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solutions2 - Assignment#2 Solutions(Chapter 4 3 Consider...

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Assignment#2 Solutions (Chapter 4) 3. Consider the training examples shown in Table 4.8 for a binary classification problem. a) What is the entropy of this collection of training examples with respect to the positive class? Answer: There are four positive examples and five negative examples. Thus, P (+) = 4 / 9 and P ( ) = 5 / 9. The entropy of the training examples is 4 / 9 log 2 (4 / 9) 5 / 9 log 2 (5 / 9) = 0 . 9911 . b) What are the information gains of a 1 and a 2 relative to these training examples? Answer: For attribute a 1 , the corresponding counts and probabilities are: a 1 + - T 3 1 F 1 4 The entropy for a 1 is given by: 4/9 [ -(3/4)log(3/4) (1/4)log(1/4)] + 5/9[ -(1/5)log(1/5) (4/5)log(4/5)] = 0.7616. Therefore, the information gain for a 1 is 0 . 9911 0 . 7616 = 0 . 2294. For attribute a 2 , the corresponding counts and probabilities are: A 2 + - T 2 3 F 2 2 The entropy for a 2 is given by: 5/9[ -(2/5)log(2/5) (3/5)log(3/5)] + 4/9 [-(2/4)log(2/4) (2/4)log(2/4)] = 0.9839. Therefore, the information gain for a 2 is 0 . 9911 0 . 9839 = 0 . 0072. c) For a 3 , which is a continuous attribute, compute the information gain for every possible split. Answer: a 3 Class label Split point Entropy Info Gain 1.0 + 2.0 0.8484 0.1427 3.0 - 3.5 0.9885 0.0026 4.0 + 4.5 0.9183 0.0728 5.0 - 5.0 - 5.5 0.9839 0.0072 6.0 + 6.5 0.9728 0.0183 7.0 + 7.0 - 7.5 0.8889 0.1022 The best split for a 3 occurs at split point equals to 2. d) What is the best split (among a 1 , a 2 , and a 3 ) according to the infor-mation gain? Answer: According to information gain, a 1 produces the best split. e) What is the best split (between a 1 and a 2 ) according to the classification error rate? Answer: For attribute a 1 , the error rate = 2 / 9. For attribute a 2 , the error rate = 4 / 9. Therefore, according to error rate, a 1 produces the best split.
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