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Fuzzy (Soft) Clustering

Example: Let cluster feature be

  • C1: “digital camera” & “lense”
  • C2: “computer”

  • 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:

\[SSE(C)=\sum_{i=1}^{n} \sum_{j=1}^{k} w_{ij}^{p}dist(o_{i},c_{j})^2\]

Speaker notes:

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