By Chin-Teng Lin, C. S. George Lee
Neural Fuzzy platforms presents a entire, updated creation to the fundamental theories of fuzzy platforms and neural networks, in addition to an exploration of ways those fields may be built-in to create Neural-Fuzzy structures. It comprises Matlab software program, with a Neural community Toolkit, and a Fuzzy process Toolkit.
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Additional resources for Neural Fuzzy Systems: A Neuro-Fuzzy Synergism to Intelligent Systems
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X n ) . Let us consider several numerical examples to illustrate this important extension principle. 9 Let U ={ 1,2,3,4,5,6,7,8,9,10}. 9/9 + l/lO. 9/81 + l/lOO. 4)}. The function/ is a squaring operation and is indicated by y=f(x) = x2. 4 1 0 1 4 From the above calculations, it is obvious that we have two points (-1 and 1) in U mapped into a single point y = 1. The membership grade of y = 1 is taken as the maximum membership grade of x = -1 and x = 1, which gives us 1. 4/4. 11 Suppose / is a function mapping ordered pairs from Ul ={-1,0, l} and U2 ={ - 2,2} to V ={-2, -1,2, 3}, and/ (x,, x2) = xf + x2.
This extension principle was applied to algebraic operations and used to define set-theoretic operations for higher-order fuzzy sets. A thorough understanding of the topics covered in this chapter is essential to understanding future topics on fuzzy relations, fuzzy relation equations, fuzzy measures, possibility theory, fuzzy logic and approximate reasoning, and fuzzy logic control systems. 1 Determine the a-cuts of the following fuzzy sets A, B, and C. 5; X = [0, oo]. 2 Using the resolution principle, express the following fuzzy sets in the form of Eq.
J _ je(0,oc) Yager[1980] 1 -min{l, (1 -a)" + (1 - 6 ) " ) + } min{l, (aw + b"y*} Dubois andPrade [1980] ab max {a, b, a} a + b-ab-rwa{a,b, 1 - a } max{l - a, 1 - 1 , a} Dombi [1982] l/l + Werners [1988] |3min{a,fe}+ Zimmermann and Zysno [1980] (a£>)(1-7) [1 — (1 — a) (1 — b) J* t rG (-oo, oo) 7e(0, \-(\-y)ab ,og|l Sec. 2 ,-J_ - l} r a + b-(2-y)ab Frank [1979] + Range + C--IK*"1) j oo) wG(0,oo) a£(0 " I) XG(0, oo) + 2 |3 maxfo fr}+0 ~ P ) (" 2 + p e [ 0 , l ] y parameter indicates compensation between intersection and union Extensions of Fuzzy Set Concepts 7 f= [0,1] 25 3.