Fuzzy Logic : Fuzzy Logic Vikas Kaduskar Asst Professor Bharati Vidyapeeth University College of Engineering, Pune (INDIA)
Discussion Points: Discussion Points Classical Sets Operations on Classical Sets Fuzzy Sets Operations on Fuzzy Sets Membership Functions Examples. Alpha Cuts
Introduction: Introduction It is the mark of an instructed mind to rest satisfied with that degree of precision which the nature of the subject admits, and not to seek exactness where only an approximation of the truth is possible. -Aristotle , 384–322 BC Precision is not truth - Henri E. B. Matisse, 1869–1954
PowerPoint Presentation: We must exploit our tolerance for imprecision. - Lotfi Zadeh , 1973
PowerPoint Presentation: Above represents the relationship between precision and uncertainty. The more uncertainty in a problem, the less precise we can be in our understanding of that problem.
Classical Sets: Classical Sets the universe of discourse is the universe of all available information on a given problem. Once this universe is defined we are able to define certain events on this information space
PowerPoint Presentation: Figure 1.1a shows an abstraction of a universe of discourse , say X, and a crisp (classical) set A somewhere in this universe. A classical set is defined by crisp boundaries, that is, there is no uncertainty in the prescription or location of the boundaries of the set, as shown in Figure 1.1a where the boundary of crisp set A is an unambiguous line.
Fig 1.1 Diagrams for (a) crisp set boundary and (b) fuzzy set boundary.: Fig 1.1 Diagrams for (a) crisp set boundary and (b) fuzzy set boundary.
PowerPoint Presentation: A fuzzy set, on the other hand, is prescribed by vague or ambiguous properties; hence, its boundaries are ambiguously specified, as shown by the fuzzy boundary for set A ∼ in Figure 1.1b. Point a in Figure 1.1a is clearly a member of crisp set A; point b is unambiguously not a member of set A. Figure 1.1b shows the vague, ambiguous boundary of a fuzzy setA ∼ on the same universeX : the shaded boundary represents the boundary region ofA ∼ .
PowerPoint Presentation: In the central (unshaded ) region of the fuzzy set, point a is clearly a full member of the set. Outside the boundary region of the fuzzy set, point b is clearly not a member of the fuzzy set. However, the membership of point c, which is on the boundary region, is ambiguous. If complete membership in a set (such as point a in Figure 1.1b ) is represented by the number 1, and no-membership in a set (such as point b in Figure 1.1b ) is represented by 0, then point c in Figure 1.1b must have some intermediate value of membership (partial membership in fuzzy setA ∼ ) on the interval [0,1].
PowerPoint Presentation: Presumably, the membership of point c inA∼ approaches a value of 1 as it moves closer to the central (unshaded) region in Figure 1.1b of A ∼ and the membership of point c in A ∼ approaches a value of 0 as it moves closer to leaving the boundary region ofA ∼
PowerPoint Presentation: A useful attribute of sets and the universes on which they are defined is a metric known as the cardinality, or the cardinal number. The total number of elements in a universe X is called its cardinal number; denoted n x , where x again is a label for individual elements in the universe
PowerPoint Presentation: Discrete universes that are composed of a countably finite collection of elements will have a finite cardinal number; continuous universes comprises an infinite collection of elements will have an infinite cardinality. Collections of elements within a universe are called sets , and collections of elements within sets are called subsets
For crisp sets A and B consisting of collections of some elements in X, the following notation is defined: For crisp sets A and B consisting of collections of some elements in X, the following notation is defined
Operations on Classical Sets: Operations on Classical Sets Let A and B be two sets on the universe X
These four operations are shown in terms of Venn diagrams in Figures 1.2: These four operations are shown in terms of Venn diagrams in Figures 1.2 Union Intersection Complement Difference
Properties of Classical (Crisp) Sets: Properties of Classical (Crisp) Sets
excluded middle axioms and De Morgan’s principles.: excluded middle axioms and De Morgan’s principles. De Morgan’s principles.
What is a fuzzy set? Randomness vs. Fuzziness: What is a fuzzy set? Randomness vs. Fuzziness Randomness refers to an event that may or may not occur. Randomness : frequency of car accidents. Fuzziness refers to the boundary of a set that is not precise. Fuzziness : seriousness of a car accident. -Prof . George J. Klir
Fuzzy Set: Fuzzy Set Two distinct notations are most commonly employed in the literature to denote membership functions. In one of them, the membership function of a fuzzy set A is denoted by µ A ; that is , µ A :X→ [0,1] In. the other one, the function is denoted by A and has of course, the same form A:X → [0,1]
PowerPoint Presentation: According to the first notation, the symbol (label, identifier, name) of the fuzzy set (A) is distinguished from the symbol of its membership function (µ A ). According to the second notation , this distinction is not made, but no ambiguity results from this double use of the same symbol.
Example 1: Example 1 let us consider four fuzzy sets whose membership functions are shown in Fig. 2 . Each of these fuzzy sets expresses, in a particular form, the general conception of a class of real numbers that are close to 2 with following properties Ai (2) = 1 and A i (x) < 1 for all x ≠2 ; . Ai is symmetric with respect to x = 2, i.e A1(2 +x) = Ai(2 — x) for all x belongs to R; A (x) decreases monotonically from 1 to 0 with the increasing difference[ 2 — x].
Example 2 : Example 2 In mathematics a set, by definition, is a collection of things that belong to some definition. Any item either belongs to that set or does not belong to that set. Let us look at another example; the set of tall men. We shall say that people taller than or equal to 6 feet are tall. This set can be represented graphically as follows:
PowerPoint Presentation: The function shown above describes the membership of the 'tall' set, you are either in it or you are not in it. This sharp edged membership functions works nicely for binary operations and mathematics, but it does not work as nicely in describing the real world. The membership function makes no distinction between somebody who is 6'1" and someone who is 7'1", they are both simply tall. Clearly there is a significant difference between the two heights. The other side of this lack of disctinction is the difference between a 5'11" and 6' man. This is only a difference of one inch, however this membership function just says one is tall and the other is not tall.
PowerPoint Presentation: The fuzzy set approach to the set of tall men provides a much better representation of the tallness of a person. The set, shown below, is defined by a continuously inclining function.
PowerPoint Presentation: The membership function defines the fuzzy set for the possible values underneath of it on the horizontal axis. The vertical axis, on a scale of 0 to 1, provides the membership value of the height in the fuzzy set. So for the two people shown above the first person has a membership of 0.3 and so is not very tall. The second person has a membership of 0.95 and so he is definitely tall. He does not, however, belong to the set of tall men in the way that bivalent sets work; he has a high degree of membership in the fuzzy set of tall men.
Membership function: M embership function The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition.
Membership function of a fuzzy set: Membership function of a fuzzy set
PowerPoint Presentation: Membership function (compare Characteristic function) The membership function M maps each element of X to a membership grade (or membership value) between 0 and 1 . A fuzzy set M, in the universal set can be presented by: -list form , - rule form, -membership function form.
PowerPoint Presentation: List form A = {<1,2>; <2,1>; <3,0.9>; <4,0.7>;<5,0.5>;... (Note : The list form can be used only for ¯ nite sets) Rule form M = (x belongs to X | x meets some condition)
Fuzzy sets of different types: Fuzzy sets of different types The membership function may be vague in itself. Interval-valued fuzzy sets A:X → ε [0,1] Fuzzy sets of type 2 A:X → F [0,1]
Basic concepts and terminology: Basic concepts and terminology An α -cut of a fuzzy set A is a crisp set α A that contains all the elements in X that have membership value in A greater than or equal to α α A = {x | A(x) ≥ α } A strong α -cut of a fuzzy set A is a crisp set α + A that contains all the elements in X that have membership value in A strictly greater than α . α + A = {x | A(x) > α }
“Example on α-Cuts”: “Example on α -Cuts”
Fuzzy Set Operations: Fuzzy Set Operations following function-theoretic operations for the set-theoretic operations of union , intersection , and complement are defined for A ∼ , B ∼ , and C∼ on X:
COMPLEMENT The absolute complement of a fuzzy set A is denoted by A and its membership function is defined by: COMPLEMENT The absolute complement of a fuzzy set A is denoted by A and its membership function is defined by
PowerPoint Presentation: The intersection of two fuzzy sets A and B is a fuzzy set whose membership function is defined by
Table: Some properties of fuzzy sets operations: Table: Some properties of fuzzy sets operations
Support: Support The support of a fuzzy set A is a crisp set supp(A) of all x belongs to X such that µA(x)>0. It is strong α -cut for α = 0. The element x belong X at which µA(x)=0.5 is referred to as cross- over point. A fuzzy set whose support is a single element in X with µA(x)=1 is referred to as a fuzzy singleton.
PowerPoint Presentation: The core of a fuzzy set A is a crisp set core(A) of all x∈X such that µA(x )= 1. The core of a fuzzy set may be an empty set. The height, h( A) of a fuzzy set A is the largest value of μA for which the α-cut is not empty. In other words, it is the largest value of the membership function attained by an element in the set. A fuzzy set with h( A) = 1 is referred to as normal, otherwise it will be referred to as sub-normal.
A graphical illustration of α-cuts, support, core, and height.: A graphical illustration of α-cuts, support, core, and height.
T-Norms and t-Conorms: Fuzzy AND and OR Operators: T-Norms and t- Conorms : Fuzzy AND and OR Operators Given fuzzy sets A, B, C, . . . all fuzzy subsets of X, we wish to compute A union B, B intersection C, and so on. What we use in fuzzy logic are the generalized AND and OR operators from classical logic. They are called t-norms (for AND) and t- conorms (for OR). We first define t-norms-
PowerPoint Presentation: A t-norm T is a function from [0, 1] × [0, 1] into [0, 1]. That is, if z =T(x, y), then x, y, and z all belong to the interval [0, 1]. All t-norms have the following four properties:
PowerPoint Presentation: T-norms generalize the AND from classical logic. This means that tv (P AND Q) = T( tv (P), tv (Q)) for any t-norm and equations are all examples of t-norms. The basic t-norms are and T*(x, y) defined as x if y = 1, y if x = 1, 0 otherwise .
PowerPoint Presentation: T m is called the standard or Zadehian intersection, and is the one most commonly employed; T L is the bounded difference intersection; T p is the algebraic product; and T is the drastic intersection. It is well known that T*<T L < T p <T m
PowerPoint Presentation: t- Conorms generalize the OR operation from classical logic. As for t-norms, a t- conorm C(x, y) = z has x, y, and z always in [0, 1]. The basic properties of any t- conorm C are
PowerPoint Presentation: The basic t- conorms are IT IS WELL KNOWN THAT Cm < Cp < CL< C*
PowerPoint Presentation: T-norms and t- conorms are only defined for two variables and in fuzzy expert systems we need to extend them to n variables
Distances on and between fuzzy sets: Distances on and between fuzzy sets Approaches generalize crisp definitions infer distance from similarity (often shape similarity of sets) deduce distance from set relations (or other relations) symbolic (language/graphs)
PowerPoint Presentation: Notions Distance relations Proximity relations Similarity relations (S : F × F →[0; 1])
PowerPoint Presentation: Distances between fuzzy sets a) Membership focused (vertical) b) Spatially focused (horizontal) c) Mix of spatial and membership (tolerance) d) Feature distances (low or high dimensional representations) e) Morphological (mixed focus)
Feature Distance: Feature Distance
Morphological Approach: Morphological Approach
References:: References: M. Benedicty and R. F. Sledge, Discrete Mathematical Structures, HBJ, Toronto, 1987 . E . Cox, The Fuzzy Systems Handbook, AP Professional, Boston, USA, 1994. A . Kaufman and M. M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhald , New York, USA, 1991. G. J. Klir and T. A. Folger , Fuzzy Sets, Uncertainty, and Information, Prentice Hall , London, UK, 1998. 5. G. J. Klir , U. H. St. Clair, and B. Yuan, Fuzzy Set Theory, Prentice Hall, Upper Saddle River, NJ, USA, 1997. 6. E. H. Mamdani , “Application of Fuzzy Algorithms for Control of a Simple Dynamic Plant”, Proceedings of IEEE, 121, 12, 1585–1588, 1974. 7. M. Sugeno,“An Introductory Survey on Fuzzy Control”, Information Sciences, 36 , 59–83, 1985.
PowerPoint Presentation: Web Resources On Fuzzy Relations, Metrics and Cluster Analysis http ://dmi.uib.es/people/valverde/gran1/GRAN1.html An article in HTML format by J. Jacas and L. Valverde . It gives an introduction to fuzzy logic and discusses: Building fuzzy transitive relations, Generators , m-cluster coverages . It provides 20 references. Modeling Linguistic Expressions Using Fuzzy Relations www.scch.at/servlet/resource.ResourceLoader?id=5 A 15-page technical report with 13 references available in PDF format. It is authored by M. De Cock, U. Bodenhofer , and E. Kerre , and appeared in The Proceedings of the 6th International Conference on Soft Computing, Iizuka , Japan , October 1-4, 2000, pp. 353–360 . http://en.wikipedia.org/wiki/Fuzzy_set