ELSEVIER Fuzzy Sets and Systems 101 (1999) 49 58 FUZZY sets and systems Membership functions in the fuzzy C-means algorithm Antonio Flores-Sintas a'*, Jos6 M. Cadenas b, Fernando Martin b ~Departamento de Nuevas Tecnologlas, Caja de Ahorros de Murcia, Gran Via-23. 30005-Murcia, Spain hDepartamento de In~brmiuica y Sistemas, Facultad de Injbrmatica, Universidad de Murcia, 30071-Espinardo, Murcia, Spain Received July 1996; received in revised form February 1997 Abstract A membership function and an objective function deduced from the geometrical properties associated to the metric defined by the covariance matrix of a sample have been proposed recently. We use these functions to determine the membership probabilities and the criterion function in the fuzzy C-means algorithm. (~ 1999 Elsevier Science B.V. All rights reserved. Keywords: Membership functions; Cluster analysis; C-means; Hyperellipsoidal clustering 1. Introduction The fuzzy c-means (FCM) algorithm is one of the most popular fuzzy clustering algorithms. How- ever, it uses a constraint on the memberships that produces relative memberships that can be inter- preted as degrees of sharing [-7, 8], while in Zadeh's formulation of fuzzy set theory [11], the member- ship value of a point in the domain of discourse in a fuzzy set does not depend on its membership values in other fuzzy sets defined over the same domain of discourse. On the other hand, when the norm used in the FCM algorithm is different from the Euclidean it is necessary to introduce more constraints, so as to limit the volume of the groups when we use a distance deduced from the covariance matrix [6, 10]. The fuzzy-minimals (FM) algorithm introduced in [5] is based *Corresponding author. Tel.: +34 68 361 769; fax: +34 68 361 811. on a membership function to a group which is independent of the existence of others groups. This membership function is conceptually the same as both the Euclidean and the Mahalanobis distance and it permits unifying their treatment and resolv- ing the problem raised by Krishnapuram. The FM algorithm treats only one group and finds the subgroups prototypes using the Euclidean distance. All the subgroups have the same metric. However, in some applications it may be necessary to use a different metric for each cluster. This can be solved with the FCM algorithm. In this paper we modify the FCM algorithm by introducing to it the membership function used by the FM algorithm and unifying the treatment of both the Euclidean and the Mahalanobis distance without introducing more constraints. Given that we propose a modification of the FCM algorithm it is convenient to present a brief description of it. The next description is taken from Bezdek [2]: 0165-0114/99/$ see front matter :('~ 1999 Elsevier Science B.V. All rights reserved. PII: S0165-01 14(97)00062-6