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GP.jl
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### A Pluto.jl notebook ###
# v0.17.7
using Markdown
using InteractiveUtils
# ╔═╡ 570a1540-87e2-11ec-3d92-c72eea2822fd
begin
using Turing, Distributions
import Turing: Variational
using Random, LinearAlgebra
using Plots, StatsPlots
using AbstractGPs, KernelFunctions
using CSV, HTTP, DataFrames
using Flux
end
# ╔═╡ c4f829c2-4e04-4555-bc09-a7c997199496
md"""
### Gaussian Process
I want to start with the definition of **Gaussian process**.
Definition : A Gaussian process is *a collection of random variables*, any finite number of which have a joint gaussian distribution.
A Gaussian process is completely specified by its mean function and covariance function. We define mean function $\mathcal{m}(x)$ and the covariance function $\mathcal{k}(x, x')$ of a real process $\mathcal{f}(x)$ as
$$\mathcal{m}(x) = \mathbb{E}[f(x)]$$
$\mathcal{k}(x, x') = \mathbb{E}[(f(x) - \mathcal{m}(x))(f(x') - \mathcal{m}(x'))]$
$f(x) \sim \mathcal{GP}(\mathcal{m}(x), \mathcal{k}(x, x'))$
Usually, GP take the mean function to be zero, so the covariance matrix(kernel) is the key ingradient. Moreover, GP has useful property, the marginalization property. That means we have submatrix of the covariance matrix without any changes.(also adding!)
As we mentioned, the kernel is the key componet of GP. Therefore there are several subjects to know about the kernel. But I am not gonna explain yet. If you are interested in that, google *kernel cookbook*.
Gaussian process is the model that can do regression and classification. See the below figure.
"""
# ╔═╡ d806f5e5-fa0f-460a-b18d-5f1ff318d77a
url = "https://raw.githubusercontent.com/delphinH/imagetoy/main/dataset/Fish.csv"
# ╔═╡ 0d6d125a-7e32-460f-8e4c-9e6cfa17b5d8
fish = CSV.read(HTTP.get(url).body, DataFrame)
# ╔═╡ 05b2b441-1b5a-4235-baf7-e33ec9e39b10
@df fish scatter(:Length1, :Weight)
# ╔═╡ 0d82bd91-28f6-43a0-ba35-506d0b123407
begin
xs = fish[:, :Length1]
ys = fish[:, :Weight]
end
# ╔═╡ d26eac35-b1b8-4d76-a240-5ccf962e14fd
size(xs)
# ╔═╡ 1b146cd3-a844-475c-bb11-f3251b255fb0
#Exponentiated quadratic kernel
function EQkernel(x1, x2, l, σ)
t₁ = sum(x1.^2, dims=2)
t₂ = sum(x2.^2, dims=2)
t₃ = 2*x1*x2'
t = (t₁ .+ t₂') - t₃
return σ^2 * exp.(-0.5/l^2 * t)
end
# ╔═╡ 0d95556d-119e-4457-9180-45664808dcfe
x = 1:60
# ╔═╡ 610eb4f0-d6b3-4b9e-94ad-4b2f406c7f58
mu = zeros(length(x))
# ╔═╡ a6a7edb1-c371-4e59-9b27-00a7e898a2be
cov = EQkernel(x, x, 1, 1)
# ╔═╡ 96af16c8-cf13-42e0-8054-1fe817ce0431
prior = rand(MvNormal(mu, Matrix(Hermitian(cov + 0.001*I))), 3)
# ╔═╡ 37c1c0a2-0450-41cc-abea-3aa8c32c813d
plot(prior[:, 1:3])
# ╔═╡ 9ca722fb-060c-4be3-9ceb-666b37289138
function post(x_n, x_t, y_t, l, σ, sigma_y)
K = EQkernel(x_t, x_t, l, σ) + sigma_y^2 * I
K_s = EQkernel(x_t, x_n, l, σ)
K_ss = EQkernel(x_n, x_n, l, σ)
mu_n = transpose(K_s) * inv(K) * y_t
cov_n = K_ss - transpose(K_s) * inv(K) * K_s
return mu_n, cov_n, K_s, K_ss
end
# ╔═╡ c1218fd1-01bd-4b36-91a0-9ae074063c3a
X_new = reshape(collect(range(-3.5, 3.5, length=100)), 100, 1)
# ╔═╡ 43d6fda7-23f7-477b-9c49-7fa28554fc6c
mu_n, cov_n, K_s, K_ss = post(x, xs, ys, 2, 1, 0.5)
# ╔═╡ 5ef8e96c-3d78-452f-8791-f7f2ba915f67
cov_n
# ╔═╡ 8cee8595-34f0-4798-99b4-0b6307bbcd3b
posterior = rand(MvNormal(mu_n, Matrix(Hermitian(cov_n + 0.001*I))), 3)
# ╔═╡ c976e59b-a691-4171-886b-7d7b611a9cc8
begin
scatter(xs, ys, legend=false)
plot!(posterior[:, 1:3])
end
# ╔═╡ 7e91e628-0d44-471c-bb95-d425ea4499f6
@model gptest(x, y) = begin
#Prior
σ ~ LogNormal(1.0, 0.5)
l ~ LogNormal(1.0, 1.0)
sigma_y ~ LogNormal(0.0, 1.0)
#covariance function
kernel = EQkernel(x, x, l, σ)
#add noise
y ~ MvNormal(kernel + sigma_y * I)
end
# ╔═╡ fe9da0ce-6282-418f-a6f0-9ef566458e5b
model = gptest(xs, ys)
# ╔═╡ e0ee7e7c-0d43-4bc8-bea8-5c23f4dd9ea7
q0 = Variational.meanfield(model)
# ╔═╡ 6d9782ce-cb04-4ef6-9e8c-624cef812f23
advi = ADVI(10, 2000)
# ╔═╡ 1039e3ad-a949-48a4-842b-2e9e2cbb8322
q = vi(model, advi, q0; optimizer = Flux.ADAM())
# ╔═╡ ade10ceb-03f0-4ae3-8ca2-f5bcc08699b7
z = rand(q, 5000);
# ╔═╡ 402d5b3f-7443-453d-bc9b-ee4b8c47f0d1
begin
_, sym2range = bijector(model, Val(true));
sym2range
end
# ╔═╡ 9946c2b6-5991-4ed0-9766-b7698e9be019
function plot_variational_marginals(z, sym2range)
ps = []
for (i, sym) in enumerate(keys(sym2range))
indices = union(sym2range[sym]...)
if sum(length.(indices)) > 1
offset = 1
for r in indices
for j in r
p = density(z[j, :], title = "$(sym)[$offset]", titlefontsize = 10, label = "")
push!(ps, p)
offset += 1
end
end
else
p = density(z[first(indices), :], title = "$(sym)", titlefontsize = 10, label = "")
push!(ps, p)
end
end
return plot(ps..., layout = (length(ps), 1), size = (500, 1500))
end
# ╔═╡ ceb6a9a9-407e-4117-9240-11b15e38d9dc
plot_variational_marginals(z, sym2range)
# ╔═╡ 200e262a-63ee-4538-9225-5a6beca7183c
avg = vec(mean(z; dims = 2))
# ╔═╡ 2b5e5118-f351-4a68-8045-ad322945a877
avg[union(sym2range[:l]...)]
# ╔═╡ 30107419-923c-4705-89b4-94779abdf6e3
avg[union(sym2range[:σ]...)]
# ╔═╡ 4c0ac354-1f98-46f2-91b0-538db3e761f0
avg[union(sym2range[:sigma_y]...)]
# ╔═╡ f67b43ef-cdaa-4eeb-81c0-4634e6b4578e
md"""
### Reference
- Gaussian Processes for Machine Learning by C. E. Rasmussen and C. K. I. Williams
- Zoubin Ghahramani NIPS 2016 slide
"""
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c"
HTTP = "cd3eb016-35fb-5094-929b-558a96fad6f3"
KernelFunctions = "ec8451be-7e33-11e9-00cf-bbf324bd1392"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"
[compat]
AbstractGPs = "~0.5.6"
CSV = "~0.10.2"
DataFrames = "~1.3.2"
Distributions = "~0.25.46"
Flux = "~0.12.9"
HTTP = "~0.9.17"
KernelFunctions = "~0.10.30"
Plots = "~1.25.8"
StatsPlots = "~0.14.33"
Turing = "~0.20.0"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
[[AbstractFFTs]]
deps = ["ChainRulesCore", "LinearAlgebra"]
git-tree-sha1 = "6f1d9bc1c08f9f4a8fa92e3ea3cb50153a1b40d4"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.1.0"
[[AbstractGPs]]
deps = ["ChainRulesCore", "Distributions", "FillArrays", "IrrationalConstants", "KernelFunctions", "LinearAlgebra", "Random", "RecipesBase", "Reexport", "Statistics", "StatsBase", "Test"]
git-tree-sha1 = "69cedd6981e20a30b5032c3551d4d7809fe51f5d"
uuid = "99985d1d-32ba-4be9-9821-2ec096f28918"
version = "0.5.6"
[[AbstractMCMC]]
deps = ["BangBang", "ConsoleProgressMonitor", "Distributed", "Logging", "LoggingExtras", "ProgressLogging", "Random", "StatsBase", "TerminalLoggers", "Transducers"]
git-tree-sha1 = "db0a7ff3fbd987055c43b4e12d2fa30aaae8749c"
uuid = "80f14c24-f653-4e6a-9b94-39d6b0f70001"
version = "3.2.1"
[[AbstractPPL]]
deps = ["AbstractMCMC", "DensityInterface", "Setfield"]
git-tree-sha1 = "ca54027a17ca3133b36166191b5faa8a404e92d3"
uuid = "7a57a42e-76ec-4ea3-a279-07e840d6d9cf"
version = "0.4.0"
[[AbstractTrees]]
git-tree-sha1 = "03e0550477d86222521d254b741d470ba17ea0b5"
uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"
version = "0.3.4"
[[Adapt]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "af92965fb30777147966f58acb05da51c5616b5f"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.3.3"
[[AdvancedHMC]]
deps = ["AbstractMCMC", "ArgCheck", "DocStringExtensions", "InplaceOps", "LinearAlgebra", "ProgressMeter", "Random", "Requires", "Setfield", "Statistics", "StatsBase", "StatsFuns", "UnPack"]
git-tree-sha1 = "189473a73d664fe2496675775b6c8a732b8dfe26"
uuid = "0bf59076-c3b1-5ca4-86bd-e02cd72cde3d"
version = "0.3.3"
[[AdvancedMH]]
deps = ["AbstractMCMC", "Distributions", "Random", "Requires"]
git-tree-sha1 = "8ad8bfddf8bb627d689ecb91599c349cbf15e971"
uuid = "5b7e9947-ddc0-4b3f-9b55-0d8042f74170"
version = "0.6.6"
[[AdvancedPS]]
deps = ["AbstractMCMC", "Distributions", "Libtask", "Random", "StatsFuns"]
git-tree-sha1 = "59c47e9525d5a807a950d8b79b5d6e60b2f6de82"
uuid = "576499cb-2369-40b2-a588-c64705576edc"
version = "0.3.3"
[[AdvancedVI]]
deps = ["Bijectors", "Distributions", "DistributionsAD", "DocStringExtensions", "ForwardDiff", "LinearAlgebra", "ProgressMeter", "Random", "Requires", "StatsBase", "StatsFuns", "Tracker"]
git-tree-sha1 = "130d6b17a3a9d420d9a6b37412cae03ffd6a64ff"
uuid = "b5ca4192-6429-45e5-a2d9-87aec30a685c"
version = "0.1.3"
[[ArgCheck]]
git-tree-sha1 = "a3a402a35a2f7e0b87828ccabbd5ebfbebe356b4"
uuid = "dce04be8-c92d-5529-be00-80e4d2c0e197"
version = "2.3.0"
[[ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
[[Arpack]]
deps = ["Arpack_jll", "Libdl", "LinearAlgebra", "Logging"]
git-tree-sha1 = "288d58589d4249a63095f3f41ece91bf34c32c19"
uuid = "7d9fca2a-8960-54d3-9f78-7d1dccf2cb97"
version = "0.5.0"
[[Arpack_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "OpenBLAS_jll", "Pkg"]
git-tree-sha1 = "4c31b0101997beb213a9e6c39116b052e73ca38c"
uuid = "68821587-b530-5797-8361-c406ea357684"
version = "3.8.0+0"
[[ArrayInterface]]
deps = ["Compat", "IfElse", "LinearAlgebra", "Requires", "SparseArrays", "Static"]
git-tree-sha1 = "745233d77146ad221629590b6d82fe7f1ddb478f"
uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
version = "4.0.3"
[[Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"
[[AxisAlgorithms]]
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