Semantics and perception of fuzzy sets and fuzzy mappings

Abstract

Fuzzy sets are constructs that come with a well defined meaning [36-38]. They capture the semantics of the framework they intend to operate within. Fuzzy sets are the building conceptual blocks (generic constructs) that are used in problem description, modeling, control, pattern classification tasks and the like. Their estimation and interpretation are of paramount relevance. Before discussing specific techniques of membership function estimation, it is worth casting the overall presentation in a certain context by emphasizing the aspect of the use of a finite number of fuzzy sets leading to some essential vocabulary reflective of the underlying domain knowledge. In particular, we are concerned with the related semantics, calibration capabilities of membership functions, and the locality of fuzzy sets. The key objectives of this study revolve around the fundamental concept of membership function estimation, calibration, perception, and reconciliation of views at information granules. Being fully cognizant of the semantics of fuzzy sets, it becomes essential to come up with a suite of effective mechanisms for the development of fuzzy sets and offer a comprehensive view of their interpretation. Given the main objectives of the study, its organization is reflective of them. We start with a discussion on the use of domain knowledge and the problem-oriented formation of fuzzy sets (Sect. 2). In Sect. 3, we focus on usercentric estimation of membership functions (including horizontal, vertical and pairwise comparison). In the following Section, we focus on the construction of fuzzy sets regarded as granular representatives of numeric data. Fuzzy clustering is covered in Sect. 5. Several design guidelines are presented in Sect. 6. Fuzzy sets are typically cast in some context, and in this regard we discuss the issue of nonlinear mappings (transformations) that provide a constructive view of the calibration of fuzzy sets. The crux of the calibration deals with a series of contractions and expansions of selected regions of the universe of discourse. Further on we discuss several ways of reconciliation of fuzzy sets and fuzzy mappings being a direct result of various views (perspectives) of the same fuzzy set/fuzzy mapping. © 2008 Springer-Verlag Berlin Heidelberg.