Gradient Descent Based Optimization of Transparent Mamdani Systems

The tradeoff between accuracy and interpretability in fuzzy modeling has shifted into focus in last few years. This paper aims at improving accuracy of linguistic models while maintaining a good interpretability. A new gradient-based method, extended version of Jager approach, is proposed for the optimization of transparent Mamdani systems. The advantage of Mamdani systems if compared to 0 order TS systems in Jager approach is that their interpolation properties allow one to obtain less complex models without loss of accuracy. Several modeling examples confirming the advantages of the chosen algorithm are included. 1 Introduction Many papers have been written about neuro-fuzzy systems. Most of these address the optimization of fuzzy systems based on gradient descent (GD) method. Perhaps the earliest work in this research line is [1] that considers 0 order TS systems with symmetric triangular antecedent membership functions (MFs). Later contributions include [2] that extends the method on various types of fuzzy systems including 1 order Takagi-Sugeno (TS) systems. The hybrid learning rule, proposed by Jang [3] combines gradient descent with least-squares estimator for consequent parameters and outperforms the previous approaches in terms of approximation error. What is common to mentioned approaches is that no attention is paid to semantic properties of fuzzy systems, thus the obtained models and controllers are non-transparent to interpretation and it can be claimed that the basic feature that distinguishes fuzzy systems from other nonlinear approximators (e.g. neural networks) is sacrificed to approximation accuracy. Jager’s work where GD is applied to 0 order TS systems in a manner that maintains fuzzy partition on antecedent variables is an interesting exception. His algorithm, published in Ph.D. thesis [4] is, however, not well known with the exception of [5] and [6] (in the latter an extension for 1 order TS systems is given). In this paper we propose a further extension of Jager algorithm for Mamdani systems. 2 System Definition We consider a MISO Mamdani system with the following rule base IF X1 is A1r AND… AND Xi is Air … AND XN is ANr THEN Y is B1r, (1) where Air and Bjr the linguistic labels of i input variable xi and output variable y (i = 1 … N), respectively, associated with the r rule (r = 1 … R). Numerical mapping of input-output variables of (1) (in discrete form) is obtained by applying the inference function (2), ∑ ∑ = = = = Q