Chapter_06.ipynb

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    "# Decision Trees\n",
    "\n",
    "tree-based methods for **Regression** and **Classification** Problems\n",
    "\n",
    " - by stratifying or segmenting the predictor space into a number of simple regions\n",
    " \n",
    "to improve prediction accuracy we  use Bagging, Random Forests and Boosting approaches"
   ]
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    "## Regression Trees\n",
    "\n",
    "**Notation:**\n",
    "\n",
    " - **internal nodes** are kind of thresholds/ conditions where the tree splits\n",
    " \n",
    " - **terminal nodes/ leaves** are at the end of a tree and consisting of observation set $$ R_K=\\{X|year\\geq 4.5 \\wedge hits < 117.5\\}$$\n",
    " \n",
    " - segments of trees that connect the nodes **branches**\n",
    " \n",
    "**Procedure:**\n",
    " 1. divide the predictor space into $J$ distinct and non-overlapping regions $R_1,R_2,\\dots,R_J$\n",
    " \n",
    " 2. For every observation in $R_j$ make the same prediction, mean of response value for the training observations in $R_j$ (e.g. $\\overline{x}(R_j)=10 \\quad \\Rightarrow \\quad \\forall x\\in R_j \\quad \\hat{y}(x)=10$)\n",
    " \n",
    "### How to construct $R_1,R_2,\\dots,R_J$?\n",
    "\n",
    "The Goal is to find boxes that minimize the RSS\n",
    "\n",
    "$$ \\sum_{j=1}^J \\sum_{i \\in R_j} (y_i-\\hat{y}_{R_i})^2 \\quad,\\hat{y}_{R_i}=\\overline{x}(R_j)$$\n",
    "\n",
    "Note: it's computational infeasible to consider every possbile partition of $J$ boxes\n",
    "\n",
    "**Solution: Recursive binary splitting**\n",
    "\n",
    " 1. consider all predictors $X_1,\\dots,X_p$ and all possible values of $s$ for each predictor\n",
    " \n",
    " 2. choose the predictor and cutpoint such that the resulting tree has the lowest RSS\n",
    " \n",
    " 3. continue until stopping criterion is reached (e.g. only 5 observations per $R_j$ are left)\n",
    " \n",
    "Problem: predictions may only be good for training data $\\Rightarrow$ **overfitting**\n",
    "\n",
    " - smaller tree (fewer splits): \n",
    " * lower variance\n",
    " * better interpretation\n",
    " But: little bias!\n",
    " \n",
    " - alternative stopping criterion: use RSS threshold\n",
    " But: there is a chance to miss an important split!\n",
    " \n",
    "### Tree Pruning\n",
    "\n",
    "For each value of $\\alpha$ there corresponds a subtree $T \\subset T_0$ such that\n",
    "\n",
    "$$ min \\sum_{m=1}^{\\mid T \\mid} \\sum_{i: x_i\\in R_m} (y_i-\\hat{y}_{R_m})^2 + \\alpha \\mid T \\mid $$\n",
    "\n",
    "we now constructed a trade-off between subtrees complexity and fit to the training data\n",
    "\n",
    "use CV or Validation set approach to select $\\alpha$"
   ]
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    "## Classification Trees\n",
    "\n",
    "using Regression/ Classification Trees to solve Classification Problems. Remember response variable is now qualitative!\n",
    "\n",
    "**Predict:** \n",
    "each observation belongs to the most commonly occuring class of training observations in the region to which it belongs\n",
    "\n",
    "**Interpretation:**\n",
    " - class prediction corresponding to a particular terminal node region\n",
    " \n",
    " - class proportions among the training observations that fall into that region\n",
    " \n",
    "Note: we need alternative measure for RSS!\n",
    " * classification error rate\n",
    "   $$ E=1-max_k (\\hat{p}_{mk})$$\n",
    "   $\\hat{p}_{mk}$ is the proportion of training observations in m'th region from class $k$\n",
    "   \n",
    " * Gini-Index\n",
    "   $$ G=\\sum_{k=1}^K \\hat{p}_{mk}(1-\\hat{p}_{mk})$$\n",
    "   \n",
    " * Cross-entropy\n",
    "   $$ D= - \\sum_{k=1}^K \\hat{p}_{mk} log(\\hat{p}_{mk}) $$\n",
    "   \n",
    "in general E is not sufficient sensitive for tree-growing, while for G \\& D the following holds:\n",
    "*the lower G \\& D, the purer the nodes are*\n",
    "\n",
    "**Summary:**\n",
    " - building tree: use Gini-Index, Cross-entropy\n",
    " \n",
    " - tree pruning: use classification error rate"
   ]
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   "source": [
    "## Linear Models vs. Trees\n",
    "\n",
    "first compare the functional form:\n",
    "\n",
    " - linear models\n",
    "   $$ f(x)=\\beta_0 + \\sum_{j=1}^p X_j \\beta_j $$\n",
    " \n",
    " - tree based methods\n",
    "   $$ f(x)= \\sum_{m=1}^M c_m* I_{x\\in R_m} $$\n",
    "   \n",
    "Advantages and Disadvantages of Trees\n",
    "\n",
    " * easy to explain, even for non-professionals\n",
    " \n",
    " * closely mirror human  decision-making\n",
    " \n",
    " * displayed graphically\n",
    " \n",
    " * easily handle qualitative predictors & missing data\n",
    " \n",
    " * not same predictive accuracy\n",
    " \n",
    " * can be very non-robust"
   ]
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    "## Bagging, Boosting and Random Forests\n",
    "\n",
    "### Bagging\n",
    "\n",
    "**Idea:** use Bootstrap to improve statistical learning tools\n",
    "\n",
    "**Procedure:**\n",
    " 1. take B seperate training sets from the population\n",
    " \n",
    " 2. build a seperate prediction model using each training set\n",
    "    $\\hat{f}^1(x), \\hat{f}^2(x),\\dots, \\hat{f}^B(x)$\n",
    " \n",
    " 3. average resulting predictors\n",
    "    $\\hat{f}_{bag}(x)=\\frac{1}{B}\\sum_{b=1}^B \\hat{f}^{*b}(x)$\n",
    "    \n",
    "#### Out-of-Bag (OOB)\n",
    "\n",
    "since Bagging uses Bootstrap, we can estimate test error without CV or validation set approach.\n",
    "\n",
    "Key to bagging is that trees are repeatedly fit to Bootstrapped subsets\n",
    "\n",
    "one can show that each bagged tree uses only $2/3$ of obserbations, thus $1/3$ are refered as OOB observations\n",
    "\n",
    "### Random Forests\n",
    "\n",
    "**Idea:** \n",
    "improve Bagging by decorrelating the trees (subtrees)\n",
    "\n",
    "**Procedure:**\n",
    " 1. build a forest of decision trees bootstraped training samples\n",
    " \n",
    " 2. each time a split in a tree is considered, random sample $m$ of the full set of $p$ predictors is choosen\n",
    " \n",
    " 3. fresh sample of $m$ predictors in taken at each split (typically $m\\approx \\sqrt{p}$)\n",
    " \n",
    "**Why?**\n",
    "suppose one very strong predictor and other moderate predictors $\\Rightarrow$ Bagged trees will all use ths strong predictor in the Top Split. Hence, averaging the correlated quantities will not reduce variance in the same magnitude as uncorrelated quantities!\n",
    "\n",
    "**Random Forests and Bagging**\n",
    "main difference is the choice of predictor subset $m$\n",
    "\n",
    "\n",
    "### Boosting\n",
    "\n",
    "**Idea:**\n",
    "similar to Bagging, except that the trees are grown **sequentially** - each tree uses information from the previous tree (no Bootstrap sampling, instead using residuals form previous tree as response to fit the current tree)\n",
    "\n",
    "**Procedure:**\n",
    " 1. set $\\hat{f}(x)=0$ and $r_i=y_i$ for all $i$ in the training set\n",
    " \n",
    " 2. for  $b=1,2,\\dots,B$ repeat:\n",
    "   \n",
    "   a) fit a tree $\\hat{f}^b$ with $d$ splits ($d+1$ terminal nodes) to training data ($X,r$)\n",
    "   \n",
    "   b) update $\\hat{f}$ by adding in a shrunken version of the new tree:\n",
    "    $$ \\hat{f}(x) \\leftarrow \\hat{f}(x) + \\lambda\\hat{f}^b(x) $$\n",
    "   \n",
    "   c) update residuals:\n",
    "    $$ r_i \\leftarrow r_i - \\lambda \\hat{f}^b(x_i) $$\n",
    "  \n",
    " 3. output the boosted model:\n",
    "    $$ \\hat{f}(x) = \\sum_{b=1}^B \\lambda \\hat{f}^b(x) $$\n",
    "    \n",
    "      \n",
    " - $B$ is the number of trees (use CV to select $B$)\n",
    " \n",
    " - $\\lambda$ shrinkage parameter controls the rate at which boosting learns (typically 0.01 or 0.001)\n",
    " \n",
    " - $d$ *interaction depth* number of splits in each tree (typically 1 or 2)"
   ]
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   "source": [
    "## Exercises"
   ]
  },
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      "package 'tree' successfully unpacked and MD5 sums checked\n",
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      "Warning message:\n",
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Woi7a4S7dOFgL7dAL5qx1c0faoi7VcU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\nBMAEIBIAE/h/lXwNTzGuqpUAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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X9P1PYoqFdtbxpWGylqU7uvvyuam6Ap+mONoM/9490UD+EeN99I13CxFeZ4W7Oa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\nQAIwEkACMBJAAjASQAIwEkAC/h9Vgqyo23F9FwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "install.packages(\"tree\")\n",
    "library(\"tree\")\n",
    "set.seed(1987)\n",
    "\n",
    "# DGP function\n",
    "dgp1 <- function(n = 250, sd = 2, beta = c(.5, -1, 1.5, .5)) {\n",
    "  X <- cbind(X <- replicate(2, rnorm(n, sd = sd)), X ^ 2)\n",
    "  y <- X %*% beta + rnorm(n)\n",
    "  list(y=y, x1=X[, 1], x2 = X[, 2])\n",
    "}\n",
    "\n",
    "\n",
    "# plot function \n",
    "dgp1_func <- function(x1, x2) .5 * x1 - 1 * x2 + 1.5 * x1 ^ 2 + .5 * x2 ^ 2\n",
    "x1 <- x2 <- seq(-2, 2, le = 50)\n",
    "y  <- outer(x1, x2, dgp1_func)\n",
    "persp(x1, x2, y, theta = 200, phi = 350)\n",
    "\n",
    "\n",
    "data_train <- dgp1()\n",
    "tree_model <- tree(y ~ x1 + x2, data_train, split = \"deviance\")\n",
    "plot(tree_model)\n",
    "text(tree_model, pretty = 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>mse_linear</dt>\n",
       "\t\t<dd>115.759421203261</dd>\n",
       "\t<dt>mse_tree</dt>\n",
       "\t\t<dd>9.71027196629251</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[mse\\textbackslash{}\\_linear] 115.759421203261\n",
       "\\item[mse\\textbackslash{}\\_tree] 9.71027196629251\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "mse_linear\n",
       ":   115.759421203261mse_tree\n",
       ":   9.71027196629251\n",
       "\n"
      ],
      "text/plain": [
       "mse_linear   mse_tree \n",
       "115.759421   9.710272 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>amse_linear</dt>\n",
       "\t\t<dd>146.225698670252</dd>\n",
       "\t<dt>amse_tree</dt>\n",
       "\t\t<dd>14.090352116384</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[amse\\textbackslash{}\\_linear] 146.225698670252\n",
       "\\item[amse\\textbackslash{}\\_tree] 14.090352116384\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "amse_linear\n",
       ":   146.225698670252amse_tree\n",
       ":   14.090352116384\n",
       "\n"
      ],
      "text/plain": [
       "amse_linear   amse_tree \n",
       "  146.22570    14.09035 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>amse_linear</dt>\n",
       "\t\t<dd>146.36993012779</dd>\n",
       "\t<dt>amse_tree</dt>\n",
       "\t\t<dd>13.1363275972553</dd>\n",
       "\t<dt>amse_linear2</dt>\n",
       "\t\t<dd>1.02183627752379</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[amse\\textbackslash{}\\_linear] 146.36993012779\n",
       "\\item[amse\\textbackslash{}\\_tree] 13.1363275972553\n",
       "\\item[amse\\textbackslash{}\\_linear2] 1.02183627752379\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "amse_linear\n",
       ":   146.36993012779amse_tree\n",
       ":   13.1363275972553amse_linear2\n",
       ":   1.02183627752379\n",
       "\n"
      ],
      "text/plain": [
       " amse_linear    amse_tree amse_linear2 \n",
       "  146.369930    13.136328     1.021836 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "compute_mse <- function(model_obj, data_test) {\n",
    "  new_df <- data.frame(x1 = data_test$x1, x2 = data_test$x2)\n",
    "  pred   <- predict(model_obj, newdata = new_df)\n",
    "  \n",
    "  mean((data_test$y - pred) ** 2)\n",
    "}\n",
    "\n",
    "linear_model  <- lm(y ~ x1 + x2 - 1, data_train)\n",
    "\n",
    "data_test <- dgp1(n = 100)\n",
    "\n",
    "mse_linear <- compute_mse(linear_model, data_test)\n",
    "mse_tree   <- compute_mse(tree_model, data_test)\n",
    "c(mse_linear=mse_linear, mse_tree=mse_tree)\n",
    "\n",
    "\n",
    "monte_carlo_simu <- function(nsim = 1000) {\n",
    "  mse_linear <- replicate(nsim, compute_mse(linear_model, dgp1(n = 100)))\n",
    "  mse_tree   <- replicate(nsim, compute_mse(tree_model, dgp1(n = 100)))\n",
    "  \n",
    "  c(amse_linear=mean(mse_linear), amse_tree=mean(mse_tree))\n",
    "}\n",
    "monte_carlo_simu()\n",
    "\n",
    "# OLS with squared covariates \n",
    "X      <- with(data_train, cbind(x1, x2, x1^2, x2^2))\n",
    "beta_2 <- solve(t(X) %*% X) %*% t(X) %*% data_train$y\n",
    "\n",
    "monte_carlo_simu2 <- function(nsim = 1000) {\n",
    "  mse_2 <- function(data) {\n",
    "    X_test <- with(data, cbind(x1, x2, x1^2, x2^2))\n",
    "    \n",
    "    mean((data$y - X_test %*% beta_2)^2)\n",
    "  }\n",
    "  \n",
    "  mean(replicate(nsim, mse_2(dgp1(n = 100))))\n",
    "}\n",
    "c(monte_carlo_simu(), amse_linear2=monte_carlo_simu2())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kR6XaC9IhHjOYVOS7L9HIanxPAAJCCS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\n14z4yiFbdchHNF/yZnb5PWgulIuyb6VhquScKdYuyiHG68X6Nq1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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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yK1eEOiLO3R8mh3Nhv0RVO5uFNKVZ06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\n6wxkUjKPmE/an7ZHBXukR+RT9X6pTik7mY+hiRtk3MiaXCWSMCg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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# DGPs well suited for analysis using regression trees\n",
    "\n",
    "#dgp2 <- function(n = 250) {\n",
    "#  x1 <- rnorm(n, sd = 2)\n",
    "#  x2 <- rnorm(n, sd = 2)\n",
    "#  y <- 5 * I(x1 < 0) + 2 * I(x2 < 0) + 3 * I(x1 > 0 & x2 > 0) + \n",
    "#          2 * I(x1 < 0 & x2 < 0) + \n",
    "#            1 * I(x1 < 1.5 & x1 > 1 & x2 < 1.5 & x2 > 1)\n",
    "#  list(y=y, x1=x1, x2=x2)\n",
    "#}\n",
    "\n",
    "dgp2_func <- function(x1, x2) {\n",
    "  5 * I(x1 < 0) + 2 * I(x2 < 0) + 3 * I(x1 > 0 & x2 > 0) + \n",
    "    2 * I(x1 < 0 & x2 < 0) + \n",
    "      1 * I(x1 < 1.5 & x1 > 1 & x2 < 1.5 & x2 > 1)\n",
    "}\n",
    "\n",
    "x1 <- seq(-3, 3, le = 50)\n",
    "x2 <- seq(-3, 3, le = 50)\n",
    "y  <- outer(x1, x2, dgp2_func)\n",
    "persp(x1, x2, y, theta = 120, phi = 20)\n",
    "\n",
    "\n",
    "# DGPs well suited for OLS \n",
    "\n",
    "#dgp3 <- function(n = 250) {\n",
    "#  x1 <- rnorm(n, sd = 2)\n",
    "#  x2 <- rnorm(n, sd = 2)\n",
    "#  y <- x1 + x2\n",
    "#  list(y=y, x1=x1, x2=x2)\n",
    "#}\n",
    "\n",
    "dgp3_func <- function(x1, x2) x1 + x2\n",
    "\n",
    "x1 <- seq(-3, 3, le = 50)\n",
    "x2 <- seq(-3, 3, le = 50)\n",
    "y  <- outer(x1, x2, dgp3_func)\n",
    "persp(x1, x2, y, theta = 45, phi = 25)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}