Fuzzy Logic and Fuzzy Systems

Transcription

1 Fuzzy Logic and Fuzzy Systems Xin Yao Some material adapted from slides by B. Vrusias, University of Surrey, and by G. Cheng, West Virginia University. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems Xin Yao

2 Outline Introduction Fuzzy Sets and Membership Functions Fuzzy Linguistic Variables Fuzzy Set Operations and Properties Fuzzy Rules Fuzzy Inference System Summary Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 2

3 Introduction As complexity rises, precise statements lose meaning, and meaningful statements lose precision Lotfi Zadeh (1965) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 3

4 Introduction (I) We rely on common sense to solve problems. How can we represent expert knowledge that uses vague and ambiguous terms in a computer? Fuzzy Logic (FL) is not logic that is fuzzy, but logic that is used to describe fuzziness. FL is the theory of fuzzy sets to calibrate vagueness. FL is based on the idea that all things admit of degrees. Temperature, height, speed, distance, beauty all come on a sliding scale. FL reflects how people think. It attempts to model our sense of words, our decision making and our common sense. For example, the motor is running really hot. Tom is a very tall guy. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 4

5 Introduction (II) FL is a set of mathematical principles for knowledge representation based on degrees of membership. Unlike Boolean logic, FL is multi-valued. It deals with degrees of membership and degrees of truth. FL uses the continuum of logical values between 0 (completely false) and 1 (completely true). Instead of just black and white, it employs the spectrum of values, accepting that things can be partly true and partly false at the same time. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 5

6 Brief History Fuzzy, or multi-valued logic, was introduced in the 1930s by Jan Lukasiewicz, a Polish philosopher. While classical logic operates with only two values 1 (true) and 0 (false), Lukasiewicz introduced logic that extended the range of truth values to all real numbers in the interval between 0 and 1. For example, the possibility that a man 181 cm tall is really tall might be set to a value of It is likely that the man is tall. This work led to an inexact reasoning technique often called possibility theory. In 1965 Lotfi Zadeh, published his famous paper Fuzzy sets. Zadeh extended the work on possibility theory into a formal system of mathematical logic, and introduced a new concept for applying natural language terms. This new logic for representing and manipulating fuzzy terms was called fuzzy logic. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 6

7 Zadeh s Paper on Fuzzy Sets (1965) L. A. Zadeh: Fuzzy sets. Information and Control (1965) 37,529 Citations

8 But Why? Why fuzzy? As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means. However, many people were repelled by the word fuzzy, because it is usually used in a negative sense. Why logic? Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 7

9 The Term Fuzzy Logic The term fuzzy logic is used in two senses: Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge. Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory. Broad Sense: fuzzy logic synonymously with fuzzy set theory Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 8

10 Fuzzy Logic Applications Theory of fuzzy sets and fuzzy logic have been applied to problems in a variety of fields: taxonomy; topology; linguistics; logic; automata theory; game theory; pattern recognition; medicine; law; decision support; Information retrieval; etc. And more recently fuzzy machines have been developed including: automatic train control; tunnel digging machinery; washing machines; rice cookers; vacuum cleaners; air conditioners, ride smoothness control, camcorder auto-focus and jiggle control, braking systems, copier quality control, high performance drives, etc. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 9

11 Fuzzy Logic Applications Advertisement: Extraklasse Washing Machine rpm. The Extraklasse machine has a number of features which will make life easier for you. Fuzzy Logic detects the type and amount of laundry in the drum and allows only as much water to enter the machine as is really needed for the loaded amount. And less water will heat up quicker - which means less energy consumption. Foam detection: Too much foam is compensated by an additional rinse cycle: If Fuzzy Logic detects the formation of too much foam in the rinsing spin cycle, it simply activates an additional rinse cycle. Fantastic! Imbalance compensation: In the event of imbalance, Fuzzy Logic immediately calculates the maximum possible speed, sets this speed and starts spinning. This provides optimum utilization of the spinning time at full speed. Washing without wasting - with automatic water level adjustment: Fuzzy automatic water level adjustment adapts water and energy consumption to the individual requirements of each wash programme, depending on the amount of laundry and type of fabric. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 10

12 Fuzzy Sets and Membership Functions Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 11

13 Fuzzy Sets The concept of a set is fundamental to mathematics. However, our own language is also the supreme expression of sets. The elements of the fuzzy set tall men are all men, but their degrees of membership depend on their height. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 12

14 Crisp and fuzzy sets of tall men Degree of Membership 1.0 Crisp Sets The y-axis represents the membership value of the fuzzy set Degree of Membership Fuzzy Sets Height, cm Height, cm The x-axis represents the universe of discourse Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 13

15 A Fuzzy Set has Fuzzy Boundaries Let X be the universe of discourse and its elements be denoted as x. In the classical set theory, crisp set A of X is defined as function f A (x) called the characteristic function of A: f A (x) : X {0, 1}, where f A ( x) 1, if 0, if x x A A This set maps universe X to a set of two elements. For any element x of universe X, characteristic function f A (x) is equal to 1 if x is an element of set A, and is equal to 0 if x is not an element of A. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 14

16 A Fuzzy Set has Fuzzy Boundaries In the fuzzy theory, fuzzy set A of universe X is defined by function µ A (x) called the membership function of set A µ A (x) : X [0, 1], where µ A (x) = 1 if x is totally in A; µ A (x) = 0 if x is not in A; 0 < µ A (x) < 1 if x is partly in A. This set allows a continuum of possible choices. For any element x of universe X, membership function µ A (x) equals the degree to which x is an element of set A. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 15

17 Classical Logic vs. Fuzzy Logic Classical Logic Element x belongs to set A or it does not: (x) {0,1} Fuzzy Logic Element x belongs to set A with a certain degree of membership: (x) [0,1] A (x) 1 A= young A (x) 1 A= young 0 x [years] 0 x [years] Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 16

18 CM 180 Very Tall Tall Fuzzy Measures 170 Medium 16 0 Short 150 Very Short Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 17

19 Membership Functions (MF) Common used Types of Membership Function Triangular Trapezoid Gaussian Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 18

20 Membership Function: An Example Very Short 1.0 Short Medium Tall Very Tall 150cm 160cm 170cm 180cm Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 19

21 Membership Function: An Example Very Short 1.0 Short Medium Tall Very Tall Short 0, 0, 0, 1, 0.5, Medium Tall Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 20 0, 0, 0 0.5, 0

22 Fuzzy Linguistic Variables (FLV) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 21

23 FLV and hedges A FLV is a fuzzy variable. For example, the statement John is tall implies that the linguistic variable John takes the linguistic value tall. A FLV carries with it the concept of fuzzy set qualifiers, called hedges. Hedges are terms that modify the shape of fuzzy sets. They include adverbs such as very, somewhat, quite, more or less and slightly. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 22

24 Examples of FLV color: red, blue, green, yellow, age: young, middle-aged, old, very old size: small, big, very big, distance: near, far, very far, not very far, µ 1 young middle-aged old very old 0 Age Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 23

25 Representation of hedges in fuzzy logic Hedge Mathematical Expression Graphical Representation A little [ A (x)] 1.3 Slightly [ A (x)] 1.7 Very [ A (x)] 2 Extremely [ A (x)] 3 Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 24

26 Representation of hedges in fuzzy logic Hedge Mathematical Expression Graphical Representation Very very [ A (x)] 4 More or less A (x) Somewhat A (x) Indeed 2 [ A (x)] 2 if 0 A [1 A (x)] 2 if 0.5 < A 1 Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 25

27 Fuzzy Set Operations and Properties Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 26

28 Characteristics of Fuzzy Sets The classical set theory developed in the late 19th century by Georg Cantor describes how crisp sets can interact. These interactions are called operations. Also fuzzy sets have well defined properties. These properties and operations are the basis on which the fuzzy sets are used to deal with uncertainty on the one hand and to represent knowledge on the other. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 27

29 Note: Membership Functions For the sake of convenience, usually a fuzzy set is denoted as: A = A (x i )/x i +. + A (x n )/x n where A (x i )/x i (a singleton) is a pair grade of membership element, that belongs to a finite universe of discourse: A = {x 1, x 2,.., x n } Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 28

30 Operations of Crisp Sets Not A B A A Complement Containment A B A B Intersection Union Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 29

31 Complement Crisp Sets: Who does not belong to the set? Fuzzy Sets: How much do elements not belong to the set? The complement of a set is an opposite of this set. For example, if we have the set of tall men, its complement is the set of NOT tall men. When we remove the tall men set from the universe of discourse, we obtain the complement. If A is a fuzzy set, its complement A can be found as follows: μ A (x) = 1 A (x) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 30

32 Containment Crisp Sets: Which sets belong to which other sets? Fuzzy Sets: Which sets belong to other sets? A set can contain other sets. The smaller set is called the subset. For example, the set of tall men contains all tall men; very tall men is a subset of tall men. However, the tall men set is just a subset of the set of men. In crisp sets, all elements of a subset entirely belong to a larger set. In fuzzy sets, however, each element can belong less to the subset than to the larger set. Elements of the fuzzy subset have smaller memberships in it than in the larger set. Tall men: (0/180, 0.25/182.5, 0.5/185, 0.75/187.5, 1/190) Very Tall Men: (0/180, 0.06/182.5, 0.25/185, 0.56/187.5, 1/190) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 32

33 Intersection Crisp Sets: Which element belongs to both sets? Fuzzy Sets: How much of an element is in both sets? In classical set theory, an intersection between two sets contains the elements shared by these sets. For example, the intersection of the set of tall men and the set of fat men is the area where these sets overlap. In fuzzy sets, an element may partly belong to both sets with different memberships. A fuzzy intersection is the lower membership in both sets of each element. The fuzzy intersection of two fuzzy sets A and B on universe of discourse X: A B (x) = min [ A (x), B (x)] = A (x) B (x), where x X Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 33

34 1 A (x) A B ( x) min [ A( x), ( x)] B x B (x) A B (x) x x Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 35

35 Union Crisp Sets: Which element belongs to either set? Fuzzy Sets: How much of the element is in either set? The union of two crisp sets consists of every element that falls into either set. For example, the union of tall men and fat men contains all men who are tall OR fat. In fuzzy sets, the union is the reverse of the intersection. That is, the union is the largest membership value of the element in either set. The fuzzy operation for forming the union of two fuzzy sets A and B on universe X can be given as: A B (x) = max [ A (x), B (x)] = A (x) B (x), where x X Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 36

36 1 A (x) A B ( x) max [ A( x), ( x)] B x A+B (x) B (x) x x Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 38

37 ( x ) A Not A x ( x ) B A B A x Operations of Fuzzy Sets 0 0 x Complement Containment ( x ) ( x ) x 1 A B 1 A B 0 x 0 x 1 A B 1 0 Intersection x 0 A B Union x Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 39

38 Properties of Fuzzy Sets Equality of two fuzzy sets Inclusion of one fuzzy set into another fuzzy set Cardinality of a fuzzy set An empty fuzzy set -cuts (alpha-cuts) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 40

39 Equality Fuzzy set A is considered equal to a fuzzy set B, IF AND ONLY IF (iff): A (x) = B (x), x X A = 0.3/ /2 + 1/3 B = 0.3/ /2 + 1/3 therefore A = B Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 41

40 Inclusion Inclusion of one fuzzy set into another fuzzy set. Fuzzy set A X is included in (is a subset of) another fuzzy set, B X: A (x) B (x), x X Consider X = {1, 2, 3} and sets A and B A = 0.3/ /2 + 1/3; B = 0.5/ /2 + 1/3 then A is a subset of B, or A B Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 42

41 Cardinality Cardinality of a non-fuzzy set, Z, is the number of elements in Z. BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A, A (x): card A = A (x 1 ) + A (x 2 ) + A (x n ) = Σ A (x i ), Consider X = {1, 2, 3} and sets A and B A = 0.3/ /2 + 1/3; B = 0.5/ /2 + 1/3 card A = 1.8 card B = 2.05 for i=1..n Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 43

42 Empty Fuzzy Set A fuzzy set A is empty IF AND ONLY IF: A (x) = 0, x X Consider X = {1, 2, 3} and set A A = 0/1 + 0/2 + 0/3 then A is empty Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 44

43 Alpha-cut An -cut or -level set of a fuzzy set A X is an ORDINARY (CRISP) SET A X, such that: A ={ A (x), x X}. Consider X = {1, 2, 3} and set A A = 0.3/ /2 + 1/3 then A 0.5 = {2, 3}, A 0.1 = {1, 2, 3}, A 1 = {3} Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 45

44 Fuzzy Set Normality A fuzzy subset of X is called normal if there exists at least one element x X such that A (x) = 1. A fuzzy subset that is not normal is called subnormal. All crisp subsets except for the null set are normal. In fuzzy set theory, the concept of nullness essentially generalises to subnormality. The height of a fuzzy subset A is the large membership grade of an element in A height(a) = max x ( A (x)) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 46

45 Fuzzy Sets Core and Support Assume A is a fuzzy subset of X: the support of A is the crisp subset of X consisting of all elements with non-zero membership grade: supp(a) = {x A (x) 0 and x X} the core of A is the crisp subset of X consisting of all elements with membership grade 1: core(a) = {x A (x) = 1 and x X} Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 47

46 Fuzzy Rules Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 48

47 Fuzzy Rules Fuzzy rules relate fuzzy sets. In a FIS (fuzzy interference system), all rules fire to some extent, or in other words they fire partially. If the antecedent is true to some degree of membership, then the consequent is also true to that same degree. In fuzzy systems, FLV are used in fuzzy rules. Ex: IF project_duration is long THEN completion_risk is high Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 49

48 Fuzzy Rules: An Example These fuzzy sets provide the basis for a weight estimation model. The model is based on a relationship between a man s height and his weight: IF height is tall THEN weight is heavy Degree of Membership Tall men Degree of Membership 1.0 Heavy men Height, cm Weight, kg Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 50

49 Fuzzy Rules: An Example The value of the output or a truth membership grade of the rule consequent can be estimated directly from a corresponding truth membership grade in the antecedent. This form of fuzzy inference uses a method called monotonic selection. Degree of Membership Tall men Degree of Membership Heavy men Height, cm Weight, kg Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 51

50 Fuzzy Rules: Examples A fuzzy rule can have multiple antecedents: IF project_duration is long AND project_staffing is large AND project_funding is inadequate THEN risk is high The consequent of a fuzzy rule can also include multiple parts: IF temperature is hot THEN hot_water is reduced; cold_water is increased Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 52

51 Fuzzy Inference System Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 53

52 Crisp Input Fuzzy System Fuzzification Input Membership Functions Fuzzy Input Rule Evaluation Rules / Inferences Fuzzy Output Defuzzification Output Membership Functions Crisp Output Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 54

53 Operation of Fuzzy Systems Step 1: Input Fuzzification Step 2: Fuzzy Rules Evaluation Step 3: Calculate Membership Step 4: Activate Fuzzy Rules Step 5: Compute Decision Function Step 6: Compute Final Decision Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 55

54 Step 1: Input Fuzzification Fuzzifier converts a crisp input into a FLV Definition of the MF must reflect the designer's knowledge provide smooth transition between member and nonmembers of a fuzzy set be simple to calculate Typical shapes of the MF are Gaussian, Trapezoidal and Triangular. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 56

55 Example Assume we want to evaluate the health state of a person based on his height and weight. The input variables are the crisp numbers of the person s height and weight. Fuzzification is a process by which the numbers (height and weight ) are changed into linguistic words. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 57

56 Fuzzification of Height Very Short Short Medium Tall Very Tall cm 160cm 170cm 180cm VS = Very Short S = Short M = Medium etc. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 58

57 Fuzzification of Weight Very Slim 1.0 Slim Medium Heavy Very Heavy 70kg 90kg 110kg 130kg VS = Very Slim S = Slim M = Medium etc. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 59

58 Step 2: Fuzzy Rules Evaluation Rules reflect expert s decisions Rules are tabulated as fuzzy words Rules can be grouped in subsets Rules can be redundant Rules can be adjusted to match desired results Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 60

59 Fuzzy Rules Function Rules are tabulated as fuzzy words (which describe consequences) Healthy (H) Somewhat healthy (SH) Less Healthy (LH) Unhealthy (U) Rule function f f = {U, LH, SH, H} Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 61

60 Fuzzified Decision f f= {U, LH, SH, H} Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 62

61 Fuzzy Rules Table Weight Very Short Very Slim Slim Mediu m Heavy Very Heavy H SH LH U U Height Short SH H SH LH U Mediu m LH H H LH U Tall U SH H SH U Very Tall U LH H SH LH Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 63

62 Step 3: Calculate Membership For a given person, compute the membership of his weight and height Example: Assume that a person s height is 172cm Assume that the person s weight is 66kg How about his health? Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 64

63 Membership of Height Very Short Short Medium Tall Very Tall cm 160cm 170cm 180cm μ height = {μ VS, μ S, μ M, μ T, μ VT } μ height = { } Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 65

64 Membership of Weight Very Slim Slim Medium Heavy Very Heavy kg 90kg 110kg 130kg μ weight = {μ VS, μ S, μ M, μ H, μ VH } μ weight = { } Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 66

65 Step 4: Activate Fuzzy Rules Weight Very Short Very Very Slim Medium Heavy Slim Heavy H SH LH U U Height Short SH H SH LH U Medium H LH U Tall H SH U Very Tall U LH H SH LH Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 67

66 Substitute Membership Values Weight Medium Heavy Very Heavy Very Short H SH LH U U Height Short 0.7 SH LH H H SH H LH LH U U 0.3 U SH H SH U Very Tall U LH H SH LH Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 68

67 Perform Min Operation Weight Height Medium Heavy V.Heavy (0) (0) (0) V.Short (0) Short (0) V.Tall (0) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 69

68 Step 5: Compute Decision Function Weight Weight V.Short(0) 0 0 V.Short(0) H SH Height Short(0) Height Short(0) SH H 0.7 LH H 0.3 U SH V.Tall(0) 0 0 V.Tall(0) U LH f = { U, L H, S H, H } f = {0.3,0.7,0.2,0.2} Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 70

69 Scaled Fuzzified Decision f = { U, L H, S H, H} f = { 0.3, 0.7, 0.2, 0.2} Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 71

70 Step 6: Compute Final Decision Use the fuzzified rules to compute the final decision. Two methods are often used: - Maximum (Sugeno method) - Centroid (Mamdani method) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 72

71 Max Method Fuzzy set with the largest membership value is selected. Fuzzy decision: f = { U, LH, SH, H } f = {0.3, 0.7, 0.2, 0.2} Final Decision (FD) = Less Healthy If two decisions have the same membership max, use the average of the two. Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 73

72 Centroid Method FD = μd μ = μ UD U + μ LH D LH + μ U + μ LH + FD = = Crisp Decision Index (D) = Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 75

73 Fuzzified Decision Crisp Decision Index (D) is the centroid D= Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 76

74 Fuzzy Decision Index Fuzzy Decision Index (D) 75% in Less Healthy group 25% in Somewhat Healthy group Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 77

75 Summary Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 78

76 Benefits Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 80

77 Strength and Weakness Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 81

78 Reading materials [Book] Ross T J. Fuzzy logic with engineering applications (Chapter 6 Logic and Fuzzy Systems). John Wiley & Sons, [Zadeh, 1965] Zadeh, L. (1965). Fuzzy sets. Information and Control, 8(3): [Madau D., 1996] Madau D., D. F. (1996). Influence value defuzzification method. Fuzzy Systems, Proceedings of the Fifth IEEE International Conference, 3: [Leekwijck and Kerre, 1999] Leekwijck, W. V. and Kerre, E. E. (1999). Defuzzification: criteria and classification. Fuzzy Sets and Systems, 108(2): Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 82

79 Homework 1 Consider two fuzzy subsets of the set X, X = {a, b, c, d, e } referred to as A and B A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e} and B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e} Calculate the following : Support, Core, Cardinality, Complement, Union, Intersection, A ( =0.5), A ( =0.2), -cut (A 0.2, A 0.3, A 0.8, A 1 ) Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 83

80 Homework 2 For A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Draw the Fuzzy Graph of A and B Then, calculate the following: - Support, Core, Cardinality, and Complement for A and B independently - Union and Intersection of A and B - the new set C, if C = A 2 - the new set D, if D = 0.5 B - the new set E, for an alpha cut at A 0.5 Fall 2017 Artificial Intelligence: Fuzzy Logic and Fuzzy Systems 84