Chapter 4 Fuzzy Logic

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1 4.1 Introduction Chapter 4 Fuzzy Logic The human brain interprets the sensory information provided by organs. Fuzzy set theory focus on processing the information. Numerical computation can be performed using linguistic labels specified by membership functions. Moreover, a selection of fuzzy if then rules forms the key component of a fuzzy inference system (FIS).Human expertise in a specific application can be effectively modelled with FIS. Although the fuzzy inference system has a structured knowledge representation of fuzzy if then rules, it lacks the adaptability to deal with varying environments. Thus we incorporate neural network learning concepts in fuzzy inference systems, resulting in neuro- fuzzy modeling, a pivotal technique in soft computing 4.2 Comparison of Classical set and fuzzy set: Classical Set In mathematical operations if we define a set, then the set is collection of items which belong to specific definition. If we take an example for classical set of items whose value is less than ten then we can defined it as A={x x<5} Where there is boundary is 5. If a number is less than 5 then it will be part of the set. The classical sets are very important for mathematics and computer science but it cannot be useful for certain applications like decision making same as human mind. As human mind take decisions from certain relative information. If we take an example for a person, whether the person is tall or not,to identify this a mind processing relatively if a person is having height more than average height then that person will consider as tall. This identification is not possible in classical set. To define classical set of tall person as A then the members of the set whose height is 34 P a g e

2 more than 6 feet are considered as tall person then the person whose height is 5 feet 11 inch will be not member of this classical set. The another limitation of classical set is if a person is having height 6.1 feet and 7.0 feet both will be tall. There will be not clear identification between both of them. It is the clear distinction between inclusion and exclusion. To overcome these situations fuzzy sets are available. The fuzzy set does not have fixed boundary means an item belongs to particular set or not that process is gradual and this is smooth transition characterized by fuzzy membership function. The fuzzy set is important is certain areas like human thinking, especially in the domain of pattern recognition. The fuzzy set does not mean randomness but it means uncertain nature of things. A = {a1, a2, a3..., an}, Here A is a set represented by classical set theory. ai (i = 1,...,n) are elements of A forms together a subset of the universal base set X. The set A can be represented for all elements x X by its characteristic function μ A (x) = { 1 if x A 0 otherwise } μ A (x) has only two values 0 (``false'') and 1 (``true'') according to the classical set theory. These type of sets are called crisp sets too. Fuzzy sets are called Non Crisp Sets. Characteristic function can be defined for fuzzy sets. Such function is a generalization of that in equation and termed as membership function. To define members in fuzzy set membership functions are used. The membership function described the membership of fuzzy set. 35 P a g e

3 Here membership function μ A (x) of A which relate to each element xo = X a grade of membership μa (xo). Unlike classical set theory the normalized closed interval [0,1] describes the membership functions μa(x) values. Therefore, each membership function maps elements of a given universal base set X which is itself a crisp set, into real numbers in [0,1]. The membership function, μa(x) notation for fuzzy set A can be described as A : X [0,1] Each fuzzy set is completely and uniquely defined by one particular membership function. Consequently symbols of membership functions are also used as labels of the associated fuzzy sets, such as big, small and others. The differences in the crisp and fuzzy sets. [19] Figure 4.1: Crisp set and a fuzzy set membership functions The classical set crisp boundary is used so there is no uncertainty in the boundaries of classical set. There are many examples of classical sets like, set of integer numbers from 1 to 10. If we refer it as universe number then the cardinal number means total 36 P a g e

4 number of elements in universe set which represented as nx, where x is a label for individual elements in universe. In discrete universe is finite collection of elements and continues universe is collection of infinite cardinality. Collection of elements within a universe are called sets and collection of elements within sets are called subset. Null set has no elements. All possible set of set A is called power set P(A). Suppose A={1,2,3,4} then the cardinal number is nx=4. The power set is P(A)={,{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}, {1,2,3},{1,2,4},{2,3,4},{1,3,4},{1,2,3,4}} np(a) is cardinality of power set. np(a) = 2 nx =2 4 =16 Operations on classical sets are: Union AUB = {x x A or x B} Intersection A B = {x x A and x B} Complement A = { x x A, x X} Difference A B = {x x A and x B} U A B ` Figure 4.2: Union of Classical Sets A and B 37 P a g e

5 A B U Figure 4.3 : Intersection of Classical Sets A and B A B U Figure 4.4: Division (A-B) of two classical sets A and B U Figure 4.5: Complement A of Classical set A 38 P a g e

6 4.2.2 Properties of Classical Sets Classical set operations The excluded middle axioms are very important because these set operations are not valid for both classical set and fuzzy set. 39 P a g e

7 4.6 (a) Crisp set A and its Complement (b) Crisp A A = X (c) Crisp A A = Fuzzy Set In contrast to classical set, fuzzy set is a set without rigid boundaries. In fuzzy set the boundaries are ambiguously specified. The fuzzy set has the flexibility in modelling commonly used linguistic expressions. The fuzzy set expresses the degree to which an element belongs to the set. The characteristic function of a fuzzy set is allowed to have the values between 0 and 1, which denotes the degree of membership of an element in the given set. Fuzzy set operations 40 P a g e

8 Figure 4.7: Union of Fuzzy Sets A and B Figure 4.8: Intersection of Fuzzy Sets A and B Figure 4.9: Complement of Fuzzy Set A 41 P a g e

9 4.2.4 Properties of Fuzzy Set Fuzzy set holds all operations same as classical set, except excluded middle axioms. The excluded middle axioms, extended for fuzzy sets, are expressed by 42 P a g e

10 4.10 (a) Fuzzy Set A and its complement (b) Fuzzy A A X (c) Fuzzy A A 4.3 Fuzzy Relations Fuzzy rules are the key strength of fuzzy inference systems, which have the most important modelling tool based on fuzzy set theory Introduction of Fuzzy Relations Binary fuzzy relations are fuzzy sets in X and Y which map each element in X x Y to a membership grade between 0 and 1. Applications of fuzzy relations include areas such as fuzzy control and decision making. The examples of binary fuzzy relations are: X is close to Y X depends on Y X and Y, looks alike If X is large then Y is small 43 P a g e

11 4.3.2 Use of fuzzy relations Different compositions suggested for fuzzy relations are best known as maxmin composition. Let R1 and R2 be two fuzzy relations defined on two Sets S1 x S2 and S2 x S3. R1 * R2 = { [ (x,z), max.min( µr1(x,y), µr1(y,z))] x ϵ X, y ϵ Y, z ϵ Z } max min (x,z) = max ( min (( µr1(x,y), µr1(y,z)) Relation-1 R1 = R2 = µr1. R2 (2,a) = max ( min (0.4,0.9), min (0.2,0.2), min (0.8,0.5)) = max (0.4,0.2,0.5) = 0.5 ( by max-min composition) 1 x 0.5 A 2 y 0.5 B 3 z 0.6 Relation-1 Figure 4.11 Fuzzy Relations Relation-2 44 P a g e

12 The maximum minimum composition of relations Let X, Y and Z be universal sets and let R be a relation that relates elements from X to Y, i.e. R = { ((x, y), μ R (x, y))} xε X, y ε Y, R X Y And Q = { ((y, z), μ Q (y, z))} yε Y, z ε Z, Q Y Z Then S will be relation that relates element in X that R contains to the element in Z that Q contains, i.e... Where o suggest the membership degree of R and Q in max min sense. 4.4 Fuzzy Rules Study or Linguistic Rules Study Fuzzy rules and fuzzy reasoning are the key strength of fuzzy inference systems, which have the most important modelling tool based on fuzzy set theory. Obtaining fuzzy relation which represents the meaning of a given fuzzy rule The linguistic variable can be characterized by a quintuple (x, T(x), X,G,M ) in which x is the name of the variable, T(x) is the term set of x- set of its linguistic values or linguistic terms. X is the universe, G is the syntactic rule and M is the semantic rule that associates with each linguistic value A. Age can be interpreted as linguistic variable, then its T(x) can be T(age) = { young, not young, very young, not very young, middle aged, not middle aged, old, not old, very old, more or less old, not very young and not very old } The universe is X = [0,100] 45 P a g e

13 Here the primary terms are: young, middle aged, old Negation: not Hedges: very, more, more or less Connectives: and, or, either, neither Let A be a linguistic value characterized by a fuzzy set with membership function µa(.) A k is represented as a modified version of the original linguistic value expressed as: A k = x [ µ A ( x)] k / x The concentration is defined as CON (A) = A 2 The dilation is defined as DIL (A) = A 0.5 NOT (A) = [ 1 µ A ( x)] x / x A and B = A B = x [ µ A( x) µ B( x)] / x A or B = A U B = x [ µ A( x) µ B( x)] / x Fuzzy if then rules The fuzzy if-then rules also known as fuzzy conditional statements It assumes the form If x is A then y is B. Where A and B are the linguistic values defined by fuzzy sets on universe of discourse X and Y. Often x is A is called the antecedent or premise, while y is B is called consequence or conclusion. 46 P a g e

14 The fuzzy rules can be: If pressure is high then volume is small. If the speed is high then apply the break. Before we can employ fuzzy if then rules to model and analyze a system, first we have to finalize what is meant by the expression If x is a then y is b which is abbreviated as AB ~ R= A B = A x B = ( x) * µ ( y) /( x, y) µ A B xxy The T-norm operator and AB is used to represent the fuzzy relation on R. The operator may represent and, or, bounded product, logical operations like {a, if b =1; b, if a=1; o, Otherwise} 4.5 Advantages and disadvantages of fuzzy logic Advantages of fuzzy logic The main advantage of fuzzy logic is it allows the use of vague linguistic terms in the rules. Below are some reasons why we need to use fuzzy logic rather than linear system: Fuzzy logic concepts are easy to understand. The concepts of mathematics used in fuzzy reasoning are simple. Fuzzy logic is flexible enough with any system and is easy to manage or add more functionality without starting again from beginning. 47 P a g e

15 Fuzzy logic tolerates inaccurate data also. Almost everything is imprecise and also most things are imprecise even on cautious review. Fuzzy reasoning creates this understanding in the process rather than moving it onto the end. It can model nonlinear functions of random complexity. data can be matched by generating a fuzzy system. Any sets of input output Adaptive techniques like Adaptive Neuro-Fuzzy Inference Systems (ANFIS) have made this process easier. Fuzzy logic can be built by the knowledge of experts. As compared to neural networks, which takes training data and generate models, fuzzy logic lets you depend on the knowledge of people who already know your system. Fuzzy logic can be combined with conventional control techniques. Fuzzy systems do not substitute conventional control methods, but in many cases fuzzy systems extend them and make their implementation simple. Fuzzy logic is based on natural language. Disadvantages of fuzzy logic In fuzzy logic, it is difficult to estimate membership function. There are many ways of interpreting fuzzy rules, combining the output of several fuzzy rules and defuzzifying the output Fuzzy logic is not a solution for all problems. Fuzzy logic should not be used in certain situations. Fuzzy logic is a suitable way to map an input space to an output space, but it may not be convenient in all circumstances. If a simpler solution already exists, fuzzy logic may not be advisable. Fuzzy logic is the modification of common sense - use common sense when it is implemented and the right decision will probably been 48 P a g e

16 made. For example, many controllers do a good job without using fuzzy logic. However, if time is taken to become familiar with fuzzy logic. Fuzzy logic is a powerful tool for dealing quickly and efficiently with uncertainties and nonlinearity. 4.6 Membership function in fuzzy logic Membership of the elements x of the base set X in the fuzzy set A is defined by the membership function μa(x), where μa(x) is a large class of functions can be taken. Functions generally used are linear functions like triangular or trapezoidal functions. The grade or degree of membership μa(x) of a membership function describes which grade, it belongs to in the fuzzy set A. The value of this membership grade or degree is in the unit interval [0, 1]. This is shown in Figure. Figure 4.12: Membership grades of a fuzzy set If X is a collection of objects denoted generically by x, then a fuzzy set A in X is defined as a set of ordered pairs: A = {(x, µ(x)) x ϵ X} whereµ(x) is called the membership function (MF) for the fuzzy set A. The membership value ranges from 0 to P a g e

17 Let X = R + be the set of possible ages of human beings. The fuzzy set B= about 50 years old may be expressed as B= {(x,,µb(x) x ϵ X } Where membership of x is defined as set. µb(x) = 1 1 x Suppose the age of person is 45, then membership value will be for the 4.7 Implementation of Fuzzy set in face recognition Set of Eyes having different colours like brown, black etc. The eye colour may be brownish black or dark brown which cannot be classified as exactly brown eye or exactly black eye. The fuzzy set provides the membership of the eye colour in the black eye colour set. 50 P a g e