Learning from Data: Decision Trees

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1 Learning from Data: Decision Trees Amos Storkey, School of Informatics University of Edinburgh Semester 1, 2004 LfD 2004

2 Decision Tree Learning - Overview Decision tree representation ID3 learning algorithm Entropy, Information gain Priors for Decision Tree Learning Overfitting and how to avoid it Reading: Mitchell, chapter 3 Acknowledgement: These slides are based on slides modified by Chris Williams and produced by Tom Mitchell, available from tom/ LfD

3 Decision trees Decision tree learning is a method for approximating discrete-valued 1 target functions, in which the learned function is represented as a decision tree Decision tree representation: Each internal node tests an attribute Each branch corresponds to attribute value Each leaf node assigns a classification Re-representation as if-then rules: disjunction of conjunctions of constraints on the attribute value instances 1 The method can be extended to learning continuous-valued functions LfD

4 Decision Tree for Play Tennis Outlook Sunny Overcast Rain Humidity Yes Wind High Normal Strong Weak No Yes No Yes Logical Formulation: (Outlook = Sunny Humidity = N ormal) (Outlook = Overcast) (Outlook = Rain W ind = W eak) LfD

5 When to Consider Decision Trees Instances describable by attribute value pairs Target function is discrete valued Disjunctive hypothesis may be required Possibly noisy training data Examples: Equipment or medical diagnosis Credit risk analysis Modelling calendar scheduling preferences LfD

6 Top-Down Induction of Decision Trees (ID3) 1. A is the best decision attribute for next node 2. Assign A as decision attribute for node 3. For each value of A, create new descendant of node 4. Sort training examples to leaf nodes 5. If training examples perfectly classified, Then STOP, Else iterate over new leaf nodes LfD

7 Which attribute is best? A1=? A2=? [29+,35-] [29+,35-] t f t f [21+,5-] [8+,30-] [18+,33-] [11+,2-] LfD

8 Entropy S is a sample of training examples p is the proportion of positive examples in S p is the proportion of negative examples in S Entropy measures the impurity of S Entropy(S) H(S) p log 2 p p log 2 p H(S) = 0 if sample is pure (all + or all -), H(S) = 1 bit if p = p = 0.5 LfD

9 Information Gain Gain(S, A) = expected reduction in entropy due to sorting on A Gain(S, A) Entropy(S) v V alues(a) S v S Entropy(S v) Information gain is also called the mutual information between A and the labels of S A1=? A2=? [29+,35-] [29+,35-] t f t f [21+,5-] [8+,30-] [18+,33-] [11+,2-] LfD

10 Training Examples Day Outlook Temperature Humidity Wind PlayTennis D1 Sunny Hot High Weak No D2 Sunny Hot High Strong No D3 Overcast Hot High Weak Yes D4 Rain Mild High Weak Yes D5 Rain Cool Normal Weak Yes D6 Rain Cool Normal Strong No D7 Overcast Cool Normal Strong Yes D8 Sunny Mild High Weak No D9 Sunny Cool Normal Weak Yes D10 Rain Mild Normal Weak Yes D11 Sunny Mild Normal Strong Yes D12 Overcast Mild High Strong Yes D13 Overcast Hot Normal Weak Yes D14 Rain Mild High Strong No LfD

11 Building the Decision Tree LfD

12 Hypothesis Space Search by ID3 Hypothesis space is complete! Target function surely in there... Outputs a single hypothesis (which one?) No back tracking Local minima... Batch rather than on-line learning More robust to noisy data... Implicit prior: approx prefer shortest tree LfD

13 Implicit priors in ID3 Searching space from simple to complex, starting with empty tree, guided by information gain heuristic Preference for short trees, and for those with high information gain attributes near the root Bias is a preference for some hypotheses, rather than a restriction of hypothesis space Occam s razor: prefer the shortest hypothesis that fits the data LfD

14 Occam s Razor Why prefer short hypotheses? Argument in favour: Fewer short hypotheses than long hypotheses a short hypothesis that fits data unlikely to be coincidence a long hypothesis that fits data might be coincidence Argument opposed: There are many ways to define small sets of hypotheses (notion of coding length(x) = log 2 P (X), Minimum Description Length...) e.g., all trees with a prime number of nodes that use attributes beginning with Z What s so special about small sets based on size of hypothesis?? LfD

15 Overfitting in Decision Trees Consider adding noisy training example #15: Sunny, Hot, Normal, Strong, P layt ennis = No What effect on earlier tree? Outlook Sunny Overcast Rain Humidity Yes Wind High Normal Strong Weak No Yes No Yes LfD

16 Overfitting in Decision Tree Learning Overfitting can occur with noisy training examples, and also when small numbers of examples are associated with leaf nodes ( coincidental or accidental regularities) Accuracy On training data On test data Size of tree (number of nodes) LfD

17 Avoiding Overfitting How can we avoid overfitting? stop growing when data split not statistically significant grow full tree, then post-prune How to select best tree: Measure performance over training data Measure performance over separate validation data set MDL: minimize size(tree) + size(misclassif ications(tree)) LfD

18 Reduced-Error Pruning Split data into training and validation set Do until further pruning is harmful: 1. Evaluate impact on validation set of pruning each possible node (plus those below it) 2. Greedily remove the one that most improves validation set accuracy produces smallest version of most accurate subtree What if data is limited? LfD

19 Effect of Reduced-Error Pruning Accuracy On training data On test data On test data (during pruning) Size of tree (number of nodes) LfD

20 Rule Post-Pruning 1. Convert tree to equivalent set of rules 2. Prune each rule independently of others, by removing any preconditions that result in improving its estimated accuracy 3. Sort final rules into desired sequence for use Perhaps most frequently used method (e.g., C4.5) LfD

21 Alternative Measures for Selecting Attributes Problem: if an attribute has many values, Gain will select it Example: use of dates in database entries One approach: use GainRatio instead GainRatio(S, A) Gain(S, A) SplitInf ormation(s, A) SplitInf ormation(s, A) c i=1 S i S log 2 S i S where S i is subset of S for which A has value v i LfD

22 Further points Dealing with continuous-valued attributes create a split, e.g. (T emperature > 72.3) = t, f. Split point can be optimized (Mitchell, 3.7.2) Handling training examples with missing data (Mitchell, 3.7.4) Handling attributes with different costs (Mitchell, 3.7.5) LfD

23 Summary Decision tree learning provides a practical method for concept learning/learning discrete-valued functions ID3 algorithm grows decision trees from the root downwards, greedily selecting the next best attribute for each new decision branch ID3 searches a complete hypothesis space, using a preference bias for smaller trees with higher information gain close to the root The overfitting problem can be tackled using post-pruning LfD

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