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A theory of fuzzy clustering

The problem of clustering a sample set on the basis of a fuzzy resemblance structure is considered. A rigorous mathematical theory is developed for general (i.e., non-necessary finite) sample sets in order to allow for future studies of an statistical nature relating the classification of a general set with that of its finite samples. The theory is presented in two major steps. The first step deals with the formalization of the concept of cluster. Unlike previous approaches which defined clusters as the fuzzy sets obtained as a result of the application of an algorithmic procedure (frequently based on an optimization goal) to the sample set, in this work clusters are characterized as fuzzy subsets which must satisfy certain conditions (independent of the classification goals being pursued) in order to qualify as potential taxonomical elements. The qualifying conditions are expressed as fuzzy relational equations derived extending simple relational equations in the conventional set domain. The problems in making such transition are discussed. Conditions for existence of fuzzy classifications are derived in terms of a class of fuzzy resemblance relations, called "likeness" relations. These conditions are shown to be much less restrictive than the corresponding conditions for the conventional case. In addition, the resulting cluster families provide richer representations of the underlying taxonomical structures. The problem of likeness relation derivation is analyzed from several viewpoints. The second step in this theoretical development deals with the problem of optimal representation of a sample set as a union of clusters. Using again the procedure of extending conventional set formulations, it is shown that approaches based on the minimization of functionals defined over all possible cluster representations, which satisfy certain desirable properties, must necessarily optimize one member of a uniparametric functional family. It is also shown that the extension of conventional concepts to the fuzzy domain provides a generalization of the concept of a number of clusters (which is no longer required to be an integer) and of "prototype element" of a cluster.