Construction of Capacities from Overlap Indexes

Abstract

In many problems, it is crucial to find a relation between groups of data. Such relation can be expressed, for instance, in terms of an appropriate fuzzy measure or capacity([10, 21]) which reflects the way the different data are connected to each other [20]. In this chapter, taking into account this fact and following the developments in [8],we introduce a method to build capacities ([20, 21]) directly from the data (inputs) of a given problem. In order to do so, we make use of the notions of overlap function and overlap index ([5, 12, 13, 7, 4, 14, 16]) for constructing capacities which reflect how different data are related to each other. This paper is organized as follows: after providing some preliminaries, we analyse, in Section 3, some properties of overlap functions and indexes. In Sections 4 we discuss a method for constructing capacities from overlap functions and overlap indexes. Finally, we present some conclusions and references.