Modeling proportional membership in fuzzy clustering

To provide feedback from a cluster structure to the data from which it has been determined, we propose a framework for mining typological structures based on a fuzzy clustering model of how the data are generated from a cluster structure. To relate data entities to cluster prototypes, we assume that the observed entities share parts of the prototypes in such a way that the membership of an entity to a cluster expresses the proportion of the cluster's prototype reflected in the entity (proportional membership). In the generic version of the model, any entity may independently relate to any prototype, which is similar to the assumption underlying the fuzzy c-means criterion. The model is referred to as fuzzy clustering with proportional membership (FCPM). Several versions of the model relaxing the generic assumptions are presented and alternating minimization techniques for them are developed. The results of experimental studies of FCPM versions and the fuzzy c-means algorithm are presented and discussed, especially addressing the issues of fitting the underlying clustering model. An example is given with data in the medical field in which our approach is shown to suit better than more conventional methods.