Fuzzy Theory Timothy J Ross 2004 Fuzzy Logic

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模糊理論 Fuzzy Theory 淡江大學 資訊管理系所 侯永昌 教科書:Timothy J. Ross (2004), Fuzzy Logic with Engineering

模糊理論 Fuzzy Theory 淡江大學 資訊管理系所 侯永昌 教科書:Timothy J. Ross (2004), Fuzzy Logic with Engineering Applications, 2 nd Edition, John Wiley & Sons, Ltd. 新月圖書代理 淡江大學 資訊管理系所 侯永昌

Preface • Fuzzy logic has come a long way since it was first subjected

Preface • Fuzzy logic has come a long way since it was first subjected to technical scrutiny in 1965, when Dr. Lotfi Zadeh published his seminal work "Fuzzy sets" in the journal Information and Control. • Unfortunately, fuzzy logic did not receive serious notice in this world until the last decade. • The attention currently being paid to fuzzy logic is most likely the result of present popular consumer products employing fuzzy logic. 淡江大學 資訊管理系所 侯永昌 1

Preface • Over the last several years, the Japanese alone have filed for well

Preface • Over the last several years, the Japanese alone have filed for well over 1000 patents in fuzzy logic technology, and they have already grossed billions of U. S. dollars in the sales of fuzzy logic-based products to consumers the world over. • The integration of fuzzy logic with neural networks and genetic algorithms is now making automated cognitive systems a reality in many disciplines. 淡江大學 資訊管理系所 侯永昌 2

Preface • the reasoning power of fuzzy system, when integrated with the learning capabilities

Preface • the reasoning power of fuzzy system, when integrated with the learning capabilities of artificial neural networks and genetic algorithms, is responsible for new commercial products and processes. • The marketing research firm of Frost & Sullivan projected that fuzzy logic, with an annual growth rate of 20 percent, would be one of the world's 10 hottest technologies going into the twenty-first century. 淡江大學 資訊管理系所 侯永昌 3

Chapter 1 Introduction 淡江大學 資訊管理系所 侯永昌 4

Chapter 1 Introduction 淡江大學 資訊管理系所 侯永昌 4

Uncertainty and Impression • Uncertainty in Information: The source of impression is the absence

Uncertainty and Impression • Uncertainty in Information: The source of impression is the absence of sharply defined criteria of class membership rather than the presence of random variable. 淡江大學 資訊管理系所 侯永昌 7

Uncertainty and Impression By chance 在學校中學的多為理想狀 況;而在真實世界中,確 定的資料是不多的。 �Inability to perform adequate measurement �Lack

Uncertainty and Impression By chance 在學校中學的多為理想狀 況;而在真實世界中,確 定的資料是不多的。 �Inability to perform adequate measurement �Lack of knowledge �Fuzziness inherent in our natural language 淡江大學 資訊管理系所 侯永昌 8

Fuzzy logic is most successful in • very complex model where understanding is strictly

Fuzzy logic is most successful in • very complex model where understanding is strictly limited or quite judgmental. • processes where human reasoning, human perception, or human decision making are inextricably involved. • Because: – Human use linguistic variables, rather than quantitative variables to represent imprecise concepts. – Human reasoning is based largely on imprecise intuition or judgment, more precision entails higher cost. 淡江大學 資訊管理系所 侯永昌 9

Uncertainty and Impression 淡江大學 資訊管理系所 侯永昌 10

Uncertainty and Impression 淡江大學 資訊管理系所 侯永昌 10

Uncertainty and Impression 淡江大學 資訊管理系所 侯永昌 11

Uncertainty and Impression 淡江大學 資訊管理系所 侯永昌 11

Uncertainty and Impression 淡江大學 資訊管理系所 侯永昌 12

Uncertainty and Impression 淡江大學 資訊管理系所 侯永昌 12

Fuzzy logic is most successful in • • • focusing and image stabilization: Fisher,

Fuzzy logic is most successful in • • • focusing and image stabilization: Fisher, Sanyo air conditioner: Mitsubishi washing machine : Matsushita subway automatic system controller : Sendai, Hitachi automatic transmission/ anti-skid braking system: Nissan golf diagnostic system toaster rice cooker vacuum cleaner pattern recognition and classification: US DOD space docking control: NASA stock-trading portfolio: Japan 淡江大學 資訊管理系所 侯永昌 13

Fuzzy Set and Membership • 例如:height around 6 feet • 多少有點自由心証,但是至少他要滿足: normal、monotonicity、symmetry 淡江大學 資訊管理系所

Fuzzy Set and Membership • 例如:height around 6 feet • 多少有點自由心証,但是至少他要滿足: normal、monotonicity、symmetry 淡江大學 資訊管理系所 侯永昌 15

Fuzzy Set and Membership • probability provides knowledge about relative frequencies • fuzzy membership

Fuzzy Set and Membership • probability provides knowledge about relative frequencies • fuzzy membership function represents similarities of objects to ambiguous properties 淡江大學 資訊管理系所 侯永昌 16

Chance versus Ambiguity • 例:two persons: 1. 95% chance of being over 7 feet

Chance versus Ambiguity • 例:two persons: 1. 95% chance of being over 7 feet tall 2. high membership in the set of very tall people • 例:two glasses of water: 1. 95% chance of being healthful and good 2. 0. 95 membership of being healthful and good • Which one will you choose? 淡江大學 資訊管理系所 侯永昌 17

Chance versus Ambiguity • The prior probability of 0. 95 in each case becomes

Chance versus Ambiguity • The prior probability of 0. 95 in each case becomes a posterior probability of 1. 0 or 0. 0. However, the membership value of 0. 95 remains 0. 95 after measuring or testing! • Fuzziness describes the ambiguity of an event, whereas randomness describes the uncertainty in the occurrence of the event. 淡江大學 資訊管理系所 侯永昌 18

Chance versus Ambiguity • 例:What is the probability of randomly selecting a circle from

Chance versus Ambiguity • 例:What is the probability of randomly selecting a circle from a bag? • fuzziness: above which membership value, say 0. 85, we would be willing to call the shape a circle? • randomness: the proportion of the shapes in the bag that have the membership value above the value (0. 85) 淡江大學 資訊管理系所 侯永昌 21

Chapter 2 Classical Sets and Fuzzy Sets 淡江大學 資訊管理系所 侯永昌 22

Chapter 2 Classical Sets and Fuzzy Sets 淡江大學 資訊管理系所 侯永昌 22

Chance versus Ambiguity Fixed set, random point 淡江大學 資訊管理系所 侯永昌 Fixed point, random set

Chance versus Ambiguity Fixed set, random point 淡江大學 資訊管理系所 侯永昌 Fixed point, random set 23

Classical Sets • Set: a collection of objects all having the same characteristics –

Classical Sets • Set: a collection of objects all having the same characteristics – x A => x belongs to A – x A => x does not belong to A • Cardinality: The total number of elements in a universe X is called its cardinal number, denoted nx – finite – infinite 淡江大學 資訊管理系所 侯永昌 24

Classical Sets • whole set: Collection of all elements in the universe • empty

Classical Sets • whole set: Collection of all elements in the universe • empty set ( ): The set containing no element • power set (P(X)): All possible sets of a universe • Example 2. 1: X = {a, b, c} • P(X) = { , {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c} } • np(x) = 2 nx = 23 = 8 淡江大學 資訊管理系所 侯永昌 25

Operations on Classical Sets • A B => A is fully contained in B

Operations on Classical Sets • A B => A is fully contained in B (if x A, then x B) • A B => A is contained in or is equivalent to B • A = B => A B and B A • Union: A ∪ B = { x | x A or x B } • Intersection: A ∩ B = { x | x A and x B } • Complement: ¯(or ~) ~ A = { x | x A, x X } • Difference: | (or -) A | B = { x | x A and x B } 淡江大學 資訊管理系所 侯永昌 26

Properties of classical crisp sets • Identity 單位元素 – A∪ = A, A∩ =

Properties of classical crisp sets • Identity 單位元素 – A∪ = A, A∩ = , A∪X = X, A∩X = A • Transitivity 遞移律 – If A B and B C, then A C • Excluded middle laws 排中律 – Law of the excluded middle A ∪ ~A = X – Law of the contradiction A ∩ ~A = • De Morgan's laws – ~(A ∩ B) = ~A ∪ ~B – ~(A ∪ B) = ~A ∩ ~B 淡江大學 資訊管理系所 侯永昌 28

Mapping of Classical Sets to Functions • If x X corresponds y Y, it

Mapping of Classical Sets to Functions • If x X corresponds y Y, it is called a mapping from X to Y. or f: X -> Y • The characteristic function A is defined by A(x) = 1, x A = 0, x A • where A expresses "membership" in set A for the element x in the universe. This is a mapping : X -> Y , where y is 0 or 1 淡江大學 資訊管理系所 侯永昌 29

Mapping of Classical Sets to Functions • Example 2 -4: A universe with three

Mapping of Classical Sets to Functions • Example 2 -4: A universe with three elements, X={a, b, c} • A mapping from the elements of the power set of X, i. e. , P(X) = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }, to a universe, Y = {0, 1} • The value set determined from the mapping are V{ P(X) } = { {0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 0}, {1, 0, 1}, {0, 1, 1}, {1, 1, 1} } 淡江大學 資訊管理系所 侯永昌 30

Mapping of Classical Sets to Functions • 因此我們可以將Set 的運算重新用Function 的形式 來表示: • Union A∪B

Mapping of Classical Sets to Functions • 因此我們可以將Set 的運算重新用Function 的形式 來表示: • Union A∪B – A∪B(x) = A(x) B(x) = max( A(x), B(x)) • Intersection A∩B – A∩B(x) = A(x) B(x) = min( A(x), B(x)) • Complement ~A • Containment A B ~A(x) = 1 - A(x) B(x) • where is the max operator and is the min operator 淡江大學 資訊管理系所 侯永昌 31

Fuzzy Sets • A fuzzy set is a set containing elements that have varying

Fuzzy Sets • A fuzzy set is a set containing elements that have varying degrees of membership in the set -> [ 0, 1 ] • Fuzzy set can be expressed by the set of pairs A = { (x, ) | x X } or, • A = where X is discrete and finite, or 淡江大學 資訊管理系所 侯永昌 32

Fuzzy Sets • Example: A = integer close to 10 = • Example: real

Fuzzy Sets • Example: A = integer close to 10 = • Example: real number close to 10 • A = { (x, ) | = [1 + ( x - 10 ) 2] 1} = 淡江大學 資訊管理系所 侯永昌 35

Fuzzy Sets • Example: Vertical line • = = 0 if | tan |

Fuzzy Sets • Example: Vertical line • = = 0 if | tan | > 1 otherwise • Example: Straight line • = if r a = 0 otherwise 淡江大學 資訊管理系所 侯永昌 36

Fuzzy Sets • Example: Circles • = r / R = 4 A /

Fuzzy Sets • Example: Circles • = r / R = 4 A / L 2 , or = where 淡江大學 資訊管理系所 侯永昌 37

Fuzzy Set Operations • The fuzzy set operations for A, B, and C on

Fuzzy Set Operations • The fuzzy set operations for A, B, and C on the universe X. • Union μA∪B(x) = μA(x) μB(x) • Intersection μA∩B(x) = μA(x) μB(x) • Complement μ~A(x) = 1 - μA(x) 淡江大學 資訊管理系所 侯永昌 38

Fuzzy Set Operations • Any fuzzy set A defined on a universe X is

Fuzzy Set Operations • Any fuzzy set A defined on a universe X is a subset of X ,因此 • A X => μA(x) μX(x) 1 • For all x X, empty set, μ (x) = 0, whole set, μX(x) = 1 • Based on the fact that all fuzzy sets can be overlap, the cardinality of the fuzzy power set is always infinite; that is, n. P(X) = ∞, no matter the cardinality of X is finite or not 淡江大學 資訊管理系所 侯永昌 40

Properties of Fuzzy Sets • Fuzzy sets follow the same properties as crisp sets,

Properties of Fuzzy Sets • Fuzzy sets follow the same properties as crisp sets, such as: • Commutativity 交換律 – A ∪ B = B ∪ A A ∩ B = B ∩ A • Associativity 結合律 – A ∪ (B ∪ C) = (A ∪ B) ∪ C – A ∩ (B ∩ C) = (A ∩ B) ∩ C • Distributivity 分配律 – A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) – A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) 淡江大學 資訊管理系所 侯永昌 41

Properties of Fuzzy Sets • Idempotency 自身律 – A ∪ A = A A

Properties of Fuzzy Sets • Idempotency 自身律 – A ∪ A = A A ∩ A = A • Identity 單位元素 – A ∪ = A A ∩ = – A ∪ X = X A ∩ X = A • Transitivity 遞移律 – If A B and B C, then A C • Involution 反身律 – ~ ( A ) ) = A 淡江大學 資訊管理系所 侯永昌 42

Properties of Fuzzy Sets • De. Morgan's laws for classical sets also hold for

Properties of Fuzzy Sets • De. Morgan's laws for classical sets also hold for fuzzy sets, – ~(A ∩ B) = ~A ∪ ~B – ~(A ∪ B) = ~A ∩ ~B • But the excluded middle laws do not hold for fuzzy sets; that is, – A ∪ ~A ≠ X – A ∩ ~A ≠ 淡江大學 資訊管理系所 侯永昌 43

Properties of Fuzzy Sets 淡江大學 資訊管理系所 侯永昌 44

Properties of Fuzzy Sets 淡江大學 資訊管理系所 侯永昌 44

Sets as Points in Hypercubes 淡江大學 資訊管理系所 侯永昌 46

Sets as Points in Hypercubes 淡江大學 資訊管理系所 侯永昌 46

Sets as Points in Hypercubes • The centroids of each of the diagrams in

Sets as Points in Hypercubes • The centroids of each of the diagrams in Fig. 2. 25 represent single points where the membership value for each element in the universe equals 1/2. This midpoint in each of the three figures is a special point - it is the set of maximum "fuzziness“ • A membership value of 1/2 indicates that the element belongs to the fuzzy sets as much as it does not. • In a geometric sense, this point is the location in the space that is farthest from any of the vertices and yet equidistant from all of them. 淡江大學 資訊管理系所 侯永昌 48

Summary In this chapter, we introduce: • basic definitions, properties, and operations on crisp

Summary In this chapter, we introduce: • basic definitions, properties, and operations on crisp sets and fuzzy sets • the difference of set membership between crisp and fuzzy sets is an infinite-valued as opposed to a binary-valued quantity • the basic axioms not common to both crisp and fuzzy sets are the two excluded middle laws • the idea that crisp sets are special forms of fuzzy sets was illustrated graphically as geometric points 淡江大學 資訊管理系所 侯永昌 49

Chapter 3 Classical Relations and Fuzzy Relations 淡江大學 資訊管理系所 侯永昌 50

Chapter 3 Classical Relations and Fuzzy Relations 淡江大學 資訊管理系所 侯永昌 50

Relations • Relations are intimately involved in logic, approximate reasoning, rule-based systems, nonlinear simulation,

Relations • Relations are intimately involved in logic, approximate reasoning, rule-based systems, nonlinear simulation, classification, pattern recognition, and control. • There are only two degrees of relationship between elements of the sets in a crisp relation: "completely related" and "not related, " in a binary sense. • However, fuzzy relations describe the relationship between elements of two or more sets to take on an infinite number of degrees of relationship between the extremes of "completely related" and "not related. " 淡江大學 資訊管理系所 侯永昌 51

Cartesian Product • For crisp sets A 1, A 2, . . . Ar,

Cartesian Product • For crisp sets A 1, A 2, . . . Ar, the set of all r-tuples (a 1, a 2, a 3, . . . ar), where a 1 A 1, a 2 A 2, . . . , and ar Ar, is called the Cartesian product of A 1, A 2, . . . Ar, and is denoted by A 1 A 2 . . . A r. • When all the Ar are identical and equal to A, the Cartesian product A 1 A 2 . . . Ar can be denoted as Ar. • Cartesian product is a means of producing ordered relationships among sets 淡江大學 資訊管理系所 侯永昌 52

Crisp Relations • The Cartesian product of two universes X and Y is determined

Crisp Relations • The Cartesian product of two universes X and Y is determined as X Y = { (x, y) | x X, y Y } which forms an ordered pair of every x X with every y Y, forming unconstrained matches between X and Y. • That is, every element in universe X is related completely to every element in universe Y. 淡江大學 資訊管理系所 侯永昌 54

Crisp Relations • When the matches between elements of two universes are constrained on

Crisp Relations • When the matches between elements of two universes are constrained on only part of elements of two universes are related, we call a binary relation R from X to Y. • R(x, y) = 1, (x, y) R complete relationship = 0, (x, y) R no relationship • r-nary relation can be represented by an rdimensional relation matrix. 淡江大學 資訊管理系所 侯永昌 55

Crisp Relations • Example : A = { 0, 1, 2 } • identity

Crisp Relations • Example : A = { 0, 1, 2 } • identity relation IA = • universal (complete) relation EA = • null relation OA = 淡江大學 資訊管理系所 侯永昌 57

Crisp Relations • Example 3. 3 R = { (x, y) | y 2

Crisp Relations • Example 3. 3 R = { (x, y) | y 2 x, x X, y Y } R(x, y) = 1, if y 2 x = 0, if y < 2 x y = 2 x 淡江大學 資訊管理系所 侯永昌 58

Cardinality of Crisp Relations • If the cardinality of X is n. X and

Cardinality of Crisp Relations • If the cardinality of X is n. X and the cardinality of Y is n. Y, then the cardinality of the relation, R, between these two universes is n. X Y = n. X*n. Y. • The cardinality of the power set, P(X Y) is n n n. P(X Y) = 2 X * Y. 淡江大學 資訊管理系所 侯永昌 59

Operations on Crisp Relations • Union R∪S -> R∪S(x, y) = max( R(x, y),

Operations on Crisp Relations • Union R∪S -> R∪S(x, y) = max( R(x, y), S(x, y)) • Intersection R∩S -> R∩S(x, y) = min( R(x, y), S(x, y)) • Complement ~R -> ~R(x, y) = 1 - R(x, y) • Containment R S -> R(x, y) S(x, y) • Identity ( O and X E) 淡江大學 資訊管理系所 侯永昌 60

Properties of Crisp Relations • The properties of commutativity, associativity, distributivity, involution, idempotency, De

Properties of Crisp Relations • The properties of commutativity, associativity, distributivity, involution, idempotency, De Morgan's laws, and the excluded middle laws all hold for crisp relations just as they do for classical set operations. 淡江大學 資訊管理系所 侯永昌 61

Composition • If R:X -> Y,and S:Y -> Z Whether we can find a

Composition • If R:X -> Y,and S:Y -> Z Whether we can find a relation T:X -> Z ? YES ! • The max-min composition is defined by T = R S T(x, z) = y Y ( R(x, y) S(y, z)) • The max-product composition is defined by T = R S T(x, z) = y Y ( R(x, y) S(y, z)) 淡江大學 資訊管理系所 侯永昌 62

Composition • Example 3. 4: max-min • T(x 1, z 1) = max[min(1, 0),

Composition • Example 3. 4: max-min • T(x 1, z 1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 T(x 1, z 2) = max[min(1, 1), min(0, 0), min(1, 1), min(0, 0)] = 1 淡江大學 資訊管理系所 侯永昌 63

Composition • Example 3. 4: max-min X 1 X 2 X 3 Y 1

Composition • Example 3. 4: max-min X 1 X 2 X 3 Y 1 Y 2 Y 3 Z 1 Z 2 Y 4 淡江大學 資訊管理系所 侯永昌 64

Fuzzy Relations • A fuzzy relation R is a mapping from the Cartesian space

Fuzzy Relations • A fuzzy relation R is a mapping from the Cartesian space X Y to the interval [0, 1], where the "strength" of the mapping is expressed by the membership function of the relation for ordered pairs from the two universes. • Since the cardinality of fuzzy sets on any universe is infinity, the cardinality of a fuzzy relation between two or more universes is also infinity. 淡江大學 資訊管理系所 侯永昌 65

Operations on Fuzzy Relations • Let R and S be fuzzy relations on the

Operations on Fuzzy Relations • Let R and S be fuzzy relations on the Cartesian space X Y • Union R∪S -> R∪S(x, y) = max( R(x, y), S(x, y)) • Intersection R∩S -> R∩S(x, y) = min( R(x, y), S(x, y)) • Complement ~R -> ~R(x, y) = 1 - R(x, y) • Containment R S -> R(x, y) S(x, y) 淡江大學 資訊管理系所 侯永昌 66

Properties of Fuzzy Relations • The properties of commutativity, associativity, distributivity, involution, idempotency, and

Properties of Fuzzy Relations • The properties of commutativity, associativity, distributivity, involution, idempotency, and De Morgan's laws hold for fuzzy relations just as they do for crisp relations. • However, the excluded middle laws does not hold. • R ∪ ~R E, where E: complete relation • R ∩ ~R O, O: null relation 淡江大學 資訊管理系所 侯永昌 67

Fuzzy Cartesian Product and Composition • Fuzzy Cartesian product and composition can be defined

Fuzzy Cartesian Product and Composition • Fuzzy Cartesian product and composition can be defined just as they are for crisp relations. • R = A B X Y μR(x, y) = μA B(x, y) = min(μA(x), μB(y)) • T = R S – The max-min composition is defined by μT(x, z) = y Y (μR(x, y) μS(y, z)) – The max-product composition is defined by μT(x, z) = y Y (μR(x, y) μS(y, z)) • It should be pointed out that neither crisp nor fuzzy compositions have converses in general; that is, R S S R 淡江大學 資訊管理系所 侯永昌 68

Fuzzy Composition • Example 3. 6 • μT(x 1, z 1) = max[min(0. 7,

Fuzzy Composition • Example 3. 6 • μT(x 1, z 1) = max[min(0. 7, 0. 9), min(0. 5, 0. 1)] = 0. 7 μT(x 2, z 2) = max[(0. 8*0. 6), (0. 4*0. 7)] = 0. 48 淡江大學 資訊管理系所 侯永昌 70

Fuzzy Composition • Example 3. 7 • Example 3. 8 • Example 3. 9

Fuzzy Composition • Example 3. 7 • Example 3. 8 • Example 3. 9 淡江大學 資訊管理系所 侯永昌 71

Noninteractive Fuzzy Sets • The fuzzy set A defined on the Cartesian space X

Noninteractive Fuzzy Sets • The fuzzy set A defined on the Cartesian space X = X 1 X 2, is separable into two noninteractive fuzzy sets, called its orthogonal projections, iff A = Pr. X 1(A) Pr. X 2(A), where • μPr. X (A)(x 1)= maxx 2 X 2 μA(x 1, x 2), x 1 X 1 1 μPr. X (A)(x 2)= maxx 1 X 1 μA(x 1, x 2), x 2 X 2 2 are the membership functions for the projections of A on universes X 1 and X 2 respectively. • Separability or noninteractivity of fuzzy set A describes a kind of independence of the components (x 1 and x 2). 淡江大學 資訊管理系所 侯永昌 72

Crisp Equivalence Relation • The relation R is an equivalence relation if it has

Crisp Equivalence Relation • The relation R is an equivalence relation if it has the following three properties: • Reflexivity: (xi, xi) R or R(xi, xi) = 1 • Symmetry: (xi , xj) R (xj , xi) R, or R(xi, xj) = R(xj, xi) • Transitivity: (xi, xj) R and (xj, xk) R (xi, xk) R or R(xi, xj) = 1 and R(xj, xk) = 1 R(xi, xk) = 1 • 例:相等,平行,相似三角形,親戚. . . 等關係都 是 equivalent relation 淡江大學 資訊管理系所 侯永昌 73

Crisp Tolerance Relation • A tolerance relation (also called proximity relation) R on a

Crisp Tolerance Relation • A tolerance relation (also called proximity relation) R on a universe X is a relation that exhibits only the properties of reflexivity and symmetry. • A tolerance relation R 1 can be reformed into an equivalence relation by at most (n-1) compositions with itself, where n is the cardinal number of the set defining R, • Equivalent relation is useful in classification. 淡江大學 資訊管理系所 侯永昌 74

Crisp Tolerance Relation • Example 3. 10 • X = {Omaha, Chicago, Rome, London,

Crisp Tolerance Relation • Example 3. 10 • X = {Omaha, Chicago, Rome, London, Detroit} • (x 1, x 2) R 1, (x 2, x 5) R 1 • (x 1, x 5) R 1, 淡江大學 資訊管理系所 侯永昌 75

Fuzzy Equivalence Relations • A fuzzy relation R is a fuzzy equivalence relation if

Fuzzy Equivalence Relations • A fuzzy relation R is a fuzzy equivalence relation if all three of the following properties for matrix relations define it; • Reflexivity μR(xi, xj) = 1 • Symmetry μR(xi, xj) = μR(xj, xi) • Transitivity μR(xi, xj) = λ 1 andμR(xj, xk) = λ 2 μR(xi, xk) = λwhereλ min[λ 1, λ 2] i. e. , the strength of the link between two elements must be at least equal to the strength of the weakest indirect chain involving other elements 淡江大學 資訊管理系所 侯永昌 76

Fuzzy Tolerance Relations • Any fuzzy tolerance relation R 1, that has properties of

Fuzzy Tolerance Relations • Any fuzzy tolerance relation R 1, that has properties of reflexivity and symmetry can be reformed into a fuzzy equivalence relation by at most (n-1) compositions. That is, 淡江大學 資訊管理系所 侯永昌 77

Fuzzy Tolerance Relation • Example 3. 11 • μR 1(x 1, x 2) =

Fuzzy Tolerance Relation • Example 3. 11 • μR 1(x 1, x 2) = 0. 8, μR 1(x 2, x 5) = 0. 9, μR 1(x 1, x 5) = 0. 2 min(0. 8, 0. 9), • μR 12(x 1, x 2) = 0. 8, μR 12 (x 2, x 4) = 0. 5, μR 12 (x 1, x 4) = 0. 2 min(0. 8, 0. 5) 淡江大學 資訊管理系所 侯永昌 78

Value Assignments • There at least six ways to develop the numerical values that

Value Assignments • There at least six ways to develop the numerical values that characterize a relation: 1. Cartesian product: form the Cartesian product of two or more fuzzy sets 2. Closed-form expression 3. Lookup table: 觀察 inputs 和outputs 之間的關 係,如果沒有固定的對應,則可找出其間之 fuzzy relation 淡江大學 資訊管理系所 侯永昌 79

Value Assignments 4. Linguistic rules of knowledge: in chapter 7 – 9 利用 if-then

Value Assignments 4. Linguistic rules of knowledge: in chapter 7 – 9 利用 if-then rules,經由專家,問卷調查, 投票法,找出知識間的關係。linguistic knowledge 多由 if-then rules 來表示 5. Classification: in chapter 11 6. Similarity methods in data manipulation: determine some sort of similar pattern or structure in data through various metric 淡江大學 資訊管理系所 侯永昌 80

Cosine Amplitude • X = {x 1, x 2, . . . , xn}

Cosine Amplitude • X = {x 1, x 2, . . . , xn} • xi = <xi 1, xi 2, . . . , xim> • n data samples m features each where i, j = 1, 2, . . . , n • This method is related to the dot product for the Cosine function. When two vectors are colinear (most similar), their dot product is unity; when the two vectors are at right angles to one another (most dissimilar), their dot product is zero. • The relation matrix will be of size n n and it will be reflexive and symmetric, hence a tolerance relation. 淡江大學 資訊管理系所 侯永昌 81

Example 3. 12: Cosine Amplitude 淡江大學 資訊管理系所 侯永昌 82

Example 3. 12: Cosine Amplitude 淡江大學 資訊管理系所 侯永昌 82

Max-min Method • X = {x 1, x 2, . . . , xn}

Max-min Method • X = {x 1, x 2, . . . , xn} n data samples • xi = <xi 1, xi 2, . . . , xim> m features each • A simpler similarity method but different from composition operation 淡江大學 資訊管理系所 侯永昌 where i, j = 1, 2, . . . , n 83

Max-min Method • Example 3. 13: same as example 3. 12 淡江大學 資訊管理系所 侯永昌

Max-min Method • Example 3. 13: same as example 3. 12 淡江大學 資訊管理系所 侯永昌 84

Other Similarity Methods • • • Absolute exponential (3. 26) Exponential similarity coefficient (3.

Other Similarity Methods • • • Absolute exponential (3. 26) Exponential similarity coefficient (3. 27) Geometric average minimum (3. 28) Scalar product (3. 29) Correlation coefficient (3. 30) Arithmetic average-minimum (3. 31) Absolute reciprocal (3. 32) Absolute subtrahend (3. 33) Nonparametric (3. 34) 淡江大學 資訊管理系所 侯永昌 85

Other Similarity Methods 淡江大學 資訊管理系所 侯永昌 86

Other Similarity Methods 淡江大學 資訊管理系所 侯永昌 86

Other Similarity Methods 淡江大學 資訊管理系所 侯永昌 87

Other Similarity Methods 淡江大學 資訊管理系所 侯永昌 87

Summary • The properties and operations of crisp and fuzzy relations are shown. •

Summary • The properties and operations of crisp and fuzzy relations are shown. • The idea of composition was introduced. Also, the principle of non-interactivity between sets was described as being analogous to the assumption of independence in probability modeling. • Tolerant and equivalent relations as well as several similarity metrics were shown to be useful in developing the relational strengths, or distances, within fuzzy relations from data sets. 淡江大學 資訊管理系所 侯永昌 88

Chapter 4 Membership Functions 淡江大學 資訊管理系所 侯永昌 89

Chapter 4 Membership Functions 淡江大學 資訊管理系所 侯永昌 89

Features of the Membership Function • Membership functions characterize the fuzziness (discrete or continuous)

Features of the Membership Function • Membership functions characterize the fuzziness (discrete or continuous) in a fuzzy set. • The core of a membership function for some fuzzy set A is defined as those elements x (or region) of the universe such that μA(x) = 1. 代表完全滿足這個模糊性質 • The boundaries of a membership function for some fuzzy set A is defined as those elements x (or region) of the universe such that 0 < μA(x) < 1. 代表有某種程度的滿足這個模糊性質 淡江大學 資訊管理系所 侯永昌 90

Features of the Membership Function • The support of a membership function for some

Features of the Membership Function • The support of a membership function for some fuzzy set A is defined as those elements x (or region) of the universe such that μA(x) > 0. 淡江大學 資訊管理系所 侯永昌 91

Features of the Membership Function • A normal fuzzy set is one whose membership

Features of the Membership Function • A normal fuzzy set is one whose membership function has at least one element x in the universe whose membership value is unity. • 一般均將非normal的fuzzy set放大/縮小至normal 淡江大學 資訊管理系所 侯永昌 92

Features of the Membership Function • A convex fuzzy set is described by a

Features of the Membership Function • A convex fuzzy set is described by a membership function whose values are strictly monotonically increasing, decreasing, or first increasing then decreasing. That is, for any elements x, y, and z in a fuzzy set A, μA(y) min [μA(x), μA(z)], for x < y < z • convex fuzzy set比較符合單一模糊性質的特性 淡江大學 資訊管理系所 侯永昌 93

Features of the Membership Function • The intersection of these two convex fuzzy sets

Features of the Membership Function • The intersection of these two convex fuzzy sets is also a convex fuzzy set. • The crossover points of a membership function are defined as the elements in the universe for which a particular fuzzy set A has value equal to 0. 5. 淡江大學 資訊管理系所 侯永昌 94

Fuzzification • the process of making a crisp quantity fuzzy • Many of the

Fuzzification • the process of making a crisp quantity fuzzy • Many of the quantities that we consider to be crisp and deterministic are actually not deterministic at all. They carry considerable uncertainty. If the form of uncertainty happens to arise because of imprecision, ambiguity, or vagueness, then the variable is probably fuzzy and can be represented by a membership function. 淡江大學 資訊管理系所 侯永昌 95

Membership Value Assignments • Intuition: derived from the contextual and semantic knowledge about an

Membership Value Assignments • Intuition: derived from the contextual and semantic knowledge about an issue and the analyst developing them. • 它的形狀並不太重要,比較重要的是放置的位置; 區分的段數;和重覆區域的大小 淡江大學 資訊管理系所 侯永昌 98

Membership Value Assignments • Inference: Given a body of facts and knowledge, we wish

Membership Value Assignments • Inference: Given a body of facts and knowledge, we wish to deduce or infer a conclusion. • Example: Identification of a triangle, let A, B, and C be the inner angles of a triangle, and let U be the universe of triangles; i. e. , U = {(A, B, C)| A B C 0; A + B + C = 180 } • We define five types of triangles: – I : Approximate isosceles triangle μI(A, B, C) = 1 - min(A-B, B-C)/60 – R: Approximate right triangle μR(A, B, C) = 1 - |A-90|/90 淡江大學 資訊管理系所 侯永昌 99

Membership Value Assignments – IR: Approximate isosceles and right triangle IR = I ∩

Membership Value Assignments – IR: Approximate isosceles and right triangle IR = I ∩ R, i. e. , μIR(A, B, C) = min[μI(A, B, C), μR(A, B, C)] = 1 - max[min(A-B, B-C)/60, |A-90|/90] – E: Approximate equilateral triangle μE(A, B, C) = 1 - |A-C|/180 – T: Other triangles = ~(I∪R∪E) = ~I ∩~R ∩~E μT(A, B, C) = min{1 -μI(A, B, C), 1 -μE(A, B, C), 1 -μR(A, B, C)} = min{3(A-B), 3(B-C), 2|A-90|, A-C}/180 淡江大學 資訊管理系所 侯永昌 100

Membership Value Assignments • if a = 85, b = 50, c = 45,

Membership Value Assignments • if a = 85, b = 50, c = 45, μR = 0. 94 μI = 0. 916 μIR = 0. 916 μE = 0. 7 μT = 0. 05 最像直角三角形 淡江大學 資訊管理系所 侯永昌 101

Membership Value Assignments • Rank Ordering: Preference, by a single individual, a committee, and

Membership Value Assignments • Rank Ordering: Preference, by a single individual, a committee, and other opinion methods, is determined by pairwise comparisons, and these determine the ordering of the membership. 淡江大學 資訊管理系所 侯永昌 102

Membership Value Assignments • Angular Fuzzy Sets: Angular fuzzy sets differ from standard fuzzy

Membership Value Assignments • Angular Fuzzy Sets: Angular fuzzy sets differ from standard fuzzy sets only in their coordinate description. Angular fuzzy sets are defined on a universe of angles, hence is useful in situations that have a natural basis in polar coordinates, or in situations where the value of a variable is cyclic. • The linguistic values vary with , and their membership values are on theμ( ) axis. The membership value of the linguistic term can be obtained from the equation μt( ) = t tan( ), where t is the horizontal projection of a radial vector. 淡江大學 資訊管理系所 侯永昌 103

Membership Value Assignments • Neural Networks: A neural network is a technique that seeks

Membership Value Assignments • Neural Networks: A neural network is a technique that seeks to build an intelligent program using models that simulate the working network of the neurons in the human brain. 淡江大學 資訊管理系所 侯永昌 104

Neural Networks • A neuron receives a set of input pulses and sends out

Neural Networks • A neuron receives a set of input pulses and sends out another pulse that is a function of the input pulses. 淡江大學 資訊管理系所 侯永昌 105

Neural Networks • xi = signal input ( i = 1, 2, . .

Neural Networks • xi = signal input ( i = 1, 2, . . , n ) • wi = weight associated with the signal input xi,代表 不同的重要性 – when wi > 0, input xi acts as an excitatory signal – when wi < 0, input xi acts as an inhibitory signal • t = threshold level(門檻值)prescribed by user, input signal 要大於門檻值才會有反應 • F(s) is a nonlinear function; e. g. , a sigmoid function F(s) = 。Other popular choices for this function are a step function, and a ramp function. 淡江大學 資訊管理系所 侯永昌 106

Neural Networks • A neural network should first be trained, then be applied to

Neural Networks • A neural network should first be trained, then be applied to solve problems. • During training stage, two sets of data: trainingdata and checking-data sets. Each elements in these sets are a pair of input and output data/signals (x, y) 淡江大學 資訊管理系所 侯永昌 108

The steps to train a neural network 1. random assignment of weights wijk 2.

The steps to train a neural network 1. random assignment of weights wijk 2. an input x from the training-data set is passed through the neural network,求出δ= f(x)actual - f(x)output 3. 利用back-propagation將errorδ分配回到hidden layers 4. 利用每一個node所分配到的error,求weight wijk的修 正量 5. step 2 to 4 are iterated until the error value is acceptable 6. step 2 to 5 are repeated for all data in the training-data set (this method makes the neural network simulate the nonlinear relation between the input-output data sets) 7. Checking-data set is used to verify how well the neural network can simulate the nonlinear relationship between the input-output data sets 淡江大學 資訊管理系所 侯永昌 109

Neural Network • Neural Network 適用的場合 – (input,output)資料量很多 – input, output 之間的關係是 unknown or

Neural Network • Neural Network 適用的場合 – (input,output)資料量很多 – input, output 之間的關係是 unknown or highly nonlinear • Neural Network 與 Expert System 的不同 – Expert system:learn by adding new rules to their knowledge base – Neural networks:learn by adding more data and modifying their overall structure. 淡江大學 資訊管理系所 侯永昌 121

Genetic Algorithms • Genetic Algorithms: use the concepts of Darwin's theory of evolution –

Genetic Algorithms • Genetic Algorithms: use the concepts of Darwin's theory of evolution – Reproduction – Crossover – Mutation 淡江大學 資訊管理系所 侯永昌 122

Steps to find a solution 1. Different possible solutions to a problem are created,

Steps to find a solution 1. Different possible solutions to a problem are created, 以 bit string 來代表各種可能的解答 2. These solutions are tested for their performance (how good a solution they provided) 3. A fraction of the good solutions is selected, and the others are eliminated(survival of the fittest,適者生存 ) 4. The selected solutions undergo the processes of reproduction, crossover, and mutation to create a new generation of possible solutions (which are expected to perform better than the previous generation) 5. steps 2 to 4 are repeated until there is convergence within a generation 淡江大學 資訊管理系所 侯永昌 123

Genetic Algorithms • Genetic Algorithms 的優點為:It searches for a solution from a broad spectrum

Genetic Algorithms • Genetic Algorithms 的優點為:It searches for a solution from a broad spectrum of possible solutions (比較有可能找到 optimal solution), rather than restrict the search to a narrow domain where the results would be normally expected (可 能找到只是 local optimal) • Genetic Algorithms 的 作重點在於: – How to code the possible solutions to the problem as finite bit strings ( Chromosomes ) – How to evaluate the fittness of each string 淡江大學 資訊管理系所 侯永昌 124

Three genetic operators • reproduction: the process by which strings with better fitness values

Three genetic operators • reproduction: the process by which strings with better fitness values receive correspondingly better copies in the new generation. • i. e. , we try to ensure that better solutions persist and contribute to better offsprings during successive generations. 淡江大學 資訊管理系所 侯永昌 125

Three genetic operators • crossover: the process in which the strings are able to

Three genetic operators • crossover: the process in which the strings are able to mix and match their desirable qualities in a random fashion. • Crossover rate 一般約 0. 5 - 1 淡江大學 資訊管理系所 侯永昌 126

Three genetic operators • mutation: the process by which the value of a randomly

Three genetic operators • mutation: the process by which the value of a randomly selected position is changed. It helps to increase the search power. • 例如:如果 selected solution 中第四個 bit 剛好 通通是 0,而實際的 solution 中第四個 bit 應該 是 1,則無論如何做 reproduction、crossover 都 無法讓第四個 bit 變成 1,mutation 能讓 bit 0, 1 互換,因此就成了唯一的 具 • mutation rate 的發生率很低,約 0. 005 bit/generation 淡江大學 資訊管理系所 侯永昌 127

Example 4. 6 • Given the following table, try to find a line y

Example 4. 6 • Given the following table, try to find a line y = C 1 x + C 2 Data number x y 1 2 3 1. 0 2. 0 3. 0 4 6. 0 • 假設 C 1、C 2 為實數,而且 -2 C 1、C 2 5 • Chromosomes: 各以 6 個 bits 來代表 C 1、C 2 這 兩個係數 • 如何將這兩個係數變成實數呢? 淡江大學 資訊管理系所 侯永昌 128

Example 4. 7 • The knowledge base has two rules: – If x is

Example 4. 7 • The knowledge base has two rules: – If x is small then y is small. – If x is large then y is very large. • Given the data set (1, 1) (2, 4) (3, 9) (4, 16) (5, 25) • Find the membership function of fuzzy variables small, large on x and small, very large on y. • Coding的方式: 4個端點座標分別用 6個bits來表示 = 0 + 7*5/63 = 0. 56 = 5 - 20*5/63 = 3. 41 淡江大學 資訊管理系所 侯永昌 131

Inductive Reasoning • Entropy 是隨機變量無約束程度的一種度量 (a measure of uncertainty of the distribution) • Entropy

Inductive Reasoning • Entropy 是隨機變量無約束程度的一種度量 (a measure of uncertainty of the distribution) • Entropy is used to determine the quantity of information in a given data set. The higher the prior estimate of the probability for an outcome to occur, the lower will be the information gained by observing it to occur. • 適合資料大,且 static 的資料使用,對於 dynamic 的資料則不適合,因為它沒有自我調整的功能, 而且它是用所有的資料去計算,因此計算量比較 大 淡江大學 資訊管理系所 侯永昌 136

Inductive Reasoning • For a simple one-dimensional case, let us assume that the probability

Inductive Reasoning • For a simple one-dimensional case, let us assume that the probability of the ith sample wi to be true is {p(wi)}. If we actually observe the sample wi in the future and discover that it is true, then we gain the following information, where k is a normalizing parameter. • If we discover that it is false, we still gain this information: 淡江大學 資訊管理系所 侯永昌 137

Inductive Reasoning • Then the entropy of all the samples (N) is defined as:

Inductive Reasoning • Then the entropy of all the samples (N) is defined as: where pi = p(wi). • 當 p = 0 or p = 1 時,entropy = 0 • 當 p = 0. 5 時,entropy 最大 淡江大學 資訊管理系所 侯永昌 138

Generation of membership function • Membership function generation is based on a partitioning. which

Generation of membership function • Membership function generation is based on a partitioning. which draws a threshold line between two classes of sample data. • 假設想要在一組數據中找一個點,將它們切割為兩 個 set,這一個切點應該使總 entropy 變得最小 (entropy minimization principle),表示愈能分的清 楚這個東西是屬於(p = 1)或不屬於(p = 0)這一個 set • 假設第一個區域稱為 p ,第二個區域稱為 q,切點 為 x 淡江大學 資訊管理系所 侯永昌 139

Generation of membership function • The entropy is calculated as follows: • where •

Generation of membership function • The entropy is calculated as follows: • where • pk(x) and qk(x) : conditional prob. that the class k sample is in the region [x 1, xl+x] and [xl+x, x 2], respectively; • p(x) and q(x) : prob. that all samples are in the region [x 1, xl+x] and [xl+x, x 2], respectively; • p(x) + q(x) = 1 淡江大學 資訊管理系所 侯永昌 140

Generation of membership function • 其中: • • • nk(x) : number of class

Generation of membership function • 其中: • • • nk(x) : number of class k samples located in [x 1, xl+x] n(x) : the total number of samples located in [x 1, xl+x] Nk(x) : number of class k samples located in [xl+x, x 2] N(x) : the total number of samples located in [xl+x, x 2] n : total number of samples in [x 1, x 2] 淡江大學 資訊管理系所 侯永昌 141

其它求 membership 的方法 • Soft partition: in Chapter 11 • Meta rules • Fuzzy

其它求 membership 的方法 • Soft partition: in Chapter 11 • Meta rules • Fuzzy statistic 淡江大學 資訊管理系所 侯永昌 146

Summary • • Neural Network: learning ability Fuzzy: reasoning system Genetic Algorithm: find optimum

Summary • • Neural Network: learning ability Fuzzy: reasoning system Genetic Algorithm: find optimum solutions Inductive Reasoning: 適合於 static data 淡江大學 資訊管理系所 侯永昌 147