Towards utilization of rule base structure to support fuzzy rule interpolation

Fuzzy rule interpolation (FRI) offers a reliable approach for providing an interpretable approximate decision with a sparse rule base, when a new observation does not match any existing rules. As the mainstream application of a fuzzy rule base is to extract valuable approximate information from each individual rules, existing FRI methods typically work by postulating that the more rules used to implement the interpolation the better the reasoning outcomes. Yet, empirical results have shown that using a large number of rules in an FRI process may adversely lead to worsening the accuracy of the inference outcomes, not just degrading efficiency. The objective of this work is to set a firm theoretical foundation for the eventual establishment of a novel FRI approach. It achieves this goal by mapping the structural patterns within a given fuzzy rule base onto a mathematically isomorphic data space, such that the essential information embedded in the original rule base can be effectively captured, represented and analysed. The resulting mathematically mapped patterns enable the production of a theorem that determines the upper limit of the number of rules required to effectively and efficiently perform FRI. The experimental investigations reported herein demonstrate that the number of required rules to perform FRI obeys the theorem discovered in this work.