Identification of fuzzy models with the aid of evolutionary data granulation

The identification of fuzzy rule-based systems is considered. By their nature, these fuzzy models are geared toward capturing relationships between information granules-fuzzy sets. The level of granularity of fuzzy sets helps establish a required level of detail that is of interest in the given modelling environment. The form of the information granules themselves (in particular their distribution and type of membership functions) becomes an important design feature of the fuzzy model, contributing to its structural as well as parametric optimisation. This, in turn, calls for a comprehensive and efficient framework of information (data) granulation, and the one introduced in the study involves a hard C-means (HCM) clustering method and genetic algorithms (GAs). HCM produces an initial collection of information granules (clusters) that are afterwards refined in a parametric way with the aid of a genetic algorithm. The rules of the fuzzy model assume the form `if x₁ is A and x₂ is B and ··· and x_n is W then y=φ (x₁ , x₂, ···, x_n, param)´ and come in two forms: a simplified one that involves conclusions that are fixed numeric values (that is, φ is a constant function), and a linear one where the conclusion part (φ) is viewed as a linear function of inputs. The parameters of the rules are optimised through a standard method of linear regression (least square error method). An aggregate objective function with weighting factor used in this study helps maintain a balance between the performance of the model for training and testing data. The proposed identification framework is illustrated with the use of two representative numerical examples