Additive Models ， Trees ， and Related Models

1. Additive Models，Trees，and Related Models Prof. Liqing Zhang Dept. Computer Science & Engineering, Shanghai Jiaotong University

2. Introduction 9.1: Generalized Additive Models 9.2: Tree-Based Methods 9.3: PRIM:Bump Hunting 9.4: MARS: Multivariate Adaptive Regression Splines 9.5: HME:Hieraechical Mixture of Experts

3. CART • Overview • Principle behind: Divide and conquer • Partition the feature space into a set of rectangles • For simplicity, use recursive binary partition • Fit a simple model (e.g. constant) for each rectangle • Classification and Regression Trees (CART) • Regress Trees • Classification Trees • Popular in medical applications

4. CART • An example (in regression case):

5. Basic Issues in Tree-based Methods • How to grow a tree? • How large should we grow the tree?

6. Regression Trees • Partition the space into M regions: R1, R2, …, RM. Note that this is still an additive model

7. Regression Trees– Grow the Tree • The best partition: to minimize the sum of squared error: • Finding the global minimum is computationally infeasible • Greedy algorithm: at each level choose variable j and value s as: • The greedy algorithm makes the tree unstable • The error made at the upper level will be propagated to the lower level

8. Regression Tree – how large? • Trade-off between bias and variance • Very large tree: overfit (low bias, high variance) • Small tree (low variance, high bias): might not capture the structure • Strategies: • 1: split only when we can decrease the error (usually short-sighted) • 2: Cost-complexity pruning (preferred)

9. Penalty on the complexity/size of the tree Cost: sum of squared errors Regression Tree - Pruning • Cost-complexity pruning: • Pruning: collapsing some internal nodes • Cost complexity: • Choose best alpha: weakest link pruning • Each time collapse an internal node which add smallest error • Choose from this tree sequence the best one by cross-validation

10. Classification Trees • Classify the observations in node m to the major class in the node: • Pmk is the proportion of observation of class k in node m • Define impurity for a node: • Misclassification error: • Entropy: • Gini index :

11. Classification Trees • Entropy and Gini are more sensitive • To grow the tree: use Entropy or Gini • To prune the tree: use Misclassification rate (or any other method) Node impurity measures versus class proportion for 2-class problem

12. Tree-based Methods: Discussions • Categorical Predictors • Problem: Consider splits of sub tree t into tL and tR based on categorical predictor x which has q possible values: 2(q-1)-1 ways ! • Treat the categorical predictor as ordered by say proportion of class 1

13. Tree-based Methods: Discussions • Linear Combination Splits • Split the node based on • Improve the predictive power • Hurt interpretability • Instability of Trees • Inherited from the hierarchical nature • Bagging (section 8.7 of [HTF]) can reduce the variance

14. Bootstrap Trees Construct B number of trees from B bootstrap samples– bootstrap trees

15. Bootstrap Trees

16. Bagging The Bootstrap Trees is computed from the bth bootstrap sample in this case a tree Bagging reduces the variance of the original tree by aggregation

17. Bagged Tree Performance Majority vote Average

22. Results for spam example

26. 9.3 PRIM: Bump Hunting • The patient rule induction method(PRIM) finds boxes in the feature space and seeks boxes in which the response average is high. Hence it looks for maxima in the target function, an exercise known as bump hunting.

27. PRIM(cont.)

28. PRIM(cont.)

29. PRIM(cont.)

30. 9.4 MARS: Multivariate Adaptive Regression Splines • In multi-dimensional spline the basis functions grow exponentially– curse of dimensionality • A partial remedy is a greedy forward search algorithm • Create a simple basis-construction dictionary • Construct basis functions on-the-fly • Choose the best-fit basis function at each step

31. Basis functions • 1-dim linear spline (t represents the knot) • Basis collections C: |C| = 2 * N * p

32. MARS(cont.) • The idea of MARS is to form reflected pairs for each input Xj with knots at each observed value Xij of that input. Therefore, the collection of basis function is: • The model has the form: where each hm(x) is a function in C or a product of two or more such functions.

33. The MARS procedure (1st stage) • Initialize basis set M with a constant function • Form candidates (cross-product of M with set C) • Add the best-fit basis pair (decrease residual error the most) into M • Repeat from step 2 (until e.g. |M| >= threshold) M (new) M (old) C

34. MARS(cont.) • We start with only the constant function h0(x)=1 in the model set M and all functions in the set C are candidate functions. At each stage we add to the model set M the term of the form: that produces the largest decrease in training error. The process is continued until the model set M contains some preset maximum number of terms.

35. MARS(cont.)

36. The MARS procedure (2nd stage) The final model M typically overfits the data =>Need to reduce the model size (# of terms) Backward deletion procedure • Remove term which causes the smallest increase in residual error • Compute • Repeat step 1 Choose the model size with minimum GCV.

37. Generalized Cross Validation (GCV) • M(.) measures effective # of parameters: • r: # of linearly independent basis functions • K: # of knots selected • c = 3

38. Discussion • Piecewise linear reflected basis • Allow operation on local region • Fitting N reflected basis pairs takes O(N) instead of O(N^2) • Left-part is zero, right-part differs by a constant X[i-1] X[i] X[i+1] X[i+2]

39. MARS & CART relationship IF • replace piecewise linear basis by step functions • keep only the newly formed product terms in M (leaf nodes of a binary tree) THEN MARS forward procedure = CART tree growing procedure