1

퍼지 이론퍼지 이론(Lecture Note #14)(Lecture Note #14)

인공지능2002 년 2 학기

이복주단국대학교 컴퓨터공학과

2

OutlineOutline

Fuzzy Logic and Intelligent Systems Progress of Fuzzy Logic Generalized Modus Ponens Fuzzy Reasoning Triangular Norms Triangular Conorms Theoretical Foundations of Fuzzy Inference

3

Saturn Fuzzy Logic TransmissionSaturn Fuzzy Logic Transmission

Saturn automobile’s smart transmission When the car is moving uphill or downhill Employs shift stabilization Fuzzy logic control is used to avoid hunting, that is fre

quent shifting of gears Shifting decisions are made by weighing many input v

ariables at once and using fuzzy if-then rules to generate an output control signal

4

Fuzzy Logic and Intelligent SystemsFuzzy Logic and Intelligent Systems

Objective: Develop a cost-effective approximate solution to a problem

Approach: Exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low cost

Preciseness

Utility (Usefulness of system)

Cost of system

Low cost, high usefulness

5

Fuzzy Logic ApplicationsFuzzy Logic Applications

Diagnosis Financial Analysis and Prediction Robotics Data Compression and Pattern Recognition Consumer electronics

– Typical KB size: 10-20 rules– Typical KB number: 1

Automobile/Transportation system– Typical KB size: 40-80 rules– Typical KB number: 1-3

Locomotive, Subways, Aircraft engines, Helicopters– Typical KB size: 60-80 rules– Typical KB number: 4-8 + supervisory (hierarchical)

6

Progress of Fuzzy LogicProgress of Fuzzy Logic

1965, Prof. Loft A. Zadeh at UCB developed fuzzy set theory and fuzzy logic

1974, Fuzzy logic controller for a steam engine (Prof. Mamdani, London University)

1980, Control of Cement-Kiln with monitor capability (Smidth, Denmark)

1987, Automatic train operation for Sendai subway (Hitachi)

1988, Stock trading expert system (Yamichi security) 1989, Laboratory for International Fuzzy Engineering 1987-90, 389 patents in U.S. regarding fuzzy system

s

7

Fuzzy Logic is Not …Fuzzy Logic is Not …

Is not a clever disguise of probability theory The behavior of a fuzzy system is not fuzzy (it is deter

ministic) The founder of fuzzy logic (Prof. Zadeh) is not a mathe

matician. Fuzzy logic is not to replace conventional techniques.

8

Generalized Modus PonensGeneralized Modus Ponens In fuzzy logic

– the truth value of a statement becomes a matter of degrees– Reasoning in fuzzy logic is based on generalized modus ponens

Modus Ponens Generalized Modus Ponens– Given A B Given A B– A A’– ----------------- -------------------------– Deduce B Deduce B’– where A’ is a fuzzy set that partially matches A

Example– If a person is self-confident then he/she has a happy life– Jack is somewhat self-confident– What can we conclude using conventional logic? – What can we conclude using generalized modus ponens? – => Jack is somewhat happy

9

Fuzzy ReasoningFuzzy Reasoning

Fuzzy Rules: If x is A1 and y is B1 then z is C1

If x is A2 and y is B2 then z is C2

– x and y are inputs, z is an output– Ai, Bi, and Ci are fuzzy sets

Input data: x is A’, y is B’– A’ and B’ are also fuzzy sets

Question: z is ?

10

Step 1: CompatibilityStep 1: Compatibility

Calculate the degree that input data (A’, B’) matches each rule premise (A1, A2, B1, B2) (A1, A’), (A2, A’), (B1, B’), (B2, B’)

Compatibility between A and A’: (A, A’) = supx min{A(x), A’(x)}

x

0.5

1

A’

0

A

11

Step 2: Combine CompatibilitiesStep 2: Combine Compatibilities

Combine the degree of matching for the inputs– for “and”, usually take min– for “or”, usually take max– min{(A1, A’), (B1, B’)}

– min{(A2, A’), (B2, B’)}

x

0.5

1

A’

0y

0.3

1

B’

0

A B

0.3

12

Step 3: Derive Output Fuzzy SetsStep 3: Derive Output Fuzzy Sets

The (combined) degree of matching i is propagated to the consequent to form an inferred fuzzy subset Ci’– Type I: C’(z) = C(z) [ usually take min ]

– Type II: C’(z) = x C(z)

– C1’ and C2’ are derived

z

1

C

0.3

0

C’

z

1

C

0.3

0

C’0.3

Type I Type II

13

Step 4: Combine Output Fuzzy SetsStep 4: Combine Output Fuzzy Sets

Combine the inferred fuzzy values (C1’ and C2’) of z– max {C1’(z), C2’(z)}

z

1

0

C2’C1’

14

Step 5: DefuzzificationStep 5: Defuzzification

Perform defuzzification to obtain z’s final value– Mean of Maximum method (MOM)

• (j=1,kwj)/k where wj is peak and k is the number of peaks

– Center of Area method (COA) • (j=1,nz(wj) x wj)/ j=1,nz(wj)

w1

1

0wk

MOM

1

0

COA

w1 wn

15

What about Crisp Input?What about Crisp Input?

Inputs are x0 and y0 rather than A’ and B’. Compatibility between A and x0 (Step 1):

(A, x0) = A(x0)

x0

0.5

1

0

A

16

Triangular Norms (Conjunction)Triangular Norms (Conjunction)

Four methods to calculate “and” Min

A and B (x) = min{A(x), B(x)} Algebraic product

A and B (x) = A(x) x B(x) Bounded difference

A and B (x) = max {0, A(x) + B(x) – 1} Drastic product

A and B (x) = A(x) if B(x) = 1 B(x) if A(x) = 1 0 otherwise

min algebraic product bounded difference drastic product

17

Triangular Conorms (Disjunction)Triangular Conorms (Disjunction)

Four methods to calculate “or” Max

A OR B (x) = max{A(x), B(x)} Bounded sum

A OR B (x) = min{1,A(x) + B(x)} Algebraic sum

A OR B (x) = A(x) + B(x) - A(x) x B(x) Drastic sum

A OR B (x) = A(x) if B(x) = 0 B(x) if A(x) = 0 1 otherwise

max bounded sum algebraic sum drastic sum

18

Theorectical Foundation of Fuzzy InferenceTheorectical Foundation of Fuzzy Inference

A fuzzy rule = A fuzzy relation Inference is a composition of relations Relation R: U X V = {0, 1}

– R(x, y) = 1: the relation holds between x and y– R(x, y) = 0: the relation does not hold between x and y– e.g., Take(Kim, ICE607) = 1 Take(Kim, ICE608) = 0

Fuzzy relation R: U X V = [0, 1]– E.g., Friendly(Jack, Joe) = 1 Friendly(Clinton, Hussain) = 0 Friendly(Clinton, Gingrich) = 0.2

19

Fuzzy Rule as Fuzzy RelationFuzzy Rule as Fuzzy Relation

Fuzzy rules are fuzzy relations over the Cartesian product of the domains of antecedent and consequent variables

Semantically the fuzzy relation captures the degree of association between a pair of antecedent variables and consequent variables

Example– If height is tall then IQ is high

4 5

6 7

8

9

80 90 100 110 120 130 140

H\I

R(6.5,135)

High

Ta

ll

R(6.5,135): The possibility that a person 6.5 tall has a 135 IQ.

20

Methods to Calculate the Degree of AssociationMethods to Calculate the Degree of Association

Two families of calculating the degree of association– Fuzzy implication operators– Fuzzy conjunction operators

21

Families of Fuzzy Implication OperatorsFamilies of Fuzzy Implication Operators

Material implication– A B = (not A) + B– e.g., Tall(6.25)=0.25, High(125)=0.5

TallHigh(6.25,125)=max{1-0.25,0.5}=0.75

Propositional calculus– A B = (not A) + (A * B) ( A (AB) = (AA) (AB) = AB )

Extended propositional calculus– A B = (A x B) + B

22

Families of Fuzzy Implication Operators (2)Families of Fuzzy Implication Operators (2)

Generalization of modus ponens– A B = sup { c[0,1] | A x c B }– e.g., Tall(6.25)=0.25, High(125)=0.5

TallHigh(6.25,125) = sup { c[0,1] | min(0.25,c) 0.5 } = 1

– If x is min, generalization of mp is– A B = 1 if A B B otherwise

Generalization of modus tollens– A B = inf { t[0,1] | B+t A }

All fuzzy implication operations share the fundamental implication property– 00=1, 01=1, 10=0, 11=1

23

Fuzzy Relation Using Conjunction OperatorFuzzy Relation Using Conjunction Operator

Use conjunction to represent relation (implication)– A B = min {A, B}– i.e., AB(x, y) = min {A(x), B(y)}

Does this hold the fundamental implication property?– 00=1, 01=1, 10=0, 11=1

24

SummarySummary

Fuzzy Logic and Intelligent Systems Progress of Fuzzy Logic Generalized Modus Ponens Fuzzy Reasoning Triangular Norms Triangular Conorms Theoretical Foundations of Fuzzy Inference