Fuzzy Rule-Based System through Granular Computing

This In this paper, we introduce a concept of granular fuzzy rule-based system, offer a motivation behind its emergence and elaborate on ensuing algorithm developments. It is shown that the granularity of the fuzzy rules is directly associated with a reduction (compression) process in which the number of rules becomes reduced in order to enhance the readability (transparency) of the resulting rule base. The retained rules are made more abstract (general) by admitting a granular form of the fuzzy sets forming their antecedents. In other words, while the original rules read as "if A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> then B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> " their reduced subset comes in the form "if G(A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) the B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> " with G(.) denoting a certain granular extension of the original fuzzy set (which can be realized e.g. in the form of interval fuzzy sets, fuzzy sets of type-2 or rough fuzzy-sets). It is shown that the optimization of the reduced set of rules is realized through an optimal distribution of information granularity among fuzzy sets forming the conditions of the reduced rules. In particular, it is shown that the distribution of information granularity, being regarded as an important design asset, is realized through a minimization of a certain objective function quantifying how well the granular fuzzy set formed by reduced rules set represents (covers) all rules. In this study, we use a technique of particle swarm optimization (PSO) as a vehicle of forming a subset of rules and the optimal allocation of information granulation to construct a granular fuzzy rule-based system. In the sequel, we introduce and idea of a granular representation of results of inferences realized in fuzzy rule-based system.