A fuzzy system index to preserve interpretability in deep tuning of fuzzy rule based classifiers

Following the successful applications of the fuzzy models in various application domains, the issue of automatic generation of Fuzzy Rule Based Systems (FRBSs) from observational data was widely studied in the literature and several approaches have been proposed. Most approaches were designed to search for the best accuracy of the generated model, neglecting the interpretability of FRBSs, which is commonly recognized as one of main reasons of the success of fuzzy linguistic models. To fill this gap, a current hot issue in linguistic fuzzy modelling area is the search for a good accuracy-interpretability trade-off. At present, despite the work done, the definition of interpretability is rather problematic. In fact there is still not an universal index widely accepted. This is mainly because the understanding of fuzzy systems is a subjective task that strongly depends on the background of the person who makes the assessment. In consequence an effective way consists of proposing a fuzzy system index instead of numerical ones. In this paper, we give our contribution proposing a fuzzy system as index to measure both fuzzy rule and set levels complexity of the system. At best of our knowledge there are not indexes to preserve interpretability of a FRBS when it is deep tuned, to this end a new fuzzy system index is formulated and an implementation is presented. To show how our fuzzy system index could be used for interpretability preservation, it is integrated in a classical Multi-Objective Evolutionary Algorithm (MOEA) and its results are presented through six comparative examples based on well-known data sets in the pattern classification field. 8