Fuzzy clustering with squared Minkowski distances

This paper presents a new fuzzy clustering model based on a root of the squared Minkowski distance which includes squared and unsquared Euclidean distances and the L-1-distance. An algorithm is presented that is based on iterative majorization and yields a convergent series of monotone nonincreasing loss function values. This algorithm coincides under some condition with the ISODATA algorithm of Dunn (J, Cybernet. 3 (1974) 32-57) and the fuzzy c-means algorithm of Bezdek (Ph,D, Thesis. Cornell University, Ithaca, 1973) for squared Euclidean distance and with an algorithm of Jajuga (Fuzzy Sets and Systems 39 (1991) 43-50) for L-1-distances. To find a global minimum we compare a special strategy called fuzzy steps with fuzzy Kohonen clustering networks (FKCN) (Pattern Recognition 27 (1994) 757-764) and multistart. Fuzzy steps and FKCN are based on finding updates for a decreasing weighting exponent, which seems to work particularly well for hard clustering. To assess the performance of the methods, two numerical experiments and a simulation study are performed. (C) 2001 Elsevier Science B.V, All rights reserved.