Fuzzy clustering with polynomial fuzzifier function in connection with m-estimators

Abstract

Fuzzy clustering approaches use membership values or weights to assign data to clusters. In standard fuzzy clustering, a parameter called fuzzifier is introduced which leads to certain disadvantages. That can be overcome by a more general approach replacing the usual fuzzifier by more general functions such as polynomial functions. Very similar to various forms of fuzzy clustering algorithms with just one prototype are M-estimators. The fuzzy clustering algorithm with polynomial fuzzifier function is used to define a new M-estimator. It is demonstrated that polynomial fuzzifier functions for M-estimators have better robustness properties than the usual fuzzifier. The special case of one prototype of fuzzy clustering algorithms can be reformulated as M-estimator. In this extend, some M-estimators can be considered to be special cases of fuzzy clustering algorithms. This is not the case for M-estimators that can not be applied with multiple prototypes. However, it is possible to extend an M-estimator in such a way that it provides an update mechanism similar to fuzzy clustering for multiple prototypes.