The idea of a VE is based on the similarity (or in this case the indistinguishability) of the considered elements. In VE the fuzzy membership function is indicating level of similarity of x to a specific element a that is a representative or prototypical element of the fuzzy set, or, equivalently, as the degree to which is indistinguishable from (Klawonn, 1994). Therefore the a-cuts of the fuzzy set are the sets which contain the elements that are -indistinguishable from a. Two values in a VE are e-distinguishable if their distance is greater than e. The distances in a VE are weighted distances. The weighting factor or function is called scaling function (factor) (Klawonn, 1994). If the VE of a fuzzy partition (the scaling function or at least the approximate scaling function (Kovács, 1996), (Kovács & Kóczy, 1997b)) exists, the member sets of the fuzzy partition can be characterized by points in that VE.
Published in Chapter:
Fuzzy Rule Interpolation
Szilveszter Kovács (University of Miskolc, Hungary)
Copyright: © 2009
|Pages: 6
DOI: 10.4018/978-1-59904-849-9.ch108
Abstract
The “fuzzy dot” (or fuzzy relation) representation of fuzzy rules in fuzzy rule based systems, in case of classical fuzzy reasoning methods (e.g. the Zadeh-Mamdani- Larsen Compositional Rule of Inference (CRI) (Zadeh, 1973) (Mamdani, 1975) (Larsen, 1980) or the Takagi - Sugeno fuzzy inference (Sugeno, 1985) (Takagi & Sugeno, 1985)), are assuming the completeness of the fuzzy rule base. If there are some rules missing i.e. the rule base is “sparse”, observations may exist which hit no rule in the rule base and therefore no conclusion can be obtained. One way of handling the “fuzzy dot” knowledge representation in case of sparse fuzzy rule bases is the application of the Fuzzy Rule Interpolation (FRI) methods, where the derivable rules are deliberately missing. Since FRI methods can provide reasonable (interpolated) conclusions even if none of the existing rules fires under the current observation. From the beginning of 1990s numerous FRI methods have been proposed. The main goal of this article is to give a brief but comprehensive introduction to the existing FRI methods.