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hw3_Solution - CS 6375 Homework 3 Chenxi Zeng UTD ID...

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CS 6375 Homework 3 Chenxi Zeng, UTD ID: 11124236 1. K=1: K=3:
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We can see from the figures above: Large k (k=3) is less sensitive to noise and can better estimate for discrete classes; Small k can capture the better structure of the space, but sometimes too exactly, and not sensitive to noise (overfitting). 2. a) When k is odd: The error happens when a data point P is in class C1, but the number of points in C1 is less than ( 1) 2 k - , such as 0,1,2,…, ( 1) 2 k - point. Since there are n points in the space, and they share equal prior probabilities, then 1 2 of them is of class C1, 1 2 is class C2. We have 2 n combination of two different points. Therefore the average probability of error is ( ) n p e = ( 1) 2 0 1 2 k n j n j - = ÷ . b) From the conclusion from a), we know that ( ) n p e is 1 2 n when k=1. When k>1, following by the same way as in a), we know that if k is even, ( ) n p e is at least ( 2) 2 0 1 2 k n j n j - = ÷ . Both the odd and even results are greater or equal than 1 2 n . So 1-nearest neighbor rule has a lower error probability than
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