Spline interpolation between hyperspaces of convex of fuzzy sets

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Authors:
Thomas Vetterlein
Martin Štĕpnička
Title: Spline interpolation between hyperspaces of convex of fuzzy sets
Journal: Fuzzy Sets and Systems
Volume 157
Number
Pages: 2472 - 2481
Year: 2006





Abstract

A fuzzy if-then rule base may be viewed as a partial function between universes of fuzzy sets. For the construction of a fuzzy inference module, this partial function needs to be extended to a total one.

Here, we propose a new method how to do so, making use of the method of smoothing splines. To this end, we identify the fuzzy sets with elements of a finite-dimensional real parameter space in an approximate way, using Perfilieva's fuzzy transforms. We then determine a function between two such parameter spaces by requiring that it reproduces the rule base as precise as possible and that it minimizes a parameter depending on its smoothness.