From Fuzzy Rule-Based Models to Granular Models

Fuzzy rule-based models constructed in the presence of numeric data are nonlinear numeric models producing for any input some numeric output. There are no ideal models so the obtained numeric output could create a false illusion of achieved accuracy. A desirable approach is to augment the results with some measure of confidence (credibility) by admitting a granular rather than numeric format of the produced output values of the model. Our focus of this study is on fuzzy Takagi–Sugeno rule-based models whose conclusions are constant. The ultimate objective is to extend such models to the generalized granular structure with the conclusions formed as information granules. We study information granules described by intervals and fuzzy sets as well as probabilistic Gaussian information granules. The original design of the granular model is realized by involving the principle of justifiable granularity. Using this principle, we also show how to determine the equivalence between information granules. The construction of probabilistic information granules of the model is completed with the aid of optimized Gaussian process models. The granular models built in this way constitute a substantial and application-oriented departure from the numeric fuzzy models by offering a comprehensive insight into the quality of the produced results. The experimental studies based on synthetic and publicly available data demonstrate the design process and discuss the quality of the obtained results.

I. Introduction

Fuzzy rule-based models are constructed on a basis of fuzzy rules. Two main categories have been studied, namely Takagi–Sugeno (TS) (functional) models and Mamdani (relational) models [1]. In the TS models, the conclusions are (numeric) local functions (f_i). The rules realize a nonlinear mapping from an n-dimensional input space to a 1-D output space, namely “if x is A_i, then y is f_i”. In Mamdani models, the conclusions are fuzzy sets with the rules in the form “if x is A_i, then y is B_i”. The TS models started to gain popularity due to a number of compelling reasons: 1) TS models realize numeric mappings owing to the nonlinear membership functions in the conditions and the nonlinearity of the local functions. 2) The data-driven design process is well established and well documented in the literature [2], [3], [4]. The models are designed in the presence of multivariable data. 3) The accuracy is achieved through the minimization of the loss function; the number of rules could be increased to reduce approximation error.