Advanced Fuzzy Systems Design and Applications (Studies in by Yaochu Jin

By Yaochu Jin

Fuzzy rule structures have discovered quite a lot of purposes in lots of fields of technological know-how and know-how. routinely, fuzzy principles are generated from human professional wisdom or human heuristics for fairly basic structures. within the previous couple of years, data-driven fuzzy rule new release has been very energetic. in comparison to heuristic fuzzy ideas, fuzzy principles generated from facts may be able to extract extra profound wisdom for extra advanced platforms. This publication offers a few ways to the iteration of fuzzy principles from facts, starting from the direct fuzzy inference established to neural web­ works and evolutionary algorithms dependent fuzzy rule iteration. in addition to the approximation accuracy, distinct consciousness has been paid to the interpretabil­ ity of the extracted fuzzy ideas. In different phrases, the bushy ideas generated from info are meant to be as understandable to people as these generated from human heuristics. To this finish, many features of interpretabil­ ity of fuzzy platforms were mentioned, which has to be taken under consideration within the data-driven fuzzy rule iteration. during this manner, fuzzy principles generated from info are intelligible to human clients and consequently, wisdom approximately unknown platforms may be extracted.

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By defining a quantitative causality measure, it was found that the representation of evolution strategies has stronger causality than the binary coding used in genetic algorithms [202]. Further, it was also shown that gray coding is of stronger causality than binary coding. This may be an explanation for why evolution strategies have exhibited better performance than genetic algorithms in parameter optimization problems that we have experienced. Meanwhile, it is well known that biological chromosomes are highly redundant [39].

69) where (integer)(v) means the integer that is closest to the value v. --'. = 3. (1. 3. 3. 5 0 0 0 0 0 PL PS 2 3 A discrete version of the defuzzification process is used in the look-up table fuzzy systems. 71) the defuzzification methods discussed above can carried out as follows. • Mean of maximum. According to this method, the crisp output is obtained by: (1. 72) If more than one element has the same maximal membership, the average value of these elements is taken as the crisp output. 2 Fuzzy Rule Systems 25 • Center of gravity.

83) is a widely used expression for the following Mamdani-type single-input single-output fuzzy rule system: 28 1. Fuzzy Sets and Fuzzy Systems Ri:If x is Ai, then y is B i , where Ai, i = 1,2, .... , N are fuzzy membership functions for x, Bi are fuzzy membership functions for y. 83) is the point where Bi(Y) reaches the maximum. Usually, all the fuzzy subsets are supposed to be normal and therefore, Yi is the kernel of the fuzzy set. The proof of the universal approximation based mainly on the StoneWeierstrass theorem popular in functional analysis and the conclusion can be extended to multiple-input single output systems [49].

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