Clustering Based on Fuzzy Tolerance Quotient Spaces ?

The quotient space theory based on fuzzy tolerance relation is put forward to solve the problem of clustering in this paper. The similarity matrix does not always satisfy ultrametric inequality, theoretically and practically. We give the method to construct the hierarchical quotient space chain if the similarity matrix is only reflexive and symmetric. We consider not only the subset of data (cluster) but also the structure of the different granular clusters. The main contributions in this paper include four parts. (1) We introduce sufficient and necessary condition that normalized distance d(x, y) is an equicrural distance; (2) The relation between equicrural distance and ultrametric inequality is discussed; (3) We propose a method to get the proximate partition of a covering from a tolerance relation; (4) An example for using fuzzy tolerance quotient spaces to clustering is given.