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1 Group M L D Machine Learning M & Data Mining Chapter 7 Decision Trees Xin-Shun SDU School of Computer Science and Technology, Shandong University

2 Top 10 Algorithm in DM #1: C4.5 #2: K-Means #3: SVM #4: Apriori #5: EM #6: PageRank #7: AdaBoost #7: knn #7: Naive Bayes #10: CART Xin-Shun SDU School of Computer Science and Technology, Shandong University 2

3 Content Introduction CLS ID3 C4.5 CART Xin-Shun SDU School of Computer Science and Technology, Shandong University 3

4 Inductive Learning dl Examples Model Prediction Generalize Instantiated for another case The general conclusion should apply to unseen examples. Xin-Shun SDU School of Computer Science and Technology, Shandong University 4

5 Decision Tree A decision tree is a tree in which each branch node represents a choice between a number of alternatives each leaf node represents a classification or decision Xin-Shun SDU School of Computer Science and Technology, Shandong University 5

6 Example I Xin-Shun SDU School of Computer Science and Technology, Shandong University 6

7 Example II Xin-Shun SDU School of Computer Science and Technology, Shandong University 7

8 Decision Rules ( Outlook sunny Humidity normal) ( Outlook overcast) ( Outlo o k rain WW in d we a k ) Xin-Shun SDU School of Computer Science and Technology, Shandong University 8

9 Decision Tree Learning We wish to be able to induce a decision tree from a set of data about instances together with the decisions or classifications for those instances. Learning Algorithms: CLS (Concept Learning System) ID3 C4 C4.5 C5 Xin-Shun SDU School of Computer Science and Technology, Shandong University 9

10 Appropriate Problems for Decision Tree Learning Instances are represented by attribute-value pairs. The target function has discrete output values. Disjunctive descriptions may be required. The training data may contain errors. The training data may contain missing attribute values. Xin-Shun SDU School of Computer Science and Technology, Shandong University 10

11 CLS Algorithm 1. T the whole training set. Create a T node. 2. If all examples in T are positive, create a P node with T as its parent and stop. 3. If all examples in T are negative, create a N node with T as its parent and stop. 4. Select an attribute X with values v 1, v 2,, v N and partition T into subsets T 1, T 2,, T N according their values on X. Create N nodes T i (i = 1,..., N) with T as their parent and X = v i as the label of the branch from T to T i. 5. For each T i do: T T i and goto step 2. Xin-Shun SDU School of Computer Science and Technology, Shandong University 11

12 Example (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e Xin-Shun SDU School of Computer Science and Technology, Shandong University 12

13 Example (tall, blond, blue) w (tall, blond, brown) w (tall, silver, blue) w blond (tall, blond, blue) w (tall, blond, brown) w silver tall (tall, silver, black) e (tall, silver, blue) w black (tall, silver, black) e (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w (tall, silver, black) e (tall, black, brown) e (tall, black, black) e black (tall, black, brown) e (tall, black, black) e blue (tall, silver, blue) w brown (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e short (short, silver, blue) w (short, black, blue) w (short, blond, blue) w black (short, black, blue) w blue (short, black, blue) w (short, blond, black) e silver (short, silver, blue) w blond (short, blond, blue) w (short, blond, black) e blue (short, blond, blue) w black (short, blond, black) e Xin-Shun SDU School of Computer Science and Technology, Shandong University 13

14 Example blue (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w brown (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e black (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, silver, blue) w (short, blond, blue) w (tall, black, brown) e (tall, blond, brown) w (tall, silver, black) e (tall, black, black) e (short, blond, black) e blond black (tall, blond, brown) w (tall, black, brown) e Xin-Shun SDU School of Computer Science and Technology, Shandong University 14

15 ID3 Iterative Dichotomizer (version) 3 developed by Quinlan Select decision sequence of the tree based on information gain. Xin-Shun SDU School of Computer Science and Technology, Shandong University 15

16 ID3 Entropy (Binary Classification) S = (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e p Ps ( C ) 1 1 p P ( s C ) 2 2 C 1 : Class 1 C 2 : Class Entropy ( S ) p log p p log p Xin-Shun SDU School of Computer Science and Technology, Shandong University 16

17 ID3 Entropy (Binary Classification) Entropy( S) p log p p log p Entropy ( S ) Xin-Shun SDU School of Computer Science and Technology, Shandong University 17 p 1

18 ID3 Entropy (Binary Classification) Entropy( S) p log p p log p Entropy ( S ) p p 1 2 Xin-Shun SDU School of Computer Science and Technology, Shandong University 18

19 ID3 Entropy (Binary Classification) Entropy( S) p log p p log p p 14 /15 p 1/15 Entropy # = 14 # = 1 Xin-Shun SDU School of Computer Science and Technology, Shandong University 19

20 Information Gain Attribute A { v,, v } 1 n S v 1 v 2 v n S 1 S 2 S n Sv Gain( S, A) Entropy( S) Entropy( Sv ) S v A Xin-Shun SDU School of Computer Science and Technology, Shandong University 20

21 Information Gain S Gain ( S, A ) Entropy ( S ) v Entropy ( Sv ) S v A (tall, blond, blue) w (tall, blond, brown) w (tall, silver, blue) w tall (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e short (short, silver, blue) w (short, black, blue) w (short, blond, blue) w Entropy( S) 1 (short, blond, black) e Et Entropy ( S ) 1 Et Entropy ( S ) 1 tall short Gi Gain ( SHeight SHih, ) 0 Xin-Shun SDU School of Computer Science and Technology, Shandong University 21

22 Information Gain S Gain ( S, A ) Entropy ( S ) v Entropy ( Sv ) S v A blond (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w silver (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e black Entropy( S) 1 (tall, blond, blue) w (tall, blond, brown) w (short, blond, blue) w (short, blond, black) e (short, silver, blue) w (tall, silver, blue) w (tall, silver, black) e Entropy ( S ) silver (short, black, blue) w (tall, black, brown) e (tall, black, black) e Entropy( Sblond ) Entropy( S ) black Gi Gain ( SHi SHair, ) Xin-Shun SDU School of Computer Science and Technology, Shandong University 22

23 Information Gain S Gain ( S, A ) Entropy ( S ) v Entropy ( Sv ) S v A blue (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w brown (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e black Entropy( S) 1 (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, silver, blue) w (short, blond, blue) w Entropy( S ) 0 blue (tall, black, brown) e (tall, blond, brown) w Entropy( S ) brown (tall, silver, black) e (tall, black, black) e (short, blond, black) e Entropy( S ) 0 black Gain ( SEye, ) Xin-Shun SDU School of Computer Science and Technology, Shandong University 23

24 Information Gain (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e Gain( S, Hair) Gain( S, Height) 0 Gain( S, Eye) Xin-Shun SDU School of Computer Science and Technology, Shandong University 24

25 ID3 (modify of CLS) 1 T the whole training set. Create a T node. 2 If all examples in T are positive, create a P node with T as its parent and stop. 3 If all examples in T are negative, create a N node with T as its parent and stop. 4 Select an attribute X with values v 1, v 2,, v N and partition T into subsets T 1, T 2,, T N according their values on X. Create N nodes T i (i = 1,..., N) with T as their parent and X = v i as the label of the branch from T to T i. 5 For each T i do: T T i and goto step 2. Xin-Shun SDU School of Computer Science and Technology, Shandong University 25

26 ID3 (modify of CLS) 1 T the whole training set. Create a T node. 2 If all examples in T are positive, create a P node By maximizing the with T as its parent and stop. information gain. 3 If all examples in T are negative, create a N node with T as its parent and stop. 4 Select an attribute X with values v 1, v 2,, v N and partition T into subsets T 1, T 2,, T N according their values on X. Create N nodes T i (i = 1,..., N) with T as their parent and X = v i as the label of the branch from T to T i. 5 For each T i do: T T i and goto step 2. Xin-Shun SDU School of Computer Science and Technology, Shandong University 26

27 Example blue (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, blond, brown) w (tall, silver, blue) w (short, blond, blue) w brown (tall, silver, black) e (tall, black, brown) e (tall, black, black) e (short, blond, black) e black (tall, blond, blue) w (short, silver, blue) w (short, black, blue) w (tall, silver, blue) w (short, blond, blue) w blond (tall, black, brown) e (tall, blond, brown) w black (tall, silver, black) e (tall, black, black) e (short, blond, black) e (tall, blond, brown) w (tall, black, brown) e Xin-Shun SDU School of Computer Science and Technology, Shandong University 27

28 Windowing Windowing Xin-Shun SDU School of Computer Science and Technology, Shandong University 28

29 Windowing ID3 can deal with very large data sets by performing induction on subsets or windows onto the data. 1. Select a random subset of the whole set of training instances. 2. Use the induction algorithm to form a rule to explain the current window. 3. Scan through all of the training instances looking for exceptions to the rule. 4. Add the exceptions to the window Repeat steps 2 to 4 until there are no exceptions left. Xin-Shun SDU School of Computer Science and Technology, Shandong University 29

30 Inductive Biases Shorter trees are preferred. Attributes with higher information gain are selected first in tree construction. Greedy Search Preference bias (relative to restriction bias as in the VS approach) Why yprefer short hypotheses? Occam's razor Generalization Xin-Shun SDU School of Computer Science and Technology, Shandong University 30

31 Overfitting to the Training Data The training error is statistically smaller than the test error for a given hypothesis. Solutions: Early stopping Validation sets Statistical criterion for continuation (of the tree) Post-pruning Minimal description length cost-function ti = error complexity Xin-Shun SDU School of Computer Science and Technology, Shandong University 31

32 Pruning Techniques Reduced error pruning (of nodes) Used by ID3 Rule post-pruning pruning Used by C4.5 Xin-Shun SDU School of Computer Science and Technology, Shandong University 32

33 Reduced Error Pruning Use a separated validation set Tree accuracy: percentage of correct classifications on validation set Method: Do until further pruning is harmful Evaluate the impact on validation set of pruning each possible node. Greedily remove the one that most improves the validation set accuracy. Xin-Shun SDU School of Computer Science and Technology, Shandong University 33

34 Reduced Error Pruning Xin-Shun SDU School of Computer Science and Technology, Shandong University 34

35 C4.5:An Extension of ID3 Some additional features of C4.5 are: Incorporation of numerical (continuous) attributes. Nominal (discrete) values of a single attribute may be grouped together, to support more complex tests. Post-pruning after induction of trees, e.g. based on test sets, in order to increase accuracy. C4.5 can deal with incomplete information (missing attribute values). Xin-Shun SDU School of Computer Science and Technology, Shandong University 35

36 Rule Post-PruningPruning Fully induce the decision tree from the training set (allowing overfitting) Convert the learned tree to rules one rule for each path from the root node to a leaf node Prune each rule by removing any preconditions that result in improving its estimated accuracy Sort the pruned rules by their estimated accuracy, and consider them in this sequence when classifying subsequent instances Xin-Shun SDU School of Computer Science and Technology, Shandong University 36

37 Converting to Rules Xin-Shun SDU School of Computer Science and Technology, Shandong University 37

38 handling numeric attributes Continuous attribute discrete attribute Example Original attribute: Temperature = 82.5 New attribute: (temperature > 72.3) = t, f Example: Temerature <54 Temerature Temerature >85 How to choose split points? Xin-Shun SDU School of Computer Science and Technology, Shandong University 38

39 handling numeric attributes Choosing split points for a continuous attribute Sort the examples according to the values of the continuous attribute. Identify adjacent examples that differ in their target labels and attribute values a set of candidate split points Calculate the gain for each split point and choose the one with the highest gain. Xin-Shun SDU School of Computer Science and Technology, Shandong University 39

40 Attributes with Many Values Information Gain biases to attribute with many values eg e.g., date One approach use GainRatio instead GainRatio ( S, A ) Gain( S, A) SplitInformation i ( S, A) ) SplitInformation( S, A) c i 1 Si Si log S S Xin-Shun SDU School of Computer Science and Technology, Shandong University 40

41 Associate Attributes of Cost The availability of attributes may vary significantly in costs, e.g., medical diagnosis. Example: Medical disease classification Temperature BiopsyResult Pulse BloodTestResult High High How to learn a consistent tree with low expected cost? Xin-Shun SDU School of Computer Science and Technology, Shandong University 41

42 Associate Attributes of Cost Tan and Schlimmer (1990) Nunez (1988) Ga i n 2 ( S, A ) Cost( S, A) Gain( S, ) 2 A 1 ( Cost( A) 1) w w [0, 1] determines the importance of cost. Xin-Shun SDU School of Computer Science and Technology, Shandong University 42

43 Unknown Attribute Values S = (, x, ) (, x, ) (, x, ) (, y, ) (, y, ) (, y, ) (, y, ) (, z, ) (, z, ) (, z, ) (,?, ) (,?, ) Attribute t A={x, { y, z} } Gain ( S, A)? Possible Approaches: Assign most common value of A to the unknown one. Assign most common value of A with the same target value to the unknown one. Assign probability to each possible value. Xin-Shun SDU School of Computer Science and Technology, Shandong University 43

44 CART Classification And Regression Trees Generates binary decision tree: only 2 children created at each node (whereas ID3 creates a child for each subcategory). Each split makes the subset more pure than that before splitting. In ID3, Entropy is used to measure the splitting; in CART, impurity is used. Xin-Shun SDU School of Computer Science and Technology, Shandong University 44

45 CART Node impurity is 0 when all patterns at the node are of the same category; it becomes maximum when all the classes at the node are equally likely. Entropy Impurity Gini Impurity i( N) P( j )log2p( j ) j i ( N) P( ) P( ) i j Misclassification impurity i( N ) 1 max P ( ) ( j j i j Xin-Shun SDU School of Computer Science and Technology, Shandong University 45

46 Group Any Question? Xin-Shun SDU School of Computer Science and Technology, Shandong University

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