计算智能 第 10 讲 : 模糊集理论 周水庚 计算机科学技术学院

Transcription

1 计算智能 第 0 讲 : 模糊集理论 周水庚 计算机科学技术学院

2 Introduction to Fuzzy Set Theory

3 Outline Fuzzy Sets Set-Theoretic Operations MF Formulation Extension Principle Fuzzy Relations Linguistic Variables Fuzzy Rules Fuzzy Reasoning Computational Intelliegence 3

4 Introduction to Fuzzy Set Theory Fuzzy Sets

5 Types of Uncertainty Stochastic uncertainty E.g., rolling a dice Linguistic uncertainty E.g., low price, tall people, young age Informational uncertainty E.g., credit worthiness, honesty Computational Intelliegence 5

6 Crisp or Fuzzy Logic ( 明确或者模糊逻辑 ) Crisp Logic A proposition can be true or false only. Bob is a student (true) Smoking is healthy (false) The degree of truth is 0 or. Fuzzy Logic The degree of truth is between 0 and. William is young (0.3 truth) Ariel is smart (0.9 truth) Computational Intelliegence 6

7 More Fuzzy Logic Fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness Fuzzy logic is the theory of fuzzy sets, sets that calibrate vagueness Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to generate decisions Computational Intelliegence 7

8 A Bit of History about Fuzzy Logic Fuzzy, or multi-valued logic, was introduced in the 930s by Jan Lukasiewicz, a Polish philosopher. Lukasiewicz introduced logic that extended the range of truth values to all real numbers in the interval between 0 and. For example, the possibility that a man 8 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 965 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 Computational Intelliegence 8

9 Fuzzy Sets L. A. Zadeh, Fuzzy sets, Information and Control, vol. 8, pp , 965. Lotfi A. Zadeh, The founder of fuzzy logic. Computational Intelliegence 9

10 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 Computational Intelliegence 0

11 Crisp Sets( 明确集 ) Classical sets are called crisp sets either an element belongs to a set or not, i.e., x A or x A Membership function of crisp set 0 x A A( x) x A ( x ) 0, A Computational Intelliegence

12 Crisp Sets P : the set of all people. Y : the set of all young people. Young y y age( x) 25, x P Young ( y) Y P 25 y Computational Intelliegence 2

13 ( x ) 0, Fuzzy Sets Crisp sets A A ( x ) [0,] Example Young ( y) y Computational Intelliegence 3

14 Definition: Fuzzy Sets and Membership Functions If U is a collection of objects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs: A ( x, ( x)) x U A membership function A : U [0,] U : universe of discourse Computational Intelliegence 4

15 Example (Discrete Universe) U {,2,3,4,5,6,7,8} # courses a student may take in a semester. (, 0.) (2, 0.3) (3, 0.8) (4,) A (5, 0.9) (6, 0.5) (7, 0.2) (8, 0.) appropriate # courses taken ( x A ) x : # courses Computational Intelliegence 5

16 Example (Discrete Universe) U {,2,3,4,5,6,7,8} # courses a student may take in a semester. (, 0.) (2, 0.3) (3, 0.8) (4,) A (5, 0.9) (6, 0.5) (7, 0.2) (8, 0.) appropriate # courses taken Alternative Representation: A 0./ 0.3/ 20.8/ 3.0/ 40.9/ 50.5/ 60.2/ 70./ 8 Computational Intelliegence 6

17 Example (Continuous Universe) U : the set of positive real numbers B ( x, ( x)) x U ( x B ) x 50 5 B Alternative Representation: B R 50 4 x 5 4 x about 50 years old ( x B ) possible ages x : age Computational Intelliegence 7

18 Alternative Notation A ( x, ( x)) x U A U : discrete universe U : continuous universe A ( x ) / x xu i A i i A ( )/ A x x U Note that and integral signs stand for the union of membership grades; / stands for a marker and does not imply division. Computational Intelliegence 8

19 Membership value Membership Functions (MF s) A fuzzy set is completely characterized by a membership function. a subjective measure. not a probability measure. tall in Asia tall in USA tall in NBA height Computational Intelliegence 9

20 More MF Membership function µ A (x) equals the degree to which x is an element of set A. This degree, a value between 0 and, represents the degree of membership, also called membership value, of element x in set A. Computational Intelliegence 20

21 Fuzzy Partition Fuzzy partitions formed by the linguistic values young, middle aged, and old : Computational Intelliegence 2

22 MF Terminology cross points MF core width -cut support x Computational Intelliegence 22

23 More Terminologies Normality core non-empty Fuzzy singleton Support= one single point Fuzzy numbers fuzzy set on real line R that satisfies convexity and normality Symmetricity ( c x) ( c x), x U A Open left or right, closed lim ( x), lim ( x) 0 x A A x A Computational Intelliegence 23

24 Normality and Fuzzy Number Normality A fuzzy set A is normal if its core is nonempty. In other words, we can always find a points x X such that μ A (x) = Fuzzy number A fuzzy number A is a fuzzy set in the real line (R) that satisfies the conditions for normality and convexity Computational Intelliegence 24

25 Open left, open right, closed A fuzzy set A is Open left if lim ( x), lim ( x) 0 x A x A Open right if lim x - μ A (x) = 0 and lim x + μ A (x) = Closed if lim x - μ A (x) = 0 and lim x + μ A (x) = 0 Computational Intelliegence 25

26 Convexity of Fuzzy Sets A fuzzy set A is convex if for any in [0, ]. ( x ( ) x ) min( ( x ), ( x )) A 2 A A 2 Computational Intelliegence 26

27 Fuzzy Applications Theory of fuzzy sets and fuzzy logic has 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 fuzzy machines have been developed including: automatic train control; tunnel digging machinery; washing machines; rice cookers; vacuum cleaners; air conditioners, etc. Computational Intelliegence 27

28 Fuzzy 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 Computational Intelliegence 28

29 Introduction to Fuzzy Set Theory Set-Theoretic Operations

30 Set-Theoretic Operations Subset Complement Union A B ( x) ( x), x U Intersection A A U A ( x) ( x) A B A C A B ( x) max( ( x), ( x)) ( x) ( x) C A B A B C A B ( x) min( ( x), ( x)) ( x) ( x) C A B A B Computational Intelliegence 30

31 Set-Theoretic Operations A B A A B A B Computational Intelliegence 3

32 Properties Involution Commutativity Associativity Distributivity Idempotence Absorption A A A B B A A B B A A BC ABC A BC ABC ABC AB AC ABC AB AC AA A AA A A A B A A A B A De Morgan s laws A B A B A B A B Computational Intelliegence 32

33 Properties The following properties are invalid for fuzzy sets: The laws of contradiction A A The laws of excluded middle AA U Computational Intelliegence 33

34 Other Definitions for Set Operations Union Intersection ( x) min, ( x) ( x) A B A B ( x) ( x) ( x) A B A B Computational Intelliegence 34

35 Other Definitions for Set Operations Union ( x) ( x) ( x) ( x) ( x) A B A B A B Intersection ( x) ( x) ( x) A B A B Computational Intelliegence 35

36 Generalized Union/Intersection Generalized Intersection t-norm Generalized Union t-conorm Computational Intelliegence 36

37 T-Norm Or called triangular norm. T :[0,] [0,] [0,]. Symmetry 2. Associativity 3. Monotonicity T( x, y) T( y, x) T( T( x, y), z) T( x, T( y, z)) x x, y y T( x, y ) T( x, y ) Border Condition T ( x,) x Computational Intelliegence 37

38 T-Conorm Or called s-norm. S :[0,] [0,] [0,]. Symmetry 2. Associativity 3. Monotonicity S( x, y) S( y, x) S( S( x, y), z) S( x, S( y, z)) x x, y y S( x, y ) S( x, y ) Border Condition S( x,0) x Computational Intelliegence 38

39 Examples: T-Norm & T-Conorm Minimum/Maximum: T( a, b) min( a, b) a b S( a, b) max( a, b) a b Lukasiewicz: T( a, b) max( a b,0) LAND( a, b) S( a, b) min( a b,) LOR( a, b) Probabilistic: T ( a, b) ab PAND( a, b) S( a, b) a b ab POR( a, b) Computational Intelliegence 39

40 Introduction to Fuzzy Set Theory MF Formulation

41 MF Formulation Triangular MF trimf x a c x ( x; a, b, c) max min,,0 b a c b Trapezoidal MF trapmf x a d x ( x; a, b, c, d) max min,,,0 b a d c Gaussian MF gaussmf ( x; a, b, c) e xc 2 2 Generalized bell MF gbellmf ( x; a, b, c) x c b 2b Computational Intelliegence 4

42 MF Formulation Computational Intelliegence 42

43 Manipulating Parameter of the Generalized Bell Function gbellmf ( x; a, b, c) 2 x c a b Computational Intelliegence 43

44 Sigmoid MF Extensions: Abs. difference of two sig. MF sig mf ( xa ;, c ) e a( x c) Product of two sig. MF Computational Intelliegence 44

45 L-R MF LR( xc ;,, ) cx FL, x c x c FR, x c Example: 2 F ( ) max(0, ) L x x F ( ) exp R x x 3 c=65 =60 =0 c=25 =0 =40 Computational Intelliegence 45

46 Introduction to Fuzzy Set Theory Extension Principle

47 Functions Applied to Crisp Sets y y = f(x) B B f ( A) B (y) A (x) A x x Computational Intelliegence 47

48 Functions Applied to Fuzzy Sets y y = f(x) B B f ( A) B (y) A (x) A x x Computational Intelliegence 48

49 Functions Applied to Fuzzy Sets y y = f(x) B B f ( A) B (y) A (x) A x x Computational Intelliegence 49

50 The Extension Principle Assume a fuzzy set A and a function f. How does the fuzzy set f(a) look like? B y y = f(x) ( y) ( y) B f ( A) max ( x) x f ( y) A B (y) A (x) A x sup ( x) x f ( y) A x Computational Intelliegence 50

51 The Extension Principle A A n fuzzy sets defined on X X n f X X V : n The extension of f operating on A,, A n gives a fuzzy set F with membership function ( v ) max min ( x ),, ( x ) F A A n n x,, x f ( v) n x,, x f ( v) sup min ( x ),, ( x ) n A A n n Computational Intelliegence 5

52 Introduction to Fuzzy Set Theory Fuzzy Relations

53 Binary Relation (R) a b A a 2 b 2 b 3 B a 3 b 4 a 4 b 5 R AB Computational Intelliegence 53

54 Binary Relation (R) R AB a b A a 2 b 2 b 3 B a 3 b 4 a 4 b 5 M R R arb a Rb a 3 2Rb5 ( a, b ),( a, b3 ),( a2, b5 ) ( a3, b ),( a3, b4 ),( a4, b2 ) a3rb a 3Rb a 4 4Rb2 Computational Intelliegence 54

55 The Real-Life Relation x is close to y x and y are numbers x depends on y x and y are events x and y look alike x and y are persons or objects If x is large, then y is small x is an observed reading and y is a corresponding action Computational Intelliegence 55

56 Fuzzy Relations A fuzzy relation R is a 2D MF: R ( x, y ), (, (, ) R xy) x y X Y Computational Intelliegence 56

57 Example (Approximate Equal) R ( x, y ), ( x, y ) X Y (, xy) R X Y U {, 2,3, 4,5} ( uv, ) R u v uv 0.3 uv 2 0 otherwise M R Computational Intelliegence 57

58 Max-Min Composition X Y Z R: fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. R S: the composition of R and S. A fuzzy relation defined on X an Z. ( x, z) max min ( x, y), ( y, z) R S y R S ( x, y) ( y, z) y R S Computational Intelliegence 58

59 Example ( x, y) max min ( x, v), ( v, y) S R v R S R a b c d S a b min c d max R S Computational Intelliegence 59

60 Max-Product Composition X Y Z R: fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. R S: the composition of R and S. A fuzzy relation defined on X an Z. ( x, z) max ( x, y) ( y, z) R S y R S Max-min composition is not mathematically tractable, therefore other compositions such as max-product composition have been suggested. Computational Intelliegence 60

61 Projection Dimension Reduction R RY R Y RX R X Computational Intelliegence 6

62 Projection Dimension Reduction R R Y R Y R X R X R RX R X Y R Y R Y Y max ( x, y) / x R y X max ( x, y) / x ( y) max ( x, y) R ( x) max R( x, y) x R X y y R Computational Intelliegence 62

63 Cylindrical Extension Dimension Expansion A : a fuzzy set in X. C(A) = [AXY] : cylindrical extension of A. C ( A ) ( x ) ( x, y ) (, ) ( ) C( A) x y A x XY A Computational Intelliegence 63

64 Introduction to Fuzzy Set Theory Linguistic Variables

65 Linguistic Variables Linguistic variable is a variable whose values are words or sentences in a natural or artificial language. Each linguistic variable may be assigned one or more linguistic values, which are in turn connected to a numeric value through the mechanism of membership functions. Computational Intelliegence 65

66 Motivation Conventional techniques for system analysis are intrinsically unsuited for dealing with systems based on human judgment, perception & emotion. Computational Intelliegence 66

67 Example if temperature is cold and oil is cheap then heating is high Computational Intelliegence 67

68 Example Linguistic Variable Linguistic Value Linguistic Variable Linguistic Value if temperature is cold and oil is cheap cold cheap high then heating is high Linguistic Variable Linguistic Value Computational Intelliegence 68

69 Definition [Zadeh 973] A linguistic variable is characterized by a quintuple x, T( x), U, G, M Name Term Set Universe Syntactic Rule Semantic Rule Computational Intelliegence 69

70 Example A linguistic variable is characterized by a quintuple x, T( x), U, G, M G(age) age old, very old, not so old, more or less young, quite young, very young old Example semantic rule: M (old) u, ( u) u [0,00] 0 u [0,50] [0, 00] old 2 ( u) u 50 u [50,00] 5 Computational Intelliegence 70

71 Example Linguistic Variable : temperature Linguistics Terms (Fuzzy Sets) : {cold, warm, hot} (x) cold warm hot x Computational Intelliegence 7

72 Introduction to Fuzzy Set Theory Fuzzy Rules

73 Classical Implication A B A B A B T T F F T F T F T F T T A B A B A B T T F F T F T F T F T T A B A B A B A B Computational Intelliegence 73

74 Classical Implication A B AB A( x) B( y) ( xy, ) B( y) otherwise A B ( x, y) max ( x), ( x) A B A B A B A B A B A B Computational Intelliegence 74

75 Modus Ponens ( 肯定前件推理 ) A B A B A B A B If A then B A A A is true B B B is true Computational Intelliegence 75

76 Fuzzy If-Than Rules A B If x is A then y is B. antecedent or premise consequence or conclusion Computational Intelliegence 76

77 Examples A B If x is A then y is B. If pressure is high, then volume is small. If the road is slippery, then driving is dangerous. If a tomato is red, then it is ripe. If the speed is high, then apply the brake a little. Computational Intelliegence 77

78 Fuzzy Rules as Relations A B R If x is A then y is B. A fuzzy rule can be defined as a binary relation with MF x, y x, y R A B Depends on how to interpret A B Computational Intelliegence 78

79 Interpretations of A B x y x y R A B,,? y A coupled with B y A entails B B B A x A x Computational Intelliegence 79

80 Interpretations of A B x y x y R A B,,? y A coupled with B y A entails B A coupled with B (A and B) R A B B B XY ( x)* ( y) /( x, y) A B A x t-norm A x Computational Intelliegence 80

81 Interpretations of A B x y x y R A B,,? y A coupled with B y R A B A entails B A coupled with B (A and B) B B XY ( x)* ( y) /( x, y) A B A x E.g., x R x, y min A( x), B( y) x Computational Intelliegence 8

82 Interpretations of A B B y A entails B (not A or B) A coupled with B Material implication R A B A B Propositional calculus R A B A ( A B) Extended propositional calculus R A B ( A B) B Generalization of modus ponens ( ) x A x B( y) R( xy, ) ( A B y) otherwise x x y x y R A B,,? B y A entails B A x Computational Intelliegence 82

83 Interpretations of A B x y x y R A B,,? A entails B (not A or B) Material implication R A B A B Propositional calculus R A B A ( A B) Extended propositional calculus R A B ( A B) B Generalization of modus ponens ( x, y) max ( x), ( x) R A B ( x, y) max ( x),min ( x), ( x) R A A B ( x, y) max max ( x), ( x), ( x) R A B B ( xy, ) R A( x) B( y) B( y) otherwise Computational Intelliegence 83

84 Introduction to Fuzzy Set Theory Fuzzy Reasoning

85 Generalized Modus Ponens Single rule with single antecedent Rule: Fact: Conclusion: if x is A then y is B x is A y is B Computational Intelliegence 85

86 Fuzzy Reasoning Single Rule with Single Antecedent Rule: if x is A then y is B Fact: x is A Conclusion: y is B ( x) A A ( y) B x y Computational Intelliegence 86

87 Fuzzy Reasoning Single Rule with Single Antecedent Rule: Fact: Conclusion: if x is A then y is B x is A y is B Firing Strength ( x, y) ( x) ( y) R A B Max-Min Composition ( y) max min ( x), ( x, y) B x A R x A( x) R( x, y) ( x) ( x) ( y) x A A B x A( x) A( x) B ( y) Firing Strength ( x) A A ( y) B x B y Computational Intelliegence 87

88 Fuzzy Reasoning Single Rule with Single Antecedent Rule: if x is A then y is B Fact: x is A Conclusion: y is B ( x, y) ( x) ( y) R A B Max-Min Composition ( y) max min ( x), ( x, y) B x A R x A( x) R( x, y) ( x) ( x) ( y) x A A B x A( x) A( x) B ( y) ( x) A A ( y) B A ( A B) B x B y Computational Intelliegence 88

89 Fuzzy Reasoning Single Rule with Multiple Antecedents Rule: Fact: Conclusion: if x is A and y is B then z is C x is A and y is B z is C Computational Intelliegence 89

90 Fuzzy Reasoning Single Rule with Multiple Antecedents Rule: Fact: Conclusion: if x is A and y is B then z is C x is A and y is B z is C ( x) ( y) ( z) A A B B C x y z Computational Intelliegence 90

91 R AB C Fuzzy Rule: Reasoning if x is A and y is B then z is C Single Fact: Rule x is A and with y is B Multiple RAntecedents x y z A B C Conclusion: z is C Max-Min Composition C x, y A, B R ( y) max min ( x, y), ( x, y, z),, ( x, y) ( x, y, z) x y A B R (,, ) ( x, y, z ) x, y A( x) B( y) A( x) B( y) C ( z) ( x) ( y) ( z) A B C x A( x) A( x) y B( y) B( y) C ( z) ( x) ( y) Firing Strength ( z) A A B B C x y C z Computational Intelliegence 9

92 Fuzzy Rule: Reasoning if x is A and y is B then z is C Single Fact: Rule x is A with and y is B Multiple Antecedents x y z Conclusion: z is C Max-Min Composition C x, y A, B R ( y) max min ( x, y), ( x, y, z),, ( x, y) ( x, y, z) x y A B R R AB C (,, ) (,, ) R x y z A B C ( x) ( y) ( z) A B C x, y A ( x) B ( y) A( x) B( y) C ( z) ( x) ( x) ( y) ( y) ( z) C A B A B C x A A y B B C ( x) ( y) Firing Strength ( z) A A B B C x y C Computational Intelliegence 92 z

93 Fuzzy Reasoning Multiple Rules with Multiple Antecedents Rule: if x is A and y is B then z is C Rule2: if x is A 2 and y is B 2 then z is C 2 Fact: x is A and y is B Conclusion: z is C Computational Intelliegence 93

94 Fuzzy Reasoning Multiple Rules with Multiple Antecedents Rule: Fact: Conclusion: if x is A and y is B then z is C Rule2: if x is A 2 and y is B 2 then z is C 2 x is A and y is B z is C ( x) A A ( y) B B ( z) C x y z ( x) A ( y) A 2 B ( z) 2 B C 2 x y z Computational Intelliegence 94

95 Fuzzy Reasoning Multiple Rules with Multiple Antecedents ( x) A A x ( y) B Rule: Fact: Conclusion: Max-Min Composition if x is A and y is B then z is C Rule2: if x is A 2 and y is B 2 then z is C 2 B y x is A and y is B z is C ( z) C C z ( x) A ( y) A 2 x 2 C A B R R B ( z) 2 B A B R A B R 2 CC 2 y ( z) C 2 Max C 2 C C C 2 Computational Intelliegence 95 z z