1. amt@cs.rit.eduProprietary and Confidential NOTICE: Proprietary and Confidential This material is proprietary to A. Teredesai and GCCIS, RIT. Slide 1 Decision Trees AMT

2. amt@cs.rit.edu Dr. Ankur M. Teredesai P2 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P3 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 Outlook sunnyovercastrainy HumidityWindy highnormal no falsetrue yes no

4. amt@cs.rit.edu Dr. Ankur M. Teredesai P4 Data Set for Playing Tennis OutlookTemp.HumidityWindyPlay SunnyHotHighFalseNo SunnyHotHighTrueNo OvercastHotHighFalseYes RainyMildHighFalseYes RainyCoolNormalFalseYes RainyCoolNormalTrueNo OvercastCoolNormalTrueYes OutlookTemp.HumidityWindyplay SunnyMildHighFalseNo SunnyCoolNormalFalseYes RainyMildNormalFalseYes SunnyMildNormalTrueYes OvercastMildHighTrueYes OvercastHotNormalFalseYes RainyMildHighTrueNo

5. amt@cs.rit.edu Dr. Ankur M. Teredesai P5 Decision Tree For Playing Tennis Outlook sunnyovercastrainy Humidity Windy highnormal no falsetrue yes no

6. amt@cs.rit.edu Dr. Ankur M. Teredesai P6 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P7 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P8 Decision Tree Induction _____________________________________ 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 Outlook ____________________________________ 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 Sunny Overcast Rain

9. amt@cs.rit.edu Dr. Ankur M. Teredesai P9 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 + log 2 p + – p - log 2 p -

10. amt@cs.rit.edu Dr. Ankur M. Teredesai P10 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 OutlookTemp.HumidityWindyPlay SunnyHotHighFalseNo SunnyHotHighTrueNo OvercastHotHighFalseYes RainyMildHighFalseYes RainyCoolNormalFalseYes RainyCoolNormalTrueNo OvercastCoolNormalTrueYes OutlookTemp.HumidityWindyplay SunnyMildHighFalseNo SunnyCoolNormalFalseYes RainyMildNormalFalseYes SunnyMildNormalTrueYes OvercastMildHighTrueYes OvercastHotNormalFalseYes RainyMildHighTrueNo

11. amt@cs.rit.edu Dr. Ankur M. Teredesai P11 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 S V = { s  S | A(s) = V}

12. amt@cs.rit.edu Dr. Ankur M. Teredesai P12 Example _____________________________________ 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 Outlook ____________________________________ 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 Sunny Overcast Rain Which attribute should be tested here? Gain (S sunny, Humidity) = =.970 - (3/5) 0.0 - (2/5) 0.0 =.970 Gain (S sunny, Temperature) =.970 - (2/5) 0.0 - (2/5) 1.0 - (1/5) 0.0 =.570 Gain (S sunny, Wind) =.970 - (2/5) 1.0 - (3/5).918 =.019

13. amt@cs.rit.edu Dr. Ankur M. Teredesai P13 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P14 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P15 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P16 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

17. amt@cs.rit.edu Dr. Ankur M. Teredesai P17

18. amt@cs.rit.edu Dr. Ankur M. Teredesai P18 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P19 Reasons for Overfitting Noisy training instances. Consider an noisy training example: Outlook = Sunny;Temperature = Hot; Humidity = Normal; Wind = Strong; PlayTennis = No Outlook sunnyovercastrainy HumidityWindy highnormal no falsetrue yes no add new test

20. amt@cs.rit.edu Dr. Ankur M. Teredesai P20 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P21 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P22 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P23 Reduced-Error Pruning 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 sunnyovercastrainy HumidityWindy highnormal no falsetrue yes no

24. amt@cs.rit.edu Dr. Ankur M. Teredesai P24 Reduced Error Pruning Example

25. amt@cs.rit.edu Dr. Ankur M. Teredesai P25 Rule Post-Pruning IF (Outlook = Sunny) & (Humidity = High) THEN PlayTennis = No IF (Outlook = Sunny) & (Humidity = Normal) THEN PlayTennis = Yes ………. 1.Convert tree to equivalent set of rules. 2.Prune each rule independently of others. 3.Sort final rules by their estimated accuracy, and consider them in this sequence when classifying subsequent instances. Outlook sunnyovercastrainy HumidityWindy normal no falsetrue yes no

26. amt@cs.rit.edu Dr. Ankur M. Teredesai P26 Continuous Valued Attributes Create a set of discrete attributes to test continuous. Apply Information Gain in order to choose the best attribute. Temperature: 40 4860728090 PlayTennis:No No Yes Yes Yes No Temp>54 Tem>85

27. amt@cs.rit.edu Dr. Ankur M. Teredesai P27 An Alternative Measure for Attribute Selection where:

28. amt@cs.rit.edu Dr. Ankur M. Teredesai P28 Missing Attribute Values Strategies: 1.Assign most common value of A among other examples belonging to the same concept. 2.If node n tests the attribute A, assign most common value of A among other examples sorted to node n. 3.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. amt@cs.rit.edu Dr. Ankur M. Teredesai P29 Summary Points 1.Decision tree learning provides a practical method for concept learning. 2.ID3-like algorithms search complete hypothesis space. 3.The inductive bias of decision trees is preference (search) bias. 4.Overfitting the training data is an important issue in decision tree learning. 5.A large number of extensions of the ID3 algorithm have been proposed for overfitting avoidance, handling missing attributes, handling numerical attributes, etc.

30. amt@cs.rit.edu Dr. Ankur M. Teredesai P30 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. amt@cs.rit.eduProprietary and Confidential NOTICE: Proprietary and Confidential This material is proprietary to A. Teredesai and GCCIS, RIT. Slide 31 RainForest 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P32 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P33 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 Decision Tree: A Classification Model

34. amt@cs.rit.edu Dr. Ankur M. Teredesai P34 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P35 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 e from 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P36 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P37 Decision Tree Terms (Cont ’ d) P { q1, q2, …, qk } n ek { qk} e2 { q2 } e1 { q1 }

38. amt@cs.rit.edu Dr. Ankur M. Teredesai P38 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P39 RainForest: Tree Induction Schema (Cont ’ d) AVC stands for Attribute ­ Value, 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 AVC ­ group of a node n to be the set of all AVC ­ sets at node n. Size of the AVC ­ set 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P40 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P41 Steps for Algorithms in RainForest Family 1. AVC ­ group 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P42 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P43 Algorithm:RF-Hybrid Combination of RF-Write and RF-Read Performance can be improved by concurrent construction of AVC-sets

44. amt@cs.rit.edu Dr. Ankur M. Teredesai P44 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P45 Experiments: Datasets

46. amt@cs.rit.edu Dr. Ankur M. Teredesai P46 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P47 Experiment Results (2) The performance of RF-Write, RF-Read and RF-Hybrid as the input database increases:

48. amt@cs.rit.edu Dr. Ankur M. Teredesai P48 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P49 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. amt@cs.rit.edu Dr. Ankur M. Teredesai P50 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