Decision Trees

1. Decision Trees AMT

2. Overview • Decision trees • Appropriate problems for decision trees • Entropy and Information Gain • The ID3 algorithm • Avoiding Overfitting via Pruning • Handling Continuous-Valued Attributes • Handling Missing Attribute Values • Alternative Measures for Selecting Attributes

3. Outlook sunny overcast rainy Humidity yes Windy high normal false true no yes yes no Decision Trees • Definition: A decision tree is a tree s.t.: • Each internal node tests an attribute • Each branch corresponds to attribute value • Each leaf node assigns a classification

4. Data Set for Playing Tennis

5. Decision Tree For Playing Tennis Outlook sunny overcast rainy Humidity yes Windy high normal false true no yes yes no

6. When to Consider Decision Trees • Each instance consists of an attribute with discrete values (e.g. outlook/sunny, etc..) • The classification is over discrete values (e.g. yes/no ) • It is okay to have disjunctive descriptions – each path in the tree represents a disjunction of attribute combinations. Any Boolean function can be represented! • It is okay for the training data to contain errors – decision trees are robust to classification errors in the training data. • It is okay for the training data to contain missing values – decision trees can be used even if instances have missing attributes.

7. Decision Tree Induction Basic Algorithm: 1. A  the “best" decision attribute for a node N. 2. Assign A as decision attribute for the node N. 3. For each value of A, create new descendant of the node N. 4. Sort training examples to leaf nodes. 5. IF training examples perfectly classified, THEN STOP. ELSE iterate over new leaf nodes

8. Decision Tree Induction Outlook Sunny Rain Overcast ____________________________________ Outlook Temp Hum Wind Play ------------------------------------------------------- Sunny Hot High Weak No Sunny Hot High Strong No Sunny Mild High Weak No Sunny Cool Normal Weak Yes Sunny Mild Normal Strong Yes _____________________________________ Outlook Temp Hum Wind Play --------------------------------------------------------- Overcast Hot High Weak Yes Overcast Cool Normal Strong Yes _____________________________________ Outlook Temp Hum Wind Play --------------------------------------------------------- Rain Mild High Weak Yes Rain Cool Normal Weak Yes Rain Cool Normal Strong No Rain Mild Normal Weak Yes Rain Mild High Strong No

9. Entropy • Let S be a sample of training examples, and • p+ is the proportion of positive examples in S and • p- is the proportion of negative examples in S. • Then:  entropy measures the impurity of S: • E( S) = - p+ log2 p+ – p- log2p-

10. Entropy Example from the Dataset • In the Play Tennis dataset we had two target classes: yes and no • Out of 14 instances, 9 classified yes, rest no

11. Information Gain • Information Gain is the expected reduction in entropy caused by partitioning the instances according to a given attribute. • Gain(S, A) = E(S) - • where SV = { sS | A(s) = V}

12. Example Outlook Sunny Rain Overcast ____________________________________ Outlook Temp Hum Wind Play ------------------------------------------------------- Sunny Hot High Weak No Sunny Hot High Strong No Sunny Mild High Weak No Sunny Cool Normal Weak Yes Sunny Mild Normal Strong Yes _____________________________________ Outlook Temp Hum Wind Play --------------------------------------------------------- Overcast Hot High Weak Yes Overcast Cool Normal Strong Yes _____________________________________ Outlook Temp Hum Wind Play --------------------------------------------------------- Rain Mild High Weak Yes Rain Cool Normal Weak Yes Rain Cool Normal Strong No Rain Mild Normal Weak Yes Rain Mild High Strong No  Which attribute should be tested here? Gain (Ssunny , Humidity) = = .970 - (3/5) 0.0 - (2/5) 0.0 = .970 Gain (Ssunny , Temperature) = .970 - (2/5) 0.0 - (2/5) 1.0 - (1/5) 0.0 = .570 Gain (Ssunny , Wind) = .970 - (2/5) 1.0 - (3/5) .918 = .019

13. ID3 Algorithm • Informally: • Determine the attribute with the highest information gain on the training set. • Use this attribute as the root, create a branch for each of the values the attribute can have. • For each branch, repeat the process with subset of the training set that is classified by that branch.

14. Hypothesis Space Search in ID3 • The hypothesis space is the set of all decision trees defined over the given set of attributes. • ID3’s hypothesis space is a compete space; i.e., the target description is there! • ID3 performs a simple-to-complex, hill climbing search through this space.

15. Hypothesis Space Search in ID3 • The evaluation function is the information gain. • ID3 maintains only a single current decision tree. • ID3 performs no backtracking in its search. • ID3 uses all training instances at each step of the search.

16. Inductive Bias in ID3 • Preference for short trees • Preference for trees with high information gain attributes near the root. • Bias is a preference to some hypotheses, not a restriction on the hypothesis space

18. Overfitting • Definition: Given a hypothesis space H, a hypothesis h  H is said to overfit the training data if there exists some hypothesis h’  H, such that h has smaller error than h’ over the training instances, but h’ has a smaller error than h over the entire distribution of instances.

19. Reasons for Overfitting • Noisy training instances. Consider an noisy training example: • Outlook = Sunny;Temperature = Hot; Humidity = Normal; Wind = Strong; PlayTennis= No Outlook sunny overcast rainy Humidity yes Windy high normal false true no yes yes no add new test

20. area with probably wrong predictions Reasons for Overfitting • Small number of instances are associated with leaf nodes. In this case it is possible that for coincidental regularities to occur that are unrelated to the actual target concept. - + + + - + - + - + - + - - + - - - - - - - - - - - -

21. Approaches to Avoiding Overfitting • Pre-pruning: stop growing the tree earlier, before it reaches the point where it perfectly classifies the training data • Post-pruning: Allow the tree to overfit the data, and then post-prune the tree.

22. Criteria for Pruning • Use a separate set of instances, distinct from the training instances, to evaluate the utility of nodes in the tree. This requires the data to be split into a training set and a validation set which is then used for pruning. The reason is that the validation set is unlikely to suffer from same errors or fluctuations as the training set. • Use all the available data for training, but apply a statistical test to estimate whether expanding/pruning a particular node is likely to produce improvement beyond the training set.

23. Reduced-ErrorPruning • Split data into training and validation sets. • Pruning a decision node d consists of: • removing the subtree rooted at d. • making d a leaf node. • assigning d the most common classification of the training instances associated with d. • Do until further pruning is harmful: • Evaluate impact on validation set of pruning each possible node (plus those below it). • Greedily remove the one that most improves validation set accuracy. Outlook sunny overcast rainy Humidity yes Windy high normal false true no yes yes no

24. Reduced Error Pruning Example

25. Rule Post-Pruning • Convert tree to equivalent set of rules. • Prune each rule independently of others. • Sort final rules by their estimated accuracy, and consider them in this sequence when classifying subsequent instances. Outlook IF (Outlook = Sunny) & (Humidity = High) THEN PlayTennis = No IF (Outlook = Sunny) & (Humidity = Normal) THEN PlayTennis = Yes ………. sunny overcast rainy Humidity yes Windy normal false true no yes yes no

26. Continuous Valued Attributes • Create a set of discrete attributes to test continuous. • Apply Information Gain in order to choose the best attribute. • Temperature: 40 48 60 72 80 90 • PlayTennis: No No Yes Yes Yes No Temp>54 Tem>85

27. An Alternative Measure for Attribute Selection where:

28. Missing Attribute Values • Strategies: • Assign most common value of A among other examples belonging to the same concept. • If node n tests the attribute A, assign most common value of A among other examples sorted to node n. • If node n tests the attribute A, assign a probability to each of possible values of A. These probabilities are estimated based on the observed frequencies of the values of A. These probabilities are used in the information gain measure.

29. Summary Points • Decision tree learning provides a practical method for concept learning. • ID3-like algorithms search complete hypothesis space. • The inductive bias of decision trees is preference (search) bias. • Overfitting the training data is an important issue in decision tree learning. • A large number of extensions of the ID3 algorithm have been proposed for overfitting avoidance, handling missing attributes, handling numerical attributes, etc.

30. References • Mitchell, Tom. M. 1997. Machine Learning. New York: McGraw-Hill • Quinlan, J. R. 1986. Induction of decision trees. Machine Learning • Stuart Russell, Peter Norvig, 1995. Artificial Intelligence: A Modern Approach. New Jersey: Prantice Hall

31. RainForest - A Framework for Fast Decision Tree Construction of Large Datasets J. Gehrke, R. Ramakrishnan, V. Ganti Dept. of Computer Sciences University of Wisconsin-Madison

32. Introduction to Classification • An important Data Mining Problem • Input: a database of training records • Class label attributes • Predictor Attributes • Goal • to build a concise model of the distribution of class label in terms of predictor attributes • Applications • scientific experiments,medical diagnosis, fraud detection, etc.

33. Decision Tree:A Classification Model • It is one of the most attractive classification models • There are a large number of algorithms to construct decision trees • E.g.: SLIQ, CART, C4.5 SPRINT • Most are main memory algorithms • Tradeoff between supporting large databases, performance and constructing more accurate decision trees

34. Motivation of RainForest • Developing a unifying framework that can be applied to most decision tree algorithms, and results in a scalable version of this algorithm without modifying the results. • Separating the scalability aspects of these algorithms from the central features that determine the quality of the decision trees

35. Decision Tree Terms • Root,Leaf, Internal Nodes • Each leaf is labeled with one class label • Each internal node is labeled with one predictor attribute called the splitting attribute • Each edge efrom node n has a predicate q associated with it, q only involves splitting attributes. • P : set of predicates on all outgoing edges of an internal node; Non-overlapping, Exhaustive • Crit(n): splitting criteria of n; combination of splitting attributes and predicates

36. Decision Tree Terms(Cont’d) • F(n) :Family of database(D) tuples of a node n • Definition: • let E={e1,e2,…,ek}, Q={q1,q2,…,qk} be the edge set and predicate set for a node n; p be the parent node of n If n=root, F(n) = D If n≠root, let q(p→n) be the predicate on e(p→n), • F(n) = {t: t€F(n),t €F(p), and q(p→ n= True}

37. n e2 { q2 } e1 { q1 } ek { qk} Decision Tree Terms (Cont’d) • P { q1, q2, … , qk }

38. RainForest Framework:Top-down Tree Induction Schema • Input: node n, partition D, classification algorithm CL • Output: decision tree for D rooted at n • Top­Down Decision Tree Induction Schema: • BuildTree(Node n, datapartition D, algorithm CL) • (1) Apply CL to D to find crit(n) • (2) let k be the number of children of n • (3) if (k > 0) • (4) Create k children c1 ; ... ; ck of n • (5) Use best split to partition D into D1 ; . . . ; Dk • (6) for (i = 1; i <= k; i++) • (7) BuildTree(ci , Di ) • (8) endfor • (9) endif • RainForest Refinement: • (1a) for each predictor attribute p • (1b) Call CL.find_best_partitioning(AVC­set of p) • (1c) endfor • (2a) k = CL:decide_splitting_criterion();

39. RainForest:Tree Induction Schema (Cont’d) • AVC stands for Attribute­Value, Classlabel • AVC-set: AVC-set of a predictor attribute a to be the projection of F(n) onto a and the class label whereby counts of the individual class labels are aggregated • AVC-group: the AVC­group of a node n to be the set of all AVC­sets at node n. • Size of the AVC­set of a predictor attribute a at node n • depends only on the number of distinct attribute values of a and the number of class labels in F(n). • AVC-group(r) is not equal to F( r ) : contains aggregated information that is sufficient for decision tree construction

40. AVC-groups and Main Memory • The memory size determines the implementation of RainForest Schema. • Case 1: AVC-group of the root node fits in the M.M. • RF-Write; RF-Read; RF-Hybrid • Case 2: each individual AVC-set of the root node fits into M.M., but the AVC-group does not. • RF-Vertical • Case 3: Other than Case 1&2

41. Steps for Algorithms in RainForest Family • 1. AVC­group Construction • 2. Choose Splitting Attribute and Predicate • This step uses the decision tree algorithm CL that is being scaled using the RainForest framework • 3. Partition D Across the Children Nodes • We must read the entire dataset and write out all records, partitioning them into child ``buckets'' according to the splitting criterion chosen in the previous step.

42. Algorithms: RF-Write/RF-Read • Prerequisite:AVC-group fits into M.M. • RF-Write: • For each level of the tree,it reads the entire database twice and writes the entire database once • RF-Read • Makes an increasing number of scans of entire database • Marks one end of the design spectrum in the RainForest framework

43. Algorithm:RF-Hybrid • Combination of RF-Write and RF-Read • Performance can be improved by concurrent construction of AVC-sets

44. Algorithm:RF-Vertical • Prerequisite: individual AVC-set can fit into M.M. • For very large sets, a temporary file is generated for each node, the large sets are constructed from this temporary file. • For small sets, construct them in M.M.

45. Experiments: Datasets

46. Experiment Results: (1) • When the overall maximum number of entries in the AVC-group of the root node is about 2.1 million, requiring a maximum memory size of 17MB.

47. Experiment Results (2) • The performance of RF-Write, RF-Read and RF-Hybrid as the input database increases:

48. Experiment Results (3) • How internal properties of the AVC-groups of the training database influence performance? • Result: the AVC-group size and Main Memory size are the two factors which determine the performance.

49. Experiment Results (4) • How performance is affected as the number of attributes is increasing? • Result: a roughly linear scaleup with the number of attributes.

50. Conclusion • A scaling decision tree algorithm that is applicable to all decision tree algorithms at that time. • AVC-group is the key idea. • Database scan at each level of the decision tree • Too much dependence over the size of available main memory