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 Ck 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 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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2 7 K-Means (continued) ± The K-Means algorithm attempts to minimize ± A more general strategy would be to try to minimize ± If we take d to be Euclidean distance and insist that m 1 , . .., m K be among the n data objects to be clustered, we are led to the K-Medoids algorithm. center. cluster a represents where ) , ( ∑∑ 1) ( k K kk i C k i k m m x d n == . || || ) ( 2 1 = = k i C k i K k k x x n 8 K Medoids Clustering 0.
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21%20K%20means%20clustering%204_15_08 - Clustering...

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