Fuzzy rule interpolation (FRI) predicts an accountable outcome of a possible course of action in sparse fuzzy rule base system (FRBS). The geometry based linear fuzzy rule interpolation (GLFRI) is extended for multi-dimensional fuzzy rule base interpolation. Expansion/contraction (EC) of triangular, trapezoidal and complex polygonal fuzzy sets has been also proposed which enables the proposed FRI method to incorporate with fuzzy rules which include triangular, trapezoidal, hexagonal or complex fuzzy sets. The study further extends to introduce the process of backward rule base interpolation. It has been shown that the scale and move transformation-based FRI method can yield a non-convex fuzzy consequent which can be avoided by using the proposed method. The proposed method performs better without any risk of obtaining non-convex fuzzy consequent. The efficiency of proposed forward and backward FRI methods is projected with several numerical examples. A detailed comparison of EC transformation with scale and move transformation is also presented here.

Fuzzy rule interpolation (FRI) predicts an accountable outcome of a possible course of action in sparse fuzzy rule base system (FRBS). However, in real life, we encounter some situations where the antecedent has to be predicted to obtain a desired consequent of FRBS. In this situation, inverse fuzzy rule interpolation (IFRI) or backward fuzzy rule interpolation (BFRI) is used to get the desired outcome. Here a geometry based inverse fuzzy rule base interpolation (GIFRI) is suggested. The mathematical detail of the proposed method is elaborated and its geometrical interpretation is given with the help of fuzzy geometry. It is to be noted that the proposed method ensures that the inverse of the inverse is the original one.

Fuzzy Rule Interpolation (FRI) provides an interpretable decision in sparse fuzzy rule based system. The objective of this work is to establish a mathematical demonstration of the pattern of existing fuzzy rule base using fuzzy geometry. Though several authors contributed on fuzzy rule base interpolation but there is a need to generate closed mathematical form of interpolating pattern. The present work is an initiative to demonstrate the same. First part of this paper presents some spatial geometrical transformation of a fuzzy point. In the second part of this paper, a new FRI scheme is suggested using fuzzy geometry with above mentioned transformation. The proposed method operates in two different steps. In the first step, all the fuzzy rules are converted into fuzzy sets or mostly fuzzy points in higher dimension by using mathematical operator on the individual of antecedent and consequent parts. All rules or fuzzy points are then joined with a class of fuzzy line segments (FLS). Second step considers the identification of mathematical pattern of the interpolated piecewise linear fuzzy polynomial which is able to compute the desired conclusion of a given observation. The presented method not only associates the FRI technique to classical interpolation technique, but also promises to provide the geometrical visualization of the behaviour of fuzzy sets during the interpolation process.