A new approach to fuzzy partitioning

Fuzzy clustering algorithms like the popular fuzzy c-means algorithm (FCM) are frequently used to automatically divide up the data space into fuzzy granules (fuzzy vector quantization). In the context of fuzzy systems, in order to be intuitive and meaningful to the user, the fuzzy membership functions of the used linguistic terms have to fulfill some requirements like boundedness of support or unimodality. By rewarding crisp membership degrees, we modify FCM and obtain different membership functions that better suit these purposes. We show that the modification can be interpreted as standard FCM using distances to the Voronoi cell of the cluster rather than using distances to the cluster prototypes. In consequence, the resulting partitions of the modified algorithm are much closer to those of the crisp original methods. The membership functions can be generalized to a fuzzified minimum function. We give some bounds on the approximation quality of this fuzzification