Decision Trees Chapter 18 From Data to Knowledge

Concerns Representational Bias: –Hyperrectangles – does it match domain Generalization Accuracy –Is the learned concept correct? Comprehensibility –Medical diagnosis Efficiency of Learning Efficiency of Learned Procedure

Simple Example: Weather Data: Four Features: windy, play, outlook: nominal Temperature: numeric outlook = sunny | humidity <= 75: yes (2.0) | humidity > 75: no (3.0) outlook = overcast: yes (4.0) outlook = rainy | windy = TRUE: no (2.0) | windy = FALSE: yes (3.0)

Dumb DT Algorithm Build tree: ( discrete features only) If all entries below node are homogenous, stop Else pick a feature at random, create a node for feature and form subtrees for each of the values of the feature. Recurse on each subtree. Will this work?

Properties of Dumb Algorithm Complexity –Homogeneity cost is O(DataSize) –Splitting is O(DataSize) –Times number of node in tree = bd on work Accuracy on training set –perfect Accuracy on test set –Not great. almost random

Many DT models Random selection worked – –If n-binary features then: –N * 2*(N-1)*2*(N-2).. = O(2^N*N!) UGH! Which trees are best? Occam’s razor: small ones (testable?) Exhaustive search impossible, so maybe Heuristic Search. But what heuristic? Goal: replace random with heuristic selection

Heuristic DT algorithm Entropy Set with mixed classes c1, c2,..ck Entropy(S) = - sum pi* lg(pi) where pi is probability of class ci. Sum weighted entropies of each subtrees, where weight is proportion of examples in the subtree. This defines a quality measure on features.

Heuristic score of a feature Say split on feature f yields: (4+, 4-) and ( 1+, 3-) quality of f = 8/12*E({4+,4-}+ 4/12*E({1+,3-}) = 8/13* 2 + 4/12* (- 1/4*log(1/4) -3/4*log(3/4)) Do this for every feature! J48 is roughly dumb + entropy heuristic

Shannon Entropy Entropy is the only function that: Is 0 when only 1 class present Is k if 2^k classes, equally present Is “additive” ie. – E(X,Y) = E(X)+E(Y) if X and Y are independent. Entropy sometimes called uncertainty and sometimes information. Uncertainty defined on RV where “draws” are from the set of classes.

Majority Function Suppose 2n boolean features. Class defined by n or more features are on. How big is the tree? At least 2n choose n leaves. Prototype Function: At least k of n are true is a common medical concept. Concepts that are prototypical do not match the representational bias of DTS.

Dts with real valued attributes Idea: convert to solved problem For each real valued attribute f with values v1, v2,… vn (sorted) and binary features: f1< (v1+v2)/2 f2 < (v2+v3/2) etc Other approaches possible. E.g. fi<any vj so no sorting needed

DTs ->Rules (Part) For each leaf, we make a rule by collecting the tests to the leaf. Number of rules = number of leaves Simplification: test each condition on a rule and see if dropping it harms accuracy. Can we go from Rules to DTs –Not easily. Hint: no root.

Summary Comprehensible if tree is not large. Effective if small number of features sufficient. Bias. Does multi-class problems naturally. Easily generates rules (expert system) –And measures of confidence (count) Can be extended for regression. Easy to implement and low complexity