A fuzzy clustering approach for fuzzy data based on a generalized distance

Most of the distances used in case of fuzzy data are based on the well-known Euclidean distance. In detail, a fuzzy number can be characterized by centers and spreads and the most common distances between fuzzy numbers are essentially defined as a weighted sum of the squared Euclidean distances between the centers and the spreads. In the multivariate case the Euclidean distance does not take into account the correlation structure between variables. For this reason, the Mahalanobis distance has been introduced which involves the corresponding covariance matrix between the variables. A generalization of that distance to the fuzzy framework is proposed. It is shown to be useful in different contexts and, in particular, in a clustering approach. As a result, non-spherical clusters, that generally are not recognized by means of Euclidean-type distances, can be recognized by means of the suggested distance. Clustering applications are reported in order to check the adequacy of the proposed approach.