# Decision Tree for Playing Tennis

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1 Decision Tree

2 Decision Tree for Playing Tennis

3 (outlook=sunny, wind=strong, humidity=normal,? )

4 DT for prediction C-section risks

5 Characteristics of Decision Trees Decision trees have many appealing properties Similar to human decision process, easy to understand Deal with both discrete and continuous features Highly flexible hypothesis space, as the # of nodes (or depth) of the tree increase, decision tree can represent increasingly complex decision boundaries

6 DT can represent arbitrarily complex decision boundaries Y N Y N Y N Y 5 N If needed, the tree can keep on growing until all examples are correctly classified! Although it may not be the best idea

7 How to learn decision trees? Possible goal: find a decision tree h that achieves minimum error on training data Trivially achievable if use a large enough tree Another possibility: find the smallest decision tree that achieves the minimum training error NP-hard

8 Greedy Learning For DT We will study a top-down, greedy search approach. Instead of trying to optimize the whole tree together, we try to find one test at a time. Basic idea: (assuming discrete features, relax later) 1. Choose the best attribute to test on at the root of the tree. 2. Create a descendant node for each possible outcome of the test 3. Training examples in training set S are sent to the appropriate descendent node 4. Recursively apply the algorithm at each descendant node to select the best attribute to test using its associated training examples If all examples in a node belong to the same class, turn it into a leaf node, label with the majority class

9

10 One possible question: is x <0.5? [13, 15] x < 0.5? [8, 0] [5, 15]?

11 Continue [13, 15] x < 0.5? [8, 0] [5, 15] y<0.5? [4, 0] [1, 15]? This could keep on going, until all examples are correctly classified.

12 Choosing the best test X X2 T F T F Which one is better?

13 Choosing the Best test: A General View S X1 S: current set of training examples T F m branches, one for each possible outcome of the test S S2,, : m subsets of training examples Uncertainty of the class label in S Total Expected Remaining Uncertainty after the test

14 Uncertainty Measure: Entropy H ( y) k i 1 p i log 1 k 2 pi log2 pi i 1 p i

15 Entropy is a concave function downward H(y) P(y=0) Minimum uncertainty occurs when p 0 =0 or 1

16 The Information Gain approach: Measuring uncertainty using entropy: 26 + t 7 - T F

17 Mutual information By measuring the reduction of entropy, we are measuring the mutual information between the feature we test on and the class label Where This is also called the information gain criterion

18 Choosing the Best Feature: Summary t Original uncertainty Total Expected Remaining Uncertainty after the test Measures of Uncertainty Error Entropy Gini Index

19 Example

20 Selecting the root test using information gain Humidity Outlook High Normal sunny Overcast Rain

21 Continue building the tree Outlook sunny Overcast Rain Yes Which test should be placed here? Humidity High Normal

22 Issues with Multi-nomial Features Multi-nomial features: more than 2 possible values Consider two features, one is binary, the other has 100 possible values, which one you expect to have higher information gain? Conditional entropy of Y given the 100-valued feature will be low why? This bias will prefer multinomial features to binary features Method 1: To avoid this, we can rescale the information gain: H ( y) H ( y x arg max j H ( x ) j Method 2: Test for one value versus all of the others Method 3: Group the values into two disjoint sets and test one set against the other j ) Information gain of

23 Dealing with Continuous Features Test against a threshold How to compute the best threshold for? Sort the examples according to. Move the threshold from the smallest to the largest value Select that gives the best information gain Trick: only need to compute information gain when class label changes Note that continuous features can be tested for multiple times on the same path in a DT

24 Considering both discrete and continuous features If a data set contains both types of features, do we need special handling? No, we simply consider all possibly splits in every step of the decision tree building process, and choose the one that gives the highest information gain This include all possible (meaningful) thresholds

25 Issue of Over-fitting Decision tree has a very flexible hypothesis space As the nodes increase, we can represent arbitrarily complex decision boundaries This can lead to over-fitting t2 t3 Possibly just noise, but the tree is grown larger to capture these examples

26 Over-fitting

27 Avoid Overfitting Early stop Stop growing the tree when data split does not offer large benefit (e.g., compare information gain to a threshold, or perform statistical testing to decide if the gain is significant) Post pruning Separate training data into training set and validating set Evaluate impact on validation set when pruning each possible node Greedily prune the node that most improves the validation set performance

28 Effect of Pruning

29 Regression Tree Similar ideas can be applied for regression problems Prediction is computed as the average of the target values of all examples in the leave node Uncertainty is measured by sum of squared errors

30 Example Regression Tree Predicting MPG of a car given its # of cylinders, horsepower, weight, and model year

31 Summary Decision tree is a very flexible classifier Can model arbitrarily complex decision boundaries By changing the depth of the tree (or # of nodes in the tree), we can increase of decrease the model complexity Handle both continuous and discrete features Handle both classification and regression problems Learning of the decision tree Greedy top-down induction Not guaranteed to find an optimal decision tree DT can overfitting to noise and outliers Can be controlled by early stopping or post pruning

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