Geometrical interpretation and applications of membership functions with fuzzy rough sets

Abstract

Fuzzy rough sets are a generalization of crisp rough sets for measuring inconsistency between conditional attributes and decision attributes for many decision systems. In many classification problems a membership function for the training sample belonging to a certain class can be computed by methods in fuzzy rough sets. In this paper, we present a geometrical interpretation and its applications of this kind of membership functions. First, we prove that every fuzzy similarity relation in fuzzy rough sets is a reproducing kernel which is related to a Krein space, thus, fuzzy similarity relations can be geometrically explained in a Krein space. Second, we will present the interpretation of several types of membership functions geometrically by using the lower approximations in fuzzy rough sets, in terms of square distances in Krein spaces. As practical applications of these membership functions, we develop a new algorithm to find reducts and reformulate soft margin support vector machines by taking the membership degree for every training sample into considerations. Experimental results also demonstrate the effectiveness of the work proposed in this paper.