Fuzzy clustering with Minkowski distance

Patrick Groenen (), Uzay Kaymak () and Joost van Rosmalen

No EI 2006-24, Econometric Institute Research Papers from Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute

Abstract: Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance. Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L_1-distance and Bobrowski and Bezdek (1991) also used the L_infty-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski distance, Groenen and Jajuga (2001) introduced a majorization algorithm to minimize the error. One of the advantages of iterative majorization is that it is a guaranteed descent algorithm, so that every iteration reduces the error until convergence is reached. However, their algorithm was limited to the case of Minkowski parameter between 1 and 2, that is, between the L_1-distance and the Euclidean distance. Here, we extend their majorization algorithm to any Minkowski distance with Minkowski parameter greater than (or equal to) 1. This extension also includes the case of the L_infty-distance. We also investigate how well this algorithm performs and present an empirical application.

Date: 2006-07-06
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