A novel and fast MIMO fuzzy inference system based on a class of fuzzy clustering algorithms with interpretability and complexity analysis

A data driven Fuzzy Inference System (FIS) employs Membership Functions (MFs) with adjustable parameters in its IF part to fuzzify the input data. The input space is partitioned simply by dividing universe of discourse of each input variable into some fuzzy subspaces. The MFs are then defined on the fuzzy subspaces of the input variables. Parameters of the MFs are tuned for maximum accuracy of the system (which demands high runtime) without considering the data structure which impairs interpretability of the FIS and degenerates the system into a black-box tool. Such a FIS does not represent actual structure of the data and its MFs are not necessarily in accord with the data distribution in the input space. In addition, the FIS suffers from exponential complexity of order O(Tr) where T is number of linguistic terms (number of subspaces on the universe of discourse of input variables) and r is number of input variables. This article presents a novel Multiple-Input and Multiple-Output Clustering based Fuzzy Inference System (MIMO CFIS) which is made directly from a class of fuzzy clustering algorithms to overcome these shortcomings. CFIS identifies dense regions of the input data using fuzzy clustering and then places a cluster on each of these regions. These fuzzy clusters represent actual structure of the data and serve as fuzzy rules in the rule base of CFIS and provide MFs that exactly fit the dense regions of the data that makes the system more interpretable and avoids redundant rules. These MFs are normal, convex, and continuous and have no parameter to be tuned (which makes CFIS much faster than other FISs) and fuzzify the input data according to their membership in the clusters. THEN part of CFIS is generalized form of THEN part of Takagi-Sugeno (TS) fuzzy system which accommodates any function of input variables. Despite less number of adjustable parameters, testing error of CFIS is less than that of TS system and its modified versions. Moreover, number of fuzzy rules in CFIS rule base is the same as the number of linguistic terms (or fuzzy clusters) and consequently its complexity is of orderO(T). Also, CFIS is a MIMO system and avoids inconsistent (contradictory) rules by generating well-separated fuzzy clusters whereas TS system is MISO and never guarantees generation of consistent rules. In addition, CFIS satisfies most of the interpretability criteria of FISs.