Improved fuzzy partitions for fuzzy regression models

Abstract

Fuzzy clustering algorithms like the popular fuzzy c-means algorithm (FCM) are frequently used to automatically divide up the data space into fuzzy granules. When the fuzzy clusters are used to derive membership functions for a fuzzy rule-based system, then the corresponding fuzzy sets should fulfill some requirements like boundedness of support or unimodality. Problems may also arise in the case, when the fuzzy partition induced by the clusters is intended as a basis for local function approximation. In this case, a local model (function) is assigned to each cluster. Taking the fuzziness of the partition into account, continuous transitions between the single local models can be obtained easily. However, unless the overlapping of the clusters is very small, the local models tend to mix and no local model is actually valid. By rewarding crisp membership degrees, we modify the objective function used in fuzzy clustering 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. We apply this modified fuzzy clustering approach to building fuzzy models of the Takagi–Sugeno (TS) type automatically from data.