A New Fast Algorithm for Fuzzy Rule Selection Barbara Pizzileo, Kang Li Abstract- This paper investigates the selection of fuzzy rules for fuzzy neural networks. The main objective is to effectively and efficiently select the rules and to optimize the associated parameters simultaneously. This is achieved by the proposal of a fast forward rule selection algorithm (FRSA), where the rules are selected one by one and a residual matrix is recursively up- dated in calculating the contribution of rules. Simulation results show that, the proposed algorithm can achieve faster selection of fuzzy rules in comparison with conventional orthogonal least squares algorithm, and better network performance than the widely used error reduction ratio method (ERR). I. INTRODUCTION Fuzzy neural networks represent a large class of neural networks that combine the advantages of associative memory networks (e.g. B-splines, radial basis functions and support vector machines) with improved transparency, a critical issue for nonlinear modelling using conventional neural networks. For associative neural networks, the advantage is that the lin- ear parameters can be trained online with good convergence and stability properties. However, they produce essentially black box models with poor interpretability. For fuzzy neural networks (FNNs), the basis functions are associated with some linguistic rules, and every numerical result can admit a linguistic interpretation [1]. One of the major issues with the FNN applications is that the network complexity can be extremely high. To tackle this problem, a number of FNN construction methods have been proposed in the literature either to determine the number of inputs, the number of membership functions or the number of rules. For example, the adaptive-network- based fuzzy inference system (ANFIS) [2], the radial basis function (RBF) based adaptive fuzzy system (RBFAFS) [3], and the self constructing neural fuzzy inference network [4], have been proposed mainly to address the problem of membership function selection. The orthogonal least squares (OLS), which is one of the most popular approaches for fast identification of nonlinear systems [5], [6], [7], [8], [9], was extended for the selection of fuzzy rules [10]. In addition, the Error Reduction Ratio [11] method has been widely used in fuzzy structure selection for the Minimal Resource Allocating Network (M-RAN) [12], Dynamic Fuzzy Neural Network (DFNN) [13] and the Generalized Dynamic Fuzzy Neural Networks (GDFNN) [14]. In [15], a Fast Recursive Algorithm (FRA) was proposed for the identification of nonlinear systems using linear-in-the- parameters models with improved efficiency and numerical stability. It was recently applied to the selection of fuzzy The authors are with the School of Electronics, Electrical Engineering & Computer Science, Queen's University Belfast, UK (Emails: bpizzileo0l, k.li@qub.ac.uk). This work was supported by the U.K. EPSRC under Grant GR/S85191/01 to K. Li. regressor terms for fuzzy neural networks [ 16]. In this paper, a fast rule selection algorithm (FRSA) is proposed by modifying and extending the FRA to the fuzzy systems, and a comparison between the FRSA, the OLS and the ERR will be provided. The paper is organized as follows. Section II is the preliminary, and section III presents the fast rule selection algorithm (FRSA). Two numerical simulation examples are given in Section IV. Section V is the conclusion. II. PRELIMINARY In Fuzzy Neural Networks, for a given set of m inputs and N samples, each input variable xi(t), i = 1,... m is classified by ki, i = 1,... ,m fuzzy sets, denoted as Ai(ji), ji = 1,... ,ki [17]. For every input value xi(t), t = 1,... N, its membership degree in Ai(ji) is denoted as 0 < Ai(ji) (t) < 1. The construction of a fuzzy neural network mainly involves the following three steps: i) fuzzification (each variable is classified into a certain number of fuzzy sets, and this involves choosing the number of the fuzzy sets and selecting the shapes of the membership functions); ii) rule evaluation (typically using the Takagi- Sugeno [17] or the Mamdani method [18]); iii) aggregation of rules and defuzzification (once the values of the basis function for each rule and for each data sample have been obtained, all the rules should be aggregated and the final value be defuzzified, i.e. to convert a fuzzy quantity to a precise, or crisp quantity [18]). The defuzzification is achieved by the centroid technique, which produces the centre of gravity NR E Wtyr (t) y(t) = r=l NR E Wtr r=l NR Ert Y (t) r=l (1) where y(t) is the crisp value for the tth sample; NR is the NR total number of rules; (D' = Wt / Wt is the fuzzy basis r=l function [19] associated to the rth rule, Vr = 1, NR. In mrAn (1) wtr H ,u> (t) where Ar is the fuzzy set of the jth variable associated with the rth rule and Ar C {Aijj),j 1, .... ki }, yr (t) is the output associated to the rth rule and tth sample whose expression depends on the particular choice of model structure. For example a linear model will have m yr(t) = E: girxi (t), Vr = 1, ...,~NR i=l (2) where g' is the consequent parameter associated to the ith input and the rth rule [17]. 1-4244-1210-2/07/$25.00 C 2007 IEEE.