Fuzzy CMeans

Fuzzy C-means [16] is a fuzzy version of k-means to include the possibility of having membership of the samples in more than one cluster. The goal is to find an optimal fuzzy c-partition that minimizes the objective function n c

where vi is the centroid of the ith cluster; uij is the membership value vector of the ith class for the jth sample; dj is the Euclidean distance between the ith class and sample xj; c and n denote the number of classes to be clustered and the total number of samples, respectively; and m is a weighting exponential parameter on each fuzzy membership with1 < m < <x>. The FCM algorithm can be described as follows:

1. Initialize membership function U(l=0) to random values.

2. Compute the centroid of the ith class with Eq. (6.3).

3. Update membership function U(l) with Eq. (6.4).

4. If ||U(l-1) — U(l) | < e or a predefined number of iteration is reached, stop. Otherwise l = l + 1 and go to step 2. e is a small positive constant.

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