# Classification and Regression Trees

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1 Classification and Regression Trees Bob Stine Dept of Statistics, School University of Pennsylvania

2 Trees Familiar metaphor Biology Decision tree Medical diagnosis Org chart Properties Recursive, partitioning items into unique leaf Increasing specialization Convey structure at-a-glance How to grow a tree from data? What rules identify the relevant variables, split rules? 2

3 Trees as Models for Data Different type of explanatory variable Decision rules replace typical predictors Implicit equation uses indicator functions X Ix c & Ix>c Software builds these from training data Process Find rule to partition data Fits are averages of subsets Use validation data to decide when to stop Models as averages All models average, just question of which cases 3

4 Old Idea Binning data Use categorical variables to define bins Each observation goes into a bin Prediction - average of cases in bin - most common category in bin Classify new case No equation: Use score for the matching bin Trade-offs Good: avoid assuming additive, transformations Bad: Some bins may be nearly empty, sparse Need lots of data to fill a contingency table with several axes Issues: Which characteristics? Which attributes? bias vs variance 4

5 Classical Example Fisher s iris data Classification tree: categorical response 50 flowers from 3 species of iris four variables: length and width of sepal and petal Classification tree G 2 = - 2 log likelihood = 2 entropy Stop? Splits are parallel to plot axes Splitting rules are not unique Tree version of collinearity 5

6 Example ANES 2008 Regression tree: numerical response Favor or oppose gay marriage X s: Obama-McCain, PresDiapproval, Econ Problem Regression tree with dummy response Node shows average of response (here percentage) for its cases Use Select Rows command in the tree nodes Stop? 6

7 Recursive Partitioning Recursive, binary splits CART Start with all cases in one group, the root node Tree grows upside down Split a current group to make homogeneous May split same group several times Continue until objective is reached Comments Recursive: once cases are split, never rejoin Greedy: immediate step rather than look ahead Very fast, even with many features Invariant of order-preserving transformations Rules are not unique (as in collinearity in regr) Interactions 7

8 Growing Tree Search for best splitting variable Numerical variable Partition cases X c and X > c, all possible c Consider only numbers c that match a data point (ie, sort cases) Categorical variable Partition cases into two mutually exclusive groups Lots of groups if the number of labels k is large (2 k-1-1 splits) Greedy search One-step look ahead (as in forward stepwise) Find next variable that maximizes search criterion, such as level of significance or R 2. Criterion depends on response: numerical or categorical 8

9 Splitting Criteria Numerous choices Log-likelihood for classification tree Recall -2 log likelihood residual SS in OLS G 2 is node s contribution to -2 log likelihood Related to the entropy of the current partition (entropy measures randomness) G 2 = 0 for node that is homogenous perfect fit, no value in trying to split further (entropy = 0) Log worth JMP version of the p-value of a split Cross-validation Use a tuning sample to decide how many splits 9

10 Common Limitations Splits are parallel to axis Binary split on an observed variable Some tree methods allow splits on linear combination slower to fit since many more possible splits Discrete fit Piecewise constant fit Lots of splits on one variable indicate trend Greedy search Vert fast but can miss the best partition Over-fitting Common advice: over-fit then prune back As used for AIC, BIC in regr 10

11 Example: ANES Classify those who did not vote Use 3-level validation variable 4000 observed Obama/Romney, exclude others 0 = training, 1 = tuning, determines tree size 2 = test sample Big assumption: same rules apply to those who voted and did not vote Predictive features to consider Avoid direct Obama/Romney specific questions Keep the problem more challenging Demographics Missing indicators sample weights? 11

12 Fitting the Tree Running options Minimum split size 25 Avoid leaves with few cases Nice interface Can force splits at any location Validation properties What happens if this feature is not used? Would be a nice feature for NN as well! 12

13 Mosaic Plot Summary of tree Thin bins have few cases Less flat means better splits Romney Obama 13

14 Estimated Tree Note variables that define first splits Feeling thermometer differences, several splits Race, but only for some Voting behavior Some leaves are very homogeneous No point in further splitting Very parallel structure 14

15 Classify Missing Majority vote Drop case into estimated tree Classify based on the preponderance of cases Results Save tree prediction formula (not predicteds) Get probabilities* as well as most likely choice Distribution of predicteds for missing cases 60/40 split among observed voters *JMP smooths probabilities using prior from parent node. 15

16 Things to Improve? So few possible values Number of leaf nodes determines the number of possible predictions; very discrete fit. Highly variable Take a different subset and split points change Fitted values, however, are likely similar Calibrated, but few possible values Example from training sample 16

17 Averaging Trees Rather than average within a model, we can average over models Model averaging borrows strength Fit collection of models Predict by majority vote or averaging Question: How to get a collection of models? Boosting Re-weight cases not fit well by current model (If numerical Y, fit next model to residuals of current model) Simple models Bagging Build trees (forest) using bootstrap samples Complicated models, different sets of variables 17

18 Random Forest Problem with trees Grainy predictions, few distinct values Each final node gives a prediction Highly variable Sharp boundaries, huge variation in fit at edges of bins Random forest Cake-and-eat-it solution to bias-variance tradeoff Complex tree has low bias, but high variance. Simple tree has high bias, but low variance. Fit ensemble of trees, each to different BS sample Average of fits of the trees Increase independence of trees by forcing different variables in the different trees Often need relatively big tree to capture interesting structure 18

19 Random Forest Fit using random forest Classification tree has only few leaves Very coarse predictions of voting behavior (though maybe enough) Forest has more branches, more variables Summary of forest More variables used bottom left of dialog 19

20 Forest Results Confusion matrix Goodness of fit summary progress as forest grows 20

21 Calibration Plot Test sample results for random forest Richer set of predictions linear, but not with slope 1 Smooth ROC b

22 Boosting General method for improving predictive model Build additive sequence of predictive models (ensemble) Final prediction is accumulated over many models. Start with initial predictive model Compute residuals from current fit Build model for residuals Repeat Implication: Use simple model at each step Weak learner: stump (one split), few splits Next response = (current response) - (learning rate) x fit Weaknesses Loss of interpretability, at what gain? Original method called Adaboost 0.1 or smaller 22

23 Boosted Trees Different way to get multiple trees Simple models Refit to training sample, but put more weight to cases not fit well so far Uses many variables without random exclusion Only in JMP Pro 23

24 Boosting Results Confusion matrix Variation in choice of test sample? Goodness of fit summary Boosted Forest Many choices for classification should have run longer! 24

25 Calibration Plots Results for test sample with boosting Similar benefits obtained by forest Boosting is a bit more predictive b

26 Comparison of Predictions Training Sample Test Sample Random Forest Boosted Tree r=0.99 Boosted Tree r=

27 Take-Aways Classification and regression trees Partition cases into homogeneous subsets Regression tree: small variation around leaf mean Classification tree: concentrate cases into one category Greedy, recursive algorithm Very fast Flexible, iterative implementation in JMP Also found in several R packages (such as tree ) Model averaging Boosting, bagging smooth predictions Borrow strength Over-fitting Control with cross-validation Analogous to use of CV in tuning Neural Net 27

28 Some questions to ponder... How does a tree indicate the presence of an interaction between factors? What does it mean when a tree splits many times on the same variable? How might you remedy this problem? Why is it important (at least 2 reasons) to avoid categorical variables with many categories in trees? What does it mean to describe a tree as defined by recursive and binary cuts? Why do it this way? 28

29 Next Time Thursday Newberry Lab day for nets and trees Friday Kernel methods and random projection Text mining Comparisons and summary 29

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