Pattern Recognition, 4th Edition

16.4.2. Fuzzy Clustering

In this section we consider indices suitable for fuzzy clustering. In this context, we seek clusterings that are not very fuzzy, that is, those whose clusters exhibit small overlap. In other words, we seek clusterings where most of the vectors of X exhibit high grade of membership in only one cluster. Recall that a fuzzy clustering is defined by the N × m matrix U = [u^ij], where u^ij denotes the grade of membership of the vector xⁱ in the j-th cluster. Also, let W = {w^j,j = 1,…,m} be the set of the cluster representatives.

The strategy followed for the hard clustering case is also adopted here. That is, we define an appropriate index q (not to be confused with the fuzzifier) and we search for the minimum or the maximum ...