Fuzzy k-Means: history and applications

The fuzzy approach to clustering arises to cope with situations where objects have not a clear assignment. Unlike the hard/standard approach where each object can only belong to exactly one cluster, in a fuzzy setting, the assignment is soft; that is, each object is assigned to all clusters with certain membership degrees varying in the unit interval. The best known fuzzy clustering algorithm is the fuzzy $k$-means (F$k$M), or fuzzy $c$-means. It is a generalization of the classical $k$-means method. Starting from the F$k$M algorithm, and in more than 40 years, several variants have been proposed. The peculiarity of such different proposals depends on the type of data to deal with, and on the cluster shape. The aim is to show fuzzy clustering alternatives to manage different kinds of data, ranging from numeric, categorical or mixed data to more complex data structures, such as interval-valued, fuzzy-valued or functional data, together with some robust methods. Furthermore, the case of two-mode clustering is illustrated in a fuzzy setting.