in the -Supplementary material 1. - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_reduce_space-1.pdf} -\caption{different type of space reduction. Each panel displays two -groups of 50\% of the data points each. Each group (orange and blue) are -generated using the following algorithm: A - randomly; B - by limit -(maximum and minimum limit); C - by density (high and low); and D - by -position (positive and negative). Panel E et F represents two typical -display of the reduction results displayed in Table 5: the dots -represent the median space occupancy values across all simulations for -each scenario of trait space change (Table 2), the solid and dashed line -respectively the 50\% and 95\% confidence intervals. Results in grey are -the random 50\% reduction (panel A). Results in blue and orange -represent the opposite scenarios from panels B, C, and D. The displayed -value is the amount of overlap (Bhattacharrya Coefficient) between the -blue or orange distributions and the grey one. Panel E and F shows -respectively the ``ideal'' and ``worst'' results for any type of -measures, where the space occupancy measurement respectively manages or -fails to captures a specific type of reduction (i.e.~size, position or -density; Table 5).} -\end{figure} - -Because occupancy measures are dependent on the space, we scaled and -centred them between -1 and 1 to make them comparable (by subtracting -the observed occupancy without reduction to all the measures of the -reduced spaces and then divided it by the maximum observed occupancy). A -value of 0 indicates no effect of the space reduction and \(>0\) and -\(<0\) respectively indicates an increase or decrease in the measure -value. We then measured the amount of overlap between the non-random -removals (limit, density and position) and the random removals using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\subsubsection{Measuring the effect of space and -dimensionality}\label{measuring-the-effect-of-space-and-dimensionality} - -Distribution differences and the number of dimensions can have an effect -on the measure results. For example, in a normally distributed space, an -increase in density can often lead to a decrease in size (though this is -not necessarily true if the space is log-normal or uniform). High -dimensional spaces (\textgreater{}10) are subject to the ``curse of -multidimensionality'' (Bellman 1957): data becomes sparser with -increasing number of dimensions. This can have two main consequences: 1) -the probability of overlap between two groups decreases as a product of -the number of dimensions; and 2) the amount of samples needed to -``fill'' the spaces increases exponentially -\href{https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners}{see -this interactive illustration by Toph Tucker}. The ``curse'' can make -the interpretation of high dimensional data counter-intuitive. For -example if a group expands in multiple dimensions (i.e.~increase in -size), the actual hypervolume (\(\prod_{i}^{d} range_{Di}\)) can -decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an -ANOVA (\(occupancy \sim distribution\) and -\(occupancy \sim dimensions\)) by using all spaces with 50 dimensions -and the uniform and normal spaces with equal variance and no correlation -with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing -respectively the effect of distribution and dimensions. The results of -the ANOVAs (F and \emph{p}-values) are reported in Table 5 (full results -in supplementary material 3). - -\subsection{Empirical examples}\label{empirical-examples} - -We analysed the effect of the different space occupancy measures on six -different empirical studies covering a range of fields that employ trait -space analyses. For each of these studies we generated trait spaces from -the data published with the papers. We divided each trait spaces into -two biologically-relevant groups and tested whether the measures -differentiated the groups in different ways. Both the grouping and the -questions were based on a simplified version of the topics of these -papers (with no intention to re-analyse the data and questions). The -procedures to generate the data and the groups varies between studies -and is detailed in the supplementary materials 2. - -\begin{longtable}[]{@{}llllllll@{}} -\toprule -\begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -study\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -field\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -taxonomic group\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -traits\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -trait space\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -groups\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -question\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Beck and Lee (2014)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Mammalia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -106*105\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -52 crown vs.~54 stem\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are crown mammals more disparate than stem mammals?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Wright (2017)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Crinoidea\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -42*41\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -16 before vs.~23 after\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Is there a difference in disparity before and after the Ordovician mass -extinction?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Marcy et al. (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Rodentia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -skull 2D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -454*134\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -225 \emph{Megascapheus} vs.~229 \emph{Thomomys}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are two genera of gopher morphologically distinct?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Hopkins and Pearson (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Trilobita\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -3D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -46*46\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -36 adults vs.~10 juveniles\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Are juvenile trilobites a subset of adult ones?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Jones et al. (2015)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Plantae\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Communities species compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Jaccard distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -48*47\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -24 aspens vs.~24 grasslands\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Is there a difference in species composition between aspens and -grasslands?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Healy et al. (2019)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Animalia\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Life history traits\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of continuous traits (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -285*6\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -83 ecthotherms vs.~202 endotherms\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -Do endotherms have more diversified life history strategies than -ectotherms?\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 4: details of the six empirical trait spaces. - -For each empirical trait space we bootstrapped each group 500 times -(Guillerme 2018) and applied the eight space occupancy measure to each -pairs of groups. We then compared the means of each groups using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\section{Results}\label{results} - -\subsection{Measure comparisons}\label{measure-comparisons-1} - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measure_correlation-1.pdf} -\caption{pairwise correlation between the scaled measures. Numbers on -the upper right corner are the Pearson correlations. The red line are -linear regressions (with the confidence intervals in grey). Av.: -average; dist.: distance; min.: minimum; span.: spanning.} -\end{figure} - -All the measures were either positively correlated (Pearson correlation -of 0.99 for the average Euclidean distance from centroid and sum of -variance or 0.97 for the average nearest neighbour distance and minimum -spanning tree average length; Fig. 3) or somewhat correlated (ranging -from 0.66 for the sum of variances and the ellipsoid volume to -0.09 -between the average displacement and the average Euclidean distance from -centroid; Fig. 3). All measures but the ellipsoid volume were normally -(or nearly normally) distributed (Fig. 3). - -\subsection{Space shifting}\label{space-shifting} - -\begin{longtable}[]{@{}llllll@{}} -\toprule -\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Size change\strut -\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut -Arrangement change\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Position change\strut -\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Distribution effect\strut -\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut -Dimensions effect\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 0.924 ; p = 0.449\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.322 ; p = 0.958\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.285 ; p = 0.274\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.478 ; p = 0.873\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 11.119 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 32.307 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 7.215 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 13.486 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.162 ; p = 0.326\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.998 ; p = 0.435\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 8.152 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 29.358 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.478 ; p = 0.207\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 0.773 ; p = 0.626\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 10.742 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -F = 26.829 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 5: Results of the effect of space reduction, space dimension -distributions and dimensions number of the different space occupancy -measures. See Fig. 2 for interpretation of the figures distributions and -values. F-values for distribution effect and dimensions effect -represents respectively the effect of the ANOVAs space occupancy -\textasciitilde{} distributions and space occupancy \textasciitilde{} -dimension represent the ratio of sum squared difference within and -between groups (the higher, the more the factor has an effect on the -measure) and associated \emph{p}-values (0 `***' 0.001 `**' 0.01 `*' -0.05 `.' 0.1 '' 1). This figure illustrates how different measures can -be influenced by different aspects of changes in the trait space. E.g. -the Average Euclidean distance from centroid (row 1) captures mainly -changes in size (column 1), but also captures changes in density (column -2) but does not capture changes in position (column 3). - -As expected, some different measures capture different aspects of space -occupancy. However, it can be hard to predict the behaviour of each -measure when 50\% of the observations are removed. We observe a clear -decrease in median metric in less than a third of the space reductions -(10/36). - -In terms of change in size, only the average Euclidean distance from -centroid and the sum of variances seem to capture a clear change in both -directions. In terms of change in density, only the minimum spanning -tree average distance and the average nearest neighbour distance seem to -capture a clear change in both directions. And in terms of change in -position, only the average displacement metric seems to capture a clear -change in direction (albeit not in both directions). This is not -surprising, since the notion of positions becomes more and more complex -to appreciate as dimensionality increases (i.e.~beyond left/right, -up/down and front/back). - -\subsection{Empirical example}\label{empirical-example} - -\begin{longtable}[]{@{}lllllll@{}} -\toprule -\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Beck and Lee 2014\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -Wright 2017\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Marcy et al. 2016\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Hopkins and Pearson 2016\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Jones et al. 2015\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Healy et al. 2019\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Comparisons (orange \emph{vs.} blue)\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -crown \emph{vs.} stem mammals morphologies\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -crinoids morphologies before \emph{vs.} after the end-Ordovician -extinction\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\emph{Megascapheus} \emph{vs.} \emph{Thomomys} skull shapes\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -adults \emph{vs.} juveniles trilobites cephalon shapes\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -aspens \emph{vs.} grasslands communities compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -ecthotherms \emph{vs.} endotherms life history traits\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-25.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-33.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-41.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-26.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-34.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-42.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-27.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-35.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-43.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-28.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-36.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-44.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-29.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-37.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-45.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-30.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-38.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-46.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-31.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-39.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-47.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-32.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-40.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-48.pdf}\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 6: Comparisons of pairs of groups in different empirical trait -spaces. NAs are used for cases where space occupancy could not be -measured due to the curse of multidimensionality. The displayed values -are the amount of overlap between both groups (Bhattacharrya -Coefficient). - -Similarly as for the simulated results, the empirical ones indicate that -there can be no perfect one-size-fit all measurement. For all eight -measures (except the ellipsoid volume) we see either one group or the -other having a bigger mean than the other and no consistent case where a -group has a bigger mean than the other for all the measures. For -example, in the Beck and Lee (2014)'s dataset, there is a clear -difference in size using the average Euclidean distance from centroid or -the sum of variances (overlaps of respectively 0.175 and 0.159) but no -overlap when measuring the size using the sum of ranges (0.966). -However, for the Hopkins and Pearson (2016)'s dataset, this pattern is -reversed (no clear differences for the average Euclidean distance from -centroid or the sum of variances - 0.701 and 0.865 respectively - but a -clear difference for the sum of ranges (0). For each dataset, the -absolute differences between each groups is not consistent depending on -the measures. For example, in Hopkins and Pearson (2016)'s dataset, the -orange group's mean is clearly higher than the blue one when measuring -the sum of ranges (0) and the inverse is true when measuring the average -displacement (0). - -\section{Discussion}\label{discussion} - -Here we tested 25 measures of trait space occupancy on simulated and -empirical datasets to assess how each measure captures changes in trait -space size, density and position. Our results show that the correlation -between measures can vary both within and between measure categories -(Fig. 3), highlighting the importance of understanding the measure -classification for the interpretation of results. Our simulations show -that different measures capture different types of trait space change -(Table 5), meaning that the use of multiple measures is important for -comprehensive interpretation of trait space occupancy. We also show that -the choice of measure impacts the interpretation of group differences in -empirical datasets (Table 6). - -\paragraph{Measures comparisons}\label{measures-comparisons} - -Measures within the same category of trait space occupancy (size, -density or position) do not have the same level of correlation with each -other. For example, the average Euclidean distance from centroid (size) -is highly correlated to the sum of variances (size - correlation of -0.99) and somewhat correlated with the minimum spanning tree average -distance (density - correlation of 0.66) but poorly with the ellipsoid -volume (size - correlation of 0.17) and the minimum spanning tree -distances evenness (density - correlation of -0.05). Furthermore, the -fact that we have such a range of correlations for normal distributions -suggests that each measure can capture different summaries of space -occupancy ranging from obvious differences (for measures not strongly -correlated) to subtle ones (for measures strongly correlated). - -\paragraph{Space shifting}\label{space-shifting-1} - -Most measures capture no changes in space occupancy for the ``null'' -(random) space reduction (in grey in Table 5). This is a desirable -behaviour for space occupancy measures since it will likely avoid false -positive errors in studies that estimate biological processes from space -occupancy patterns (e.g.~convergence Marcy et al. 2016, life history -traits Healy et al. (2019)). However, the average nearest neighbour -distance and the sum of ranges have a respectively positive and negative -``null'' median. This is not especially a bad property but it should be -kept in mind that even random processes can increase or decrease these -measures' values. - -For changes in size, the sum of variances and the average Euclidean -distance from centroid are good descriptors (Table 5). However, as -illustrated in the 2D examples in Fig. 2-B only the blue change results -(Table 5) should not result in a direct change in overall size because -the trait space is merely ``hollowed'' out. That said, ``hollowing'' is -harder to conceptualise in many dimensions and the measures can still be -interpreted for comparing groups (orange has a smaller volume than -blue). - -The average nearest neigbhour distance and the minimum spanning tree -average distance consistently detect changes in density with more -precision for low density trait spaces (in blue in Table 5). However, we -can observe some degree of correlation between the changes in density -and the changes in size for most measure picking either signal. This -could be due to the use of normally distributed spaces where a change in -density often leads to a change in size. This is not necessarily the -case with empirical data. - -Regarding the changes in position, only the average displacement measure -seems able to distinguish between a random change and a displacement of -the trait space (Table 5). However, the average displacement measure -does not distinguish between positive or negative displacement: this -might be due to the inherent complexity of \emph{position} in a -multidimensional trait space. - -\paragraph{Empirical examples}\label{empirical-examples-1} - -Although most differences are fairly consistent within each dataset with -one group having a higher space occupancy score than the other for -multiple measures, this difference can be more or less pronounced within -each dataset (ranging from no to nearly full overlap - BC -\(\in(0;0.995)\)) and sometimes even reversed. This indicates that -opposite conclusions can be drawn from a dataset depending on which -space occupancy measure is considered. For example, in Wright (2017), -crinoids after the Ordovician mass extinction have a higher median -measure value for all measures but for the average displacement. - -These differences depending on the measures are also more pronounced in -the empirical datasets where the observations per group are unequal -(Hopkins and Pearson 2016; Healy et al. 2019). - -\subsubsection{Caveats}\label{caveats} - -While our simulations are useful to illustrate the behaviour of diverse -space occupancy measures, they have several caveats. First, the -simulated observations in the trait spaces are independent. This is not -the case in biology where observations can be spatially (Jones et al. -2015) or phylogenetically correlated (e.g. Beck and Lee 2014). Second, -the algorithm used to reduce the trait spacesmight not always accurately -reflect changes. This might favour some specific measures over others, -in particular for the changes in density that modify the nearest -neighbour density rather than changing the global density. This -algorithmic choice was made in order to not confound changes in density -along with changes in size. However, the results presented here probably -capture the general behaviour of each measure since results are -consistent between the simulated and empirical analysis. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -workers to test the caveats mentioned above by uploading empirical trait -spaces. - -\subsubsection{Conclusions}\label{conclusions} - -We insist that no measure is better than the next one and that workers -should identify the most appropriate measures based on their trait space -properties as well as their specific biological question. However, -following the findings of this study we make several suggestions: - -First, we suggest using multiple measures to tackle different aspects of -the trait space. This follows the same logical thinking that the mean -might not be sufficient to describe a distribution (e.g.~the variance -might be a good additional descriptor). Although using multiple measures -is not uncommon in macroevolutionary studies (e.g. Halliday and Goswami -2016) or in ecology (Mammola 2019), they often do no cover more than one -of the three categories of trait space measures. - -Second, we suggest selecting the measures that best address the -biological question at hand. If one studies an adaptive radiation in a -group of organisms, it is worth thinking what would be the expected null -model: would the group's size increase (radiation in all directions), -would it increase in density (niche specialisation) or would it shift in -position (radiation into a new set of niches)? - -Third, we suggest not naming measures after the biological aspect they -describe which can be vague (e.g. ``disparity'' or ``functional -dispersion'') but rather after what they are measuring and why (e.g. -``we used sum of ranges to measure the space size''). We believe this -will support both a clearer understanding of what \emph{is} measured as -well as better communication between ecology and evolution research -where measures can be similar but have different names. - -Multidimensional analyses have been acknowledged as essential tools in -modern biology but they can often be counter-intuitive (Bellman 1957). -It is thus crucial to accurately describe patterns in multidimensional -trait spaces to be able to link them to biological processes. When -summarising trait spaces, it is important to remember that a pattern -captured by a specific space occupancy measure is often dependent on the -properties of the space and of the particular biological question of -interest. We believe that having a clearer understanding of both the -properties of the trait space and the associated space occupancy -measures (e.g.~using -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}) as well as -using novel space occupancy measures to answer specific questions will -be of great use to study biological processes in a multidimensional -world. - -\section{Acknowledgements}\label{acknowledgements} - -We thank Natalie Jones and Kevin Healy for helping with the empirical -datasets and two anonymous reviewer for their comments. We acknowledge -funding from the Australian Research Council DP170103227 and FT180100634 -awarded to VW. - -\section{Authors contributions}\label{authors-contributions} - -TG, MNP, AEM and VW designed the project. TG and AEM collected the -empirical dataset. TG ran the analyses and designed the software. TG, -MNP, AEM and VW wrote the manuscript. - -\section{Data Availability, repeatability and -reproducibility}\label{data-availability-repeatability-and-reproducibility} - -The raw empirical data is available from the original papers (Beck and -Lee 2014; Jones et al. 2015, Marcy et al. (2016); Hopkins and Pearson -2016; Wright 2017; Healy et al. 2019). The subsets of the empirical data -used in this analysis are available on figshare -\href{https://doi.org/10.6084/m9.figshare.9943181.v1}{DOI: -10.6084/m9.figshare.9943181.v1}. The modified empirical data are -available in the package accompanying this manuscript -(\texttt{data(moms::demo\_data)}). This manuscript (including the -figures, tables and supplementary material) is repeatable and -reproducible by compiling the vignette of the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. - -\section*{References}\label{references} -\addcontentsline{toc}{section}{References} - -\hypertarget{refs}{} -\hypertarget{ref-beck2014}{} -Beck R.M.D., Lee M.S.Y. 2014. Ancient dates or accelerated rates? -Morphological clocks and the antiquity of placental mammals. Proceedings -of the Royal Society B: Biological Sciences. 281:20141278. - -\hypertarget{ref-cursedimensionality}{} -Bellman R.E. 1957. Dynamic programming. Princeton University Press. - -\hypertarget{ref-bhattacharyya1943}{} -Bhattacharyya A. 1943. On a measure of divergence between two -statistical populations defined by their probability distributions. -Bulletin of the Calcutta Mathematical Society. 35:99--109. - -\hypertarget{ref-blonder2018}{} -Blonder B. 2018. Hypervolume concepts in niche-and trait-based ecology. -Ecography. 41:1441--1455. - -\hypertarget{ref-momocs}{} -Bonhomme V., Picq S., Gaucherel C., Claude J. 2014. Momocs: Outline -analysis using R. Journal of Statistical Software. 56:1--24. - -\hypertarget{ref-ciampaglio2001}{} -Ciampaglio C.N., Kemp M., McShea D.W. 2001. Detecting changes in -morphospace occupation patterns in the fossil record: Characterization -and analysis of measures of disparity. Paleobiology. 71:695--715. - -\hypertarget{ref-close2015}{} -Close R.A., Friedman M., Lloyd G.T., Benson R.B. 2015. Evidence for a -mid-Jurassic adaptive radiation in mammals. Current Biology. - -\hypertarget{ref-diaz2016}{} -Díaz S., Kattge J., Cornelissen J.H., Wright I.J., Lavorel S., Dray S., -Reu B., Kleyer M., Wirth C., Prentice I.C., others. 2016. The global -spectrum of plant form and function. Nature. 529:167. - -\hypertarget{ref-donohue2013}{} -Donohue I., Petchey O.L., Montoya J.M., Jackson A.L., McNally L., Viana -M., Healy K., Lurgi M., O'Connor N.E., Emmerson M.C. 2013. On the -dimensionality of ecological stability. Ecology Letters. 16:421--429. - -\hypertarget{ref-endler2005}{} -Endler J.A., Westcott D.A., Madden J.R., Robson T. 2005. Animal visual -systems and the evolution of color patterns: Sensory processing -illuminates signal evolution. Evolution. 59:1795--1818. - -\hypertarget{ref-foote1992}{} -Foote M. 1992. Rarefaction analysis of morphological and taxonomic -diversity. Paleobiology. 18:1--16. - -\hypertarget{ref-grant2006}{} -Grant P.R., Grant B.R. 2006. Evolution of character displacement in -darwins finches. Science. 313:224--226. - -\hypertarget{ref-disprity}{} -Guillerme T. 2018. dispRity: A modular R package for measuring -disparity. Methods in Ecology and Evolution. 9:1755--1763. - -\hypertarget{ref-halliday2015}{} -Halliday T.J.D., Goswami A. 2016. Eutherian morphological disparity -across the end-cretaceous mass extinction. Biological Journal of the -Linnean Society. 118:152--168. - -\hypertarget{ref-geiger2008}{} -Harmon L.J., Weir J.T., Brock C.D., Glor R.E., Challenger W. 2008. -GEIGER: Investigating evolutionary radiations. Bioinformatics. -24:129--131. - -\hypertarget{ref-healy2019}{} -Healy K., Ezard T.H.G., Jones O.R., Salguero-G'omez R., Buckley Y.M. -2019. Animal life history is shaped by the pace of life and the -distribution of age-specific mortality and reproduction. Nature Ecology -\& Evolution. 2397-334X. - -\hypertarget{ref-hopkins2016}{} -Hopkins M., Pearson K. 2016. Non-linear ontogenetic shape change in -cryptolithus tesselatus (trilobita) using three-dimensional geometric -morphometrics. Palaeontologia Electronica. 19:1--54. - -\hypertarget{ref-hopkins2017}{} -Hopkins M.J., Gerber S. 2017. Morphological disparity. In: Nuno de la -Rosa L., Müller G., editors. Evolutionary developmental biology: A -reference guide. Cham: Springer International Publishing. p. 1--12. - -\hypertarget{ref-jones2015}{} -Jones N.T., Germain R.M., Grainger T.N., Hall A.M., Baldwin L., Gilbert -B. 2015. Dispersal mode mediates the effect of patch size and patch -connectivity on metacommunity diversity. Journal of Ecology. -103:935--944. - -\hypertarget{ref-lalibertuxe92010}{} -Laliberté É., Legendre P. 2010. A distance-based framework for measuring -functional diversity from multiple traits. Ecology. 91:299--305. - -\hypertarget{ref-legendre2012}{} -Legendre P., Legendre L.F. 2012. Numerical ecology. Elsevier. - -\hypertarget{ref-mammola2019}{} -Mammola S. 2019. Assessing similarity of n-dimensional hypervolumes: -Which metric to use? Journal of Biogeography. 0. - -\hypertarget{ref-marcy2016}{} -Marcy A.E., Hadly E.A., Sherratt E., Garland K., Weisbecker V. 2016. -Getting a head in hard soils: Convergent skull evolution and divergent -allometric patterns explain shape variation in a highly diverse genus of -pocket gophers (thomomys). BMC evolutionary biology. 16:207. - -\hypertarget{ref-oksanen2007vegan}{} -Oksanen J., Kindt R., Legendre P., O'Hara B., Stevens M.H.H., Oksanen -M.J., Suggests M. 2007. The vegan package. Community ecology package. -10:631--637. - -\hypertarget{ref-psych}{} -Revelle W. 2018. Psych: Procedures for psychological, psychometric, and -personality research. Evanston, Illinois: Northwestern University. - -\hypertarget{ref-ruta2013}{} -Ruta M., Angielczyk K.D., Fröbisch J., Benton M.J. 2013. Decoupling of -morphological disparity and taxic diversity during the adaptive -radiation of anomodont therapsids. Proceedings of the Royal Society of -London B: Biological Sciences. 280. - -\hypertarget{ref-sedgewick1990}{} -Sedgewick R. 1990. Algorithms in c. Addison-Wesley, Reading. - -\hypertarget{ref-villuxe9ger2008}{} -Villéger S., Mason N.W.H., Mouillot D. 2008. New multidimensional -functional diversity indices for a multifaceted framework in functional -ecology. Ecology. 89:2290--2301. - -\hypertarget{ref-wills2001}{} -Wills M.A. 2001. Morphological disparity: A primer. In: Adrain J.M., -Edgecombe G.D., Lieberman B.S., editors. Fossils, phylogeny, and form. -Springer US. p. 55--144. - -\hypertarget{ref-wright2017}{} -Wright D.F. 2017. Phenotypic innovation and adaptive constraints in the -evolutionary radiation of palaeozoic crinoids. Scientific Reports. -7:13745. - - -\end{document} diff --git a/inst/shiftingspace_bkp.Rmd b/inst/shiftingspace_bkp.Rmd deleted file mode 100644 index 4b15b5d..0000000 --- a/inst/shiftingspace_bkp.Rmd +++ /dev/null @@ -1,975 +0,0 @@ ---- -title: "Shifting spaces: which disparity or dissimilarity metrics best summarise occupancy in multidimensional spaces?" -author: "Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker" -bibliography: references.bib -date: "`r Sys.Date()`" -output: - html_document: - fig_width: 8 - fig_height: 8 ---- - - - - - - - - - - -```{r header, echo = FALSE, results = 'hide', message = FALSE, warning = FALSE} -## Repeatability note: -## This whole paper is entirely reproducible and compiles as a single document. The data, figures -## and tables are all generated through the code snippets in this file. Note that the first time -## that this paper is compiled, it will generate the data which will take substantial time -## as the shiftingspace_supplementary.Rmd file completion takes ~10 minutes. Subsequent compilations -## will be much faster! - -## Loading the packages -if(!require(devtools)) install.packages("devtools") -if(!require(knitr)) install.packages("knitr"); library(knitr) -if(!require(rmarkdown)) install.packages("rmarkdown"); library(rmarkdown) -if(!require(ape)) install.packages("ape"); library(ape) -if(!require(dispRity)) install.packages("dispRity"); library(dispRity) -if(packageVersion("dispRity") < "1.2.4") { - ## dispRity must be above v1.2.3 - devtools::install_github("TGuillerme/dispRity"); library(dispRity) -} -if(!require(moms)) devtools::install_github("TGuillerme/moms"); library(moms) - -## Setting the datapath -DATA_PATH <- "../data/processed/" - -## Create data directory -if(!dir.exists(DATA_PATH)) { - dir.create(path = DATA_PATH, showWarnings = FALSE) -} - -## Setting the default parameters for the space plots -defaults <- list(pch = 20, - xlim = c(-3, 3), - ylim = c(-3, 3), - col1 = "grey", - col2 = "black", - xlab = "Trait", - ylab = "Trait", - cex = 1) -## Generating the default palette -default.palette <- function(n) { - hues <- seq(15, 375, length = n + 1) - grDevices::hcl(h <- hues, l = 65, c = 100)[1:n] -} - -## Checking whether the data exists -if(!all(c("remove_05.Rda") %in% list.files(DATA_PATH))) { - ## Run The supplementary material (simulation) ~ 30 minutes - rmarkdown::render("shiftingspace_supplementary_simulation.Rmd", "html_document") -} -if(!all(c("empirical_results.Rda") %in% list.files(DATA_PATH))) { - ## Run The supplementary material ~ 15 minutes - rmarkdown::render("shiftingspace_supplementary_empirical.Rmd", "html_document") -} -``` - -```{r table_counter, echo = FALSE} -## Table and figure counter -element.counter <- function(name, all_names, increment = FALSE) { - if(increment) { - ## Numbering a table or a figure - all_names <- c(all_names[[2]], name) - ## Print the table number - # length(all_names) - return(list(print = length(all_names), list = all_names)) - } else { - ## Printing the number of the table of figure - # cat(which(all_names %in% name)) - return(list(print = which(all_names[[2]] %in% name), list = all_names[[2]])) - } -} -## Initialising the elements counters -tables <- list(print = NULL, list = character()) -figures <- list(print = NULL, list = character()) - -## Function wrappers -table <- function(name, add = FALSE, all_names = tables) { - return(element.counter(name, all_names, increment = add)) -} -figure <- function(name, add = FALSE, all_names = figures) { - return(element.counter(name, all_names, increment = add)) -} - -``` - - -```{r compilation_html, echo = FALSE, eval = TRUE} -## Changing defaults -body(plot.id)[[2]] <- substitute(type <- ".png") -``` - -# Abstract - - - - - - - - -Multidimensional analysis of traits are now a common toolkit in ecology and evolution. -Such analyses are based on multidimensional datasets in which each dimension represents a trait that summarise all observed and potential trait combinations in a multidimensional trait-space (e.g. a morphospace or an ecospace). -Observations of interest will typically only occupy a subset of this space, so researchers apply one or more metrics to quantify the way in which organisms "inhabit" that trait-space. -Researchers use these metrics to investigate how space occupancy changes through time, in relation to other groups of organisms, and in response to global environmental changes, such as global warming events or mass extinctions. - -In macroevolutionary and ecological studies these metrics are often referred to as disparity or dissimilarity metrics, respectively, and can be generalised as "space occupancy metrics", a term that is applicable to both disciplines. -The mathematical and biological meaning of most space occupancy metrics is vague. -The majority of widely-used space-occupancy metrics in ecology and evolution lack formal description; this problem is acutely true in studies of macroevolution disparity. -This has consequences when interpreting changes in space occupancy and linking them to biological phenomena. -For example, does a reduction in volume in the trait-space correspond to episodes of extinction or changes in traits distribution? -Does an overlap in trait-space correspond to an overlap in ecological niches? - -Here we propose a broad classification of space occupancy metrics into three categories that capture changes in volume, density, or position of the trait-space. -We analyse the behaviour of eight space occupancy metrics including two novel ones to study changes in density and position on a series of simulated and empirical datasets. -We find no one metric describes all of trait space but that some metrics are better at capturing certain aspects compared to other approaches. -As no one metric is sufficient to describe the changes in trait space it is necessary for researchers to compare many metrics. This can be difficult as different softwares to do so will require different input data types and may require specialist skills. -Here we introduced [`moms`](@@@), a user-friendly tool based on a graphical interface that allows users to both visualise and measure changes space occupancy for any metric in simulated or imported trait-spaces. -Users are also provided with tools to transform their data in space (e.g. contraction, displacement, etc.). -This tool is designed to help researchers choose the right space occupancy metrics, given the properties of their trait-space and their biological question. - - -# Introduction - -Groups of species and environments share specific, easily recognisable, correlated characteristics of many kinds: guilds or biomes with shared phenotypic, physiological, phylogenetic or behavioural traits. -Organisms or environments should therefore be studied as a set of traits rather than some specific traits in isolation [e.g. @donohue2013; @hopkins2017]. -Increased computational power and statistical methods, as well as their implementation in `R` packages has facilitated a recent explosion of multiple traits studies approaches [@oksanen2007vegan; @adams2013geomorph; @momocs; @blonder2018; @disprity]. -Ordination techniques (or dimensionality reduction) are a popular way to analyse sets of multiple traits and create a multidimensional [see @legendre2012 for a summary]. -These trait-space approaches have been successfully used across many sub-disciplines in biology to either explore properties of the data or test hypotheses. -For example, in palaeobiology, workers use trait-spaces to study how groups of species' characteristics change through time [e.g. @wright2017]; in ecology, researchers study evidence of processes such as competition by the comparison of two populations to show whether they overlap in traits [e.g. @jones2015]. -However, different fields use a different set of terms and approaches for such trait-space approaches (Table 1). -In macroevolution, for example, such trait-spaces are commonly referred to as "morphospaces" with the whole procedure often being often called "disparity analysis" [e.g. @adams2013geomorph; @marcy2016; @wright2017], while in ecology, such trait-spaces can be called "functional spaces" and or "dissimilarity analysis" [e.g. @diaz2016; @mammola2019]. -Nonetheless, they are the same mathematical objects: matrices with columns representing an original or transformed trait value and rows representing observations, such as taxon, field site, etc. [@disprity]. - -In mathematics | In ecology | In macroevolution | In this paper ----------------|------------|-------------------|--------------- -Matrix ($n \times d$) | Function-space, Eco-space, etc. | Morphospace, traitspace, etc. | trait-space -Rows (*n*) | Taxa, field sites, environments, etc. | Taxa, specimen, populations, etc. | observations -Columns (*d*) | Traits, Ordination scores, distances, etc. | Traits, Ordination scores, distances, etc. | dimensions -Matrix subset ($m \times d$; $m \leq n$) | Treatments, phylogenetic group (clade), etc. | Clades, geological stratum, etc. | group -Statistic | Dissimilarity index or metric, hypervolume, functional diversity | Disparity metric or index | space occupancy metric -Multidimensional analysis | Dissimilarity analysis, trait analysis, etc. | Disparity analysis, disparity-through-time, etc. | multidimensional analysis -Table 1@@@: terms and equivalence between mathematics, ecology and macroevolution. - -Ecologists and evolutionary biologists are also using trait-spaces for similar reasons. -Across many sub-disciplines they will look at how their observations occupy the trait-space, often with respect to the same fundamental questions: are two groups overlapping in the trait-space? -Are some regions of the trait-space not occupied? -Or how do specific factors influence the occupancy of the trait-space? -Studying the occupancy of trait-spaces is achieved in various ways such as using disparity metrics in macroevolution [@wills2001; @hopkins2017; @disprity] or comparing hypervolumes in ecology [@donohue2013; @diaz2016; @blonder2018; @mammola2019]. -Although these space occupancy metrics are common in ecology and evolution, surprisingly little work has been published on their behaviour [but see @ciampaglio2001; @villéger2008; @mammola2019]. - -Different space occupancy metrics capture different aspects of the trait-space [ciampaglio2001; @villéger2008; @mammola2019]. -It may be widely-known by many in the field that distinct occupancy metrics elucidate different signals of the underlying space, but to our knowledge this is infrequently mentioned in peer-reviewed papers. -First, space occupancy metrics are often named as the biological aspect they are describing (e.g. "disparity" or "functional dispersion") rather than what they are measuring (e.g. the sum of dimensions variance, etc.) which obscures the differences or similarities in space occupancy metrics between studies in both ecology and evolution. -Second, in many studies in ecology and evolution, authors have focused on measuring the volume of the trait-space with different metrics [e.g. ellipsoid volume @donohue2013; hypervolume @diaz2016; product of variance @wright2017; Procrustes variance @marcy2016]. -However, volume only represents a single aspects of multidimensional occupancy that disregards other aspects, such as the density [@geiger2008], the position [@wills2001; @ciampaglio2001], and the dissimilarity between groups [@oksanen2007vegan; @close2015]. -For example, any analysis of volume occupancy through trait space may remain constant, and this will lead to supporting a certain biological conclusion. -Yet, an alternative aspect of multidimensional occupancy may indicate the position is changing while the volume is constant; this would likely lead to a different biological conclusion. -Measuring the volume would miss position information that might be of interest to biologists (e.g. a group of species maintains the same diversity of traits while also shifting towards a new ecological niche). -Using metrics that only measure one aspect of the multidimensional trait space may restrain the potential of multidimensional analysis [@villéger2008]. - -Here we propose a broad classification of space occupancy metrics as used across ecology [@villéger2008] and evolution [@wills2001] and analyse their power to detect changes in multidimensional space occupancy in empirical and simulated data. -We provide a review of each broad type of space occupancy metrics and propose a unified terminology aiming to foster communication between ecology and evolution. -Unsurprisingly, we found no one metric describes all changes through a trait-space and the results from each metric are dependent on the characteristics of the space and the hypotheses. -Furtheremore, because there can potentially be as many metrics as there are trait-spaces and studies, it would be a daunting task to propose clear generalities to space occupancy metrics behavior. -Therefore, we propose [`moms`](@@@) user-friendly tool allowing researchers to design, experiment and visual their own space occupancy metric tailored for their specific project and helping them understanding the "null" behavior of their metrics of interest. - - - -## Space occupancy metrics - -In this paper, we define trait-spaces as any matrix where rows are observations and columns are traits. -In practice, these traits can widely vary in number and types: they could be coded as discrete [e.g. presence of absence of a bone; @beck2014; @wright2017], continuous measurements [e.g. leaf area; @diaz2016] or more sophisticated measures of 2- or 3-dimensions [e.g. landmark position; @marcy2016]. -Traits can also be measured by using relative observations such as community composition [e.g. @jones2015] or even some distance between observations [e.g. @close2015]. -However, regardless of the methodology used to build a trait-space, three broad occupancy metrics can be measured to investigate how observations are distributed in the trait-space: the volume which will approximate the amount of space occupied, the density which will approximate the distribution in space and the position which will approximate the location in space [Fig. 1@@@; @villéger2008]. -Of course any combination of these three aspects is always possible. - -```{r fig_metrics_types, echo = FALSE, fig.height = 3, fig.width = 9, results = 'hide', fig.cap = paste("Figure 1: different type of information captured by space occupancy metrics. A - Volume (e.g. sum of ranges); B - Density (e.g. average squared pairwise distances); C - Position (e.g. median distance from centroid).") } - -set.seed(11) -## Plot space function (utility shortcut) -## The elements -elements <- 10 - -## Trait space -trait_space <- space.maker(elements, 2, distribution = rnorm) - -## Graphical parameters -op <- par(mfrow = c(1,3), bty = "n") - -## The volume -plot(trait_space, pch = defaults$pch, main = "A - Volume", xlab = "Trait 1", ylab = "Trait 2") -## The range lines -lines(x = range(trait_space[,1]), y = rep(mean(trait_space[,2]), 2), col = defaults$col1) -lines(x = rep(mean(trait_space[,1]), 2), y = range(trait_space[,2]), col = defaults$col1) - -## The density -plot(trait_space, pch = defaults$pch, main = "B - Density", xlab = "Trait 1", ylab = "") -## Plotting the pairwise lines -pair.line <- function(line, trait_space, defaults) { - lines(x = trait_space[line,1], y = trait_space[line,2], col = defaults$col1) -} -apply(combn(1:elements, 2), 2, pair.line, trait_space, defaults) -points(trait_space, pch = defaults$pch) - -## The position -plot(trait_space, pch = defaults$pch, main = "C - Position", xlab = "Trait 1", ylab = "") -## Plotting the center of the space -points(0,0, pch = 13, cex = 2) -arrows(x0 = 0, y0 = 0, x1 = trait_space[,1], y1 = trait_space[,2], code = 0, col = defaults$col1) -points(trait_space, pch = defaults$pch) - -par(op) -``` - -#### 1. Volume - -Volume metrics measure the spread of a group in the trait-space. -They can be interpreted as the amount of the trait-space that is occupied by observations. -Typically, larger values for such metrics indicate the presence of more extreme trait combinations in the sample. -For example, if group A has a bigger volume than group B, the observations in group A achieve more extreme trait combinations than in group B. -This type of metric is widely used in both ecology [the hypervolume; @blonder2018; for example to compare group's niche overlap @mammola2019] and in evolution [the sum or product of ranges or variances @wills2001; @ciampaglio2001 or the Procrustes variance @adams2013geomorph]. - -Although volume metrics are a highly suitable indicator for comparing a group's trait-space occupancy, it is limited to comparing the range of extreme trait-combinations between groups. -Typically, volume metrics do not take into account the spread of the observations within a group. -In other words, they can make it difficult to determine whether all the observations are on the edge of the volume or whether the volume is simply driven by small number of extreme observations. - -This aspect of space occupancy is used routinely in ecology and macroevolution [@wills2001; @blonder2018] because disparity or dissimilarity analysis often look at the "spread" of observations in multidimensional space. -For example @wright2017 uses the volume of trait-space to analyse how crinoid ecology changes through time relative to all the observed ecologies in the the crinoid trait-space. -A larger "spread" at a specific point in time [e.g. in the Early Carboniferous - 360 to 330 million years ago (Mya); @wright2017] indicates a larger diversity of ecological feeding strategies. -In general the volume of a group in the trait-space indicates how much trait combinations that group achieves in the trait-space. - -#### 2. Density - -Density metrics measure the distribution of a group in the trait-space. -They can be interpreted as an indication of the distribution of the observations *within* a group in the trait-space; this can be considered as a measure of the "packedness", so spaces with higher density have more observations "packed" into that region. -For example, if group A is more dense than group B, observations in group A tend to be more similar between each other relatively than within group B. -Density is less commonly measured compared to trait-space volume, but it is still used in both ecology [e.g. the minimum spanning tree length; @oksanen2007vegan] and evolution [e.g. the average pairwise distance; @geiger2008]. -For example, if group A has a greater volume than group B but both have the same density, similar mechanisms could be driving both groups' trait-space occupancy. -However, this might suggest that A is older and had more time to achieve more extreme trait combinations under essentially the same process. - -#### 3. Position - -Position metrics measure where a group lies in trait-space. -They can be interpreted as an indication of where a group lies in the trait-space either relative to the space itself or relative to another group. -For example, if group A has a different position than group B, observations in group A will have a different trait-combination than group B. - -Position metrics may be harder to interpret in multidimensional spaces (i.e. beyond left/right and above/below) and are hence less often used. -However, when thinking about unidimensional data (one trait), this metric is obvious: for example, two groups A or B could have the same variance (i.e. "volume" or spread) with the same number of observations (i.e. the same density) but could have a totally different mean and thus be in different positions. -Although these metrics are rarer in macroevolution, they have been used in ecology to compare the position of two groups relatively to each other [@mammola2019]. - -## No metric to rule them all: benefits of considering multiple metrics - -The use of multiple metrics to assess trait-space occupancy has the benefit of providing a more detailed characterization of occupancy changes. -For example, if the question of interest is, say, to look at how space occupancy changes in response to mass extinction, using a single metric to describe one aspect of occupancy (usually the volume) can miss part of the picture. -In this mass extinction example, a change in volume (or the absence thereof) could be decoupled from a change in the observations’ position or density in trait-space. -The Cretaceous-Palaeogene extinction (66 Mya) has been linked to an increase in volume of the mammalian trait-space [the mark of an adaptive radiation; @halliday2015] but more specific questions can be answered by looking at other aspects of trait-occupancy: does the radiation expands on previously existing morphologies [elaboration, increase in density; @endler2005] or does it explore entire new regions of the trait-space [innovation, change in position; @endler2005]? -Similarly, in ecology, if two groups occupy the same volume in the trait-space, it can be interesting to look at differences in density within these two hypothetical groups: different selection pressure can lead to different density within equal volume groups. -Such examples can be found throughout the literature in both evolution and ecology on different aspects of space occupancy [e.g. @close2015; @diaz2016]. - - -```{r fig_metric_captures, echo = FALSE, fig.height = 3, fig.width = 9, results = 'hide', fig.cap = paste("Figure 2: Illustration how three different volume (vol. - Product of ranges), density (den. - Average nearest neighbour distance) and position (pos. - Average displacement) metrics capture different occupancy aspects in this simplified trait-space. This illustrates how using only one specific space occupancy metric (e.g. the volume) will fail to capture changes in B (no change in volume), or partial changes in C (also change in density and position).") } - -## Making the volume/density/position space -vdp_space <- moms::vdp.make() - -## Measuring disparity -vdp_disp <- moms::vdp.dispRity(vdp_space, - volume = c(prod, ranges), - density = c(mean, neighbours), - position = c(mean, displacements)) - -## Checking the table of volume changes -silent <- moms::vdp.check.table(vdp_disp, vdp_space) - -## Plotting the results -moms::vdp.plot(vdp_space, plots = c(1, 4, 8), disparity = vdp_disp, mfrow = c(1, 3), - plot.names = c("A - Base (no changes)", #A - "B - Change in position", #D - "C - Change in vol. den. & pos.")) #I -``` - - - -In this paper, we provide the first interdisciplinary review of 25 space occupancy metrics that uses the broad classification of metrics into volume, density and position to capture pattern changes in trait-space. -We assess the behaviour of metrics using simulations and a selection of interdisciplinary empirical datasets; these cover a wide range of potential data types and evolutionary/ecological questions. -We also introduce a tool for Measuring Occupancy in Multidimensional Space ([`moms`](@@@)), which is a user-friendly, open-source graphical interface to allow the tailored testing of metric behaviour for any use case. -[`moms`](@@@) will allow workers to comprehensively assess the properties of their trait-space and the metrics associated with their specific biological question. - - - - - - - - - - - - - - - - - - - - - - -# Methods - -```{r metrics_list, echo = FALSE} -## Loading the list of metrics -source("list.of.metrics.R") -``` - -Briefly, we tested how `r metric.names()` different space occupancy metrics relate to each other, are affected by modifications of traits space and affect group comparisons in empirical data. -To do so, we performed the following steps (explained in more detail below): - -1. We simulated 13 different spaces with different sets of parameters (dimensions, distributions, etc); -2. We transformed these spaces by removing 50% of the observations following four different scenarios corresponding to different empirical scenarios: randomly, by limit (e.g. expansion or reduction of niches), by density (e.g. different degrees of competition within a guild) and by position (e.g. ecological niche shift). -3. We measured occupancy on the resulting transformed spaces using `r metric.names()` different space occupancy metrics; -4. We applied the same space occupancy metrics to six empirical datasets (covering a range of disciplines and a range of dataset properties). - - -| Test | Methods section | Results -|------|-----------------|------------------ -| How do the metrics compare to each other? | Metrics comparisons | Figure 4@@@ -| How are metrics affected by space shifts? | Space shifts | Table 6@@@ -| How is that reflected in the real world? | Empirical examples | Table 7@@@ -Table 2@@@: summary of the tests in this paper. - -Note that the results of these four steps are described in the supplementary material 4@@@ for an additional 17 metrics not mentioned in the main text. - -## Generating spaces - -We generated trait spaces using the following combinations of size, distributions, variance and correlation: - -| space name | size | distribution(s) | dimensions variance | correlation | -|------------|-------|--------------------------------|---------------------|-------------| -| Uniform3 |200*3 | Uniform (min = -0.5, max = 0.5)| Equal | None | -| Uniform15 |200*15 | Uniform | Equal | None | -| Uniform50 |200*50 | Uniform | Equal | None | -| Uniform150 |200*150| Uniform | Equal | None | -| Uniform50c |200*50 | Uniform | Equal | Random (between 0.1 and 0.9) | -| Normal3 |200*3 | Normal (mean = 0, sd = 1) | Equal | None | -| Normal15 |200*15 | Normal | Equal | None | -| Normal50 |200*50 | Normal | Equal | None | -| Normal150 |200*150| Normal | Equal | None | -| Normal50c |200*50 | Normal | Equal | Random (between 0.1 and 0.9) | -| Random |200*50 | Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)| Equal | None | -| PCA-like |200*50 | Normal | Multiplicative | None | -| PCO-like |200*50 | Normal | Additive | None | -Table 3@@@: different simulated space distributions. - -The different in space sizes (200 $\times$ 3, 200 $\times$ 15, 200 $\times$ 50 or 200 $\times$ 150) reflects the range of trait-space dimensions in literature: "low-dimension" spaces ($<15$) are common in ecology [@mammola2019] whereas high dimension spaces ($>100$) are common in macroevolution [@hopkins2017]. -We used a distinct range of distributions (uniform, normal or random) to test the effect of observation distributions on the metrics. -We used different levels of variance for each dimensions in the spaces by making the variance on each dimension either equal (i.e. $\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}$) or decreasing (i.e. $\sigma_{D1} < \sigma_{D2} < \sigma_{Di}$) with the decreasing factor being either multiplicative (using the cumulative product of the inverse of the number of dimensions: $\prod_i^d(1/d)$) or additive (using the cumulative sum of the inverse of the number of dimensions: $\sum_i^d(1/d)$). -Both multiplicative and cumulative reductions of variance are used to illustrate the properties of ordinations where the variance decreases per dimensions [in a lognormal way in principal components analysis - PCA; e.g. @marcy2016; healy2019; and in a normal way in Multidimensional Scaling - MDS, PCO or PCoA; e.g. @close2015; @wright2017]. -Finally, we added a correlation parameter to take into account the potential correlation between different traits. -We repeated the simulation of each trait-space 20 times (resulting in 260 trait-spaces). - -## Spatial occupancy metrics - -We then measured `r metric.names()` different metrics on the resulting transformed spaces, including a new metric we produced, the average displacement, which we expect to be mainly influenced by changes in trait-space position. - - -| Name | Definition | Captures | Source | Notes | -|------|------------|----------|--------|--------| -| Average distance from centroid | $\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}$ | Volume | @laliberté2010 | equivalent to the functional dispersion (FDis) in @laliberté2010 (but not weighted by observation's abundance) | -| Sum of variances | $\sum_{i}^{d}{\sigma^{2}{k_i}}$ | Volume | @wills2001 | common metric used in palaeobiology [@ciampaglio2001] | -| Sum of ranges | $\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}$ | Volume | @wills2001 | more sensitive to outliers than the sum of variances | -| Ellipsoid volume | $\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})$ | Volume | @donohue2013 | less sensitive to outliers than the convex hull hypervolume [@diaz2016; @blonder2018] | -| Minimum spanning tree average distance | $\frac{\sum(\text{branch length})}{n}$ | Density | @oksanen2007vegan | similar to the unscaled functional evenness [@villéger2008] | -| Minimum spanning tree distances evenness | $\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}$ | Density | @villéger2008 | the functional evenness without weighted abundance [FEve; @villéger2008] | -| Average nearest neighbour distance | $min\left(\sqrt{\sum_{i}^{n}{({q}_{i}-p_{i})^2}}\right)\times \frac{1}{n}$ | Density | @foote1990 | the density of pairs of observations in the trait-space (c.f. the Minimum spanning tree average distance above) | -| Average displacement | $\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}$ | Position | This paper | the ratio between the observations' position from their centroid and the centre of the trait-space. A value of 1 indicates that the observations' centroid is the centre of the trait-space | -Table 4@@@: List of metrics with *n* being the number of observations, *d* the total number of dimensions, *k* any specific row in the matrix, *Centroid* being their mean and $\sigma^{2}$ their variance. $\Gamma$ is the Gamma distribution and $\lambda_{i}$ the eigen value of each dimension and ${q}_{i}$ and $p_{i}$ are any pairs of coordinates. - - -We selected these `r metric.names()` space occupancy metrics to illustrate how they capture different aspects of space occupancy (but note that this is not an expression of our preference). -The supplementary material 4@@@ contains the same analysis as described below, performed on a total of 25 metrics. -Furthermore, [`moms`](@@@) allows exploring the effect of many more metrics as well as the customisation of metrics by combining them or using user-designed functions. - - - - - - - - - - - -## Metric comparisons - -We compared the space occupancy correlations across all simulations between each pair of metrics to assess they captured similar signal [@villéger2008; @laliberté2010]. -First, we used the metrics on the full 13 trait-spaces described above. -We then scaled the results and measured the pairwise Pearson correlation to test whether metrics were capturing a similar signal (high positive correlation), a different signal (correlation close to 0) or an opposite signal (high negative correlations). -We performed all the tests using the `psych` package [@psych]. - - - - - - - - - -## Changing space {#changing-spaces} - -To measure how the metrics responded to changes within trait-spaces, we reduced the spaces by removing 50% of elements each time using the following algorithms: - -* **Randomly:** by randomly removing 50% of elements (Fig. 3@@@-A). -This reflects a "null" biological model of changes in trait-space: the case when observations (e.g. species) are removed regardless of their intrinsic characteristics. -For example, if diversity (i.e. number of species) is reduced by 50% but the trait-space volume remains the same, there is a decoupling between diversity and space occupancy [@ruta2013; @hopkins2013]. -Our selected metrics are expected to not be affected by this change. - -* **Limit:** by removing all elements with a distance from the centre (mean point) of the space lower or greater than a radius $\rho$ (where \$rho$ is selected such that 50% elements are selected) generating two limit removals: *maximum* and *minimum* (respectively in orange and blue; Fig. 3@@@-B). -This can reflect a strict selection model where observations with trait values below or above a threshold are removed leading to an expansion or a contraction of the trait-space. -Volume metrics are expected to be most affected by this change. - -* **Density:** by removing any pairs of point with a distance $D$ from each other where (where $D$ is selected such that 50% elements are selected) generating two density removals: *high* and *low* (respectively in orange and blue; Fig. 3@@@-C). -This can reflect changes within groups in the trait-space due to ecological factors [e.g. competition can generate niche repulsion - lower density; @grant2006]. -Density metrics are expected to be most affected by this change. - -* **Position:** by removing points similarly as for **Limit** but using the distance from the furthest point from the centre (max point) generating two position removals: *positive* and *negative* (respectively in orange and blue; Fig. 3@@@-D). -This can reflect global changes in trait-space due, for example, to an entire group remaining diverse but occupying a different niche (e.g. in the transition from non-avian dinosaur to avian dinosaurs). -Position metrics are expected to be most affected by this change. - -The algorithm to select $\rho$ or $D$ is described in greater detail in in the [Appendix](#Appendix_algorithm_reduce). - - - - -```{r fig_reduce_space, echo = FALSE, fig.height = 12, fig.width = 8, results = 'hide', fig.cap = paste("Figure 3: different type of space reduction. Each panel displays two groups (orange and blue) of 50% of the data points each. Each group are generated using the following algorithm: A - randomly; B - by limit (maximum and minimum limit in respectively orange and blue); C - by density (high and low in respectively orange and blue); and D - by position (positive and negative in respectively orange and blue). Panel E represents a typical display of the reduction results displayed in Table 6@@@: the dots represent the median space occupancy values across all simulations for each scenario of trait-space change, the solid and dashed line respectively the 50% and 95% confidence intervals. Results in grey are the random 50% reduction (panel A). Results in blue and orange represent the blue and orange opposite scenarios from panels B, and D. The displayed numerical value is the probability of overlap (Bhattacharrya Coefficient) between the blue and orange distributions and the grey one.")} - -set.seed(42) - -## Change colors -defaults$col1 <- "blue" -defaults$col2 <- "orange" - -op <- par(mfrow = c(3,2), bty = "n") -## The elements -elements <- 300 -## The amount to remove -remove <- 0.5 - -## Trait space -trait_space <- space.maker(elements, 2, distribution = rnorm) - -## Random removal colours -defaults$col1 <- "black" -defaults$col2 <- "grey" - -## The randomly removed space -random_rm <- reduce.space(trait_space, type = "random", remove = 0.5) -## Plotting the reduction -plot.space(trait_space, random_rm, main = "A - Random removal", defaults) - -## Back to normal colours -defaults$col1 <- "blue" -defaults$col2 <- "orange" - -## The limit removal -limit_rm <- reduce.space(trait_space, type = "limit", remove = 0.5) -## Plotting the reduction -plot.space(trait_space, limit_rm, main = "B - Limit", defaults) - -## The density removal -density_rm <- reduce.space(trait_space, type = "density", remove = 0.5) -## Plotting the reduction -plot.space(trait_space, density_rm, main = "C - Density", defaults) - -## The displacement removal -displace_rm <- reduce.space(trait_space, type = "displacement", remove = 0.5) -## Moving the space to the upper right corner (positive) -plot.space(trait_space, displace_rm, main = "D - Position", defaults) - -## Test example -set.seed(42) -colours <- c("grey", "orange", "blue") -par(bty = "n") -## Plot size -plot(NULL, ylim = c(0.8, 3.2), pch = 19, xlim = c(-1,1), xlab = "Centred and scaled space occupancy", ylab = "", xaxt = "n", yaxt = "n", main = "E - Space reduction results example (Table 6)") -## Adding lines -abline(v = 0, lty = 2, col = colours, lwd = 2) -## Adding the x axis -axis(1, at = c(-1, -0.5, 0, 0.5, 1), labels = TRUE, tick = TRUE, col.ticks = "black", col = "black", lwd = 2) -## Adding the values -values <- cbind(rnorm(30, mean = 0, sd = 0.2), - rnorm(30, mean = -0.75, sd = 0.13), - rnorm(30, mean = 0.75, sd = 0.2)) -quantile_vals <- apply(values, 2, quantile, probs = c(0.025, 0.250, 0.750, 0.975)) -centtend_vals <- apply(values, 2, median) - -## Loop through the lines -for(column in 1:3) { - ## Get the x y values - line_x_vals <- quantile_vals[, column] - line_y_vals <- rep(column, 2) - - ## Add the lines - n_cis <- 4 - for(ci in 1:(n_cis/2)) { - lines(x = line_x_vals[c(ci, n_cis-(ci-1))], y = line_y_vals, lty = (n_cis/2 - ci + 1), lwd = ci * 1.5 * 2, col = colours[column]) - } -} -## Add the points -points(x = centtend_vals, y = 1:ncol(values), pch = 19, col = colours, cex = 1.5 + 2) - -bc_1 <- dispRity::bhatt.coeff(values[,2], values[,1]) -bc_2 <- dispRity::bhatt.coeff(values[,3], values[,1]) - -text(x = 0.2, y = 3, labels = round(bc_1, 3), cex = 1.4) -text(x = 0.2, y = 2.9, labels = paste0("\n(Probability of overlap between\nthe blue and the grey distributions)"), cex = 0.8) -text(x = -0.2, y = 2, labels = round(bc_2, 3), cex = 1.4) -text(x = - 0.2, y = 1.9, labels = paste0("\n(Probability of overlap between\nthe orange and the grey distributions)"), cex = 0.8) - -## Some arrows -arrows(x1 = centtend_vals[3], y1 = 2.5, - x0 = centtend_vals[3], y0 = 2.9, - col = "black", lwd = 1, length = 0.1, code = 1) -text(labels = "Metric score from the\nremoval displayed in blue above\n(Limit, Density or Position).", - x = centtend_vals[3]-0.35, y = 2.35, cex = 0.8) - -arrows(x1 = centtend_vals[2], y1 = 2.1, - x0 = centtend_vals[2], y0 = 2.5, - col = "black", lwd = 1, length = 0.1, code = 2) -text(labels = "Metric score from\nthe removal displayed\nin orange above\n(Limit, Density or Position).", - x = centtend_vals[2]+0.05, y = 2.75, cex = 0.8) - -arrows(x1 = centtend_vals[1]+0.4, y1 = 1.2, - x0 = centtend_vals[1]+0.05, y0 = 1.05, - col = "black", lwd = 1, length = 0.1, code = 1) -text(labels = "Metric score from\nthe random removal.", - x = centtend_vals[1]+0.55, y = 1.3, cex = 0.8) -``` - -To measure the effect of space reduction, distribution and dimensionality on the metric, we first scaled the metric to be relative to the non-reduced space for each dimension distribution or number of dimensions. -To do so, we subtracted the observed occupancy with no space reduction (i.e. base occupancy) to all the occupancy measurements of the reduced spaces and then divided it by the resulting maximum observed occupancy. -This way, our occupancy metrics where scaled between -1 and 1 with a value of 0 indicating no effect of the space reduction and $>0$ and $<0$ respectively indicating an increase or decrease in the occupancy metric value. -We then measured the probability of overlap of the between the non-random removals (limit (max/min), density (high/low) and displacement (positive/negative)) and the random removals using the Bhattacharrya Coefficient [i.e. the probability of overlap between two distributions; @bhattacharyya1943; @guillerme2016]. - -### Measuring the effect of space and dimensionality on shifting spaces - -In some situations, distribution differences and the number of dimensions can have an effect on the metric results. -For example, in a normally distributed space, a decrease in density can often lead to an increase in volume. -This is not necessarily true in log-normal spaces or in uniform spaces for certain metrics (e.g. the convex hull of a lognormal space will be less sensitive to changes in density). -Furthermore, high dimensional spaces (>10) are subject to the "curse of multidimensionality" [@cursedimensionality]: data becomes sparser with increasing number of dimensions, such that the probability of two points A and B overlapping in *n* dimensions is the product of the probability of the two points overlapping on each dimensions($\prod_{i}^{d} P(A = B)_{Di}$). -This probability decreases as a product of the number of dimensions. -Therefore, the "curse" can make the interpretation of high dimensional data counter-intuitive. -For example if a group expands in multiple dimensions (i.e. increase in volume), the actual hypervolume can decrease (Fig. 4@@@ and Tables 6@@@, 7@@@). - -We measured the effect of space distribution and dimensionality using an ANOVA ($occupancy \sim distribution$ and $occupancy \sim dimensions$) by using all spaces with 50 dimensions and the uniform and normal spaces with equal variance and no correlation with 3, 15, 50, 100 and 150 dimensions (Table 3@@@) for testing respectively the effect of distribution and dimensions. -The results of the ANOVAs (*p*-values) are reported in Table 6@@@ (see supplementary material for the full ANOVA results). - - - - - - - - - - -## Empirical examples - -To address the degree to which the simulations can be applied to real biological problems, we analysed the effect of the different space occupancy metrics on six different empirical studies covering a broad range of fields that employ trait-space analyses (palaeobiology, macroevolution, evo-devo, ecology, etc.). -For each of these six studies we generated trait-spaces from the data published with the papers. -We then divided the trait-spaces into two biologically-relevant groups and tested whether the metrics differentiated the groups in different ways. -Both the grouping and the questions where based on a simplified version of the biological questions addressed in these papers (with no intention to re-analyse the data but to be representative of a sample of the diversity of questions in ecology and evolution). -The procedures to generate the data and the groups varies from one study to the other but is detailed and fully reproducible in the supplementary materials. - - -study | field | taxonomic Group | traits (data) | trait-space | size | groups (orange/blue in Table 7) | type of question | -------|-------|-----------------|---------------|-------------|------|--------|------------------| -@beck2014 | Palaeontology | Mammalia | discrete morphological phylogenetic data | Ordination of a distance matrix (PCO) | 106*105 | 52 crown vs. 54 stem | Are crown mammals more disparate than stem mammals?| -@wright2017 | Palaeontology | Crinoidea | discrete morphological phylogenetic data | Ordination of a distance matrix (PCO) | 42*41 | 16 before vs. 23 after | Is there a difference in disparity before and after the Ordovician mass extinction?| -@marcy2016 | Evolution | Rodentia | skull 2D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 454*134 | 225 *Megascapheus* vs. 229 *Thomomys* | Are two genera of gopher morphologically distinct? | -@hopkins2016 | Evolution | Trilobita | 3D landmark coordinates | Ordination of a Procrustes Superimposition (PCA) | 46*46 | 36 adults vs. 10 adults | Are juvenile trilobites a subset of adult ones? | -@jones2015 | Ecology | Plantae | Communities species compositions | Ordination of a Jaccard distance matrix (PCO) | 48*47 | 24 aspens vs. 24 grasslands | Is there a difference in species composition between aspens and grasslands? | -@healy2019 | Ecology | Animalia | Life history traits | Ordination (PCA) | 285*6 | 83 ecthotherms vs. 202 endotherms | Do endotherms have more diversified life history strategies than ectotherms? | -Table 5@@@: details of the six empirical trait-spaces. - -For each empirical trait-space we bootstrapped each group 500 times [@disprity] and applied the `r metric.names()` space occupancy metric to each pairs of groups. -Note that when space occupancy could not be calculated (e.g. due to the curse of multidimensionality), we collapsed the metric value to 0 (i.e. no space occupancy was captured). -We then compared the means of each groups using the Bhattacharrya Coefficient [i.e. the probability of overlap between two distributions; @bhattacharyya1943; @guillerme2016]. -The are displayed in Table 7@@@. - - - - - - - - - - - - - - - - -# Results - -```{r loading_results, echo = FALSE} -## Loading the results -remove_05 <- load.results("remove_05") -``` - -```{r running_tests, echo = FALSE} -## Anova function -anova.fun <- function(data) {return(aov(glm(disparity ~ factor, data = data)))} - -## Running the tests -all_test <- test.simulation(remove_05, test = anova.fun, scale = TRUE) -all_dim_test <- test.simulation(remove_05, test = anova.fun, scale = TRUE, - factors = c("uniform3", "uniform15", "uniform50", "uniform100", "uniform150", - "normal3", "normal15", "normal50", "normal100", "normal150")) -space_test <- test.simulation(remove_05, test = anova.fun, scale = TRUE, - factors = c("uniform50", "uniform50c", "normal50", "normal50c", - "random50", "pca_like", "pco_like")) -``` - -## Metric comparisons - - -```{r fig_metric_correlation, echo = FALSE, fig.height = 8, fig.width = 8, results = 'hide', fig.cap = paste("Figure 4: pairwise correlation between the scaled metrics. Numbers on the upper right corner are the Pearson correlations. The red line are linear regressions (with the confidence intervals in grey).")} - -## Adding proper names -results_pairwise <- remove_05 -names(results_pairwise[[1]]) <- metric_names - -## Plotting the pairwise results -pairwise.plot(results_pairwise, scale = TRUE, type = "base", plot = "cor") -``` - -All the metrics selected were either positively correlated (Pearson correlation of 0.99 for the Average distance from centroid and Sum of variance or 0.97 for the Average nearest neighbour distance and Minimum spanning tree average length; Fig. 4@@@) or somewhat correlated (ranging from 0.66 for the sum of variances and the ellipsoid volume to -0.09 between the average displacement and the average distance from centroid; Fig. 4@@@). -Note that all metrics but the ellipsoid volume were normally (or nearly normally) distributed (Fig. 4@@@). -This is due to the ellipsoid volume being more sensitive to the curse of multidimensionality (many values close to 0). -More comparisons between metrics are available in the supplementary materials. - -## Space shifting - -```{r fable_results, fig.show='hide', echo=FALSE, fig.height=3, fig.width=3} -## The metrics names (shortened vector) -name <- metric_names - -## Making a list of parameters for each mini plot -plot.param <- list(scaler = 3, - bg.col = "black", - col = c("grey", "orange", "blue"), - quantiles = c(95, 50), - cent.tend = median, - pch = 19, - metric.max = length(metrics_list), - cex = 2) - -## Looping through each mini plot for every metric -for(metric in 1:length(metrics_list)) { - generate.fable.plot(data = remove_05, metric = metric, what = "limits", plot.param = plot.param, overlap = TRUE) - generate.fable.plot(data = remove_05, metric = metric, what = "densit", plot.param = plot.param, overlap = TRUE) - generate.fable.plot(data = remove_05, metric = metric, what = "displa", plot.param = plot.param, overlap = TRUE) -} - -``` - -```{r, echo = FALSE, eval = FALSE} -## Function for printing the table in the R console -print.fable <- function(n_metrics, byrow, ncol, test) { - ## Make the plot.id table - ids <- matrix(1:(n_metrics*ncol), ncol = ncol, byrow = byrow) - - for(one_metric in 1:n_metrics) { - ## Print all the rows one by one - text <- c(paste0("`r name[", one_metric, "]`"), paste0("`r plot.id(", ids[one_metric, ], ")`")) - ## Add some tests? - if(test) { - text <- c(text, paste0("`r s.test(", one_metric, ", \"s\")`"), paste0("`r s.test(", one_metric, ", \"r\")`")) - } - ## Print the line - cat(paste(text, collapse = " | ")) - cat("|\n") - } -} -## Getting the fable to copy paste under the table header below. -print.fable(length(metrics_list), byrow = TRUE, ncol = 3, test = TRUE) -``` - -Metric | Volume change | Density change | Position change | Distribution effect | Dimensions effect | -:-----------|----------------|-----------------|----------------|---------------------|-------------------| -`r name[1]` | `r plot.id(1)` | `r plot.id(2)` | `r plot.id(3)` | `r s.test(1, "s")` | `r s.test(1, "r")`| -`r name[2]` | `r plot.id(4)` | `r plot.id(5)` | `r plot.id(6)` | `r s.test(2, "s")` | `r s.test(2, "r")`| -`r name[3]` | `r plot.id(7)` | `r plot.id(8)` | `r plot.id(9)` | `r s.test(3, "s")` | `r s.test(3, "r")`| -`r name[4]` | `r plot.id(10)` | `r plot.id(11)` | `r plot.id(12)` | `r s.test(4, "s")` | `r s.test(4, "r")`| -`r name[5]` | `r plot.id(13)` | `r plot.id(14)` | `r plot.id(15)` | `r s.test(5, "s")` | `r s.test(5, "r")`| -`r name[6]` | `r plot.id(16)` | `r plot.id(17)` | `r plot.id(18)` | `r s.test(6, "s")` | `r s.test(6, "r")`| -`r name[7]` | `r plot.id(19)` | `r plot.id(20)` | `r plot.id(21)` | `r s.test(7, "s")` | `r s.test(7, "r")`| -`r name[8]` | `r plot.id(22)` | `r plot.id(23)` | `r plot.id(24)` | `r s.test(8, "s")` | `r s.test(8, "r")`| -Table 6@@@: Results of the effect of space reduction, space dimension distributions and dimensions number of the different space occupancy metrics. See Fig. 3@@@ for interpretation of the figures. _p_-values for distribution effect and dimensions effect represents respectively the effect of the ANOVAs space occupancy ~ distributions and space occupancy ~ dimensions (Signif. codes: 0 '\*\*\*' 0.001 '\*\*' 0.01 '\*' 0.05 '.' 0.1 '' 1). - -As expected (Fig. 2@@@), some different metrics capture different aspects of space occupancy. -In contrast to the clear hypothetical example in Fig. 2@@@, it can be hard to predict the behaviour of each metric when 50% of the observations are removed. -In fact we observe a clear decrease in median metric in less than a third of the space reductions (10/36). - -In terms of change in volume (increase or decrease), only the average distance from centroid and the sum of variances seem to capture a clear change in both directions. -Note however that the increase in volume does not correspond to an *actual* increase in volume in the trait-space (i.e. the volume from the blue observations in Fig. 3@@@-B is equivalent to the one in Fig. 3@@@-A). -In terms of change in density (increase or decrease), only the minimum spanning tree average distance and the average nearest neighbour distance seem to capture a clear change in both directions. -In terms of change in position (increase or decrease), only the average displacement metric seems to capture a change a clear change in direction (albeit not in both directions). -This is not surprising however, since the notion of positions becomes more and more complex to appreciate as dimensionality increases (i.e. beyond left/right (1D), up/down (2D) and front/back (3D)). - -## Empirical example - -```{r test_empirical, echo = FALSE} -## Loading the results -empirical_results <- load.results("empirical_results") - -## Testing the differences for each distributions -bhatt.coeff.safe <- function(x, y, tol = 1e-16, ...) { - if(all(x < tol) | all(y < tol)) { - bhatt.coeff(1, 0) - } else { - bhatt.coeff(x, y, ...) - } -} -disparity_test <- lapply(empirical_results, lapply, test.dispRity, test = bhatt.coeff.safe) -``` - -```{r fable_results_empirical, fig.show='hide', echo=FALSE, fig.height=3, fig.width=3} -## Plotting parameters -plot.param <- list(cex = 2, - col = c("#F7B27E", "#BFE4E3"), - border = c("#F65205", "#3E9CBA"), - na.cex = 3, - scaler = 3) - -## Looping through each mini plot for every metric -data_names <- c("Beck and Lee 2014", "Wright 2017", "Marcy et al. 2016", - "Hopkins and Pearson 2016", "Jones et al. 2015", "Healy et al. 2019") - -for(dataset in 1:length(data_names)){ - for(metric in 1:length(metrics_list)){ - ## Plotting the results - generate.fable.empirical(data = empirical_results[[dataset]][[metric]], - test = disparity_test[[dataset]][[metric]], - precision = 1e-5, plot.param, dataset = dataset) - } -} -``` - -```{r change_plotid_chain, echo = FALSE, eval = TRUE} -## Changing defaults -body(plot.id)[[3]] <- substitute(chain <- "fable_results_empirical") -``` - -```{r, echo = FALSE, eval = FALSE} -print.fable(length(metrics_list), byrow = FALSE, ncol = 6, test = FALSE) -``` - -Metric | Beck and Lee 2014 | Wright 2017 | Marcy et al. 2016 | Hopkins and Pearson 2016 | Jones et al. 2015 | Healy et al. 2019 | -:-----------|----------------|-----------------|------------------|----------------|------------------|----------------| -`r name[1]` | `r plot.id(1)` | `r plot.id(9)` | `r plot.id(17)` | `r plot.id(25)` | `r plot.id(33)` | `r plot.id(41)`| -`r name[2]` | `r plot.id(2)` | `r plot.id(10)` | `r plot.id(18)` | `r plot.id(26)` | `r plot.id(34)` | `r plot.id(42)`| -`r name[3]` | `r plot.id(3)` | `r plot.id(11)` | `r plot.id(19)` | `r plot.id(27)` | `r plot.id(35)` | `r plot.id(43)`| -`r name[4]` | `r plot.id(4)` | `r plot.id(12)` | `r plot.id(20)` | `r plot.id(28)` | `r plot.id(36)` | `r plot.id(44)`| -`r name[5]` | `r plot.id(5)` | `r plot.id(13)` | `r plot.id(21)` | `r plot.id(29)` | `r plot.id(37)` | `r plot.id(45)`| -`r name[6]` | `r plot.id(6)` | `r plot.id(14)` | `r plot.id(22)` | `r plot.id(30)` | `r plot.id(38)` | `r plot.id(46)`| -`r name[7]` | `r plot.id(7)` | `r plot.id(15)` | `r plot.id(23)` | `r plot.id(31)` | `r plot.id(39)` | `r plot.id(47)`| -`r name[8]` | `r plot.id(8)` | `r plot.id(16)` | `r plot.id(24)` | `r plot.id(32)` | `r plot.id(40)` | `r plot.id(48)`| -Table 7@@@: Comparisons of different pairs of groups in different empirical trait-spaces. NAs are used for cases where space occupancy could not be measured due to the curse of multidimensionality. The displayed values are the probability of overlap between both groups (Bhattacharrya Coefficient). - -Similarly as for the simulated results, the empirical ones indicate that there is no perfect one-size-fit all metric. -For all `r metric.names()` metrics (expect the ellipsoid volume) we see either one group or the other having a bigger mean than the other and no consistent case where a group has a bigger mean than the other for all the metrics. -For example, in the @beck2014's dataset, there is a clear non-overlap in space occupancy volume using the average distance from centroid or the sum of variances (overlaps of respectively 0.175 and 0.159) but no overlap when measuring the volume using the sum of ranges (overlap of 0.966). -However, for the @hopkins2016's dataset, this pattern is reversed (no clear differences for the average distance from centroid or the sum of variances - overlap of 0.701 and 0.865 respectively) but a clear difference for the sum of ranges (overlap of 0). -Furthermore, for each dataset, the absolute differences between each groups (i.e. mean of the orange group higher or lower than the one of the blue group) is not consistent depending on the metrics. -For example, in @hopkins2016's dataset, the orange group's mean is clearly higher than the blue one when measuring the sum of ranges (overlap of 0) and the inverse is true when measuring the average displacement (overlap of 0). - - - - - - - - - - - - - - - - - -# Discussion - -In this study, we tested twenty-five metrics of trait-space occupancy on simulated and empirical datasets to assess how each metric captures changes in trait-space volume, density and position. -Our results show that the correlation between metrics can vary both within and between metric categories (Fig. 4@@@), highlighting the importance of understanding the metric classification for the interpretation of results. -Furthermore, our simulations show that different metrics capture different types of trait-space change (Table 6@@@), meaning that the use of multiple metrics is important for comprehensive interpretation of trait-space occupancy. -We also show that the choice of metric impacts the interpretation of group differences in empirical datasets (Table 7@@@), again emphasizing that metric choice has a real impact on the interpretation of specific biological questions - - -#### Metrics comparisons - -Metrics within the same category of trait-space occupancy (volume, density or position) do not have the same level of correlation with each other. -For volume metrics, only the average distance from centroid and the sum of variances are strongly correlated (Pearson correlation of 0.99; Fig. 4@@@) whereas the sum of ranges and the ellipsoid volume are more weekly correlated to them (ranging from 0.07 to 0.40; Fig. 4@@@). -This is due to the sum of ranges and the ellipsoid volume being more affected by the number of dimensions and the type of trait-spaces [Table 6@@@; @cursedimensionality]. -This does not preclude using such metrics but raises caution about what they capture in which scenarios [@donohue2013; @Butler2012]. -We see the same pattern for the density metrics with the minimum spanning tree average distance and the average nearest neighbour distance being strongly correlated (correlation of 0.97; Table 6@@@), conversely to the minimum spanning tree distances evenness (correlation of respectively 0.18 and 0.20; Table 6@@@) that is also more affected by the number of dimensions and the type trait-space (Table 6@@@). -Some metrics can capture different aspects of trait-space occupancy at the same time: the average distance from centroid (volume proxy) is correlated to the sum of variance (volume proxy) but not to the ellipsoid volume (volume proxy) but still to the average nearest neigbhour distance (density proxy). - - -Furthermore, the space occupancy values captured by all metrics behave following a normal distribution except for the ellipsoid volume which demonstrates a classic "curse of multidimensionality" with values increasing logarithmically first and then quickly decreasing to 0 [Fig. 4@@@; @cursedimensionality]. -Overall, the fact that we have such a range of correlations for normal distributions suggests that each metric can capture different summaries of space occupancy ranging from obvious differences (for metrics not strongly correlated) to subtle ones (for metrics strongly correlated). -Comparisons between more metrics are available in the Supplementary Material 4@@@ and show the same wide range of patterns. - -Analysing the properties of metrics in a trait-space prior to using them to answer macroevolutionary or ecological questions can bring valuable insights into what aspect of space occupancy they are capturing. -For example, if some space occupancy metrics captures the same pattern using the sum of variances and the sum of ranges, it is possible that the trait-space have really similar properties (i.e. space with equal normal dimensions). - -#### Space shifting - -Most metrics capture a normal "null" space shift with no changes in space occupancy on average (i.e. randomly removing 50% of the data - in grey in Table 6@@@). -This is a desirable behaviour for space occupancy metrics since most studies using space occupancy look at potential processes resulting in the observed space occupancy patterns [e.g. competition @brusatte2008, convergence @marcy2016, life history traits @healy2019, etc.]. -In this context, random changes in the space occupancy should not be picked up as a pattern to avoid false positive errors. -However, the average nearest neighbour distance and the sum of ranges have a respectively positive and negative "null" median meaning that any decrease of elements will always lead to respectively and increase and a decrease of these metrics even when the process is random (Table 6@@@). -This is not especially a bad property for both metrics (i.e. they can still be used to analysis changes in space occupancy) but it should be kept in mind that even random processes will increase or decrease the metric value which can lead to false positives or negatives. - -Regarding the changes in volume of the trait-spaces, the sum of variances and the average distance from centroid are excellent descriptors, with both maximum and minimum limit changes leading to a clear increase or decrease of the metrics values. -However, as illustrated in the 2D examples in Fig. 3@@@-B, only the minimum limit changes (orange) should actually lead to a decrease of volume of the trait-space; the volume of the maximum limit shifted trait-spaces should have the same volume as the non-modified trait-space but with a hole in its centre ("hollowing out"). -Therefore, an increase in "volume" of the trait-space does not necessary always corresponds to an *actual* 3D volume increase but could also reflect a "hollowing" of the trait-space. - -Regarding changes in density of the trait-spaces, the average nearest neigbhour distance and the minimum spanning tree average distance consistently detect changes in density with more precision for low density trait-spaces (in blue in Table 6@@@). -However, these metrics seem to be also associated to a "null" increase: a small increase in the metric value could be due to a random change (null) or a small decrease in density (see above). -In general, we can observe some degree of correlation between the changes in density and the changes in volume for most metric picking either signal (i.e. the average nearest neighbour distance picks both changes in density and in volume). -This could be due to the generation of normally distributed spaces where a change in density often leads to a change in volume (i.e. to increase density in a normal distribution, one can simply reduce its tails). -This is not necessary the case with empirical data. - -Regarding the changes in position of the trait-space, all but the average displacement metric seems to not be able to distinguish between a random change in trait-space (grey) and a positive or a negative displacement of the trait-space (blue and orange - Table 6@@@). -Furthermore, the average displacement metric does not distinguish between and positive or a negative displacement of the trait-space: in both cases, the metric increases. -This might be due to the inherent complexity of "position" in a trait-space that increases with the number of dimensions. - -Regarding the effect of dimensions and trait-space distribution, metrics seems to be either affected by neither of these factors or by both (Table 6@@@). -Again, this should not preclude using metrics that are sensitive to trait-spaces' dimensions and distributions [e.g. the ellispoid volume is a good volume descriptor up until a moderate number of dimensions; @jackson2011; @donohue2013] but should be kept in mind. - -#### Empirical examples - -Although most differences are fairly consistent within each dataset with one group having a higher space occupancy score than the other for multiple metrics, this difference can be more or less pronounced within each dataset (ranging from no to nearly full overlap - BC $\in(0;0.995)$) and sometimes even reversed. -This indicates that opposite conclusions can be drawn from a dataset depending on which space occupancy metric is considered. -For example, in @wright2017 dataset where the blue group has a greater occupancy metric median value but for the average displacement metric showing a higher median displacement in the red group (Table 7@@@). -These differences in group's space occupancy depending on the metrics are also more pronounced in the empirical datasets where the elements per group is highly different [@hopkins2016; @healy2019]. -This also highlights the influence of different groups size on top of the differences in trait-spaces and metrics properties. -Finally, it is worth noting that the ellispoid volume only picks up accurate differences between groups in the @healy2019 dataset which is yet another illustration of the curse of multidimensionality: the @healy2019 dataset has only 6 dimensions compared to all the other ones having more than 41 dimensions [and the @marcy2016 dataset having 136 dimensions for which the ellipsoid volume can't be calculated; Table 7@@@; @cursedimensionality]. - -### Caveats - -While our simulations have been useful to illustrate the behavior of diverse space occupancy metrics, they have several caveats. For example, the trait-spaces are simulated by treating each observation as independent. -This is not often the case in biology since observations might be spatially [@jones2015] or phylogenetically auto-correlated [e.g. @beck2014]. -However, we can note that the we get expected results from the empirical data where the observations are more realistically auto-correlated. -Furthermore, the algorithm used to reduce the multi-dimensional spaces might not always accurately reflect how trait-spaces are changed in nature. -They might indeed favour the response of specific metrics, in particular for the changes in density that only modifies the density between pairs of points rather than changing the global density. -This is a specific algorithmic choice was made in order to not confound changes in density along with changes in volume (i.e. a general change in density can be obtain by uniformly increasing/decreasing the volume). -However, the results probably capture the general behaviour of each space occupancy metric since results are consistent between the simulated and emipirical analysis. -Furthermore, the [`moms`](@@@) shiny app allows to test the caveats mentioned above by uploading empirical trait-space and personalised space occupancy metrics. - -### Suggestions - -We insist that no metric is better than the next one but that researchers should use the most appropriate metrics based on their question, the specific trait-space, and knowledge of the metric’s properties. -However, the findings of this study might suggest several points: - -First, we suggest using multiple metrics to tackle different aspects of the trait-space. -This is suggestion is in the same logical thinking line that the mean can not be sufficient to describe a distribution (the variance and the type of distribution might be good additional indicators). -Although using multiple metrics is not uncommon in macroevolutionary studies [e.g. @halliday2015] or in ecology [@mammola2019], they often do not cover contrasted aspects of the trait-space (and are often different proxies of the volume of a trait-space). - -Second, we suggest selecting a metric (or a series of metrics) that best help answering the biological question a hand. -In fact, if one studies an adaptive radiation in a group of organisms, it is worth thinking what would be the expected null model: would the group's volume increase (radiation in all directions), would it increase in density (niche specialisation) or would it shift in position (radiation into a new set of niches)? - -Third, we suggest to not name metrics as the biological aspect they are potentially describing (e.g. disparity or functional dispersion) but rather what they are measuring (e.g. sum of dimensions variance or the average distance from centroids). -We believe this point will allow both a clearer understanding of what is measured (c.f. what is approximated) and an easier understanding between ecology and evolution for which the metrics can be the similar but named differently and used for approximating different types of trait occupancy (Fig. 4@@@). - -Multidimensional analysis has been acknowledged to be an essential toolkit modern biology [@adams2019] but can often be counter-intuitive [@cursedimensionality]. -It is thus crucial to accurately describe patterns in multidimensional trait-spaces to be able to link them to biological processes of interest. -Mainly, it is important to remember that there are no one-size-fits-all trait-space occupancy metric. -And that what a pattern captured by a specific space occupancy metric is often dependent on the properties of the trait-space and of the particular biological question of interest. -We thus believe that having a clearer understanding of both the properties of the trait-space and the associated space occupancy metrics (e.g. using [`moms`](@@@)) as well as using novel space occupancy metrics to answer specific questions (e.g. the average displacement metric to capture changes in group’s position) will be of great use to study biological processes in a multidimensional world. - - -# Acknowledgements - -We thank Natalie Jones for pointing out to the empirical ecological datasets. -We acknowledge funding from the Australian Research Council DP170103227 and FT180100634 awarded to VW. - -# Authors contributions - -TG, MNP, AEM and VW designed the project. -TG and AEM collected the empirical dataset. -TG ran the analyses and designed the software. -TG, MNP, AEM and VW wrote the manuscript. - -# Data Availability, repeatability and reproducibility - -The raw empirical data is available from the original papers @beck2014, @jones2015, @marcy2016, @hopkins2016, @wright2017 and @healy2019. -The subset of the empirical data used in this analysis is available on figshare @@@. -The modified empirical data is available in the package accompanying this manuscript (`data(moms::demo_data)`). -The entirety of this manuscript (including the figures, tables and supplementary material) is repeatable and reproducible by compiling the vignette of the [GitHub `moms R` package](https://github/TGuillerme/moms) (see the `README` file for more information). -The code for the `moms` shiny app open source and available from the [GitHub `moms R` package](https://github/TGuillerme/moms). - -# References - - - -# Supplementary material - -## Algorithm for selecting the parameters to reduce the space ($radius$, $displacement$, $density$) {#Appendix_algorithm_reduce} - -We used a recursive algorithm for selecting the parameter that removes $P$ = 50% elements. - -1. Select a random reduction parameters $R$. -2. Remove elements from the space using $R$ resulting in $P'$ removed elements. -If the remaining number of elements is to the required proportion $P$ ; exit the algorithm; -Else go to 3. -3. Get the different between the proportion of removed elements $P'$ and $P$. -If the difference is positive set the increment parameter $R$ to $R = 1.1 \times R$, then go to 2. -Else set $R = 0.9 \times R$, then go to 2. - -The algorithm is implemented in the `optimise.parameter` function in [reduce.space_fun.R](@@@). diff --git a/inst/shiftingspace_resubmit.tex b/inst/shiftingspace_resubmit.tex deleted file mode 100644 index 0b85d21..0000000 --- a/inst/shiftingspace_resubmit.tex +++ /dev/null @@ -1,1753 +0,0 @@ -\documentclass[]{article} -\usepackage{xcolor} -\usepackage{lineno} -\usepackage{lmodern} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\usepackage{fixltx2e} % provides \textsubscript -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} -\fi -% use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -% use microtype if available -\IfFileExists{microtype.sty}{% -\usepackage{microtype} -\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\usepackage[margin=1in]{geometry} -\usepackage{hyperref} -\hypersetup{unicode=true, - pdftitle={Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?}, - pdfauthor={Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker}, - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage{longtable,booktabs} -\usepackage{graphicx,grffile} -\makeatletter -\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} -\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} -\makeatother -% Scale images if necessary, so that they will not overflow the page -% margins by default, and it is still possible to overwrite the defaults -% using explicit options in \includegraphics[width, height, ...]{} -\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -%%% Use protect on footnotes to avoid problems with footnotes in titles -\let\rmarkdownfootnote\footnote% -\def\footnote{\protect\rmarkdownfootnote} - -%%% Change title format to be more compact -\usepackage{titling} - -% Create subtitle command for use in maketitle -\providecommand{\subtitle}[1]{ - \posttitle{ - \begin{center}\large#1\end{center} - } -} - -\setlength{\droptitle}{-2em} - - \title{Shifting spaces: which disparity or dissimilarity measurement best -summarise occupancy in multidimensional spaces?} - \pretitle{\vspace{\droptitle}\centering\huge} - \posttitle{\par} - \author{Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker} - \preauthor{\centering\large\emph} - \postauthor{\par} - \predate{\centering\large\emph} - \postdate{\par} - \date{2020-03-02} - -\linespread{1.6} %in the document header - -\begin{document} -\maketitle -\modulolinenumbers[1] % just after the \begin{document} tag -\linenumbers -\section{Abstract}\label{abstract} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Multidimensional analysis of traits are now a common toolkit in - ecology and evolution and are based on - \textcolor{black}{trait spaces} in which each dimension - summarises the observed trait combination (a morphospace or an - ecospace). Observations of interest will typically occupy a - \textcolor{black}{subspace of this space, and researchers will calculate one or more measures to quantify the way in which organisms "inhabit" that space.} - In macroevolution and ecology these measures are referred to as - disparity or dissimilarity metrics and can be generalised as - \textcolor{black}{space occupancy measures (the distribution of the data in space).} - Researchers use these measures to investigate how space occupancy - changes through time, in relation to other groups of organisms, and in - response to global environmental changes. However, the mathematical - and biological meaning of most space occupancy measures is vague with - the majority of widely-used measures lacking formal description. -\item - Here we propose a broad classification of space occupancy measures - into three categories that capture changes in - \textcolor{black}{size}, density, or position. We analyse - the behaviour of 25 measures to study changes in - \textcolor{black}{trait space size}, density and position - on a series of simulated and empirical datasets. -\item - We find no one measure describes all of - \textcolor{black}{trait space} but that some measures are - better at capturing certain aspects and that their performance depends - on both the \textcolor{black}{trait space} and the - hypothesis analysed. Our results confirm the three broad categories - (\textcolor{black}{size}, density and position) and allow - us to relate changes in any of these categories to biological - phenomena. -\item - Since the choice of space occupancy measures is specific to the data - and question, we introduced - \href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool - based allowing users to both visualise and capture changes in space - occupancy for any measurement. - \href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} is - designed to help workers choose the right space occupancy measures, - given the properties of their - \textcolor{black}{trait space} and their biological - question. - \textcolor{black}{By providing guidelines and common vocabulary for space occupancy analysis, we hope to help bridging the gap in multidimensional research between ecology and evolution.} -\end{enumerate} - -\section{Introduction}\label{introduction} - -Groups of species and environments share specific, recognisable, -correlated characteristics: guilds or biomes with shared phenotypic, -physiological, phylogenetic or behavioural traits. Organisms or -environments should therefore be studied as a set of traits rather than -some specific traits in isolation (Donohue et al. 2013; Hopkins and -Gerber 2017). Biologists have increasingly been using ordination -techniques (see Legendre and Legendre 2012 for a summary) to create -multidimensional \textcolor{black}{trait spaces} to either -explore properties of the data or test hypotheses (e.g. Oksanen et al. -2007; Blonder 2018; Guillerme 2018). For example, in palaeobiology, -Wright (2017) \textcolor{black}{used trait spaces} to study -how groups of species' characteristics change through time; in ecology, -Jones et al. (2015) study evidence of competition by looking at trait -overlap between two populations. However, different fields use a -different set of terms for such approaches (Table 1). Nonetheless, they -use the same mathematical objects: matrices with columns representing an -original or transformed trait value and rows representing observations -(taxon, field site, etc.; Guillerme 2018). - - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{llll} -\toprule -\begin{minipage}[b]{0.2111\columnwidth}\raggedright\strut -Mathematics\strut -\end{minipage} & \begin{minipage}[b]{0.2111\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[b]{0.2111\columnwidth}\raggedright\strut -Macroevolution\strut -\end{minipage} & \begin{minipage}[b]{0.2111\columnwidth}\raggedright\strut -This paper\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Matrix (\(n \times d\)) -\textcolor{black}{with a structural relation between rows and columns}\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Function-space, Eco-space, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Morphospace, traitspace, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -trait space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Rows (\emph{n})\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Taxa, field sites, environments, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Taxa, specimen, populations, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Columns (\emph{d})\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -dimensions\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Matrix subset (\(m \times d\); \(m \leq n\))\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Treatments, phylogenetic group (clade), etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Clades, geological stratum, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -group\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Statistic\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Dissimilarity index or metric, hypervolume, functional diversity\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Disparity metric or index\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -space occupancy measure\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Multidimensional analysis\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Dissimilarity analysis, trait analysis, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -Disparity analysis, disparity-through-time, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.2111\columnwidth}\raggedright\strut -multidimensional analysis\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{terms and equivalence between mathematics, ecology and -macroevolution.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -Ecologists and evolutionary biologists often use -\textcolor{black}{trait spaces} with respect to the same -fundamental questions: -\textcolor{black}{are groups occupying the same amount of trait space? -Do some groups contain more species than others in the same amount of trait space? -Are some specific factors correlated with different patterns of trait space occupancy? -Because of the multidimensional nature of these trait spaces, it is often not possible to study them using bi- or tri-variate techniques (Díaz et al. -2016; Hopkins and Gerber 2017; Mammola 2019).} -Studying the occupancy of -\textcolor{black}{trait spaces is done using disparity indices} -in macroevolution (Wills 2001; Hopkins and Gerber 2017; Guillerme 2018) -or comparing hypervolumes in ecology (Donohue et al. 2013; Díaz et al. -2016; Blonder 2018; Mammola 2019). Despite the commonalities between -\textcolor{black}{the measures used in ecology and evolution (which are often metric but don't necessarily need to be)}, -surprisingly little work has been published on their behaviour (but see -Ciampaglio et al. 2001; Villéger et al. 2008; Mammola 2019). - -Different -\textcolor{black}{occupancy measures capture different aspects of trait space} -(ciampaglio2001; Villéger et al. 2008; Mammola 2019). It may be -widely-known, but to our knowledge is infrequently mentioned in -peer-reviewed papers. First, -\textcolor{black}{space occupancy measures} are often named -as the biological aspect they are describing (``disparity'', -``functional diversity'') rather than what they are measuring (e.g.~the -product of ranges), which obscures the differences and similarities -between studies. Second, in many studies in ecology and evolution, -authors have focused on measuring the -\textcolor{black}{size of the trait space} (e.g.~ellipsoid -volume Donohue et al. 2013; hypervolume Díaz et al. 2016; Procrustes -variance Marcy et al. 2016; product of variance Wright 2017). However, -\textcolor{black}{the size of the trait space} only -represents one aspects of occupancy, disregarding others such as the -density (Harmon et al. 2008) or position (Wills 2001; Ciampaglio et al. -2001). For example, if two groups -\textcolor{black}{have the same size}, this can support -certain biological conclusions. Yet, an alternative aspect of space -occupancy may indicate that the groups' position are different, leading -to a different biological conclusion (e.g.~the groups are equally -diverse but occupy different niches). -\textcolor{black}{Using measures that only capture one aspect of the trait space} -may restrain the potential of multidimensional analysis (Villéger et al. -2008). - -Here we propose a broad classification of space occupancy measures as -used across ecology and evolution and study their power to detect -changes in \textcolor{black}{trait space} occupancy in -simulated and empirical data -(\textcolor{black}{regardless of whether spaces are are truly "occupiable" which might be important in some cases - e.g. if the space is infinite or if some regions inapplicable)}. -We provide an assessment of each broad type of space occupancy measures -along with a unified terminology to foster communication between ecology -and evolution. Unsurprisingly, we found no one measure describes all -changes and that the results from each measures are dependent on the -characteristics of the space and the hypotheses. Furthermore, because -there can be an infinite number of measures, it would be impossible to -propose clear generalities to space occupancy measures behaviour. -Therefore, we propose -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a tool -allowing researchers to design, experiment and visualise their own space -occupancy measure tailored for their specific project and helping them -understanding the ``null'' behaviour of the measures of interest. - -\subsection{Space occupancy measures}\label{space-occupancy-measures} - -\textcolor{black}{Here we define trait spaces as any matrix where rows are observations and columns are their related traits.} -These traits can vary in number and types: they could be discrete -(e.g.~presence or absence of a bone; Beck and Lee 2014; Wright 2017), -continuous measurements (e.g.~leaf area; Díaz et al. 2016) or more -sophisticated ones (e.g.~landmark position; Marcy et al. 2016). Traits -can also be measured by using relative observations (e.g.~community -compositions; Jones et al. 2015) or distance between observations (e.g. -Close et al. 2015). However, regardless of the methodology used to build -a \textcolor{black}{trait space}, three broad occupancy -measures can be used: the \textcolor{black}{size} which -approximates the amount of space occupied, the density which -approximates the distribution in space and the position which -approximates the location in space (Fig. 1; Villéger et al. 2008). Of -course any combination of these three aspects is always possible. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measures_types-1.pdf} -\caption{different type of information captured by space -occupancy measures: (A) size, (B) density and (C() position.} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - -\paragraph{1. Size}\label{size} - -\textcolor{black}{Size captures the spread of a group in the trait space}. -They can be interpreted as the amount of the -\textcolor{black}{trait space} that is occupied by -observations. Typically, larger values for such measures indicate the -presence of more extreme trait combinations. For example, if group A is -bigger than B, the observations in A achieve more extreme trait -combinations than in B. This type of measure is widely used in both -ecology (e.g.~the hypervolume; Blonder 2018) and in evolution (e.g.~the -sum or product of ranges or variances; Wills 2001). - -Although -\textcolor{black}{size measures are suitable indicators of a group's trait space} -occupancy, they are limited to comparing the range of trait-combinations -between groups. -\textcolor{black}{Size measures do not take into account the distribution of the observations within a group and can often be insensitive to unoccupied "holes" in the trait space (overstimating the size; Blonder 2018).} -They can make it difficult to determine whether all the observations are -on the edge of the group's distribution or whether the -\textcolor{black}{size} is simply driven by outliers. - -\paragraph{2. Density}\label{density} - -\textcolor{black}{Density gives an indication of the quantity of observations in the trait space. -They can be interpreted as the distribution of the observations *within* a group in the trait space. -Groups with higher density contain more observations that will tend to be more similar to each other. -For example, if group A is greater is size than group B and both have the same density, similar mechanisms could be driving both groups' trait space occupancy. -This pattern could suggest that A is older and has had more time to achieve extreme trait combinations under essentially the same process as younger, smaller group B (Endler et al. 2005). -Note that density based measures can be sensitive to sampling (e.g. if only living species are present). -Density measures are less common compared to size measures, but they are still used in both ecology (e.g. the minimum spanning tree length; Oksanen et al. 2007) and evolution (e.g. the average pairwise distance; Harmon et al. 2008). } - -\paragraph{3. Position}\label{position} - -Position captures where a group lies in -\textcolor{black}{trait space}. They can be interpreted as -where a group lies in the \textcolor{black}{trait space} -either relative to the space itself or relative to another group. For -example, if group A has a different position than group B, A will have a -different trait-combination than in B. - -Position measures may be harder to interpret in multidimensional spaces. -\textcolor{black}{In a 2D space, two groups can be equally distant from a fixed point but in different parts of the space (left, right, up, or down - with the amount of parts of space increasing with dimensions).} -However, when thinking about unidimensional data, this measure is -obvious: two groups A or B could have the same variance -(\textcolor{black}{size}) with the same number of -observations (density) but could have two different means and thus be in -different positions. These measures are used in ecology to compare the -position of two groups relative to each other (Mammola 2019). - -\subsection{No measure to rule them all: benefits of considering -multiple -measures}\label{no-measure-to-rule-them-all-benefits-of-considering-multiple-measures} - -The use of multiple measurements to assess -\textcolor{black}{trait space} occupancy has the benefit of -providing a more detailed characterisation of occupancy changes. If the -question is to look at how space occupancy changes in response to mass -extinction, using a single space occupancy measure can miss part of the -picture: a change in -\textcolor{black}{size could be decoupled from a change in position or density in trait space}. -For example, the -\textcolor{black}{Cretaceous-Paleogene extinction (66 million years ago) shows an increase in size of the mammalian trait space} -(adaptive radiation; Halliday and Goswami 2016) but more specific -questions can be answered by looking at other aspects of -\textcolor{black}{trait space occupancy: does the radiation expand} -on previously existing morphologies (elaboration, increase in density; -Endler et al. 2005) or does it explore new regions of the -\textcolor{black}{trait space} (innovation, change in -position; Endler et al. 2005)? Similarly, in ecology, if two groups have -the same \textcolor{black}{trait space size}, it can be -interesting to look at differences in density within these two groups: -different selection pressure can lead to different density within -equally \textcolor{black}{sized} groups. - -Here, we provide the first interdisciplinary review of 25 space -occupancy measures that uses the broad classification into -\textcolor{black}{size, density and position to capture pattern changes in trait space}. -We assess the behaviour of measures using simulations and six -interdisciplinary empirical datasets covering a range of potential data -types and biological questions. We also introduce a tool for measuring -occupancy in multidimensional space -(\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}), which is -tailored to test the behaviour of measures for any use case. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -workers to comprehensively assess the properties of their -\textcolor{black}{trait space} and the measures associated -with their specific biological question. - -\section{Methods}\label{methods} - -We tested how 25 space occupancy measures relate to each other, are -affected by modifications of traits space and affect group comparisons -in empirical data: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - We simulated 13 different spaces with different sets of parameters; -\item - We transformed these spaces by removing 50\% of the observations - following four different scenarios corresponding to different - empirical scenarios: randomly, by limit (e.g.~expansion or reduction - of niches), by density (e.g.~different degrees of competition within a - guild) and by position (e.g.~ecological niche shift). -\item - We measured occupancy on the resulting transformed spaces using eight - different space occupancy measures; -\item - We applied the same space occupancy measures to six empirical datasets - (covering a range of disciplines and a range of dataset properties). -\end{enumerate} - -\textcolor{black}{Note that the paper contains the results for only eight measures which were selected as representative of common measures covering the size, density and position trait space aspects.} -The results for an additional 17 measures is available in the -supplementary material 4. - -\subsection{Generating spaces}\label{generating-spaces} - -We generated \textcolor{black}{trait spaces} using the -following combinations of size, distributions, variance and correlation: - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -space name\strut -\end{minipage} & \begin{minipage}[b]{0.12333\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.31\columnwidth}\raggedright\strut -distribution(s)\strut -\end{minipage} & \begin{minipage}[b]{0.21\columnwidth}\raggedright\strut -dimensions variance\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -correlation\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform (min = -0.5, max = 0.5)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D uniform\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D uniform correlated\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Uniform\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -3D normal\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*3\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal (mean = 0, sd = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -15D normal\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*15\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -150D normal\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*150\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D normal correlated\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Random (between 0.1 and 0.9)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D with random distributions\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal, Uniform, Lognormal (meanlog = 0, sdlog = 1)\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Equal\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCA-like\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Multiplicative\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -50D PCO-like\strut -\end{minipage} & \begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -200*50\strut -\end{minipage} & \begin{minipage}[t]{0.31\columnwidth}\raggedright\strut -Normal\strut -\end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright\strut -Additive\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -None\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{different simulated space distribution. -\textcolor{black}{ \textit{Name} of the simulated space; \textit{dimensions} of the matrix (row * columns); \textit{distribution(s)} of the data on each dimensions (for the 'Random', dimensions are randomly chosen between Normal, Uniform or Lognormal); \textit{dimension variance}: distribution of the variance between dimensions (when equal, the dimensions have the same variance); \textit{correlation} between dimensions.}} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -The differences in \textcolor{black}{trait space} sizes (200 -elemeents for 3, 15, 50 or 150 dimensions) reflects the range found in -literature (e.g.~hopkins2017; Mammola 2019). We used a range of -distributions (uniform, normal or -\textcolor{black}{a random combination of uniform, normal and lognormal}) -to test the effect of observation distributions on the measurements. We -used different levels of variance for each dimensions in the spaces by -making the variance on each dimension either equal -(\(\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}\)) or decreasing -(\(\sigma_{D1} < \sigma_{D2} < \sigma_{Di}\)) with the decreasing factor -being either multiplicative (using the cumulative product of the inverse -of the number of dimensions: \(\prod_i^d(1/d)\)) or additive -(\(\sum_i^d(1/d)\)). Both reductions of variance are used to illustrate -the properties of ordinations where the variance decreases per -dimensions (and normal win Multidimensional Scaling - MDS, PCO or PCoA; -e.g. Close et al. 2015; lognormal in principal components analysis - -PCA; e.g. Marcy et al. 2016; Wright 2017; Healy et al. 2019). Finally, -we added a correlation parameter -\textcolor{black}{to illustrate the effect of co-linearity between traits (especially in non-ordinated trait spaces).} -We repeated the simulation of each -\textcolor{black}{trait space} 20 times (resulting in 260 -spaces). - -\subsection{Spatial occupancy -measures}\label{spatial-occupancy-measures} - -We then -\textcolor{black}{ calculated eight different measures } on -the resulting transformed spaces, including a new one, the average -displacement, which we expect to be influenced by changes in -\textcolor{black}{trait space} position. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -\begin{minipage}[b]{0.12333\columnwidth}\raggedright\strut -Name\strut -\end{minipage} & \begin{minipage}[b]{0.23333\columnwidth}\raggedright\strut -Definition\strut -\end{minipage} & \begin{minipage}[b]{0.08333\columnwidth}\raggedright\strut -Captures\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Source\strut -\end{minipage} & \begin{minipage}[b]{0.26\columnwidth}\raggedright\strut -Notes\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Laliberté and Legendre (2010)\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -the functional dispersion (FDis - without abundance)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\sigma^{2}{k_i}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright\strut -common measure used in palaeobiology (Ciampaglio et al. 2001; Wills -2001)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright\strut -more sensitive to outliers than the sum of variances\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Size\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Donohue et al. (2013)\strut -\end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright\strut -less sensitive to outliers than the convex hull hypervolume (Díaz et al. -2016; Blonder 2018)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\frac{\sum(\text{branch length})}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Sedgewick (1990)\strut -\end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright\strut -similar to the unscaled functional evenness (Villéger et al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Villéger et al. (2008)\strut -\end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright\strut -the functional evenness without weighted abundance (FEve; Villéger et -al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\sqrt{\sum_{i}^{n}{min({q}_{i}-p_{i})^2}})\times \frac{1}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Foote (1992)\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -the density of pairs of observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.12333\columnwidth}\raggedright\strut -Average displacement\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.08333\columnwidth}\raggedright\strut -Position\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -This paper\strut -\end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright\strut -the ratio between the observations' position from their centroid and the -centre of the trait space (coordinates: 0, 0, 0, \ldots{}). A value of 1 -indicates that the observations' centroid is the centre of the trait -space\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{List of measures with \emph{n} being the number of -observations, \emph{d} the total number of dimensions, \emph{k} any -specific row in the matrix, \emph{Centroid} being their mean and -\(\sigma^{2}\) their variance. \(\Gamma\) is the Gamma distribution and -\(\lambda_{i}\) the \textcolor{black}{eigenvalue} of each -dimension and \({q}_{i}\) and \(p_{i}\) are any pairs of coordinates.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -We selected these eight space occupancy measures to illustrate how they -capture different aspects of space occupancy (not as an expression of -our preference). -\textcolor{black}{These measures are specific to Euclidean and isotropic trait spaces (which is not necessary for all measures).} -The supplementary material 4 contains the same analysis as described -below, performed on 17 measures. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -exploration into the effect of many more measures as well as the -customisation of measures by combining them or using user-designed -functions. - -\subsection{Measure comparisons}\label{measure-comparisons} - -We compared the space occupancy measures correlations across all -simulations between each pair of measures to assess -\textcolor{black}{their} captured signal (Villéger et al. -2008; Laliberté and Legendre 2010). We used the measures on the full 13 -\textcolor{black}{trait spaces} described above. We then -scaled the results and measured the pairwise Pearson correlation to test -whether measures were capturing a similar signals or not using the -\texttt{psych} package (Revelle 2018). - -\subsection{Changing space}\label{changing-spaces} - -To assess how the measures responded to changes within -\textcolor{black}{trait spaces}, we removed 50\% of -observations each time using the following algorithms: - -\begin{itemize} -\item - \textbf{Randomly:} by randomly removing 50\% of observations (Fig. - 2-A). This reflects a ``null'' biological model of changes in - \textcolor{black}{trait space}: the case when observations - are removed regardless of their intrinsic characteristics. For - example, if diversity is reduced by 50\% but the - \textcolor{black}{space size} remains the same, there is a - decoupling between diversity and space occupancy (Ruta et al. 2013). - Our selected measures are expected to not be affected by this change. -\item - \textbf{Limit:} by removing observations within a distance from the - centre of the \textcolor{black}{trait space} lower or - greater than a radius \(\rho\) (where \(\rho\) is chosen such that - 50\% observations are selected) generating two limit removals: - \emph{maximum} and \emph{minimum} (respectively in orange and blue; - Fig. 2-B). This can reflect a strict selection model where - observations with trait values below or above a threshold are removed - leading to an expansion or a contraction of the - \textcolor{black}{trait space}. - \textcolor{black}{Size} measures are expected to be most - affected by this change. -\item - \textbf{Density:} by removing any pairs of point with a distance \(D\) - from each other where (where \(D\) is chosen such that 50\% - observations are selected) generating two density removals: - \emph{high} and \emph{low} (respectively in orange and blue; Fig. - 2-C). This can reflect changes within groups in the - \textcolor{black}{trait space} due to ecological factors - (e.g.~niche repulsion resulting in lower density; Grant and Grant - 2006). Density measures are expected to be most affected by this - change. -\item - \textbf{Position:} by removing points similarly as for \textbf{Limit} - but using the distance from the furthest point from the centre - generating two position removals: \emph{positive} and \emph{negative} - (respectively in orange and blue; Fig. 2-D). This can reflect global - changes in \textcolor{black}{trait space} (e.g.~if an - entire group remaining diverse but occupying a different niche). - Position measures are expected to be most affected by this change. -\end{itemize} - -The algorithm to select \(\rho\) or \(D\) is described in the -Supplementary material 1. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{figure} -\centering -\includegraphics[width=0.7\textwidth]{shiftingspace_files/figure-latex/fig_reduce_space-1.pdf} -\caption{\small{this figure illustrates the different type of space -reduction and how this could affect the measures for the simulation -results displayed in table 5. Each panel displays two groups of 50\% of -the points each. Each group (orange and blue) are generated using the -following algorithm: A - randomly; B - by limit (maximum and minimum -limit); C - by density (high and low); and D - by position (positive and -negative). Panel E et F represents two typical display of the reduction -results displayed in Table 5: the dots represent the median space -occupancy values across all simulations for each scenario of trait space -change (Table 2), the solid and dashed line respectively the 50\% and -95\% confidence intervals. Results in grey are the random 50\% reduction -(panel A). Results in blue and orange represent the opposite scenarios -from panels B, C, and D. The displayed value is the amount of overlap -(Bhattacharrya Coefficient) between the blue or orange distributions and -the grey one. Panel E and F shows respectively the ``ideal'' and -``worst'' results for any type of measures, where the space occupancy -measurement respectively manages or fails to captures a specific type of -reduction (size, position or density; Table 5).}} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - - -\textcolor{black}{Because occupancy measures are dependent on the space, we scaled and centred them between -1 and 1 to make them comparable (by subtracting the observed occupancy without reduction to all the measures of the reduced spaces and then divided it by the maximum observed occupancy).} -A value of 0 indicates no effect of the space reduction and \(>0\) and -\(<0\) respectively indicates an increase or decrease in the measure -value. We then measured the \textcolor{black}{amount} of -overlap between the non-random removals (limit, density and position) -and the random removals using the Bhattacharrya Coefficient -(Bhattacharyya 1943). - -\subsubsection{Measuring the effect of space and -dimensionality}\label{measuring-the-effect-of-space-and-dimensionality} - -Distribution differences and the number of dimensions can have an effect -on the measure results. For example, in a normally distributed space, an -\textcolor{black}{increase in density can often lead to a decrease in size (though this is not necessarily true if the space is log-normal or uniform)}. -High dimensional spaces (\textgreater{}10) are subject to the ``curse of -multidimensionality'' (Bellman 1957): data becomes sparser with -increasing number of dimensions. -\textcolor{black}{This can have two main consequences: 1) the probability of overlap between two groups decreases as a product of the number of dimensions; and 2) the amount of samples needed to "fill" the spaces increases exponentially [see this interactive illustration by Toph Tucker](https://observablehq.com/@tophtucker/theres-plenty-of-room-in-the-corners).} -The ``curse'' can make the interpretation of high dimensional data -counter-intuitive. For example if a group expands in multiple dimensions -(i.e.~increase in \textcolor{black}{size}), the actual -hypervolume \textcolor{black}{($\prod_{i}^{d} range_{Di}$)} -can decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an -ANOVA (\(occupancy \sim distribution\) and -\(occupancy \sim dimensions\)) by using all spaces with 50 dimensions -and the uniform and normal spaces with equal variance and no correlation -with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing -respectively the effect of distribution and dimensions. The results of -the ANOVAs (\textcolor{black}{F and *p*-values}) are reported -in Table 5 (full results in supplementary material 3). - -\subsection{Empirical examples}\label{empirical-examples} - -We analysed the effect of the different space occupancy measures on six -different empirical studies covering a range of fields that employ -\textcolor{black}{trait space} analyses. For each of these -studies we generated \textcolor{black}{trait spaces} from the -data published with the papers. We divided each -\textcolor{black}{trait spaces} into two -biologically-relevant groups and tested whether the measures -differentiated the groups in different ways. Both the grouping and the -questions \textcolor{black}{were} based on a simplified -version of the topics of these papers (with no intention to re-analyse -the data and questions). The procedures to generate the data and the -groups varies between studies and is detailed in the supplementary -materials 2. - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}llllllll@{}} -\toprule -\begin{minipage}[b]{0.08444\columnwidth}\raggedright\strut -study\strut -\end{minipage} & \begin{minipage}[b]{0.08444\columnwidth}\raggedright\strut -field\strut -\end{minipage} & \begin{minipage}[b]{0.08444\columnwidth}\raggedright\strut -taxonomic group\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -traits\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -trait space\strut -\end{minipage} & \begin{minipage}[b]{0.08444\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut -groups\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -question\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Beck and Lee (2014)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Mammalia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -106*105\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -52 crown vs.~54 stem\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are crown mammals more disparate than stem mammals?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Wright (2017)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Crinoidea\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -42*41\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -16 before vs.~23 after\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Is there a difference in disparity before and after the Ordovician mass -extinction?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Marcy et al. (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Rodentia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -skull 2D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -454*134\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -225 \emph{Megascapheus} vs.~229 \emph{Thomomys}\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are two genera of gopher morphologically distinct?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Hopkins and Pearson (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Trilobita\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -3D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -46*46\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -36 adults vs.~10 juveniles\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are juvenile trilobites a subset of adult ones?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Jones et al. (2015)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Plantae\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Communities species compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Jaccard distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -48*47\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -24 aspens vs.~24 grasslands\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Is there a difference in species composition between aspens and -grasslands?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Healy et al. (2019)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -Animalia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Life history traits\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of continuous traits (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -285*6\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -83 ecthotherms vs.~202 endotherms\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Do endotherms have more diversified life history strategies than -ectotherms?\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{details of the six empirical trait spaces.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -For each empirical \textcolor{black}{trait space} we -bootstrapped each group 500 times (Guillerme 2018) and applied the eight -space occupancy measure to each pairs of groups. We then compared the -means of each groups using the Bhattacharrya Coefficient (Bhattacharyya -1943). - -\section{Results}\label{results} - -\subsection{Measure comparisons}\label{measure-comparisons-1} - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_measure_correlation-1.pdf} -\caption{pairwise correlation between the scaled measures. -Numbers on the upper right corner are the Pearson correlations. The red -line are linear regressions (with the confidence intervals in grey). -Av.: average; dist.: distance; min.: minimum; span.: spanning.} -\end{figure} - -\renewcommand\baselinestretch{1.6}\selectfont - - -All the measures were either positively correlated (Pearson correlation -of 0.99 for the average Euclidean distance from centroid and sum of -variance or 0.97 for the average nearest neighbour distance and minimum -spanning tree average length; Fig. 3) or somewhat correlated (ranging -from 0.66 for the sum of variances and the ellipsoid volume to -0.09 -between the average displacement and the average Euclidean distance from -centroid; Fig. 3). All measures but the ellipsoid volume were normally -(or nearly normally) distributed (Fig. 3). - -\subsection{Space shifting}\label{space-shifting} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llllll@{}} -\toprule -\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Size change\strut -\end{minipage} & \begin{minipage}[b]{0.08444\columnwidth}\raggedright\strut -Density change\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Position change\strut -\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Distribution effect\strut -\end{minipage} & \begin{minipage}[b]{0.23333\columnwidth}\raggedright\strut -Dimensions effect\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 0.924 ; p = 0.449\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 0.322 ; p = 0.958\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.285 ; p = 0.274\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 0.478 ; p = 0.873\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 11.119 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 32.307 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 7.215 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 13.486 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.162 ; p = 0.326\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 0.998 ; p = 0.435\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 8.152 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 29.358 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 1.478 ; p = 0.207\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 0.773 ; p = 0.626\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.08444\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -F = 10.742 ; p = \textless{}1e-3***\strut -\end{minipage} & \begin{minipage}[t]{0.23333\columnwidth}\raggedright\strut -F = 26.829 ; p = \textless{}1e-3***\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{Results of the effect of space reduction, space dimension -distributions and dimensions number of the different space occupancy -measures. See Fig. 2 for interpretation of the figures distributions and values. F-values for distribution effect and dimensions effect represents respectively the effect of the ANOVAs space occupancy ~ distributions and space occupancy ~ dimension represent the ratio of sum squared difference within and between groups (the higher, the more the factor has an effect on the measure) and associated \textit{p}-values (0 '\*\*\*' 0.001 '\*\*' 0.01 '\*' 0.05 '.' 0.1 '' 1). This figure illustrates how different measures can be influenced by different aspects of changes in the trait space.\textcolor{black}{ E.g. the Average Euclidean distance from centroid (row 1) captures mainly changes in size (column 1), but also captures changes in density (column 2) but does not capture changes in position (column 3)}.} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - - -As expected, some different measures capture different aspects of space -occupancy. However, it can be hard to predict the behaviour of each -measure when 50\% of the observations are removed. We observe a clear -decrease in median metric in less than a third of the space reductions -(10/36). - -In terms of change in \textcolor{black}{size}, only the -average Euclidean distance from centroid and the sum of variances seem -to capture a clear change in both directions. In terms of change in -density, only the minimum spanning tree average distance and the average -nearest neighbour distance seem to capture a clear change in both -directions. And in terms of change in position, only the average -displacement metric seems to capture a clear change in direction (albeit -not in both directions). - -\subsection{Empirical examples}\label{empirical-example} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllllll@{}} -\toprule -\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Measure\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Beck and Lee 2014\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -Wright 2017\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Marcy et al. 2016\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Hopkins and Pearson 2016\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Jones et al. 2015\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Healy et al. 2019\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -\textcolor{black}{Comparisons (orange *vs.* blue)}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\textcolor{black}{crown *vs.* stem mammals morphologies}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\textcolor{black}{crinoids morphologies before *vs.* after the end-Ordovician extinction}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\textcolor{black}{*Megascapheus* *vs.* *Thomomys* skull shapes}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\textcolor{black}{adults *vs.* juveniles trilobites cephalon shapes}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\textcolor{black}{aspens *vs.* grasslands communities compositions}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\textcolor{black}{ecthotherms *vs.* endotherms life history traits}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average Euclidean distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-25.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-33.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-41.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-26.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-34.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-42.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-27.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-35.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-43.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-28.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-36.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-44.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-29.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-37.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-45.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-30.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-38.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-46.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-31.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-39.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-47.pdf}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-32.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-40.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-48.pdf}\strut -\end{minipage}\tabularnewline -\bottomrule -\caption{Comparisons of pairs of groups in different empirical trait -spaces. NAs are used for cases where space occupancy could not be -measured due to the curse of multidimensionality. The displayed values -are the \textcolor{black}{amount} of overlap between both -groups (Bhattacharrya Coefficient).} -\end{longtable} - -\renewcommand\baselinestretch{1.6}\selectfont - -Similarly as for the simulated results, the empirical ones indicate that -there \textcolor{black}{can be} no perfect one-size-fit all -measurement. For all eight measures -(\textcolor{black}{except} the ellipsoid volume) we see -either one group or the other having a bigger mean than the other and no -consistent case where a group has a bigger mean than the other for all -the measures. For example, in the Beck and Lee (2014)'s dataset, there -is a clear \textcolor{black}{difference in size} using the -average Euclidean distance from centroid or the sum of variances -(overlaps of respectively 0.175 and 0.159) but no overlap when measuring -the \textcolor{black}{size} using the sum of ranges (0.966). -However, for the Hopkins and Pearson (2016)'s dataset, this pattern is -reversed (no clear differences for the average Euclidean distance from -centroid or the sum of variances - 0.701 and 0.865 respectively - but a -clear difference for the sum of ranges (0). For each dataset, the -absolute differences between each groups is not consistent depending on -the measures. For example, in Hopkins and Pearson (2016)'s dataset, the -orange group's mean is clearly higher than the blue one when measuring -the sum of ranges (0) and the inverse is true when measuring the average -displacement (0). - -\section{Discussion}\label{discussion} - -Here we tested 25 measures of \textcolor{black}{trait space} -occupancy on simulated and empirical datasets to assess how each measure -captures changes in \textcolor{black}{trait space size}, -density and position. Our results show that the correlation between -measures can vary both within and between measure categories (Fig. 3), -highlighting the importance of understanding the measure classification -for the interpretation of results. Our simulations show that different -measures capture different types of -\textcolor{black}{trait space} change (Table 5), meaning that -the use of multiple measures is important for comprehensive -interpretation of \textcolor{black}{trait space} occupancy. -We also show that the choice of measure impacts the interpretation of -group differences in empirical datasets (Table 6). - -\paragraph{Measures comparisons}\label{measures-comparisons} - -Measures within the same category of -\textcolor{black}{trait space} occupancy -(\textcolor{black}{size}, density or position) do not have -the same level of correlation with each other. For example, the average -Euclidean distance from centroid (\textcolor{black}{size}) is -highly correlated to the sum of variances -(\textcolor{black}{size} - correlation of 0.99) and somewhat -correlated with the minimum spanning tree average distance (density - -correlation of 0.66) but poorly with the ellipsoid volume -(\textcolor{black}{size} - correlation of 0.17) and the -minimum spanning tree distances evenness (density - correlation of --0.05). - -\paragraph{Space shifting}\label{caveats} - -Most measures capture no changes in space occupancy for the ``null'' -(random) space reduction (in grey in Table 5). This is a desirable -behaviour for space occupancy measures since it will likely avoid false -positive errors in studies that estimate biological processes from space -occupancy patterns (e.g.~convergence Marcy et al. 2016, life history -traits Healy et al. (2019)). However, the average nearest neighbour -distance and the sum of ranges have a respectively positive and negative -``null'' median. This is not especially a bad property but it should be -kept in mind that even random processes can increase or decrease these -measures' values\}. - -For changes in \textcolor{black}{size}, the sum of variances -and the average Euclidean distance from centroid are good descriptors -(Table 5). However, as illustrated in the 2D examples in Fig. 2-B only -the blue change results (Table 5) should not result in a direct change -in \textcolor{black}{overall size because the trait space} is -merely ``hollowed'' out. . - -The average nearest neigbhour distance and the minimum spanning tree -average distance consistently detect changes in density with more -precision for low density \textcolor{black}{trait spaces} (in -blue in Table 5). However, we can observe some degree of correlation -between the changes in density and the changes in -\textcolor{black}{size} for most measure picking either -signal. This could be due to the use of normally distributed spaces -where a change in density often leads to a change in -\textcolor{black}{size}. This is not necessarily the case -with empirical data. - -Regarding the changes in position, only the average displacement measure -seems able to distinguish between a random change and a displacement of -the \textcolor{black}{trait space} (Table 5). -\textcolor{black}{However}, the average displacement measure -does not distinguish between positive or negative displacement: this -might be due to the inherent complexity of \emph{position} in a -multidimensional \textcolor{black}{trait space}. - -\paragraph{Empirical examples}\label{empirical-examples-1} - -Although most differences are fairly consistent within each dataset with -one group having a higher space occupancy score than the other for -multiple measures, this difference can be more or less pronounced within -each dataset (ranging from no to nearly full overlap - BC -\(\in(0;0.995)\)) and sometimes even reversed. This indicates that -opposite conclusions can be drawn from a dataset depending on which -space occupancy measure is considered. These differences depending on -the measures are also more pronounced in the empirical datasets where -the observations per group are unequal (Hopkins and Pearson 2016; Healy -et al. 2019). - -\subsubsection{Caveats}\label{caveats} - -While our simulations are useful to illustrate the behaviour of diverse -space occupancy measures, they have several caveats. First, the -simulated observations in the \textcolor{black}{trait spaces} -are independent. This is not the case in biology where observations can -be spatially (Jones et al. 2015) or phylogenetically correlated (e.g. -Beck and Lee 2014). Second, the algorithm used to reduce the -\textcolor{black}{trait spacesmight not always accurately reflect changes}. -This might favour some specific measures over others, in particular for -the changes in density that modify the nearest neighbour density rather -than changing the global density. This algorithmic choice was made in -order to not confound changes in density along with changes in -\textcolor{black}{size}. However, the results presented here -probably capture the general behaviour of each measure since results are -consistent between the simulated and empirical analysis. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows -workers to test the caveats mentioned above by uploading empirical -\textcolor{black}{trait spaces}. - -\subsubsection{Conclusions}\label{conclusions} - -We insist that no measure is better than the next one and that workers -should identify the most appropriate measures based on their -\textcolor{black}{trait space} properties as well as their -specific biological question. However, following the findings of this -study we make several suggestions: - -First, we suggest using multiple measures to tackle different aspects of -the \textcolor{black}{trait space}. Although using multiple -measures is not uncommon in macroevolutionary studies (e.g. Halliday and -Goswami 2016) or in ecology (Mammola 2019), -\textcolor{black}{they often do no cover more than one of the three categories of trait space measures}. - -Second, we suggest selecting the measures that best address the -biological question \textcolor{black}{at} hand. If one -studies an adaptive radiation in a group of organisms, it is worth -thinking what would be the expected null model: would the group's -\textcolor{black}{size} increase (radiation in all -directions), would it increase in density (niche specialisation) or -would it shift in position (radiation into a new set of niches)? - -Third, we suggest not naming measures after the biological aspect they -describe -\textcolor{black}{which can be vague (e.g. "disparity" or "functional dispersion") but rather after what they are measuring and why (e.g. "we used sum of ranges to measure the space size").} -We believe this will support both a clearer understanding of what -\emph{is} measured as well as better communication between ecology and -evolution research where measures can be similar but have different -names. - -Multidimensional analyses have been acknowledged as essential tools in -modern biology but they can often be counter-intuitive (Bellman 1957). -It is thus crucial to accurately describe patterns in multidimensional -\textcolor{black}{trait spaces} to be able to link them to -biological processes. When summarising -\textcolor{black}{trait spaces}, it is important to remember -that a pattern captured by a specific space occupancy measure is often -dependent on the properties of the space and of the particular -biological question of interest. We believe that having a clearer -understanding of both the properties of the -\textcolor{black}{trait space} and the associated space -occupancy measures (e.g.~using -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}) as well as -using novel space occupancy measures to answer specific questions will -be of great use to study biological processes in a multidimensional -world. - -\section{Acknowledgements}\label{acknowledgements} - -We thank Natalie Jones and Kevin Healy for helping with the empirical -datasets and two anonymous reviewer for their comments. We acknowledge -funding from the Australian Research Council DP170103227 and FT180100634 -awarded to VW. - -\section{Authors contributions}\label{authors-contributions} - -TG, MNP, AEM and VW designed the project. TG and AEM collected the -empirical dataset. TG ran the analyses and designed the software. TG, -MNP, AEM and VW wrote the manuscript. - -\section{Data Availability, repeatability and -reproducibility}\label{data-availability-repeatability-and-reproducibility} - -The raw empirical data is available from the original papers (Beck and -Lee 2014; Jones et al. 2015, Marcy et al. (2016); Hopkins and Pearson -2016; Wright 2017; Healy et al. 2019). The subsets of the empirical data -used in this analysis are available on figshare -\href{https://doi.org/10.6084/m9.figshare.9943181.v1}{DOI: -10.6084/m9.figshare.9943181.v1}. The modified empirical data are -available in the package accompanying this manuscript -(\texttt{data(moms::demo\_data)}). This manuscript (including the -figures, tables and supplementary material) is repeatable and -reproducible by compiling the vignette of the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. - -\section*{References}\label{references} -\addcontentsline{toc}{section}{References} - -\hypertarget{refs}{} -\hypertarget{ref-beck2014}{} -Beck R.M.D., Lee M.S.Y. 2014. Ancient dates or accelerated rates? -Morphological clocks and the antiquity of placental mammals. Proceedings -of the Royal Society B: Biological Sciences. 281:20141278. - -\hypertarget{ref-cursedimensionality}{} -Bellman R.E. 1957. Dynamic programming. Princeton University Press. - -\hypertarget{ref-bhattacharyya1943}{} -Bhattacharyya A. 1943. On a measure of divergence between two -statistical populations defined by their probability distributions. -Bulletin of the Calcutta Mathematical Society. 35:99--109. - -\hypertarget{ref-blonder2018}{} -Blonder B. 2018. Hypervolume concepts in niche-and trait-based ecology. -Ecography. 41:1441--1455. - -\hypertarget{ref-ciampaglio2001}{} -Ciampaglio C.N., Kemp M., McShea D.W. 2001. Detecting changes in -morphospace occupation patterns in the fossil record: Characterization -and analysis of measures of disparity. Paleobiology. 71:695--715. - -\hypertarget{ref-close2015}{} -Close R.A., Friedman M., Lloyd G.T., Benson R.B. 2015. Evidence for a -mid-Jurassic adaptive radiation in mammals. Current Biology. - -\hypertarget{ref-diaz2016}{} -Díaz S., Kattge J., Cornelissen J.H., Wright I.J., Lavorel S., Dray S., -Reu B., Kleyer M., Wirth C., Prentice I.C., others. 2016. The global -spectrum of plant form and function. Nature. 529:167. - -\hypertarget{ref-donohue2013}{} -Donohue I., Petchey O.L., Montoya J.M., Jackson A.L., McNally L., Viana -M., Healy K., Lurgi M., O'Connor N.E., Emmerson M.C. 2013. On the -dimensionality of ecological stability. Ecology Letters. 16:421--429. - -\hypertarget{ref-endler2005}{} -Endler J.A., Westcott D.A., Madden J.R., Robson T. 2005. Animal visual -systems and the evolution of color patterns: Sensory processing -illuminates signal evolution. Evolution. 59:1795--1818. - -\hypertarget{ref-foote1992}{} -Foote M. 1992. Rarefaction analysis of morphological and taxonomic -diversity. Paleobiology. 18:1--16. - -\hypertarget{ref-grant2006}{} -Grant P.R., Grant B.R. 2006. Evolution of character displacement in -darwins finches. Science. 313:224--226. - -\hypertarget{ref-disprity}{} -Guillerme T. 2018. dispRity: A modular R package for measuring -disparity. Methods in Ecology and Evolution. 9:1755--1763. - -\hypertarget{ref-halliday2015}{} -Halliday T.J.D., Goswami A. 2016. Eutherian morphological disparity -across the end-cretaceous mass extinction. Biological Journal of the -Linnean Society. 118:152--168. - -\hypertarget{ref-geiger2008}{} -Harmon L.J., Weir J.T., Brock C.D., Glor R.E., Challenger W. 2008. -GEIGER: Investigating evolutionary radiations. Bioinformatics. -24:129--131. - -\hypertarget{ref-healy2019}{} -Healy K., Ezard T.H.G., Jones O.R., Salguero-G'omez R., Buckley Y.M. -2019. Animal life history is shaped by the pace of life and the -distribution of age-specific mortality and reproduction. Nature Ecology -\& Evolution. 2397-334X. - -\hypertarget{ref-hopkins2016}{} -Hopkins M., Pearson K. 2016. Non-linear ontogenetic shape change in -cryptolithus tesselatus (trilobita) using three-dimensional geometric -morphometrics. Palaeontologia Electronica. 19:1--54. - -\hypertarget{ref-hopkins2017}{} -Hopkins M.J., Gerber S. 2017. Morphological disparity. In: Nuno de la -Rosa L., Müller G., editors. Evolutionary developmental biology: A -reference guide. Cham: Springer International Publishing. p. 1--12. - -\hypertarget{ref-jones2015}{} -Jones N.T., Germain R.M., Grainger T.N., Hall A.M., Baldwin L., Gilbert -B. 2015. Dispersal mode mediates the effect of patch size and patch -connectivity on metacommunity diversity. Journal of Ecology. -103:935--944. - -\hypertarget{ref-lalibertuxe92010}{} -Laliberté É., Legendre P. 2010. A distance-based framework for measuring -functional diversity from multiple traits. Ecology. 91:299--305. - -\hypertarget{ref-legendre2012}{} -Legendre P., Legendre L.F. 2012. Numerical ecology. Elsevier. - -\hypertarget{ref-mammola2019}{} -Mammola S. 2019. Assessing similarity of n-dimensional hypervolumes: -Which metric to use? Journal of Biogeography. 0. - -\hypertarget{ref-marcy2016}{} -Marcy A.E., Hadly E.A., Sherratt E., Garland K., Weisbecker V. 2016. -Getting a head in hard soils: Convergent skull evolution and divergent -allometric patterns explain shape variation in a highly diverse genus of -pocket gophers (thomomys). BMC evolutionary biology. 16:207. - -\hypertarget{ref-oksanen2007vegan}{} -Oksanen J., Kindt R., Legendre P., O'Hara B., Stevens M.H.H., Oksanen -M.J., Suggests M. 2007. The vegan package. Community ecology package. -10:631--637. - -\hypertarget{ref-psych}{} -Revelle W. 2018. Psych: Procedures for psychological, psychometric, and -personality research. Evanston, Illinois: Northwestern University. - -\hypertarget{ref-ruta2013}{} -Ruta M., Angielczyk K.D., Fröbisch J., Benton M.J. 2013. Decoupling of -morphological disparity and taxic diversity during the adaptive -radiation of anomodont therapsids. Proceedings of the Royal Society of -London B: Biological Sciences. 280. - -\hypertarget{ref-sedgewick1990}{} -Sedgewick R. 1990. Algorithms in c. Addison-Wesley, Reading. - -\hypertarget{ref-villuxe9ger2008}{} -Villéger S., Mason N.W.H., Mouillot D. 2008. New multidimensional -functional diversity indices for a multifaceted framework in functional -ecology. Ecology. 89:2290--2301. - -\hypertarget{ref-wills2001}{} -Wills M.A. 2001. Morphological disparity: A primer. In: Adrain J.M., -Edgecombe G.D., Lieberman B.S., editors. Fossils, phylogeny, and form. -Springer US. p. 55--144. - -\hypertarget{ref-wright2017}{} -Wright D.F. 2017. Phenotypic innovation and adaptive constraints in the -evolutionary radiation of palaeozoic crinoids. Scientific Reports. -7:13745. - - -\end{document} diff --git a/inst/shiftingspace_supplementary_all_results.Rmd b/inst/shiftingspace_supplementary_all_results.Rmd index 12c0b97..f7fcac1 100644 --- a/inst/shiftingspace_supplementary_all_results.Rmd +++ b/inst/shiftingspace_supplementary_all_results.Rmd @@ -347,7 +347,7 @@ print.fable <- function(n_metrics, byrow, ncol, test) { print.fable(length(all_metrics), byrow = TRUE, ncol = 3, test = TRUE) ``` -### All measures results 20% removal +# Table 1: All measures results 20% removal ```{r fable_all_metrics_02, echo = FALSE, eval = TRUE} ## Changing defaults @@ -384,7 +384,7 @@ Metric | Size change | Density change | Position change | Distribution eff `r name[25]` | `r plot.id(73)` | `r plot.id(74)` | `r plot.id(75)` | `r s.test(25, "s", res)` | `r s.test(25, "r", res)`| -### All measures results 50% removal +# Table 2: All measures results 50% removal ```{r fable_all_metrics_05, echo = FALSE, eval = TRUE} ## Changing defaults @@ -420,7 +420,7 @@ Measurements | Size change | Density change | Position change | Distributi `r name[24]` | `r plot.id(70)` | `r plot.id(71)` | `r plot.id(72)` | `r s.test(24, "s", res)` | `r s.test(24, "r", res)`| `r name[25]` | `r plot.id(73)` | `r plot.id(74)` | `r plot.id(75)` | `r s.test(25, "s", res)` | `r s.test(25, "r", res)`| -### All measures results 80% removal +# Table 3: All measures results 80% removal ```{r fable_all_metrics_08, echo = FALSE, eval = TRUE} ## Changing defaults diff --git a/inst/shiftingspace_supplementary_sampling.Rmd b/inst/shiftingspace_supplementary_sampling.Rmd deleted file mode 100644 index 73a85ac..0000000 --- a/inst/shiftingspace_supplementary_sampling.Rmd +++ /dev/null @@ -1,172 +0,0 @@ ---- -title: "Shifting spaces: which disparity or dissimilarity measurement best summarise occupancy in multidimensional spaces?" -author: "Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker" -date: "`r Sys.Date()`" -output: - pdf_document: - fig_width: 12 - fig_height: 6 ---- - -# Supplementary material 5: effect of sampling on space occupancy measurements - -```{r, echo = FALSE, message = FALSE, warning = FALSE, results = 'hide'} -## Loading the packages -if(!require(devtools)) install.packages("devtools") -if(!require(knitr)) install.packages("knitr"); library(knitr) -if(!require(rmarkdown)) install.packages("rmarkdown"); library(rmarkdown) -if(!require(ape)) install.packages("ape"); library(ape) -if(!require(dispRity)) install.packages("dispRity"); library(dispRity) -if(packageVersion("dispRity") < "1.2.4") { - ## dispRity must be above v1.2.3 - devtools::install_github("TGuillerme/dispRity"); library(dispRity) -} -if(!require(moms)) devtools::install_github("TGuillerm/moms"); library(moms) - -## Setting the default parameters for the space plots -defaults <- list(pch = 20, - xlim = c(-3, 3), - ylim = c(-3, 3), - col1 = "grey", - col2 = "black", - xlab = "Trait", - ylab = "Trait", - cex = 1) -## Generating the default palette -default.palette <- function(n) { - hues = seq(15, 375, length = n + 1) - grDevices::hcl(h = hues, l = 65, c = 100)[1:n] -} -``` - -In this section, we look at the effect of sampling on the 25 selected space occupancy measurements. -We modified the protocol presented in the main text as follows: - -1. We simulated 13 different spaces with different sets of parameters; -2. We transformed these spaces by removing 50% of the observations following four different scenarios corresponding to different empirical scenarios: randomly, by limit (e.g. expansion or reduction of niches), by density (e.g. different degrees of competition within a guild) and by position (e.g. ecological niche shift). -2-bis. We reduced the sampling of the 50% selected elements by selecting 100% of them (50% of the total), 80% (40% of the total), 60% (30% of the total), 40% (20% of the total) and 20% (10% of the total). -3. We measured occupancy on the resulting transformed spaces using `r metric.names()` different space occupancy measures; -4. We applied the same space occupancy measures to six empirical datasets with the similar re-sampling strategy as in point 2-bis. - -# Data - -## Simulations - -```{r start_simulation, echo = TRUE, eval = FALSE} -## Set the overall number of replicates -n_replicates <- 20 - -## Set the overall number of elements -#elements <- function() sample(20:200) #possibility to use a variable number of elements -elements <- function() 200 - -## Simulation seed -set.seed(42) -``` - -```{r generate_spaces_simulation, echo = TRUE, warning = FALSE, results = 'hide', eval = FALSE} -## List of distributions -distributions_list <- list( - "unifor" = list(distribution = runif, arguments = list(list("min" = -0.5, - "max" = 0.5))), - "normal" = list(distribution = rnorm), - "random" = list(distribution = "random"), - "unicor" = list(distribution = runif, arguments = list(list("min" = -0.5, - "max" = 0.5)), - cor.matrix = "random"), - "norcor" = list(distribution = rnorm, cor.matrix = "random"), - "pcalik" = list(distribution = rnorm, scree = "lognormal"), - "pcolik" = list(distribution = rnorm, scree = "normal") - ) - -## Generate all 13 spaces -all_spaces <- list(uniform3 = space.simulation(distributions_list$unifor, - dimensions = 3, - elements = elements(), - replicates = n_replicates), - uniform15 = space.simulation(distributions_list$unifor, - dimensions = 15, - elements = elements(), - replicates = n_replicates), - uniform50 = space.simulation(distributions_list$unifor, - dimensions = 50, - elements = elements(), - replicates = n_replicates), - uniform100 = space.simulation(distributions_list$unifor, - dimensions = 100, - elements = elements(), - replicates = n_replicates), - uniform150 = space.simulation(distributions_list$unifor, - dimensions = 150, - elements = elements(), - replicates = n_replicates), - uniformc50 = space.simulation(distributions_list$unicor, - dimensions = 50, - elements = elements(), - replicates = n_replicates), - normal3 = space.simulation(distributions_list$normal, - dimensions = 3, - elements = elements(), - replicates = n_replicates), - normal15 = space.simulation(distributions_list$normal, - dimensions = 15, - elements = elements(), - replicates = n_replicates), - normal50 = space.simulation(distributions_list$normal, - dimensions = 50, - elements = elements(), - replicates = n_replicates), - uniform100 = space.simulation(distributions_list$unifor, - dimensions = 100, - elements = elements(), - replicates = n_replicates), - normal150 = space.simulation(distributions_list$normal, - dimensions = 150, - elements = elements(), - replicates = n_replicates), - normalc50 = space.simulation(distributions_list$norcor, - dimensions = 50, - elements = elements(), - replicates = n_replicates), - random50 = space.simulation(distributions_list$random, - dimensions = 50, - elements = elements(), - replicates = n_replicates), - pca_like = space.simulation(distributions_list$pcalik, - dimensions = 50, - elements = elements(), - replicates = n_replicates), - pco_like = space.simulation(distributions_list$pcolik, - dimensions = 50, - elements = elements(), - replicates = n_replicates)) - -#TG: Warnings are due to inexact correlations -``` - - -```{r shift_spaces_simulation, echo = TRUE, warning = FALSE, results = 'hide', eval = FALSE} -## Shifting each space -shift_groups <- lapply(all_spaces, lapply, shift.group.simulation, remove = 0.5) -``` - -```{r get_disparity_simulation, echo = TRUE, warning = FALSE, results = 'hide', eval = FALSE} -## Measuring disparity -remove_05 <- mapply(metrics.simulation, all_spaces, shift_groups, - MoreArgs = list(metrics = metrics_list, rare.dim = NULL, - verbose = TRUE), - SIMPLIFY = FALSE) - -## Saving the results -save.results(remove_05) -``` - -## Empirical data - - -```{r} -## Results list -save.results(empirical_results) -``` - -# Results diff --git a/inst/shiftingspace_text_v.tex b/inst/shiftingspace_text_v.tex deleted file mode 100644 index 1624285..0000000 --- a/inst/shiftingspace_text_v.tex +++ /dev/null @@ -1,1622 +0,0 @@ -\documentclass[]{article} -\usepackage{lmodern} -\usepackage{lineno} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\usepackage{fixltx2e} % provides \textsubscript -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} -\fi -% use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -% use microtype if available -\IfFileExists{microtype.sty}{% -\usepackage{microtype} -\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\usepackage[margin=1in]{geometry} -\usepackage{hyperref} -\hypersetup{unicode=true, - pdftitle={Shifting spaces: which disparity or dissimilarity metrics best summarise occupancy in multidimensional spaces?}, - pdfauthor={Thomas Guillerme, Mark N. Puttick, Ariel E. Marcy, Vera Weisbecker}, - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage{longtable,booktabs} -\usepackage{graphicx,grffile} -\makeatletter -\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} -\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} -\makeatother -% Scale images if necessary, so that they will not overflow the page -% margins by default, and it is still possible to overwrite the defaults -% using explicit options in \includegraphics[width, height, ...]{} -\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -%%% Use protect on footnotes to avoid problems with footnotes in titles -\let\rmarkdownfootnote\footnote% -\def\footnote{\protect\rmarkdownfootnote} - -%%% Change title format to be more compact -\usepackage{titling} - -% Create subtitle command for use in maketitle -\providecommand{\subtitle}[1]{ - \posttitle{ - \begin{center}\large#1\end{center} - } -} - - -\linespread{2} - - -\setlength{\droptitle}{-2em} - - \title{Shifting spaces: which disparity or dissimilarity metrics best summarise -occupancy in multidimensional spaces?} - \pretitle{\vspace{\droptitle}\centering\huge} - \posttitle{\par} - \author{Thomas Guillerme$^{1,*}$, Mark N. Puttick$^{2}$, Ariel E. Marcy$^{1}$, Vera Weisbecker$^{1}$} - \preauthor{\centering\large\emph} - \postauthor{\par} - \predate{\centering\large\emph} - \postdate{\par} - \date{2019-10-06} - - -\begin{document} - -\modulolinenumbers[1] % just after the \begin{document} tag -\linenumbers - -\maketitle - -\noindent {\small \it -$^1$School of Biological Sciences, University of Queensland, St. Lucia, Queensland, Australia.\\ -$^2$School of Earth Sciences, University of Bristol, Wills Memorial Building, Queen's Road, Bristol BS8 1RJ, UK.\\} - - -\section{Abstract}\label{abstract} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Multidimensional analysis of traits are now a common toolkit in - ecology and evolution and are based on trait-spaces in which each - dimension summarise the observed trait combination (a morphospace or - an ecospace). Observations of interest will typically occupy a subset - of this trait-space, and researchers will apply one or more metrics to - quantify the way in which organisms ``inhabit'' that trait-space. In - macroevolution and ecology these metrics are referred to as disparity - or dissimilarity metrics and can be generalised as space occupancy - metrics. Researchers use these metrics to investigate how space - occupancy changes through time, in relation to other groups of - organisms, and in response to global environmental changes, such as - global warming events or mass extinctions. However, the mathematical - and biological meaning of most space occupancy metrics is vague with - the majority of widely-used metrics lacking formal description. -\item - Here we propose a broad classification of space occupancy metrics into - three categories that capture changes in volume, density, or position. - We analyse the behaviour of 25 metrics to study changes in trait-space - volume, density and position on a series of simulated and empirical - datasets. -\item - We find no one metric describes all of trait-space but that some - metrics are better at capturing certain aspects compared to other - approaches and that their performance depends on both the trait-space - and the hypothesis analysed. However, our results confirm the three - broad categories (volume, density and position) and allow to relate - changes in any of these categories to biological phenomena. -\item - Since the choice of space occupancy metric should be specific to the - data and question at had, we introduced - \href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a - user-friendly tool based on a graphical interface that allows users to - both visualise and measure changes space occupancy for any metric in - simulated or imported trait-spaces. Users are also provided with tools - to transform their data in space (e.g.~contraction, displacement, - etc.). This tool is designed to help researchers choose the right - space occupancy metrics, given the properties of their trait-space and - their biological question. -\end{enumerate} - -\section{Introduction}\label{introduction} - -Groups of species and environments share specific, easily recognisable, -correlated characteristics of many kinds: guilds or biomes with shared -phenotypic, physiological, phylogenetic or behavioural traits. Organisms -or environments should therefore be studied as a set of traits rather -than some specific traits in isolation (Donohue et al. 2013; Hopkins and -Gerber 2017). Biologists have increasingly been using ordination -techniques (see Legendre and Legendre 2012 for a summary) to create -multidimensional trait-spaces to either explore properties of the data -or test hypotheses (Oksanen et al. 2007; Adams and Otárola-Castillo -2013; Bonhomme et al. 2014; Blonder 2018; Guillerme 2018). For example, -in palaeobiology, Wright (2017) use trait-spaces to study how groups of -species' characteristics change through time; in ecology, Jones et al. -(2015) study evidence of competition by looking at trait overlap between -two populations. However, different fields use a different set of terms -for such approaches (Table 1). Nonetheless, they are the same -mathematical objects: matrices with columns representing an original or -transformed trait value and rows representing observations, such as -taxon, field site, etc. (Guillerme 2018). - - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}llll@{}} -\toprule -\begin{minipage}[b]{0.20\columnwidth}\raggedright\strut -In mathematics\strut -\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut -In ecology\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -In macroevolution\strut -\end{minipage} & \begin{minipage}[b]{0.20\columnwidth}\raggedright\strut -In this paper\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -Matrix (\(n \times d\))\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -Function-space, Eco-space, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -Morphospace, traitspace, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -trait-space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -Rows (\emph{n})\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -Taxa, field sites, environments, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -Taxa, specimen, populations, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -observations\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -Columns (\emph{d})\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -Traits, Ordination scores, distances, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -dimensions\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -Matrix subset (\(m \times d\); \(m \leq n\))\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -Treatments, phylogenetic group (clade), etc.\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -Clades, geological stratum, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -group\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -Statistic\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -Dissimilarity index or metric, hypervolume, functional diversity\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -Disparity metric or index\strut -\end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -space occupancy metric\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -Multidimensional analysis\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -Dissimilarity analysis, trait analysis, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -Disparity analysis, disparity-through-time, etc.\strut -\end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright\strut -multidimensional analysis\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 1: terms and equivalence between mathematics, ecology and -macroevolution. - -\renewcommand\baselinestretch{2}\selectfont - -Ecologists and evolutionary biologists also often use trait-spaces with -respect to the same fundamental questions: are groups overlapping in the -trait-space? Are some regions of the trait-space not occupied? How do -specific factors influence the occupancy of the trait-space? Studying -the occupancy of trait-spaces can be achieved using disparity metrics in -macroevolution (Wills 2001; Hopkins and Gerber 2017; Guillerme 2018) or -comparing hypervolumes in ecology (Donohue et al. 2013; Díaz et al. -2016; Blonder 2018; Mammola 2019). However, although these space -occupancy metrics are common in ecology and evolution, surprisingly -little work has been published on their behaviour (but see Ciampaglio et -al. 2001; Villéger et al. 2008; Mammola 2019). - -Different space occupancy metrics capture different aspects of the -trait-space (ciampaglio2001; Villéger et al. 2008; Mammola 2019). It may -be widely-known by many in the field, but to our knowledge this is -infrequently mentioned in peer-reviewed papers. First, space occupancy -metrics are often named as the biological aspect they are describing -(e.g. ``disparity'' or ``functional diversity'') rather than what they -are measuring (e.g.~the average pairwise distances) which obscures the -differences or similarities between studies. Second, in many studies in -ecology and evolution, authors have focused on measuring the volume of -the trait-space with different metrics (e.g.~ellipsoid volume Donohue et -al. 2013; hypervolume Díaz et al. 2016; Procrustes variance Marcy et al. -2016; product of variance Wright 2017). However, volume only represents -a single aspects of space occupancy, disregarding others such as the -density (Harmon et al. 2008) or position (Wills 2001; Ciampaglio et al. -2001). For example, if two groups occupy the same volume in trait-space, -this will lead to supporting certain biological conclusions. Yet, an -alternative aspect of space occupancy may indicate that the groups' -position are different; this would likely lead to a different biological -conclusion (e.g.~the groups are equally diverse but occupy different -niches). Using metrics that only measure one aspect of the -multidimensional trait-space may restrain the potential of -multidimensional analysis (Villéger et al. 2008). - -Here we propose a broad classification of space occupancy metrics as -used across ecology and evolution and analyse their power to detect -changes in trait-space occupancy in simulated and empirical data. We -provide an assessment of each broad type of space occupancy metrics -along with a unified terminology to foster communication between ecology -and evolution. Unsurprisingly, we found no one metric describes all -changes through a trait-space and the results from each metric are -dependent on the characteristics of the space and the hypotheses. -Furthermore, because there can potentially be an infinite number of -metrics, it would be impossible to propose clear generalities to space -occupancy metrics behavior. Therefore, we propose -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}, a -user-friendly tool allowing researchers to design, experiment and -visualise their own space occupancy metric tailored for their specific -project and helping them understanding the ``null'' behavior of their -metrics of interest. - -\subsection{Space occupancy metrics}\label{space-occupancy-metrics} - -In this paper, we define trait-spaces as any matrix where rows are -observations and columns are traits. These traits can widely vary in -number and types: they could be coded as discrete (e.g.~presence or -absence of a bone; Beck and Lee 2014; Wright 2017), continuous -measurements (e.g.~leaf area; Díaz et al. 2016) or more sophisticated -measures (Fourier ellipses; Bonhomme et al. 2014; e.g.~landmark -position; Marcy et al. 2016). Traits can also be measured by using -relative observations (e.g.~community compositions; Jones et al. 2015) -or distance between observations (e.g. Close et al. 2015). However, -regardless of the methodology used to build a trait-space, three broad -occupancy metrics can be measured: the volume which will approximate the -amount of space occupied, the density which will approximate the -distribution in space and the position which will approximate the -location in space (Fig. 1; Villéger et al. 2008). Of course any -combination of these three aspects is always possible. - -\renewcommand\baselinestretch{1}\selectfont -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_metrics_types-1.pdf} -\caption{different type of information captured by space -occupancy metrics. A - Volume (e.g.~sum of ranges); B - Density -(e.g.~average squared pairwise distances); C - Position (e.g.~median -distance from centroid).} -\end{figure} -\renewcommand\baselinestretch{2}\selectfont - - - -\paragraph{1. Volume}\label{volume} - -Volume metrics measure the spread of a group in the trait-space. They -can be interpreted as the amount of the trait-space that is occupied by -observations. Typically, larger values for such metrics indicate the -presence of more extreme trait combinations. For example, if group A has -a bigger volume than group B, the observations in A achieve more extreme -trait combinations than in B. This type of metric is widely used in both -ecology (e.g.~the hypervolume; Blonder 2018) and in evolution (e.g.~the -sum or product of ranges or variances; Wills 2001). - -Although volume metrics are a suitable indicator for comparing a group's -trait-space occupancy, it is limited to comparing the range of -trait-combinations between groups. Volume metrics do not take into -account the distribution of the observations within a group. In other -words, they can make it difficult to determine whether all the -observations are on the edge of the volume or whether the volume is -simply driven by small number of extreme observations. - -\paragraph{2. Density}\label{density} - -Density metrics measure the distribution of a group in the trait-space. -They can be interpreted as the distribution of the observations -\emph{within} a group in the trait-space. Groups with higher density -have observations within it that tend to be more similar to each other. -For example, if group A has a greater volume than group B but both have -the same density, similar mechanisms could be driving both groups' -trait-space occupancy. However, this might suggest that A is older and -had more time to achieve more extreme trait combinations under -essentially the same process (Endler et al. 2005). Density is less -commonly measured compared to volume, but it is still used in both -ecology (e.g.~the minimum spanning tree length; Oksanen et al. 2007) and -evolution (e.g.~the average pairwise distance; Harmon et al. 2008). - -\paragraph{3. Position}\label{position} - -Position metrics measure where a group lies in trait-space. They can be -interpreted as where a group lies in the trait-space either relative to -the space itself or relative to another group. For example, if group A -has a different position than group B, observations in A will have a -different trait-combination than B. - -Position metrics may be harder to interpret in multidimensional spaces -(i.e.~beyond left/right, up/down and front/back). However, when thinking -about unidimensional data (one trait), this metric is obvious: two -groups A or B could have the same variance (i.e. ``volume'' or spread) -with the same number of observations (i.e.~density) but could have a -different mean and thus be in different positions. These metrics have -been used in ecology to compare the position of two groups relatively to -each other (Mammola 2019). - -\subsection{No metric to rule them all: benefits of considering multiple -metrics}\label{no-metric-to-rule-them-all-benefits-of-considering-multiple-metrics} - -The use of multiple metrics to assess trait-space occupancy has the -benefit of providing a more detailed characterisation of occupancy -changes. If the question of interest is, say, to look at how space -occupancy changes in response to mass extinction, using a single space -occupancy metric can miss part of the picture: a change in volume could -be decoupled from a change in position or density in trait-space. For -example, the Cretaceous-Palaeogene extinction (66 million years ago - -Mya) has been linked to an increase in volume of the mammalian -trait-space (adaptive radiation; Halliday and Goswami 2016) but more -specific questions can be answered by looking at other aspects of -trait-space occupancy: does the radiation expands on previously existing -morphologies (elaboration, increase in density; Endler et al. 2005) or -does it explore new regions of the trait-space (innovation, change in -position; Endler et al. 2005)? Similarly, in ecology, if two groups -occupy the same volume in the trait-space, it can be interesting to look -at differences in density within these two groups: different selection -pressure can lead to different density within equal volume groups. - -Here, we provide the first interdisciplinary review of 25 space -occupancy metrics that uses the broad classification of metrics into -volume, density and position to capture pattern changes in trait-space. -We assess the behaviour of metrics using simulations and six -interdisciplinary empirical datasets covering a wide range of potential -data types and biological questions. We also introduce a tool for -measuring occupancy in multidimensional space -(\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}), which is -a user-friendly, open-source, graphical interface to allow the tailored -testing of metric behaviour for any use case. -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} will allow -workers to comprehensively assess the properties of their trait-space -and the metrics associated with their specific biological question. - -\section{Methods}\label{methods} - -We tested how 25 different space occupancy metrics relate to each other, -are affected by modifications of traits space and affect group -comparisons in empirical data. To do so, we performed the following -steps (explained in more detail below): - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - We simulated 13 different spaces with different sets of parameters; -\item - We transformed these spaces by removing 50\% of the observations - following four different scenarios corresponding to different - empirical scenarios: randomly, by limit (e.g.~expansion or reduction - of niches), by density (e.g.~different degrees of competition within a - guild) and by position (e.g.~ecological niche shift). -\item - We measured occupancy on the resulting transformed spaces using eight - different space occupancy metrics; -\item - We applied the same space occupancy metrics to six empirical datasets - (covering a range of disciplines and a range of dataset properties). -\end{enumerate} - -Note that the paper contains the results for only eight metrics, the -results for the additional 17 metrics is available in the supplementary -material 4. - -\subsection{Generating spaces}\label{generating-spaces} - -We generated trait-spaces using the following combinations of size, -distributions, variance and correlation: - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lllll@{}} -\toprule -space name & size & distribution(s) & dimensions variance & -correlation\tabularnewline -\midrule -\endhead -Uniform3 & 200*3 & Uniform & Equal & None\tabularnewline - & &(min = -0.5, max = 0.5)& & \tabularnewline -Uniform15 & 200*15 & Uniform & Equal & None\tabularnewline -Uniform50 & 200*50 & Uniform & Equal & None\tabularnewline -Uniform150 & 200*150 & Uniform & Equal & None\tabularnewline -Uniform50c & 200*50 & Uniform & Equal & Random\tabularnewline - & & & & (between 0.1 and 0.9)\tabularnewline -Normal3 & 200*3 & Normal & Equal & None\tabularnewline - & & (mean = 0, sd = 1) & & \tabularnewline -Normal15 & 200*15 & Normal & Equal & None\tabularnewline -Normal50 & 200*50 & Normal & Equal & None\tabularnewline -Normal150 & 200*150 & Normal & Equal & None\tabularnewline -Normal50c & 200*50 & Normal & Equal & Random\tabularnewline - & & & & (between 0.1 and 0.9)\tabularnewline -Random & 200*50 & Normal, Uniform, Lognormal & Equal & None\tabularnewline - & & (meanlog = 0, sdlog = 1) & & \tabularnewline -PCA-like & 200*50 & Normal & Multiplicative & None\tabularnewline -PCO-like & 200*50 & Normal & Additive & None\tabularnewline -\bottomrule -\end{longtable} - -Table 2: different simulated space distributions. - -\renewcommand\baselinestretch{2}\selectfont - -The differences in trait-space sizes (200 \(\times\) 3, 200 \(\times\) -15, 200 \(\times\) 50 or 200 \(\times\) 150) reflects the range of -dimensions in literature: ``low-dimension'' spaces (\(<15\)) are common -in ecology (Mammola 2019) whereas high dimension spaces (\(>100\)) are -common in macroevolution (Hopkins and Gerber 2017). We used a range of -distributions (uniform, normal or random) to test the effect of -observation distributions on the metrics. We used different levels of -variance for each dimensions in the spaces by making the variance on -each dimension either equal -(\(\sigma_{D1} \simeq \sigma_{D2} \simeq \sigma_{Di}\)) or decreasing -(\(\sigma_{D1} < \sigma_{D2} < \sigma_{Di}\)) with the decreasing factor -being either multiplicative (using the cumulative product of the inverse -of the number of dimensions: \(\prod_i^d(1/d)\)) or additive -(\(\sum_i^d(1/d)\)). Both multiplicative and cumulative reductions of -variance are used to illustrate the properties of ordinations where the -variance decreases per dimensions (healy2019; and in a normal way in -Multidimensional Scaling - MDS, PCO or PCoA; e.g. Close et al. 2015; in -a lognormal way in principal components analysis - PCA; e.g. Marcy et -al. 2016; Wright 2017). Finally, we added a correlation parameter to -take into account the potential correlation between different traits. We -repeated the simulation of each trait-space 20 times (resulting in 260 -trait-spaces). - -\subsection{Spatial occupancy metrics}\label{spatial-occupancy-metrics} - -We then measured eight different metrics on the resulting transformed -spaces, including a new metric we produced, the average displacement, -which we expect to be influenced by changes in trait-space position. - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}lccll@{}} -\toprule -\begin{minipage}[b]{0.1\columnwidth}\raggedright\strut -Name\strut -\end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright\strut -Definition\strut -\end{minipage} & \begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Captures\strut -\end{minipage} & \begin{minipage}[b]{0.1\columnwidth}\raggedright\strut -Source\strut -\end{minipage} & \begin{minipage}[b]{0.4\columnwidth}\raggedright\strut -Notes\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Average distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}{d}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Volume\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Laliberté and Legendre (2010)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -equivalent to the functional dispersion (FDis) in Laliberté and Legendre -(2010) (without abundance)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\sigma^{2}{k_i}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Volume\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Wills (2001)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -common metric used in palaeobiology (Ciampaglio et al. 2001)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\sum_{i}^{d}{\|\text{max}(d_{i})-\text{min}(d_{i})\|}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Volume\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Wills (2001)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -more sensitive to outliers than the sum of variances\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\displaystyle\prod_{i}^{d} (\lambda_{i}^{0.5})\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Volume\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Donohue et al. (2013)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -less sensitive to outliers than the convex hull hypervolume (Díaz et al. -2016; Blonder 2018)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum(\text{branch length})}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Oksanen et al. (2007)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -similar to the unscaled functional evenness (Villéger et al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sum\text{min}\left(\frac{\text{branch length}}{\sum\text{branch length}}\right)-\frac{1}{n-1}}{1-\frac{1}{n-1}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Villéger et al. (2008)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -the functional evenness without weighted abundance (FEve; Villéger et -al. 2008)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(min\left(\sqrt{\sum_{i}^{n}{({q}_{i}-p_{i})^2}}\right)\times \frac{1}{n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Density\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Foote (1990)\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -the density of pairs of observations in the trait-space\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Average displacement\strut -\end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright\strut -\(\frac{\sqrt{\sum_{i}^{n}{({k}_{n})^2}}}{\sqrt{\sum_{i}^{n}{({k}_{n}-Centroid_{k})^2}}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Position\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -This paper\strut -\end{minipage} & \begin{minipage}[t]{0.4\columnwidth}\raggedright\strut -the ratio between the observations' position from their centroid and the -centre of the trait-space. A value of 1 indicates that the observations' -centroid is the centre of the trait-space\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Table 3: List of metrics with \emph{n} being the number of observations, -\emph{d} the total number of dimensions, \emph{k} any specific row in -the matrix, \emph{Centroid} being their mean and \(\sigma^{2}\) their -variance. \(\Gamma\) is the Gamma distribution and \(\lambda_{i}\) the -eigen value of each dimension and \({q}_{i}\) and \(p_{i}\) are any -pairs of coordinates. - -\renewcommand\baselinestretch{2}\selectfont - - -\subsection{Metric comparisons}\label{metric-comparisons} - -We compared the space occupancy metrics correlations across all -simulations between each pair of metrics to assess they captured signal -(Villéger et al. 2008; Laliberté and Legendre 2010). We used the metrics -on the full 13 trait-spaces described above. We then scaled the results -and measured the pairwise Pearson correlation to test whether metrics -were capturing a similar signal (high positive correlation), a different -signal (correlation close to 0) or an opposite signal (high negative -correlations) using the \texttt{psych} package (Revelle 2018). - -\subsection{Changing space}\label{changing-spaces} - -To measure how the metrics responded to changes within trait-spaces, we -removed 50\% of observations each time using the following algorithms: - -\begin{itemize} -\item - \textbf{Randomly:} by randomly removing 50\% of observations (Fig. - 2-A). This reflects a ``null'' biological model of changes in - trait-space: the case when observations are removed regardless of - their intrinsic characteristics. For example, if diversity is reduced - by 50\% but the trait-space volume remains the same, there is a - decoupling between diversity and space occupancy (Ruta et al. 2013). - Our selected metrics are expected to not be affected by this change. -\item - \textbf{Limit:} by removing all observations with a distance from the - centre of the trait-space lower or greater than a radius \(\rho\) - (where \(\rho\) is chosen such that 50\% observations are selected) - generating two limit removals: \emph{maximum} and \emph{minimum} - (respectively in orange and blue; Fig. 2-B). This can reflect a strict - selection model where observations with trait values below or above a - threshold are removed leading to an expansion or a contraction of the - trait-space. Volume metrics are expected to be most affected by this - change. -\item - \textbf{Density:} by removing any pairs of point with a distance \(D\) - from each other where (where \(D\) is chosen such that 50\% - observations are selected) generating two density removals: - \emph{high} and \emph{low} (respectively in orange and blue; Fig. - 2-C). This can reflect changes within groups in the trait-space due to - ecological factors (e.g.~competition can generate niche repulsion - - lower density; Grant and Grant 2006). Density metrics are expected to - be most affected by this change. -\item - \textbf{Position:} by removing points similarly as for \textbf{Limit} - but using the distance from the furthest point from the centre - generating two position removals: \emph{positive} and \emph{negative} - (respectively in orange and blue; Fig. 2-D). This can reflect global - changes in trait-space due, for example, to an entire group remaining - diverse but occupying a different niche. Position metrics are expected - to be most affected by this change. -\end{itemize} - -The algorithm to select \(\rho\) or \(D\) is described in greater detail -in in the Supplementary material 1. - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_reduce_space-1.pdf} -\caption{} -\end{figure} - - -To measure the effect of space reduction, distribution and -dimensionality on the metric, we scaled the metric to be relative to the -non-reduced space for each dimension distribution or number of -dimensions. We subtracted the observed occupancy with no space reduction -to all the occupancy measurements of the reduced spaces and then divided -it by the resulting maximum observed occupancy. Our occupancy metrics -where scaled between -1 and 1 with a value of 0 indicating no effect of -the space reduction and \(>0\) and \(<0\) respectively indicating an -increase or decrease in the occupancy metric value. We then measured the -probability of overlap of the between the non-random removals (limit, -density and position) and the random removals using the Bhattacharrya -Coefficient (probability of overlap between two distributions; -Bhattacharyya 1943). - -\newpage -\renewcommand\baselinestretch{1}\selectfont - -Figure 2: different type of space reduction. Each panel -displays two groups of 50\% of the data points each. Each group (orange -and blue) are generated using the following algorithm: A - randomly; B - -by limit (maximum and minimum limit); C - by density (high and low); and -D - by position (positive and negative). Panel E represents a typical -display of the reduction results displayed in Table 5: the dots -represent the median space occupancy values across all simulations for -each scenario of trait-space change (Table 2), the solid and dashed line -respectively the 50\% and 95\% confidence intervals. Results in grey are -the random 50\% reduction (panel A). Results in blue and orange -represent the opposite scenarios from panels B, C, and D. The displayed -value is the probability of overlap (Bhattacharrya Coefficient) between -the blue or orange distributions and the grey one. - -\renewcommand\baselinestretch{2}\selectfont - -\subsubsection{Measuring the effect of space and dimensionality on -shifting -spaces}\label{measuring-the-effect-of-space-and-dimensionality-on-shifting-spaces} - -Distribution differences and the number of dimensions can have an effect -on the metric results. For example, in a normally distributed space, a -decrease in density can often lead to an increase in volume. This is not -necessarily true in log-normal spaces or in uniform spaces for certain -metrics. Furthermore, high dimensional spaces (\textgreater{}10) are -subject to the ``curse of multidimensionality'' (Chávez et al. 2001): -data becomes sparser with increasing number of dimensions, such that the -probability of two points A and B overlapping in \emph{n} dimensions is -the product of the probability of the two points overlapping on each -dimensions(\(\prod_{i}^{d} P(A = B)_{Di}\)). This probability decreases -as a product of the number of dimensions. Therefore, the ``curse'' can -make the interpretation of high dimensional data counter-intuitive. For -example if a group expands in multiple dimensions (i.e.~increase in -volume), the actual hypervolume can decrease (Fig. 3 and Tables 6, 7). - -We measured the effect of space distribution and dimensionality using an -ANOVA (\(occupancy \sim distribution\) and -\(occupancy \sim dimensions\)) by using all spaces with 50 dimensions -and the uniform and normal spaces with equal variance and no correlation -with 3, 15, 50, 100 and 150 dimensions (Table 2) for testing -respectively the effect of distribution and dimensions. The results of -the ANOVAs (\emph{p}-values) are reported in Table 5 (see supplementary -material 1 for the full ANOVA result tables). - -\subsection{Empirical examples}\label{empirical-examples} - -We analysed the effect of the different space occupancy metrics on six -different empirical studies covering a broad range of fields that employ -trait-space analyses (palaeobiology, macroevolution, evo-devo, ecology, -etc.). For each of these studies we generated trait-spaces from the data -published with the papers. We divided each trait-spaces into two -biologically-relevant groups and tested whether the metrics -differentiated the groups in different ways. Both the grouping and the -questions where based on a simplified version of the topics of these -papers (with no intention to re-analyse the data but to be -representative of the diversity of questions in ecology and evolution). -The procedures to generate the data and the groups varies from one study -to the other but is detailed and fully reproducible in the supplementary -materials 2. - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}llllllll@{}} -\toprule -\begin{minipage}[b]{0.06\columnwidth}\raggedright\strut -study\strut -\end{minipage} & \begin{minipage}[b]{0.08\columnwidth}\raggedright\strut -field\strut -\end{minipage} & \begin{minipage}[b]{0.1\columnwidth}\raggedright\strut -taxonomic Group\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -traits (data)\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -trait-space\strut -\end{minipage} & \begin{minipage}[b]{0.06\columnwidth}\raggedright\strut -size\strut -\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut -groups (orange/blue in Table 6)\strut -\end{minipage} & \begin{minipage}[b]{0.15\columnwidth}\raggedright\strut -type of question\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Beck and Lee (2014)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Mammalia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -106*105\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -52 crown vs.~54 stem\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are crown mammals more disparate than stem mammals?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Wright (2017)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Palaeontology\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Crinoidea\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -discrete morphological phylogenetic data\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -42*41\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -16 before vs.~23 after\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Is there a difference in disparity before and after the Ordovician mass -extinction?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Marcy et al. (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Rodentia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -skull 2D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -454*134\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -225 \emph{Megascapheus} vs.~229 \emph{Thomomys}\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are two genera of gopher morphologically distinct?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Hopkins and Pearson (2016)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Evolution\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Trilobita\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -3D landmark coordinates\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Procrustes Superimposition (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -46*46\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -36 adults vs.~10 adults\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Are juvenile trilobites a subset of adult ones?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Jones et al. (2015)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Plantae\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Communities species compositions\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of a Jaccard distance matrix (PCO)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -48*47\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -24 aspens vs.~24 grasslands\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Is there a difference in species composition between aspens and -grasslands?\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -Healy et al. (2019)\strut -\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut -Ecology\strut -\end{minipage} & \begin{minipage}[t]{0.1\columnwidth}\raggedright\strut -Animalia\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -Life history traits\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -Ordination of continuous traits (PCA)\strut -\end{minipage} & \begin{minipage}[t]{0.06\columnwidth}\raggedright\strut -285*6\strut -\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut -83 ecthotherms vs.~202 endotherms\strut -\end{minipage} & \begin{minipage}[t]{0.15\columnwidth}\raggedright\strut -Do endotherms have more diversified life history strategies than -ectotherms?\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -\renewcommand\baselinestretch{2}\selectfont - -Table 4: details of the six empirical trait-spaces. - -For each empirical trait-space we bootstrapped each group 500 times -(Guillerme 2018) and applied the eight space occupancy metric to each -pairs of groups. We then compared the means of each groups using the -Bhattacharrya Coefficient (Bhattacharyya 1943). - -\section{Results}\label{results} - -\subsection{Metric comparisons}\label{metric-comparisons-1} - -\begin{figure} -\centering -\includegraphics{shiftingspace_files/figure-latex/fig_metric_correlation-1.pdf} -\caption{pairwise correlation between the scaled metrics. -Numbers on the upper right corner are the Pearson correlations. The red -line are linear regressions (with the confidence intervals in grey).} -\end{figure} - -All the metrics were either positively correlated (Pearson correlation -of 0.99 for the average distance from centroid and sum of variance or -0.97 for the average nearest neighbour distance and minimum spanning -tree average length; Fig. 3) or somewhat correlated (ranging from 0.66 -for the sum of variances and the ellipsoid volume to -0.09 between the -average displacement and the average distance from centroid; Fig. 3). -All metrics but the ellipsoid volume were normally (or nearly normally) -distributed (Fig. 3). More comparisons between metrics are available in -the supplementary materials 3. - -\subsection{Space shifting}\label{space-shifting} - -\renewcommand\baselinestretch{1}\selectfont - -\begin{longtable}[]{@{}llllll@{}} -\toprule -\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut -Metric\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Volume change\strut -\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut -Density change\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Position change\strut -\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut -Distribution effect\strut -\end{minipage} & \begin{minipage}[b]{0.16\columnwidth}\raggedright\strut -Dimensions effect\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0.449\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0.958\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0.274\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0.873\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0.326\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0.435\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0.207\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0.626\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut -p = 0 ***\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - - -Table 5: Results of the effect of space reduction, space dimension -distributions and dimensions number of the different space occupancy -metrics. See Fig. 2 for interpretation of the figures. \emph{p}-values -for distribution effect and dimensions effect represents respectively -the effect of the ANOVAs space occupancy \textasciitilde{} distributions -and space occupancy \textasciitilde{} dimensions (0 `***' 0.001 `**' -0.01 `*' 0.05 `.' 0.1 '' 1). - -\renewcommand\baselinestretch{2}\selectfont - - -As expected, some different metrics capture different aspects of space -occupancy. However, it can be hard to predict the behaviour of each -metric when 50\% of the observations are removed. We observe a clear -decrease in median metric in less than a third of the space reductions -(10/36). - -In terms of change in volume, only the average distance from centroid -and the sum of variances seem to capture a clear change in both -directions. However, the increase in volume does not correspond to an -\emph{actual} increase in volume in the trait-space (i.e.~the volume -from the blue observations in Fig. 2-B is equivalent to the one in Fig. -2-A). In terms of change in density, only the minimum spanning tree -average distance and the average nearest neighbour distance seem to -capture a clear change in both directions. In terms of change in -position, only the average displacement metric seems to capture a change -a clear change in direction (albeit not in both directions). This is not -surprising, since the notion of positions becomes more and more complex -to appreciate as dimensionality increases (i.e.~beyond left/right, -up/down and front/back). - -\subsection{Empirical example}\label{empirical-example} - -\renewcommand\baselinestretch{1}\selectfont - - -\begin{longtable}[]{@{}lllllll@{}} -\toprule -\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut -Metric\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Beck and Lee 2014\strut -\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut -Wright 2017\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Marcy et al. 2016\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Hopkins and Pearson 2016\strut -\end{minipage} & \begin{minipage}[b]{0.13\columnwidth}\raggedright\strut -Jones et al. 2015\strut -\end{minipage} & \begin{minipage}[b]{0.11\columnwidth}\raggedright\strut -Healy et al. 2019\strut -\end{minipage}\tabularnewline -\hline -\midrule -\endhead -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average distance from centroid\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-1.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-9.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-17.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-25.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-33.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-41.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of variances\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-2.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-10.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-18.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-26.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-34.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-42.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Sum of ranges\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-3.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-11.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-19.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-27.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-35.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-43.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Ellipsoid volume\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-4.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-12.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-20.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-28.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-36.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-44.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree average distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-5.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-13.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-21.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-29.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-37.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-45.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Minimum spanning tree distances evenness\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-6.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-14.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-22.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-30.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-38.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-46.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average nearest neighbour distance\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-7.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-15.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-23.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-31.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-39.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-47.pdf}\strut -\end{minipage}\tabularnewline -\hline -\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut -Average displacements\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-8.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-16.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-24.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-32.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.13\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-40.pdf}\strut -\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut -\includegraphics{shiftingspace_files/figure-latex/fable_results_empirical-48.pdf}\strut -\end{minipage}\tabularnewline -\hline -\bottomrule -\end{longtable} - -Table 6: Comparisons of pairs of groups in different empirical -trait-spaces. NAs are used for cases where space occupancy could not be -measured due to the curse of multidimensionality. The displayed values -are the probability of overlap between both groups (Bhattacharrya -Coefficient). - -\renewcommand\baselinestretch{2}\selectfont - - -Similarly as for the simulated results, the empirical ones indicate that -there is no perfect one-size-fit all metric. For all eight metrics -(expect the ellipsoid volume) we see either one group or the other -having a bigger mean than the other and no consistent case where a group -has a bigger mean than the other for all the metrics. For example, in -the Beck and Lee (2014)'s dataset, there is a clear non-overlap in space -occupancy volume using the average distance from centroid or the sum of -variances (overlaps of respectively 0.175 and 0.159) but no overlap when -measuring the volume using the sum of ranges (0.966). However, for the -Hopkins and Pearson (2016)'s dataset, this pattern is reversed (no clear -differences for the average distance from centroid or the sum of -variances - 0.701 and 0.865 respectively) but a clear difference for the -sum of ranges (0). Furthermore, for each dataset, the absolute -differences between each groups is not consistent depending on the -metrics. For example, in Hopkins and Pearson (2016)'s dataset, the -orange group's mean is clearly higher than the blue one when measuring -the sum of ranges (0) and the inverse is true when measuring the average -displacement (0). - -\section{Discussion}\label{discussion} - -Here we tested 25 metrics of trait-space occupancy on simulated and -empirical datasets to assess how each metric captures changes in -trait-space volume, density and position. Our results show that the -correlation between metrics can vary both within and between metric -categories (Fig. 3), highlighting the importance of understanding the -metric classification for the interpretation of results. Furthermore, -our simulations show that different metrics capture different types of -trait-space change (Table 5), meaning that the use of multiple metrics -is important for comprehensive interpretation of trait-space occupancy. -We also show that the choice of metric impacts the interpretation of -group differences in empirical datasets (Table 6), again emphasizing -that metric choice has a real impact on the interpretation of specific -biological questions - -\paragraph{Metrics comparisons}\label{metrics-comparisons} - -Metrics within the same category of trait-space occupancy (volume, -density or position) do not have the same level of correlation with each -other. For example, the average distance from centroid (volume) is -highly correlated to the sum of variances (volume - correlation of 0.99) -and somewhat correlated with the minimum spanning tree average distance -(density - correlation of 0.66) but poorly with the ellipsoid volume -(volume - correlation of 0.17) and the minimum spanning tree distances -evenness (density - correlation of -0.05). Furthermore, the fact that we -have such a range of correlations for normal distributions suggests that -each metric can capture different summaries of space occupancy ranging -from obvious differences (for metrics not strongly correlated) to subtle -ones (for metrics strongly correlated). - -\paragraph{Space shifting}\label{space-shifting-1} - -Most metrics capture no changes in space occupancy for the ``null'' -(random) space reduction (in grey in Table 5). This is a desirable -behaviour for space occupancy metrics since it will likely avoid false -positive errors in empirical studies that estimate biological processes -from space occupancy patterns (e.g.~competition Brusatte et al. 2008, -convergence Marcy et al. (2016), life history traits Healy et al. -(2019)). However, the average nearest neighbour distance and the sum of -ranges have a respectively positive and negative ``null'' median. This -is not especially a bad property but it should be kept in mind that even -random processes can increase or decrease these metric value. - -Regarding the changes in volume, the sum of variances and the average -distance from centroid are good descriptors (Table 5). However, as -illustrated in the 2D examples in Fig. 2-B only the blue change results -(maximum limit - Table 5) should not result in a direct change in volume -since the trait-space is merely ``hollowed'' out. That said, -``hollowing'' is more hard to conceptualise in many dimensions and the -metrics can still be interpreted for comparing groups (orange has a -smaller volume than blue). - -Regarding changes in density, the average nearest neigbhour distance and -the minimum spanning tree average distance consistently detect changes -in density with more precision for low density trait-spaces (in blue in -Table 5). However, we can observe some degree of correlation between the -changes in density and the changes in volume for most metric picking -either signal. This could be due to the use of normally distributed -spaces where a change in density often leads to a change in volume. This -is not necessary the case with empirical data. - -Regarding the changes in position of the trait-space, all but the -average displacement metric seems to not be able to distinguish between -a random change and a displacement of the trait-space (Table 5). -Furthermore, the average displacement metric does not distinguish -between and positive or a negative displacement of the trait-space: this -might be due to the inherent complexity of \emph{position} in a -multidimensional trait-space. - -\paragraph{Empirical examples}\label{empirical-examples-1} - -Although most differences are fairly consistent within each dataset with -one group having a higher space occupancy score than the other for -multiple metrics, this difference can be more or less pronounced within -each dataset (ranging from no to nearly full overlap - BC -\(\in(0;0.995)\)) and sometimes even reversed. This indicates that -opposite conclusions can be drawn from a dataset depending on which -space occupancy metric is considered. For example, in Wright (2017), -crinoids after the Ordovician mass extinction have a higher median -metric value for all metrics but for the average displacement. These -differences depending on the metrics are also more pronounced in the -empirical datasets where the observations per group are unequal (Hopkins -and Pearson 2016; Healy et al. 2019). \#\#\# Caveats - -While our simulations have been useful to illustrate the behavior of -diverse space occupancy metrics, they have several caveats. First, the -simulated observations in the trait-spaces are independent. This is not -the case in biology where observations can be spatially (Jones et al. -2015) or phylogenetically correlated (e.g. Beck and Lee 2014). Second, -the algorithm used to reduce the trait-spaces might not always -accurately empirical changes. This might favour some specific metrics -over others, in particular for the changes in density that modifies the -nearest neighbour density rather than changing the global density. This -algorithmic choice was made in order to not confound changes in density -along with changes in volume. However, the results presented here -probably capture the general behaviour of each metric since results are -consistent between the simulated and empirical analysis. Furthermore, -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}} allows to -test the caveats mentioned above by uploading empirical trait-space. - -\subsubsection{Suggestions}\label{suggestions} - -We insist that no metric is better than the next one and that -researchers should use the most appropriate metrics based on the metric -and trait-space properties as well as their specific biological -question. However, following the findings of this study we suggest -several points: - -First, we suggest using multiple metrics to tackle different aspects of -the trait-space. This follows the same logical thinking that the mean -might not be sufficient to describe a distribution (e.g.~the variance -might be good additional descriptor). Although using multiple metrics is -not uncommon in macroevolutionary studies (e.g. Halliday and Goswami -2016) or in ecology (Mammola 2019), they often do not cover contrasted -aspects of the trait-space. - -Second, we suggest selecting the metrics that best help answering the -biological question a hand. If one studies an adaptive radiation in a -group of organisms, it is worth thinking what would be the expected null -model: would the group's volume increase (radiation in all directions), -would it increase in density (niche specialisation) or would it shift in -position (radiation into a new set of niches)? - -Third, we suggest to not name metrics as the biological aspect they are -describing (e.g. ``disparity'' or ``functional dispersion'') but rather -what they are measuring (e.g. ``sum of dimensions variance''). We -believe this will allow both a clearer understanding of what \emph{is} -measured and a better communication between ecology and evolution -research where metrics can be similar but have different names (Fig. 3). - -Multidimensional analyses have been acknowledged to be an essential -tool-kit modern biology but can often be counter-intuitive (Chávez et -al. 2001). It is thus crucial to accurately describe patterns in -multidimensional trait-spaces to be able to link them to biological -processes. When summarising trait-spaces, it is important to remember -that a pattern captured by a specific space occupancy metric is often -dependent on the properties of the trait-space and of the particular -biological question of interest. We believe that having a clearer -understanding of both the properties of the trait-space and the -associated space occupancy metrics (e.g.~using -\href{https://tguillerme.shinyapps.io/moms/}{\texttt{moms}}) as well as -using novel space occupancy metrics to answer specific questions will be -of great use to study biological processes in a multidimensional world. - -\section{Acknowledgements}\label{acknowledgements} - -We thank Natalie Jones and Kevin Healy for helping with the empirical -ecological datasets. We acknowledge funding from the Australian Research -Council DP170103227 and FT180100634 awarded to VW. - -\section{Authors contributions}\label{authors-contributions} - -TG, MNP, AEM and VW designed the project. TG and AEM collected the -empirical dataset. TG ran the analyses and designed the software. TG, -MNP, AEM and VW wrote the manuscript. - -\section{Data Availability, repeatability and -reproducibility}\label{data-availability-repeatability-and-reproducibility} - -The raw empirical data is available from the original papers (Beck and -Lee 2014; Jones et al. 2015, Marcy et al. (2016); Hopkins and Pearson -2016; Wright 2017; Healy et al. 2019). The subsets of the empirical data -used in this analysis are available on figshare -\href{https://doi.org/10.6084/m9.figshare.9943181.v1}{DOI: -10.6084/m9.figshare.9943181.v1}. The modified empirical data are -available in the package accompanying this manuscript -(\texttt{data(moms::demo\_data)}). This manuscript (including the -figures, tables and supplementary material) is repeatable and -reproducible by compiling the vignette of the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. -The code for the \texttt{moms} shiny app is available from the -\href{https://github/TGuillerme/moms}{GitHub \texttt{moms\ R} package}. - -\section*{References}\label{references} -\addcontentsline{toc}{section}{References} - -\hypertarget{refs}{} -\hypertarget{ref-adams2013geomorph}{} -Adams D.C., Otárola-Castillo E. 2013. Geomorph: An R package for the -collection and analysis of geometric morphometric shape data. Methods in -Ecology and Evolution. 4:393--399. - -\hypertarget{ref-beck2014}{} -Beck R.M.D., Lee M.S.Y. 2014. Ancient dates or accelerated rates? -Morphological clocks and the antiquity of placental mammals. Proceedings -of the Royal Society B: Biological Sciences. 281:20141278. - -\hypertarget{ref-bhattacharyya1943}{} -Bhattacharyya A. 1943. On a measure of divergence between two -statistical populations defined by their probability distributions. -Bulletin of the Calcutta Mathematical Society. 35:99--109. - -\hypertarget{ref-blonder2018}{} -Blonder B. 2018. Hypervolume concepts in niche-and trait-based ecology. -Ecography. 41:1441--1455. - -\hypertarget{ref-momocs}{} -Bonhomme V., Picq S., Gaucherel C., Claude J. 2014. Momocs: Outline -analysis using R. Journal of Statistical Software. 56:1--24. - -\hypertarget{ref-brusatte2008}{} -Brusatte S.L., Benton M.J., Ruta M., Lloyd G.T. 2008. Superiority, -competition, and opportunism in the evolutionary radiation of dinosaurs. -Science. 321:1485--1488. - -\hypertarget{ref-cursedimensionality}{} -Chávez E., Navarro G., Baeza-Yates R., Marroquín J.L. 2001. Searching in -metric spaces. ACM Comput. Surv. 33:273--321. - -\hypertarget{ref-ciampaglio2001}{} -Ciampaglio C.N., Kemp M., McShea D.W. 2001. Detecting changes in -morphospace occupation patterns in the fossil record: Characterization -and analysis of measures of disparity. Paleobiology. 71:695--715. - -\hypertarget{ref-close2015}{} -Close R.A., Friedman M., Lloyd G.T., Benson R.B. 2015. Evidence for a -mid-Jurassic adaptive radiation in mammals. Current Biology. - -\hypertarget{ref-diaz2016}{} -Díaz S., Kattge J., Cornelissen J.H., Wright I.J., Lavorel S., Dray S., -Reu B., Kleyer M., Wirth C., Prentice I.C., others. 2016. The global -spectrum of plant form and function. Nature. 529:167. - -\hypertarget{ref-donohue2013}{} -Donohue I., Petchey O.L., Montoya J.M., Jackson A.L., McNally L., Viana -M., Healy K., Lurgi M., O'Connor N.E., Emmerson M.C. 2013. On the -dimensionality of ecological stability. Ecology Letters. 16:421--429. - -\hypertarget{ref-endler2005}{} -Endler J.A., Westcott D.A., Madden J.R., Robson T. 2005. Animal visual -systems and the evolution of color patterns: Sensory processing -illuminates signal evolution. Evolution. 59:1795--1818. - -\hypertarget{ref-foote1990}{} -Foote M. 1990. Nearest-neighbor analysis of trilobite morphospace. -Systematic Zoology. 39:371--382. - -\hypertarget{ref-grant2006}{} -Grant P.R., Grant B.R. 2006. Evolution of character displacement in -darwins finches. Science. 313:224--226. - -\hypertarget{ref-disprity}{} -Guillerme T. 2018. dispRity: A modular R package for measuring -disparity. Methods in Ecology and Evolution. 9:1755--1763. - -\hypertarget{ref-halliday2015}{} -Halliday T.J.D., Goswami A. 2016. Eutherian morphological disparity -across the end-cretaceous mass extinction. Biological Journal of the -Linnean Society. 118:152--168. - -\hypertarget{ref-geiger2008}{} -Harmon L.J., Weir J.T., Brock C.D., Glor R.E., Challenger W. 2008. -GEIGER: Investigating evolutionary radiations. Bioinformatics. -24:129--131. - -\hypertarget{ref-healy2019}{} -Healy K., Ezard T.H.G., Jones O.R., Salguero-G'omez R., Buckley Y.M. -2019. Animal life history is shaped by the pace of life and the -distribution of age-specific mortality and reproduction. Nature Ecology -\& Evolution. 2397-334X. - -\hypertarget{ref-hopkins2016}{} -Hopkins M., Pearson K. 2016. Non-linear ontogenetic shape change in -cryptolithus tesselatus (trilobita) using three-dimensional geometric -morphometrics. Palaeontologia Electronica. 19:1--54. - -\hypertarget{ref-hopkins2017}{} -Hopkins M.J., Gerber S. 2017. Morphological disparity. In: Nuno de la -Rosa L., Müller G., editors. Evolutionary developmental biology: A -reference guide. Cham: Springer International Publishing. p. 1--12. - -\hypertarget{ref-jones2015}{} -Jones N.T., Germain R.M., Grainger T.N., Hall A.M., Baldwin L., Gilbert -B. 2015. Dispersal mode mediates the effect of patch size and patch -connectivity on metacommunity diversity. Journal of Ecology. -103:935--944. - -\hypertarget{ref-lalibertuxe92010}{} -Laliberté É., Legendre P. 2010. A distance-based framework for measuring -functional diversity from multiple traits. Ecology. 91:299--305. - -\hypertarget{ref-legendre2012}{} -Legendre P., Legendre L.F. 2012. Numerical ecology. Elsevier. - -\hypertarget{ref-mammola2019}{} -Mammola S. 2019. Assessing similarity of n-dimensional hypervolumes: -Which metric to use? Journal of Biogeography. 0. - -\hypertarget{ref-marcy2016}{} -Marcy A.E., Hadly E.A., Sherratt E., Garland K., Weisbecker V. 2016. -Getting a head in hard soils: Convergent skull evolution and divergent -allometric patterns explain shape variation in a highly diverse genus of -pocket gophers (thomomys). BMC evolutionary biology. 16:207. - -\hypertarget{ref-oksanen2007vegan}{} -Oksanen J., Kindt R., Legendre P., O'Hara B., Stevens M.H.H., Oksanen -M.J., Suggests M. 2007. The vegan package. Community ecology package. -10:631--637. - -\hypertarget{ref-psych}{} -Revelle W. 2018. Psych: Procedures for psychological, psychometric, and -personality research. Evanston, Illinois: Northwestern University. - -\hypertarget{ref-ruta2013}{} -Ruta M., Angielczyk K.D., Fröbisch J., Benton M.J. 2013. Decoupling of -morphological disparity and taxic diversity during the adaptive -radiation of anomodont therapsids. Proceedings of the Royal Society of -London B: Biological Sciences. 280. - -\hypertarget{ref-villuxe9ger2008}{} -Villéger S., Mason N.W.H., Mouillot D. 2008. New multidimensional -functional diversity indices for a multifaceted framework in functional -ecology. Ecology. 89:2290--2301. - -\hypertarget{ref-wills2001}{} -Wills M.A. 2001. Morphological disparity: A primer. In: Adrain J.M., -Edgecombe G.D., Lieberman B.S., editors. Fossils, phylogeny, and form. -Springer US. p. 55--144. - -\hypertarget{ref-wright2017}{} -Wright D.F. 2017. Phenotypic innovation and adaptive constraints in the -evolutionary radiation of palaeozoic crinoids. Scientific Reports. -7:13745. - - -\end{document} diff --git a/rsconnect/shinyapps.io/tguillerme/moms.dcf b/rsconnect/shinyapps.io/tguillerme/moms.dcf index 0ff22d5..257516e 100644 --- a/rsconnect/shinyapps.io/tguillerme/moms.dcf +++ b/rsconnect/shinyapps.io/tguillerme/moms.dcf @@ -5,6 +5,6 @@ account: tguillerme server: shinyapps.io hostUrl: https://api.shinyapps.io/v1 appId: 1266739 -bundleId: 2457663 +bundleId: 3105274 url: https://tguillerme.shinyapps.io/moms/ -when: 1569902668.87661 +when: 1588671213.57724 diff --git a/space.test b/space.test deleted file mode 100644 index b28530d..0000000 --- a/space.test +++ /dev/null @@ -1,10 +0,0 @@ --0.576493860326921,0.917565036097387,0.719525368039258,0.702623213111991,0.80183650709144 --0.409512884087109,0.392848048700531,0.96252306757752,0.868277100869212,-0.369248200285772 -0.881719467085642,-0.45703941516627,0.455260870574095,0.492429838417246,-0.0171691541151069 --0.504266758003973,0.495573882320549,1.31679537688417,-1.61550594735134,0.753891636101004 -1.10231622362998,0.958340431464062,-0.174580328992963,0.0788439525088021,-2.0504421568868 -0.047266000825037,1.2447428184496,0.997799815475882,0.943970421984684,-0.919989181181447 --0.147289840521016,1.34857566002024,-0.22977908194329,1.1745286354936,0.119233046040596 -0.302151219472938,0.920040676594574,0.883923849073921,0.42193532077646,-0.844342982714898 -0.465345816647428,-0.144843742657328,0.510692506563388,0.13895190259143,-0.264621840968631 -1.18789509407304,-0.774452372578483,0.497141063862592,-0.561919038259494,1.41360648639229