Analyzing interpretability of fuzzy rule-based systems by means of fuzzy inference-grams

Since the proposal of Zadeh and Mamdani’s seminal ideas, interpretability is acknowledged as one of the most appreciated and valuable characteristics of fuzzy system identification methodologies. It represents the ability of fuzzy systems to formalize the behavior of a real system in a human understandable way. Interpretability analysis involves two main points of view: readability of the knowledge base description (regarding complexity of fuzzy partitions and rules) and comprehensibility of the fuzzy system (regarding implicit and explicit semantics embedded in fuzzy partitions and rules, but also the fuzzy reasoning method). Readability has been thoroughly treated by many authors who have proposed several criteria and metrics. Unfortunately, comprehensibility has almost never been considered because it involves some cognitive aspects related to the human reasoning which are very hard to formalize and to deal with. This paper proposes the creation of fuzzy systems’ inference maps, so-called fuzzy inference-grams (fingrams) by analogy with scientograms used for visualizing the structure of science. Fingrams show graphically the interaction between rules at the inference level in terms of co-fired rules, i.e., rules fired at the same time by a given input vector. The analysis of fingrams offers many possibilities: measuring the comprehensibility of fuzzy systems, detecting redundancies and/or inconsistencies among fuzzy rules, discovering the most significant rules, etc. Some of these capabilities are explored in this initial work.