By M. Mizumoto (auth.)

One of the sights of fuzzy good judgment is its software in fixing many actual engineering difficulties. As many have realised, the main stumbling blocks in development a true clever computing device contain facing random disturbances, processing quite a lot of obscure information, interacting with a dynamically altering setting, and dealing with uncertainty. Neural-fuzzy options aid one to resolve a lot of those difficulties.
Fuzzy good judgment and clever Systems displays the latest advancements in neural networks and fuzzy good judgment, and their program in clever platforms. moreover, the stability among theoretical paintings and functions makes the e-book appropriate for either researchers and engineers, in addition to for graduate scholars.

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This issue has been investigated by many a u t h o r s including Carroll and Dickinson [6], Cybenko [5], Funahashi [7], Gallant and W h i t e [8], Heclit-Nielsen [10], and Hornik [15]. For function approximation, b o t h the series expansion approach and the Stone-Weierstrass theorem are very effective analytic tools. However, Hedit-Nielsen [10,11] found the relationship between the Kolmogorov's theorem and the approximation principle of the feedforward networks. Indeed, functional analytic m e t h o d s have been successfully used to show t h a t feedforward neural structures with at-Ieast one hidden layer are capable of simultaneously approximating continuous functions in several variables and their derivatives if the neural activation functions of the hidden neural imits are differentiable [15].

Based on this idea, t h e fuzzy system may be trained to realize desired i n p u t - o u t p u t relationship using various learning algorithms stich as fuzzy back-propagation algorithm. As pointed out by Wang and Mendel [36], the most i m p o r t a n t advantage of using fuzzy basis functions, rather t h a n polynomials, or radial basis fimctions, etc, is t h a t a linguistic fuzzy I F - T H E N rule is naturally related to a fuzzy basis function. A fuzzy system whose basic configuration is depicted in Figure 10 has four principal elements: fuzzifier, fuzzy rule base, fuzzy inference engine, a n d defuzzifier.

Cos nx, and sin nx. All these functions have the common period 27r. 1) 22 CHAPTER 2 which is simply obtained by replacing the sigmoid function with a trigonometric function in the conventional two-layered neural network as shown in Figure 1. The trigonometric activation function 4> may be chosen as: i) all 4>i{x) = cos{x), (cosine network); ii) all 4>i{x) = sin{x), (sine network); iii) (t>i(x) = cos{x), or sin{x), (trigonometric network). ) ^V^TV Figure 1 Block diagram of the trigonometric network.

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