Dra. Ma. del Pilar Gómez Gil Primavera 2014

Transcription

1 C Tópicos Avanzados: Inteligencia Computacional I Introducción a la Lógica Difusa Dra. Ma. del Pilar Gómez Gil Primavera 2014 pgomez@inaoep.mx Ver: 08-Mar

2 Este material ha sido tomado de varias fuentes, y forma parte del Tutorial en Lógica Difusa impartido por el Dr. Juan Manuel Ramírez C. Coordinación de Electrónica,

3 FUZZY LOGIC 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. 3

4 Introduction Many decision-making and problemsolving 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. 4

5 Example: driving decisions 5

6 Introduction (2) Since knowledge can be expressed in a more natural way by using fuzzy sets, many engineering and decision problems can be greatly simplified. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. Fuzzy theory may be seen as an extension to classical set theory, where a membership function is added to a set. This function is defined in [0,1] 6

7 Membership function Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership. Unlike two-valued Boolean logic, fuzzy logic is multivalued. It deals with degrees of membership and degrees of truth. Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1(completely true). Instead of just black and white it accepts that things can be partly true and partly false at the same time (a) Boolean Logic. (b) Multi-valued Logic. 7

8 Membership function (2) A membership function A (t) represents the degree in that variable t is included in the concept represented by the tag A. 8

9 Example: fuzzy variable temperatura Linguistic value/variable: temperatura Fuzzy subsets: {frío, fresco, normal, tibio, caliente} Membership functions: { ( t), ( t), ( t), ( t), ( t)} frío fresco normal tibio caliente 9

10 Example: fuzzy variable temperatura (2) A (t) Each colour line is a membership function t The t-axis represents the universe of discourse 10

11 The membership function of a Crisp set 11

12 The membership function of a Fuzzy set 12

13 Crisp sets and characteristic function 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 A x 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. 13

14 Fuzzy sets and membership functions 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. 14

15 Fuzzy Set Representation First, we determine the membership functions. In our height example, we can obtain fuzzy sets of tall, short and average men. The universe of discourse the person s heights is represented by three sets: short, average and tall men. As you will see, a man who is 184 cm tall is a member of the average men set with a degree of membership of 0.1, and at the same time, he is also a member of the tall men set with a degree of 0.4. (see graph on next page) 15

16 Fuzzy Set Representation Degree of Membership Degree of Membership 1.0 Crisp Sets Short Average Short Tall Tall Men Fuzzy Sets Height, cm Short Average Tall Tall

17 Fuzzy Set Representation Typical functions that can be used to represent a fuzzy set are sigmoid, gaussian and pi. However, these functions increase the time of computation. Therefore, in practice, most applications use linear fit functions. 17

18 Funciones de membresía (1) Función de membresía triangular 18

19 Funciones de membresía (2) Función de membresía trapezoidal 19

20 Funciones de membresía (3) Función de membresía gaussiana 20

21 Funciones de membresía (4) Funciones de membresía sigmoide 21

22 Linguistic Variables and Hedges At the root of fuzzy set theory lies the idea of linguistic variables. A linguistic variable is a fuzzy variable. For example, the statement John is tall implies that the linguistic variable John takes the linguistic value tall. In fuzzy expert systems, linguistic variables are used in fuzzy rules. For example: IF wind is strong THEN sailing is good IF project_duration is long THEN completion_risk is high IF speed is slow THEN stopping_distance is short 22

23 Linguistic Variables and Hedges The range of possible values of a linguistic variable represents the universe of discourse of that variable. For example, the universe of discourse of the linguistic variable speed might have the range between 0 and 220 km/h and may include such fuzzy subsets as very slow, slow, medium, fast, and very fast. A linguistic variable 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. 23

24 Linguistic Variables and Hedges Degree of Membership Short Average Short Tall Very Short Very Tall Tall Height, cm 24

25 Linguistic Variables and Hedges 25

26 Linguistic Variables and Hedges 26

27 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 } 27

28 Logic operators (1/2) (Matlab, 2015) 28

29 Operations of Crisp Sets Not A B A A Complement Containment A B A B Intersection Union 29

30 Operations of Fuzzy Sets ( x ) A Not A x ( x ) B A B A x Blue line is the resulting membership function 0 Complement x 0 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 30

31 Logic operators (2/2) (Matlab, 2015) 31

32 Complement Crisp Sets: Who does not belong to the set? Fuzzy Sets: How much do elements not belong to the set? If A is the fuzzy set, its complement A can be found as follows: A (x) = 1 A (x) 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. 32

33 Containment Crisp Sets: Which sets belong to which other sets? Fuzzy Sets: How much sets belong to other sets? Similar to a Chinese box, 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. 33

34 Intersection Crisp Sets: Which element belongs to both sets? Fuzzy Sets: How much of the 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 34

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 35

36 Crisp and Fuzzy union and intersection A B A B A B A A A = U A A = max (μ A, μ B ) min (μ A, μ B ) max (μ A, μ A ) min (μ A, μ A ) = max (0,0) 0 = min (0,0) 1 1 = max (0,1) 0 = min (0,1) 0 1 1= max (0,1) 0 = min (0,1) 1 0 1= max (1,0) 0 = min (1,0) 0 1 = max (1,0) 0 = min (1,0) 1 1 1= max (1,1) 1 = min (1,1) 36

37 Crisp and Fuzzy containment (subset) If A B then A B = B A B = A, A B A B A B max (μ A, μ B ) min (μ A, μ B ) = max (0,0) 0 = min (0,0) 0 1 1= max (0,1) 0 = min (0,1) 1 0 1= max (1,0) 0 = min (1,0) 1 1 1= max (1,1) 1 = min (1,1) Example: A = very tall B = Tall You can t be very tall and not tall at the same time! 37

38 Properties of Fuzzy Sets Equality of two fuzzy sets Inclusion of one set into another fuzzy set Cardinality of a fuzzy set An empty fuzzy set -cuts (alpha-cuts) 38

39 Equality Fuzzy set A is considered equal to a fuzzy set B, IF AND ONLY IF (iff): A (x) = B (x), x X 39

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 40

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 ), for i=1..n 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 =

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 42

43 Alpha-cut An -cut or -level set of a fuzzy set A X is an ORDINARY 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} 43

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)) 44

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 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: core(a) = {x A (x) = 1 and x X} 45

46 Fuzzy Set Math Operations aa = {a A (x), x X} Let a =0.5, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then A a = {0.25/a, 0.15/b, 0.1/c, 0.5/d} A a = { A (x) a, x X} Let a =2, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then A a = {0.25/a, 0.09/b, 0.04/c, 1/d} 46

47 Fuzzy Sets Examples 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} 47

48 Fuzzy Sets Examples Support: supp(a) = {a, b, c, d } supp(b) = {a, b, c, d, e } Core: core(a) = {a} core(b) = {o} Cardinality: card(a) = = 2.3 card(b) = =

49 Fuzzy Sets Examples Complement: A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e} A = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e} Union: A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e} Intersection: A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e} 49

50 Fuzzy Sets Examples aa: for a=0.5 aa = {0.5/a, 0.15/b, 0.1/c, 0.4/d, 0/e} A a : for a=2 A a = {1/a, 0.09/b, 0.04/c, 0.64/d, 0/e} a-cut: A 0.2 = {a, b, c, d} A 0.3 = {a, b, d} A 0.8 = {a, d} A 1 = {a} 50

51 Exercises 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

52 Solutions 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} Support Supp(A) = {a, b, c, d} Supp(B) = {b, c, d, e} Core Core(A) = {c} Core(B) = {} Cardinality Card(A) = = 2.4 Card(B) = = 1.5 Complement Comp(A) = {0.8/a, 0.6/b, 0/c, 0.2/d, 1/e} Comp(B) = {1/a, 0.1/b, 0.7/c, 0.8/d, 0.9/e} 52

53 Solutions 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} Union A B = {0.2/a, 0.9/b, 1/c, 0.8/d, 0.1/e} Intersection A B = {0/a, 0.4/b, 0.3/c, 0.2/d, 0/e} C=A 2 C = {0.04/a, 0.16/b, 1/c, 0.64/d, 0/e} D = 0.5 B D = {0/a, 0.45/b, 0.15/c, 0.1/d, 0.05/e} E = A 0.5 E = {c, d} 53

54 Inference Fuzzy Rules In 1973, Dr. Lotfi Zadeh published his second most influential paper. This paper outlined a new approach to analysis of complex systems, in which Zadeh suggested capturing human knowledge in fuzzy rules. A fuzzy rule can be defined as a conditional statement in the form: IF x is A THEN y is B where x and y are linguistic variables; and A and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively. 54

55 Classical Vs Fuzzy Rules A classical IF-THEN rule uses binary logic, for example, Rule: 1 Rule: 2 IF speed is > 90 km/h IF speed is < 40 km/h THEN THEN stopping_distance> 40 m stopping_distance < 15 m The variable speed can have any numerical value between 0 and 150 km/h, and the variable stopping_distance can take any numerical value between 0 and 60 m. In other words, classical rules are expressed in the black-and-white language of Boolean logic. 55

56 Classical Vs Fuzzy Rules A fuzzy IF-THEN rule uses fuzzy logic : Rule: 1 Rule: 2 IF speed is fast IF speed is slow THEN stopping_distance is long THEN stopping_distance is short In fuzzy rules, the linguistic variable speed also has the range (the universe of discourse) between 0 and 150 km/h, but this range includes fuzzy sets, such as slow, medium and fast. The universe of discourse of the linguistic variable stopping_distance can be between 0 and 60 m and may include such fuzzy sets as short, medium and long. 56

57 Classical Vs Fuzzy Rules Fuzzy rules relate fuzzy sets. In a fuzzy 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 the corresponding degree. 57

58 Firing Fuzzy Rules 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 120 Weight, kg 58

59 Firing Fuzzy Rules 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 Height, cm Degree of Membership Heavy men Weight, kg

60 Firing Fuzzy Rules A fuzzy rule can have multiple antecedents, for example: IF service is excellent OR food is delicious THEN tip is generous The consequent of a fuzzy rule can also include multiple parts, for instance: IF temperature is hot THEN hot_water is reduced; cold_water is increased 60

61 Example: Air Conditioner 1a. Specify the problem Air-conditioning involves the delivery of air, which can be warmed or cooled and have its humidity raised or lowered. An air-conditioner is an apparatus for controlling, especially lowering, the temperature and humidity of an enclosed space. An air-conditioner typically has a fan which blows/cools/circulates fresh air and has a cooler. The cooler is controlled by a thermostat. Generally, the amount of air being compressed is proportional to the ambient temperature. 1b. Define linguistic variables Ambient Temperature Air-conditioner Fan Speed 61

62 Example: Air Conditioner 2. Determine Fuzzy Sets Fuzzy sets can have a variety of shapes. However, a triangle or a trapezoid can often provide an adequate representation of the expert knowledge, and at the same time, significantly simplifies the process of computation. 62

63 0< (T)<1 (T)=0 Example: Air Conditioner 2. Determine Fuzzy Sets: Temperature Temp COLD COOL PLEASANT WARM HOT ( 0 C). D T T 0 Y* N N N N 5 Y Y N N N 10 N Y N N N 12.5 N Y* N N N 15 N Y N N N 17.5 N N Y* N N 20 N N N Y N 22.5 N N N Y* N 25 N N N Y N 27.5 N N N N Y 30 N N N N Y* (T)=1 63

64 Truth Value Truth Value Example: Air Conditioner 2. Determine Fuzzy Sets: Temperature (cont.) Temperature Fuzzy Sets Cold Cold Cool Cool Pleasent Warm Hot Hot Temperature Degrees C 64

65 2. Determine Fuzzy Sets: Fan Speed Rev/sec (RPM) Example: Air Conditioner MINIMAL SLOW MEDIUM FAST BLAST 0 Y* N N N N 10 Y N N N N 20 Y Y N N N 30 N Y* N N N 40 N Y N N N 50 N N Y* N N 60 N N N Y N 70 N N N Y* N 80 N N N Y Y 90 N N N N Y 100 N N N N Y* 65

66 Truth Value Example: Air Conditioner 2. Determine Fuzzy Sets: Fan Speed (cont.) Speed Fuzzy Sets Speed MINIMAL SLOW MEDIUM FAST BLAST 66

67 Example: Air Conditioner 3. Bring out and construct fuzzy rules To accomplish this task, we might ask the expert to describe how the problem can be solved using the fuzzy linguistic variables defined previously. Required knowledge also can be collected from other sources such as books, computer databases, flow diagrams and observed human behaviour. RULE 1: IF temp is cold THEN speed is minimal RULE 2: IF temp is cool THEN speed is slow RULE 3: IF temp is pleasant THEN speed is medium RULE 4: IF temp is warm THEN speed is fast RULE 5: IF temp is hot THEN speed is blast 67

68 Example: Air Conditioner 4. Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system To accomplish this task, we may choose an option: to build our system using a programming language such as C/C++ or Pascal, or to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox or Fuzzy Knowledge Builder. etc 68

69 Example: Air Conditioner 5. Evaluate and tune the system The last, and the most laborious, task is to evaluate and tune the system. We want to see whether our fuzzy system meets the requirements specified at the beginning. Evaluation of the system output is performed for test situations on the several representative values of input variables. Fuzzy Logic development tools often can generate surface to help us evaluate and analyze the system s performance. Tuning of the system consists of reviewing, adding and/or changing the membership functions and rules in order to increase the performance of the system. 69

70 Example: Air Conditioner 5a. Evaluate the system Consider a temperature of 16 o C, use the system to compute the optimal fan speed. RECALL: Operation of a fuzzy expert system: Fuzzification: determination of the degree of membership of crisp inputs in appropriate fuzzy sets. Inference: evaluation of fuzzy rules to produce an output for each rule. Aggregation: combination of the outputs of all rules. Defuzzification: computation of crisp output 70

71 Example: Air Conditioner Fuzzification Affected fuzzy sets: COOL and PLEASANT COOL (T) = T / = 16 / = 0.3 PLSNT (T) = T /2.5-6 = 16 /2.5-6 = 0.4 Temp=16 COLD COOL PLEASANT WARM HOT

72 Example: Air Conditioner Inference RULE 1: IF temp is cold THEN speed is minimal RULE 2: IF temp is cool THEN speed is slow RULE 3: IF temp is pleasant THEN speed is medium RULE 4: IF temp is warm THEN speed is fast RULE 5: IF temp is hot THEN speed is blast RULE 2: IF temp is cool (0.3) THEN speed is slow (0.3) RULE 3: IF temp is pleasant (0.4) THEN speed is medium (0.4) 72

73 Example: Air Conditioner Aggregation speed is slow (0.3) speed is medium (0.4) + 73

74 Example: Air Conditioner Defuzzification COG = 0.125(12.5) (15) + 0.3( ) + 0.4( ) (57.5) (11) + 0.4(5) = 45.54rpm 74

75 Centroid of area z COA z COA (z)zdz, (z)dz where A (z) is the aggregated output MF. Z Z A A

76 number_of_spares fan-speed Input Output Plot temp Example: Air Conditioner one input one output gives nonlinear transfer characteristic mean_delay number_of_servers More general example: two inputs one output gives 3D transfer surface 1 76