Functions, Fuzzification and Defuzzification-Fuzzy Intelligence-Lecture Slides, Slides of Artificial Intelligence

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Department of Computer & Information Sciences Pakistan Institute of Engineering and Applied Sciences Functions Fuzzification, and Defuzzification CIS-524 Fuzzy Intelligence Lecture 04 Umar Faiz http://www.pieas.edu.pk/umarfaiz/cis524 CIS-524 Fuzzy Intelligence docsity.com Classical Relations and Fuzzy Relations • Outline • Cartesian product and crisp relations • Composition of relations • E i l ti f l tiqu va ence proper es o re a ons 2Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Boundaries of a Membership Function • The boundaries of a membership function for some fuzzy set A are defined as that region of the universe containing elements that have a nonzero membership but not complete membership. • The boundaries comprise those elements x of the universe such that 0 < μA (x) < 1. 5Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Normal Fuzzy Set • A normal fuzzy set is one whose membership function has at least one element x in the universe whose membership value is unity. For fuzzy sets where one and only one element has a membership equal to one, this element is typically referred to as the prototype of the set, or the prototypical element. 6Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Convex Fuzzy Set • A convex fuzzy set is described by a membership function whose membership values are strictly monotonically increasing, or whose membership values are strictly monotonically decreasing, or whose membership values are strictly monotonically increasing then strictly monotonically decreasing with increasing values for elements in the universe. 7Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Cross-over Points of a Membership Function • The crossover points of a membership function are defined as the elements in the universe for which a particular fuzzy set A has values equal to 0.5, i.e., for which μA (x) = 0.5. 10Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Height of a Fuzzy Set • The height of a fuzzy set A is the largest membership grade obtained by any element in that set., i.e., hgt(A) = max{μA(x)}. • If the hgt(A) < 1, the fuzzy set is said to be subnormal. • The hgt(A) may be viewed as the degree of validity or credibility of information expressed by A. • If A is a convex single-point normal fuzzy set defined on the real line then A is often termed a fuzzy number, . 11Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Various Forms of Membership Functions • The most common forms of membership functions are those that are normal and convex. However, many operations on fuzzy sets, hence operations on membership functions, result in fuzzy sets that are subnormal and non-convex. 12Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Fuzzification • Fuzzification is the process of making a crisp quantity fuzzy . • Many of the quantities that we consider to be crisp and deterministic are actually not deterministic at all They carry . considerable uncertainty. • If the form of uncertainty happens to arise because of imprecision, ambiguity, or vagueness, then the variable is probably fuzzy and can be represented by a membership function. 15Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Fuzzification: Example • In the real world hardware such as a digital voltmeter , generates crisp data, but these data are subject to experimental error. The information shown in figure shows one possible range of errors for a typical voltage reading and the associated membership function that might represent such imprecision. 16Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Fuzzification: Example • In the real world hardware such as a digital voltmeter , generates crisp data, but these data are subject to experimental error. The information shown in Figure shows one possible range of errors for a typical voltage reading and the associated membership function that might represent such imprecision. 17Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification to Crisp Sets • Example Let us consider the discrete fuzzy set using Zadeh’s: , notation, defined on universe X = {a, b, c, d, e, f } • We can reduce this fuzzy set into several λ cut sets all of - , which are crisp. For example, we can define λ-cut sets for the values of λ = 1, 0.9, 0.6, 0.3, 0+, and 0. 20Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification to Crisp Sets 21Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Special Properties of λ-cut Crisp Sets 22Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • λ-Cuts for Fuzzy Relations • Example : • Suppose, in a biotechnology experiment, five potentially new strains of bacteria have been detected in the area around an anaerobic corrosion pit on a new aluminum–lithium alloy used in the fuel tanks of a new experimental aircraft. In order to propose methods to eliminate the biocorrosion caused by these bacteria, the five strains must first be categorized. One way to categorize them is to compare them to one another. In a pairwise comparison, the following ‘‘similarity’’ relation R is developed , 1, . 25Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • λ-Cuts for Fuzzy Relations • Example : If f λ t ti f th l f λ 1 0 9 • we per orm -cu opera ons or e va ues o = , . , then the following crisp relations are obtained: 26Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Special Properties for Fuzzy Relations • λ cuts on fuzzy relations obey certain properties- 27Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification: Example 30Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification: Example 31Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification • A general fuzzy output process can involve many output parts (more than two), and the membership function representing each part of the output can have shapes other than triangles and trapezoids. 32Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 2 Weighted Average Method Example. : • The two functions shown in Fig would result in the following general form for the defuzzified value. 35Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 3 Mean Max Membership . : • This method (also called middle-of-maxima) is closely related to the first method, except that the locations of the maximum membership can be nonunique (i.e., the maximum membership can be a plateau rather than a single point). • This method is given by the expression where a and b are as defined in Fig. 36Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 4 Centroid Method . : • This procedure (also called center of area, center of gravity) is the most prevalent and physically appealing of all the defuzzification methods [Sugeno, 1985; Lee, 1990]; it is given by the algebraic expression 37Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods • Mean Max Method z* = 40Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods • Weighted Average Method 41Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods Centroid Method 42Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods Center of Sums : • The defuzzified value z* is given by the following equation: where z-bar is the distance to the centroid of each of the respective membership functions . • This method is similar to the weighted average method, except in the center of sums method the weights are the areas of the respective membership functions whereas in the weighted average method the weights are individual membership values. 45Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 5 Center of Sums . : 46Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 6 Center of Largest Area . : • If the output fuzzy set has at least two convex subregions, then the center of gravity (i.e., z is calculated using the centroid method) of the convex fuzzy subregion with the largest area is used to obtain the defuzzified value z of the output. where Cm is the convex subregion that has the largest area making up Ck Thi diti li i th h th ll t t C i . s con on app es n e case w en e overa ou pu k s nonconvex; and in the case when Ck is convex, z* is the same quantity as determined by the centroid method or the center of largest area method (because then there is only one convex region). 47Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods • Centre of Sums Example: 50Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods • Centre of the largest Area Example: 51Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 7 First (or last) of Maxima . : • This method uses the overall output or union of all individual output fuzzy sets Ck to determine the smallest value of the domain with maximized membership degree in Ck. The equations for z* are as follows. • First the largest height in the union (denoted hgt(Ck)) is determined, , • Then the first of the maxima is found, • An alternative to this method is called the last of maxima, and it is given by 52Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • DeFuzzification Methods 7 First (or last) of Maxima Example . : 55Functions, Fuzzification and Defuzzification docsity.com Functions, Fuzzification and Defuzzification • Summary • We introduced the various features and forms of a membership function and the idea of fuzzyifying scalar quantities to make them fuzzy sets. • It also explains the process of converting from fuzzy membership functions to crisp formats – a process called defuzzification. • Defuzzification is a natural and necessary process. In fact, there is an analogous form of defuzzification in mathematics where we solve a complicated problem in the complex plane, find the real and imaginary parts of the solution then , decomplexify the imaginary solution back to the real numbers space. 56Functions, Fuzzification and Defuzzification docsity.com