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### ML Decision Tree_2.pptx

• 1. BCSE209L Machine Learning Module- : Decision Trees Dr. R. Jothi
• 2. Decision Tree Induction Algorithms ■ ID3 ■ Can handle both numerical and categorical features ■ Feature selection – Entropy ■ CART (continuous features and continuous label) ■ Can handle both numerical and categorical features ■ Feature selection – Gini ■ Generally used for both regression and classification
• 3. Measure of Impurity: GINI • The Gini Index is the probability that a variable will not be classified correctly if it was chosen randomly. • Gini Index for a given node t with classes j NOTE: p( j | t) is computed as the relative frequency of class j at node t    j t j p t GINI 2 )] | ( [ 1 ) ( 3
• 4. GINI Index : Example • Example: Two classes C1 & C2 and node t has 5 C1 and 5 C2 examples. Compute Gini(t) • 1 – [p(C1|t) + p(C2|t)] = 1 – [(5/10)2 + [(5/10)2 ] • 1 – [¼ + ¼] = ½.    j t j p t GINI 2 )] | ( [ 1 ) ( 4 The Gini index will always be between [0, 0.5], where 0 is a selection that perfectly splits each class in your dataset (pure), and 0.5 means that neither of the classes was correctly classified (impure).
• 5. More on Gini • Worst Gini corresponds to probabilities of 1/nc, where nc is the number of classes. • For 2-class problems the worst Gini will be ½ • How do we get the best Gini? Come up with an example for node t with 10 examples for classes C1 and C2 • 10 C1 and 0 C2 • Now what is the Gini? • 1 – [(10/10)2 + (0/10)2 = 1 – [1 + 0] = 0 • So 0 is the best Gini • So for 2-class problems: • Gini varies from 0 (best) to ½ (worst). 5
• 6. Some More Examples • Below we see the Gini values for 4 nodes with different distributions. They are ordered from best to worst. See next slide for details • Note that thus far we are only computing GINI for one node. We need to compute it for a split and then compute the change in Gini from the parent node. C1 0 C2 6 Gini=0.000 C1 2 C2 4 Gini=0.444 C1 3 C2 3 Gini=0.500 C1 1 C2 5 Gini=0.278 6
• 7. Examples for computing GINI C1 0 C2 6 C1 2 C2 4 C1 1 C2 5 P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0    j t j p t GINI 2 )] | ( [ 1 ) ( P(C1) = 1/6 P(C2) = 5/6 Gini = 1 – (1/6)2 – (5/6)2 = 0.278 P(C1) = 2/6 P(C2) = 4/6 Gini = 1 – (2/6)2 – (4/6)2 = 0.444
• 8. Examples for Computing Error C1 0 C2 6 C1 2 C2 4 C1 1 C2 5 P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Error = 1 – max (0, 1) = 1 – 1 = 0 P(C1) = 1/6 P(C2) = 5/6 Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6 P(C1) = 2/6 P(C2) = 4/6 Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3 ) | ( max 1 ) ( t i P t Error i   8 CSEDIU
• 9. Comparison among Splitting Criteria For a 2-class problem: 9 CSEDIU
• 10. Example: Construct Decision tree using Gini index
• 11. Example: Construct Decision tree using Gini index Therefore, attribute B will be chosen to split the node.
• 12. Gini vs Entropy •Computationally, entropy is more complex since it makes use of logarithms and consequently, the calculation of the Gini Index will be faster. • Accuracy using the entropy criterion are slightly better (not always).
• 13. Table 11.6 Algorithm Splitting Criteria Remark ID3 Information Gain 𝛼 𝐴, 𝐷 = 𝐸 𝐷 − 𝐸𝐴(D) Where 𝐸 𝐷 = Entropy of D (a measure of uncertainty) = − 𝑖=1 𝑘 𝑝𝑖 log 2𝑝𝑖 where D is with set of k classes 𝑐1, 𝑐2, … , 𝑐𝑘 and 𝑝𝑖 = |𝐶𝑖,𝐷| |𝐷| ; Here, 𝐶𝑖,𝐷 is the set of tuples with class 𝑐𝑖 in D. 𝐸𝐴 (D) = Weighted average entropy when D is partitioned on the values of attribute A = 𝑗=1 𝑚 |𝐷𝑗| |𝐷| 𝐸(𝐷𝑗) Here, m denotes the distinct values of attribute A. • The algorithm calculates 𝛼(𝐴𝑖,D) for all 𝐴𝑖 in D and choose that attribute which has maximum 𝛼(𝐴𝑖,D). • The algorithm can handle both categorical and numerical attributes. • It favors splitting those attributes, which has a large number of distinct values. 13
• 14. Algorithm Splitting Criteria Remark CART Gini Index 𝛾 𝐴, 𝐷 = 𝐺 𝐷 − 𝐺𝐴(D) where 𝐺 𝐷 = Gini index (a measure of impurity) = 1 − 𝑖=1 𝑘 𝑝𝑖 2 Here, 𝑝𝑖 = |𝐶𝑖,𝐷| |𝐷| and D is with k number of classes and GA(D) = |𝐷1| |𝐷| 𝐺(𝐷1) + |𝐷2| |𝐷| 𝐺(𝐷2), when D is partitioned into two data sets 𝐷1 and 𝐷2 based on some values of attribute A. • The algorithm calculates all binary partitions for all possible values of attribute A and choose that binary partition which has the maximum 𝛾 𝐴, 𝐷 . • The algorithm is computationally very expensive when the attribute A has a large number of values. 14
• 15. Algorithm Splitting Criteria Remark C4.5 Gain Ratio 𝛽 𝐴, 𝐷 = 𝛼 𝐴, 𝐷 𝐸𝐴 ∗ (D) where 𝛼 𝐴, 𝐷 = Information gain of D (same as in ID3, and 𝐸𝐴 ∗ (D) = splitting information = − 𝑗=1 𝑚 |𝐷𝑗| |𝐷| 𝑙𝑜𝑔2 |𝐷𝑗| |𝐷| when D is partitioned into 𝐷1, 𝐷2, … , 𝐷𝑚 partitions corresponding to m distinct attribute values of A. • The attribute A with maximum value of 𝛽 𝐴, 𝐷 is selected for splitting. • Splitting information is a kind of normalization, so that it can check the biasness of information gain towards the choosing attributes with a large number of distinct values. In addition to this, we also highlight few important characteristics of decision tree induction algorithms in the following. 15
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