Classification and Regression Trees

Transcription

1 Classification and Regression Trees Matthew S. Shotwell, Ph.D. Department of Biostatistics Vanderbilt University School of Medicine Nashville, TN, USA March 16, 2018

2 Introduction trees partition feature space into a set of rectangles fit simple model in each partition (e.g., constant) start with CART: Classification And Regression Trees quantitative response Y and inputs X 1 and X 2, all with support in [0, 1]

3 Elements of Statistical Learning (2nd Ed.) c Hastie, Tibshirani & Friedman 2009 Chap 9 R2 R5 t4 R3 t2 R4 R1 top-left partition lines parallel to axes (e.g., X 1 = c), but can t be described by tree top-right partitions are recursive, can be described by tree at bottom-left trees have (root, internal, and terminal) nodes and branches terminal nodes also called leaf node t1 t2 t3 t4 R1 R2 R3 t1 t3 R4 R5 FIGURE 9.2. Partitions and CART. Top right panel shows a partition of a two-dimensional feature space by recursive binary splitting, as used in CART, applied to some fake data. Top left panel shows a general partition that cannot be obtained from recursive binary splitting. Bottom left panel shows the tree corresponding to the partition in the top right panel, and a perspective plot of the prediction surface appears in the bottom right panel.

4 Fitting recursive binary trees first split space into two regions model the response by the mean of Y in each region choose feature and split-point to achieve the best fit one or both resulting regions are split into two more regions repeat until some stopping rule is applied

5 Elements of Statistical Learning (2nd Ed.) c Hastie, Tibshirani & Friedman 2009 Chap 9 Top-right panel tree constructed as follows: 1. split at X 1 = t 1 R2 R5 t4 2. for X 1 t 1 split at X 2 = t 2 R3 3. for X 1 > t 1 split at X 1 = t 3 t2 R4 4. for X 1 > t 3, split at X 2 = 4 R1 Partition into 5 regions: R 1,..., R 5 ; predictor is mean of Y in each region: t1 t3 5 ˆf(x) = ĉ mi(x R m) m=1 t1 t2 t3 where t4 Ni=1 I(x i R m)y i ĉ m = Ni=1 I(x i R m) R1 R2 R3 R4 R5 Bottom-left is tree representation of top-right, and bottom-right is surface of ˆf(x) FIGURE 9.2. Partitions and CART. Top right panel shows a partition of a two-dimensional feature space by recursive binary splitting, as used in CART, applied to some fake data. Top left panel shows a general partition that cannot be obtained from recursive binary splitting. Bottom left panel shows the tree corresponding to the partition in the top right panel, and a perspective plot of the prediction surface appears in the bottom right panel.

6 Fitting (growing) regression trees Regression trees have this form: f(x) = M c m I(x R m ) m=1 Under squared-error loss, conditional on R m, ĉ m is the sample mean of Y in region R m. Finding partitions R m is more difficult. Greedy algorithm: consider a splitting variable j and split point s and define resulting regions as follows: R 1 (j, s) = {X : X j s} and R 2 (j, s) = {X : X j > s} The splitting task is then to find j and s that minimize: RSS(j, s) = (y i ĉ 1 ) 2 + (y i ĉ 2 ) 2 x i R 1 (j,s) x i R 2 (j,s)

7 Fitting (growing) regression trees The splitting task is then to find j and s that minimize: RSS(j, s) = (y i ĉ 1 ) 2 + (y i ĉ 2 ) 2 x i R 1 (j,s) x i R 2 (j,s) this may seem difficult at first, but s is only identifiable for up to N 1 distinct values for given j, RSS(j, s) is constant between X j1 and X j2 for tractable N p, can simply enumerate all RSS(j, s) split at optimal j and s, then repeat within each split proceed this way until stopping rule triggered

8 Stopping rules tree complexity is a tuning parameter one approach: grow a large tree, stopping only when minimum node size (i.e., minimum number of x i R m ) is reached, then prune tree back using a cost-complexity criterion

9 Pruning Elements of Statistical Learning (2nd Ed.) c Hastie, Tibshirani & Friedman 2009 Chap 9 R2 R5 t4 R3 t2 R4 R1 t1 t3 t1 t2 t3 t4 R1 R2 R3 R4 R5 FIGURE 9.2. Partitions and CART. Top right panel shows a partition of a two-dimensional feature space by recursive binary splitting, as used in CART, applied to some fake data. Top left panel shows a general partition that cannot be obtained from recursive binary splitting. Bottom left panel shows the tree corresponding to the partition in the top right panel, and a perspective plot of the prediction surface appears in the bottom right panel.

10 Cost-complexity pruning let T T 0 be a sub-tree obtained by pruning T 0 let T be the number of terminal nodes let N m be the node size N i=1 I(x i R m ) let ĉ m = 1 N m x i R m y i let Q m (T ) = 1 N m x i R m (y i ĉ m ) 2 be lack of fit (LOF) the cost-complexity criterion is C α (T ) = T m=1 N m Q m (T ) + α T

11 Cost-complexity pruning C α (T ) = T m=1 N m Q m (T ) + α T pruning increases cost, lowers complexity for given α, find T α that minimizes C α (T ) greedy algorithm (weakest-link pruning): 1. collapse internal node that produces smallest increase in C α (T ) 2. continue until just one node 3. select among sequence of trees; T α must be part of this sequence α selected using, e.g., cross validation

12 Classification trees for k = 1,..., K classes let ˆp mk = 1 N m x i R m I(y i = k) apply loss function-specific classification rule k(m) need new metric of node impurity and LOF (for cost-complexity pruning): misclassification rate - 1 ˆpmk(m) Gini index - K k=1 ˆp mk(1 ˆp mk ) cross-entropy deviance - K k=1 ˆp mk log ˆp mk

13 Elements of Statistical Learning (2nd Ed.) c Hastie, Tibshirani & Friedman 2009 Chap Gini index Misclassification error Entropy p FIGURE 9.3. Node impurity measures for two-class classification, as a function of the proportion p in class 2. Cross-entropy has been scaled to pass through (0.5, 0.5).

14 Why prefer misclassification vs. Gini vs. cross-entropy? say we have a two-class problem with 400 in each class, denoted (400,400) consider two candidate splits: s1 : (300,100) and (100,300) s 2 : (200,400) and (200,0) both splits have misclassification rate 0.25 (assuming zero-one loss classifier), but second split produces a pure node, which is preferable in many cases both Gini and cross-entropy give preference to split with pure node; thus often used in growing trees often misclassification is used for pruning

15 Gini index consider classifying training observations (x i R m ) to class k with probability ˆp mk then the training error rate of this rule in node m is k k ˆp mk ˆp mk - the Gini index

16 Problems with trees instability; sample variability in tree structure lack of smoothness of prediction surface in feature space categorical features; 2 q 1 1 partitions of categorical predictor with q values into 2 parts; can be simplified for binary outcomes using Gini or entropy loss, and quantitative outcomes using squared-error loss

17 R package rpart R package rpart implements recursive partitioning for classification and regressoin trees great vignette written my Terry Therneau and others