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1. This problem involves the K-means clustering algorithm. (a) Prove (10.12). (b) On the basis of this identity, argue that the K-means clustering algorithm (Algorithm 10.1) decreases the objective (10.11) at each iteration. P P 1 |CA| 1,1' C, j=1 ( Air j)2 = 2 (11) - )?, 1 (lij Ikj (10.12) ieCk j=1 Algorithm 10.1 K-Means Clustering 1. Randomly assign a number, from 1 to K, to each of the observations. These serve as initial cluster assignments for the observations. 2. Iterate until the cluster assignments stop changing: (a) For each of the K clusters, compute the cluster centroid. The kth cluster centroid is the vector of the p feature means for the observations in the kth cluster. (b) Assign each observation to the cluster whose centroid is closest (where closest is defined using Euclidean distance). K minimize C1,...,CK 1 Ck (; tv j)? . ( .. ' (10.11) k- 1,1' ECk j=1