Fuzzy Rules & Fuzzy Reasoning - Universitas Negeri...

Fuzzy Rules & Fuzzy Reasoning

Jyh-Shing Roger Jang et al., Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence, First Edition, Prentice Hall, 1997

Pengampu: Fatchul Arifin

Referensi:

Sistem Cerdas : PTK – Pasca Sarjana - UNY

Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch. 3: Fuzzy rules & fuzzy reasoning

22

– Orthogonality

A term set T = t1,…, tn of a linguistic variable x on the universe X is orthogonal if:

Where the ti’s are convex & normal fuzzy sets defined on X.

n

1iit Xx ,1)x(

Fuzzy if-then rules (3.3) (cont.)


23

General format:

– If x is A then y is B (where A & B are linguistic values defined by fuzzy sets on universes of discourse X & Y).

• “x is A” is called the antecedent or premise• “y is B” is called the consequence or conclusion

– Examples:

• 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.



24

– Meaning of fuzzy if-then-rules (A B)

• It is a relation between two variables x & y; therefore it is a binary fuzzy relation R defined on X * Y

• There are two ways to interpret A B:

– A coupled with B– A entails B

if A is coupled with B then:

or.normoperat-T a is * where

)y,x/()y(*)x(B*ABAR

~Y*X

B

~

A



25

If A entails B then:

R = A B = A B ( material implication)

R = A B = A (A B) (propositional calculus)

R = A B = ( A B) B (extended propositional calculus)

1c0),y(c*)x(;csup)y,x( B

~

AR



26

Two ways to interpret “If x is A then y is B”:

A

B

A entails By

x

A coupled with B

A

B

x

y



27

• Note that R can be viewed as a fuzzy set with a two-dimensional MF

R(x, y) = f(A(x), B(y)) = f(a, b)

With a = A(x), b = B(y) and f called the fuzzy implication function provides the membership value of (x, y)



28

– Case of “A coupled with B”

(minimum operator proposed by Mamdani, 1975)

(product proposed by Larsen, 1980)

(bounded product operator)

YX

BAm yxyxBAR*

),/()()(*

)y,x/()y( )x(B*ARY*X

BAp

)y,x/()1)y()x((0

)y,x/()y()x(B*AR

BAY*X

Y*XBAbp



29

– Case of “A coupled with B” (cont.)

(Drastic operator)

)10,3,4;()( and )10,3,4;()(for otherwise if 0

1a if 1b if

.̂ab)f(a, :

),/()(.̂)(*

BA

*

ybellyxbellxExample

ba

bwhere

yxyxBARYX

BAdp



30

A coupled with B



31

– Case of “A entails B”

(Zadeh’s arithmetic rule by using bounded sum operator for union)

(Zadeh’s max-min rule)

)ba1(1)b,a(f :where

)y,x/()y()x(1(1BAR

a

Y*XBAa

)ba()a1()b,a(f :where

)y,x/())y()x(())x(1()BA(AR

m

Y*XBAAmm



32

– Case of “A entails B” (cont.)

(Boolean fuzzy implication with max for union)

(Goguen’s fuzzy implication with algebraic product for T-norm)

b)a1()b,a(f :where

)y,x/()y())x(1(BAR

s

Y*XBAs

otherwise a/bba if 1

b~a :where

)y,x/())y(~)x((RY*X

BA



33

A entails B


34

Fuzzy Reasoning (3.4)

Definition

– Known also as approximate reasoning

– It is an inference procedure that derives conclusions from a set of fuzzy if-then-rules & known facts


35

Fuzzy Reasoning (3.4) (cont.)

Compositional rule of inference

– Idea of composition (cylindrical extension & projection)

• Computation of b given a & f is the goal of the composition

– Image of a point is a point

– Image of an interval is an interval


36

Derivation of y = b from x = a and y = f(x):

a and b: pointsy = f(x) : a curve

x

y

a

b

y = f(x)

a

b

y

x

a and b: intervalsy = f(x) : an interval-valued

function

y = f(x)



37

– The extension principle is a special case of the compositional rule of inference

• F is a fuzzy relation on X*Y, A is a fuzzy set of X & the goal is to determine the resulting fuzzy set B

– Construct a cylindrical extension c(A) with base A

– Determine c(A) F (using minimum operator)

– Project c(A) F onto the y-axis which provides B



38

a is a fuzzy set and y = f(x) is a fuzzy relation:

cri.m



39

Given A, A B, infer B

A = “today is sunny”A B: day = sunny then sky = blueinfer: “sky is blue”

• illustration

Premise 1 (fact): x is APremise 2 (rule): if x is A then y is B

Consequence: y is B



40

Approximation

A’ = “ today is more or less sunny”B’ = “ sky is more or less blue”

• iIlustration

Premise 1 (fact): x is A’Premise 2 (rule): if x is A then y is B

Consequence: y is B’

(approximate reasoning or fuzzy reasoning!)



41

Definition of fuzzy reasoning

Let A, A’ and B be fuzzy sets of X, X, and Y, respectively. Assume that the fuzzy implication A B is expressed as a fuzzy relation R on X*Y. Then the fuzzy set B induced by “x is A’ ” and the fuzzy rule “if x is A then y is B’ is defined by:

Single rule with single antecedentRule : if x is A then y is BFact: x is A’Conclusion: y is B’ (B’(y) = [x (A’(x) A(x)] B(y))

)y,x(),x(minmax)y( R'Ax

'B



42

A

X

w

A’ B

Y

x is A’

B’

Y

A’

Xy is B’



43

– Single rule with multiple antecedentsPremise 1 (fact): x is A’ and y is B’Premise 2 (rule): if x is A and y is B then z is C

Conclusion: z is C’

Premise 2: A*B C

)z()w(w

)z()y()y()x()x(

)z()y()x()y()x(

)z()y()x()y()x()z(

)CB*A()'B'*A('C

)z,y,x/()z()y()x(C*)B*A()C,B,A(R

C21

C

2w

B'By

1w

A'Ax

CBA'B'Ay,x

CBA'B'Ay,x'C

2premise1premise

CBZ*Y*X

Amamdani


44

A BT-norm

X Y

w

A’ B’ C2

Z

C’

ZX Y

A’ B’

x is A’ y is B’ z is C’

w1

w2


45

– Multiple rules with multiple antecedents

Premise 1 (fact): x is A’ and y is B’Premise 2 (rule 1): if x is A1 and y is B1 then z is C1

Premise 3 (rule 2): If x is A2 and y is B2 then z is C2

Consequence (conclusion): z is C’

R1 = A1 * B1 C1

R2 = A2 * B2 C2

Since the max-min composition operator o is distributive over the union operator, it follows:

C’ = (A’ * B’) o (R1 R2) = [(A’ * B’) o R1] [(A’ * B’) o R2] = C’1 C’2Where C’1 & C’2 are the inferred fuzzy set for rules 1 & 2 respectively



46

A1 B1

A2 B2

T-norm

X

X

Y

Y

w1

w2

A’

A’ B’

B’ C1

C2

Z

Z

C’

ZX Y

A’ B’

x is A’ y is B’ z is C’

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