A Self-Constructing Compensatory Neural Fuzzy and Its Applications System

K e y w o r d s C o m p e n s a t o r y , Fuzzy similarity measure, Inverted wedge system, Backpropagation algorithm. 1. I N T R O D U C T I O N Recently, the neural fuzzy approach to system modeling has become a popular research topic [110]. Moreover, the neural fuzzy method possesses the advantages of both the pure neural and the fuzzy methods; it brings the low-level learning and computational power of neural networks into fuzzy systems and incorporates the high-level human-like thinking and reasoning of fuzzy systems into neural networks. Many papers [4-10] have dealt with optimal fuzzy membership functions and defuzzification schemes for applications by using learning algorithms to adjust the parameter of fuzzy membership functions and defuzzification functions. Unfortunately, for optimal fuzzy logic reasoning and selected optimal fuzzy operators, only static fuzzy operators are often used for fuzzy reasoning, such that the conventional neural fuzzy system can only adjust the fuzzy membership functions by using fixed fuzzy operations, such as Min and Max. The compensatory neural fuzzy system [11] with adaptive fuzzy reasoning is more effective and adaptive than the conventional This research is supported by the National Science Council of the I~.O.C. under Grant NSC 90-2213-E-324-011. 0895-7177/05/$ see front mat ter (~) 2005 Elsevier Ltd. All rights reserved. Typeset by A.A48-TEX doi: 10.1016/j.mcm.2004.07.017 340 C.-J. LIN AND C.-H. CHEN neural fuzzy system with nonadaptive fuzzy reasoning [4]. Therefore, an effective neural fuzzy system should be able not only to adaptively adjust fuzzy membership functions, but also to dynamically optimize adaptive fuzzy operators. In this paper, a self-constructing compensatory neural fuzzy system (SCCNFS) is proposed. The compensatory fuzzy reasoning method uses adaptive fuzzy operations of a neural fuzzy network to make the fuzzy logic system more adaptive and effective. An online learning algorithm is proposed to automatically construct the SCCNFS. It consists of structure learning and parameter learning. The structure learning algorithm determines whether to add a new node which satisfies the fuzzy partition of the input data. The similarity measure of symmetric Gaussian membership functions is used. The backpropagation learning algorithm is then used for tuning membership functions. The proposed learning algorithm has four advantages. First, it does not require human assistance. Second, its structure is obtained from the input data. Third, it converges quickly. Fourth, the obtained fuzzy rules are more precise than other learning algorithms. This paper is organized as follows. Section 2 describes the basic structure and functions of the SCCNFS. The online structure and parameter learning algorithms of the SCCNFS is presented in Section 3. In Section 4, the SCCNFS is applied to solve several problems. Finally, conclusions are given in the last section. 2. T H E S T R U C T U R E O F S C C N F S In this section, the structure of the SCCNFS is introduced. This four-layer network [6] realizes a fuzzy model in the following form: Rj : IF xl is Alj and x2 is A2j. . . and x,~ is Any THEN y' = bj, (1) where xi is the input variable, y~ is the output variable, A,~j is the linguistic term of the precondition part, bj is the constant consequent part, and n is the number of input variables. The structure of the SCCNFS is shown in Figure 1. The functions of the nodes in each layer of the SCCNFS model are described as follows. Y Layer 3 1 / / t / 2 ~ / (2)