Title: Generalization of the Mahalanobis distance for fuzzy data: An application to robust fuzzy clustering
Authors: Ana Belen Ramos-Guajardo - University of Oviedo (Spain) [presenting]
Maria Brigida Ferraro - Sapienza University of Rome (Italy)
Abstract: Most of the distances used in case of fuzzy data are based on the well-known Euclidean distance. A fuzzy set can be characterized by centers and spreads and the distances between fuzzy sets 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, one possibility to avoid this drawback is to consider the Mahalanobis distance since it involves the covariance matrix between the variables. A generalization of that distance in the fuzzy framework is proposed. It is shown to be very useful in different contexts as, for instance, in the robust fuzzy clustering approach. Non-spherical clusters are not generally recognized by means of Euclidean-type distances whereas they are shown to be recognized if the generalized Mahalanobis distance is taken into account. Some theoretical properties are addressed and clustering applications are reported in order to check its adequacy.