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Fuzzy (Soft) Clustering
Example: Let cluster feature be

 Fuzzy clustering:
 k fuzzy clusters C1, …, Ck, represented as a partition matrix M = [wij]
 P1: For each object oi and cluster Cj, 0 ≤ wij ≤ 1(fuzzy set)
 P2: For each object oi,
\[\sum_{i=1}^{k} w_{ij}=1\]
(equal participation in clustering)  P3: For each cluster Cj,
\[0<\sum_{i=1}^{n} w_{ij}
(ensure no empty cluster)  Let c1, …, ck as the centers of k clusters
 For each object oi, sum of square error SSE, p is a parameter:
\[SSE(C_{j})=\sum_{i=1}^{n} w_{ij}^{p}dist(o_{i},c_{j})^2\]
\[SSE(o_{j})=\sum_{j=1}^{k} w_{ij}^{p}dist(o_{i},c_{j})^2\]
 For each cluster C, sum of square error:
\[SSE(C)=\sum_{i=1}^{n} \sum_{j=1}^{k} w_{ij}^{p}dist(o_{i},c_{j})^2\]
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