Representing membership functions as points in high-dimensional spaces forfuzzy interpolation and extrapolation

This paper introduces a new approach for fuzzy interpolation and extrapolat ion of sparse rule base comprising of membership functions with finite numb er of characteristic points. The approach calls for representing membership functions as points in high-dimensional Cartesian spaces using the locatio ns of their characteristic points as coordinates. Hence, a fuzzy rule base can be viewed as a set of mappings between the antecedent and consequent sp aces and the interpolation and extrapolation problem becomes searching for an image in the consequent space upon given an antecedent observation. Anal ysis of well-defined membership functions can also be readily incorporated with the approach. Furthermore, the Cartesian representation enables separa tion between membership functions to be quantitatively measured by the Eucl idean distance between their representing points, thereby allowing the inte rpolation and extrapolation problems to be treated using various scaling eq uations. The present approach divides observations into two groups. Observa tions within the antecedent spanning set contain the same geometric propert ies as the given antecedents. Interpolation and extrapolation can be conduc ted based on the given rules using a weighted-sum-averaging formula. On the other hand, observations lying outside the antecedent spanning set contain new geometric properties beyond those of the given rules. Heuristic reason ing must therefore be applied. In this case, a two-step approach with certa in flexibility to accommodate additional criteria and design objectives is formulated.