Fuzzy Rule Interpolation Based on Polar Cuts

Systems applying fuzzy logic are rule based ones. The collection of the rules the so called rule base can be characterized as dense or sparse depending on whether there exist rules for all the possible observations. In the sparse case for some observations there are no rules whose antecedent part would overlap the observation at least partially. Therefore the classical compositional reasoning methods can not produce an acceptable conclusion. The inference techniques based on fuzzy rule interpolation are developed for especially this purpose. This paper proposes a new fuzzy rule interpolation based inference technique applying the concept of linguistic term shifting and polar cut. It is called FRIPOC (Fuzzy Rule Interpolation based in POlar Cuts) and it is applicable in the case of sparse and dense rule bases, too. Its main advantages are its comprehensibility, extrapolation capability and its applicability even if the height of one or more fuzzy sets is smaller than one. The rest of this paper is organized as follows. Section 1 gives a brief overview on the relevant fuzzy rule interpolation techniques grouping them depending on the main steps they are following. Section 2 presents the main structure and the steps and stages that characterize the method FRIPOC. Section 3 introduces the concept of the polar cut and a fuzzy set interpolation technique called FEAT-p based on it as a possible implementation for the first and third stage of the first step. In section 4 the authors propose a technique for the determination of the position of the consequent sets that is an extension and adaptation of the Shepard 2D interpolation [15]. Section 5 introduces a new polar cut based single rule inference method for the determination of the conclusion. In section 6 some relevant features of the new method are outlined through some numerical examples.