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21%20K%20means%20clustering%204_15_08

# 21%20K%20means%20clustering%204_15_08 - Clustering...

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1 1 K-means Clustering Analysis Peng Liu 4/15/2008 2 Clustering Algorithms For a given dissimilarity measure, the algorithms of clustering fall into 2 categories: Partitioning methods that attempt to optimally separate n objects into K clusters. Hierarchical methods that produce a nested sequence of clusters. 3 Some Partitioning Methods 1. K-Means 2. K-Medoids 3. Self-Organizing Maps (SOM) (Kohonen, 1990; Tomayo, P. et al., 1998) 4 K-Means Let x 1 , x 2 , ..., x n denote the objects to be clustered (each x i is an m-dimensional vector). Let C(i) denote the cluster assignment for the i th object. For a given K, the K-Means algorithm attempts to find a clustering of objects that minimizes || - || 2 1 1 ) ( ) ( 2 K k k i C k j C j i x x = = = ∑ ∑ 5 K-Means (continued) k k i C i k n x x / ) ( = = It is straightforward to show that ∑ ∑ = = = = = = k i C k i K k k i C k j C j i x x x x ) ( 2 K 1 k k 1 ) ( ) ( 2 || || n || - || 2 1 where n k is the number of objects in the k th cluster, and . Thus the K-Means algorithm . || || n ) ( 2 K 1 k k = = k i C k i x x attempts to minimize 6 K-Means Clustering Algorithm 0. Choose K points in m-dimensional space as K cluster means. 1. Given a current set of K means, assign each object to the nearest mean to produce an assignment of objects to K clusters. 2. For a given assignment of objects to K clusters, find the new mean of each cluster by averaging the objects in the each cluster. 3. Repeat steps 1 and 2 until the cluster assignments do not change.

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