Fuzzy Rule Interpolation with A General Representation of Fuzzy Sets

Fuzzy interpolative reasoning offers an important method to perform fuzzy inference in sparse fuzzy rule-based systems and helps to reduce the complexity of completed fuzzy systems. A number of fuzzy interpolative reasoning methods have been proposed for sparse fuzzy rule-based systems in the literature. However, the existing methods consider fuzzy interpolative reasoning with popular fuzzy sets such as triangular, trapezoidal, polygonal or bell-shaped fuzzy sets, but there is no work on fuzzy sets represented by non-piece-wise-linear functions. There are also only few fuzzy interpolative reasoning methods available in the form of functions. This paper presents a fuzzy interpolative reasoning framework for sparse fuzzy rule-based systems based on the general representation of fuzzy sets, i.e., the inverse of the membership function of fuzzy sets, to address the aforementioned limitations. The proposed approach not only is able to represent complex fuzzy sets in the form of functions, but also makes fuzzy interpolation with complicated membership functions possible. Meanwhile, the interpolated results inferred by the proposed method holds the properties of normality and convexity. A comparative study in reference to the popular fuzzy interpolation approaches confirms the effectiveness of the proposed method.