IF-THEN RULES AND FUZZY INFERENCEsite.iugaza.edu.ps › mahir › files › 2010 › 02 ›...

IFIF--THEN RULES THEN RULES ANDAND

FUZZY INFERENCEFUZZY INFERENCE

inference\In"fer*ence\, n. [From Infer.]

1. The act or process of inferring by deduction or induction.

2. That which inferred; a truth or proposition drawn from another which is admitted or supposed to be true; a conclusion; a deduction. --Milton.

Inference is a process of obtaining new knowledge through existing knowledge.

inferenceinference\\In"ferIn"fer**enceence\\, n. [From , n. [From InferInfer.] .]

1. The act or process of inferring by 1. The act or process of inferring by deduction or induction.deduction or induction.

2. That which inferred; a truth or 2. That which inferred; a truth or proposition drawn from another which is proposition drawn from another which is admitted or supposed to be true; a admitted or supposed to be true; a conclusion; a deduction. conclusion; a deduction. ----Milton. Milton.

Inference is a process of obtaining new Inference is a process of obtaining new knowledge through existing knowledge.knowledge through existing knowledge.

InferenceInference

Representation of knowledgeRepresentation of knowledgeTo perform inference, knowledge should To perform inference, knowledge should be represented in some form be represented in some form

Representation of knowledge as rules is the most Representation of knowledge as rules is the most popular form. popular form.

if x is A then y is Bif x is A then y is B(where (where AA and and BB are linguistic values defined by fuzzy are linguistic values defined by fuzzy sets on universes of discourse sets on universes of discourse XX and and YY).).

A rule is also called a A rule is also called a fuzzy implication fuzzy implication ““x is Ax is A”” is called the is called the antecedentantecedent or or premise premise ““y is By is B”” is called the is called the consequenceconsequence or or conclusionconclusion

http://if.kaist.ac.kr/lecture/cs670/textbook/

Representation of knowledgeRepresentation of knowledge

Examples:Examples:

If pressure is high, then volume is small. If pressure is high, then volume is small. If the road is slippery, then driving is If the road is slippery, then driving is dangerous. dangerous. If an apple is red, then it is ripe. If an apple is red, then it is ripe. If the speed is high, then apply the brake aIf the speed is high, then apply the brake alittle.little.


Knowledge as RulesKnowledge as RulesHow do you reason? How do you reason? –– You want to play golf on Saturday or Sunday You want to play golf on Saturday or Sunday

and you donand you don’’t want to get wet when you play. t want to get wet when you play. Use rules! Use rules! –– If it rains, you get wet! If it rains, you get wet! –– If you get wet, you canIf you get wet, you can’’t play golf t play golf If it rains on Saturday and wonIf it rains on Saturday and won’’t rain on t rain on Sunday Sunday –– You play golf on SundayYou play golf on Sunday!!*Fuzzy Thinking:The new Science of Fuzzy Logic, Bart Kosko

Knowledge is rules Knowledge is rules Rules are in blackRules are in black--andand--white language white language –– Bivalent rules Bivalent rules AI has so far, after over 30 years of AI has so far, after over 30 years of research, not produced smart machines! research, not produced smart machines! –– Because they canBecause they can’’t yet put enough rules in the t yet put enough rules in the

computer (use 100computer (use 100--1000 rules, need >100k} 1000 rules, need >100k} –– Throwing more rules at the problem Throwing more rules at the problem

*Fuzzy Thinking:The new Science of Fuzzy Logic, Bart Kosko

Knowledge as RulesKnowledge as Rules

Forms of reasoningForms of reasoning

Generalized Modus Generalized Modus PonensPonens::

Premise:Premise: xx is Ais A’’Implication:Implication: if if xx is A then is A then yy is Bis BConsequence:Consequence: yy is Bis B’’

Where Where A, AA, A’’, B, B, B, B’’ are fuzzy sets and are fuzzy sets and xx and and yyare symbolic names for objects.are symbolic names for objects.


Forms of reasoningForms of reasoning

Generalized Modus Generalized Modus TolensTolens::

Premise:Premise: yy is Bis B’’Implication:Implication: if if xx is A then is A then yy is Bis BConsequence:Consequence: xx is Ais A’’

Where Where A, AA, A’’, B, B, B, B’’ are fuzzy sets andare fuzzy sets and xx and and yyare symbolic names for objects.are symbolic names for objects.


Fuzzy rule as a relationFuzzy rule as a relation

if x is A then y is Bif x is A then y is B““x is Ax is A””, , ““y is By is B”” –– fuzzy predicates fuzzy predicates A(xA(x), ), B(yB(y))

if if A(xA(x) then ) then B(yB(y) ) can be represented as a relation can be represented as a relation

R(x,yR(x,y): ): A(xA(x) ) →→ B(yB(y) ) where where R(x,yR(x,y)) can be considered a fuzzy set can be considered a fuzzy set

with 2with 2--dimentional membership function dimentional membership function µµRR(x,y(x,y)=)=f(f(µµAA(x),(x),µµBB(y(y)) ))

where where ff is is fuzzy implication functionfuzzy implication functionhttp://if.kaist.ac.kr/lecture/cs670/textbook/

MIN fuzzy implicationMIN fuzzy implication

Interprets the fuzzy implication as the minimum operation [Mamdani]. Interprets the fuzzy implication as the minimum operation [Mamdani].


PRODUCT fuzzy implicationPRODUCT fuzzy implication

Interprets the fuzzy implication as the product operation [Larsen]. Interprets the fuzzy implication as the product operation [Larsen].


EXAMPLE OF FUZZY EXAMPLE OF FUZZY IMPLICATIONIMPLICATION

Fuzzy rule: “If temperature is high, then humidity is fairly high”

Lets define: T – universe of discourse for temperature H – universe of discourse for humidity t∈T, h∈H – variables for temperature and humudityDenote “high” as A, A⊆T Denote “fairly high” as B, B⊆H

Then the rule becomes: R(t,h): if t is A then h is B or R(t,h): R(t) → R(h)

Fuzzy rule: “If temperature is high, then humidity is fairly high”

Lets define: T – universe of discourse for temperature H – universe of discourse for humidity t∈T, h∈H – variables for temperature and humudityDenote “high” as A, A⊆T Denote “fairly high” as B, B⊆H

Then the rule becomes: R(t,h): if t is A then h is B or R(t,h): R(t) → R(h)



if we know A and B, we can find R(t,h)=A×B

RC(t, h) = A × B = ∫µA(t) ∧ µB(h) / (t, h)

Mamdani (min) implication

if we know A and B, we can find R(t,h)=A×B

RRCC(t(t, h) = A , h) = A ×× B B = = ∫∫µµAA(t(t) ) ∧∧ µµBB(h(h) / (t, h) ) / (t, h)

MamdaniMamdani (min) (min) implication implication


t t 20 20 30 30 40 40

µA(t) µA(t) 0.1 0.1 0.5 0.5 0.9 0.9

h h 20 20 50 50 70 70 90 90

µB(h) µB(h) 0.2 0.2 0.6 0.6 0.7 0.7 1 1

20 20 50 50 70 70 90 90

20 20 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

30 30 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5

40 40 0.2 0.2 0.6 0.6 0.7 0.7 0.9 0.9

tthh


we know RC(t, h) for fuzzy rule “If temperature is high, then humidity is fairly high”

According to this rule, what is the humidity when “temperature is fairly high” or t is A’, A’⊆T ?

dgtgz

we know RRCC(t(t, h), h) for fuzzy rule for fuzzy rule “If temperature is high, then humidity is fairly high”

According to this rule, what is the humidity when “temperature is fairly high” or t is A’, A’⊆T ?

dgtgzhttp://if.kaist.ac.kr/lecture/cs670/textbook/

t 20 30 40

µA′(t) 0.01 0.25 0.81


We can use composition of fuzzy relations to find R(h)!

R(t)

RC(t, h)

R(h) = R(t) ο RC(t, h)

We can use composition of fuzzy relations to find R(h)!

R(t)

RRCC(t(t, h), h)

R(hR(h) = ) = R(tR(t) ) οο RRCC(t(t, h), h)


t 20 30 40 µA′(t) 0.01 0.25 0.81

20 20 50 50 70 70 90 90

20 20 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

30 30 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5

40 40 0.2 0.2 0.6 0.6 0.7 0.7 0.9 0.9

tthh

h h 20 20 50 50 70 70 90 90

µB′(h)µB′(h) 0.2 0.2 0.6 0.6 0.7 0.7 0.81 0.81

COMPOSITIONAL RULE OF COMPOSITIONAL RULE OF INFERENCEINFERENCE

In order to draw conclusions from a set of rules (rule base) oneneeds a mechanism that can produce an output from a collection of rules. This is done using the compositional rule of inference.

Consider a single fuzzy rule and its inference Rule: if v is A then w is C Input: v is A′Result: C′A ⊂ U, C ⊂ W, v ∈ U, and w ∈ C.

The fuzzy rule is interpreted as an implication R:A→C or R=A×C

When input A′ is given to the inference system, the output C′ = A′ ο R

In order to draw conclusions from a set of rules (rule base) oneneeds a mechanism that can produce an output from a collection of rules. This is done using the compositional rule of inference.

Consider a single fuzzy rule and its inference Rule: if v is A then w is C Input: v is A′Result: C′A ⊂ U, C ⊂ W, v ∈ U, and w ∈ C.

The fuzzy rule is interpreted as an implication R:A→C or R=A×C

When input A′ is given to the inference system, the output C′ = A′ ο R



C′ = A′ ο R “ο“ is the composition operator. The inference

procedure is called “compositional rule of inference”.The inference mechanism is determined by two factors:

1. Implication operators: Mamdani: min Larsen: algebraic product

2. Composition operators: Mamdani: max-min Larsen: max-product

C′ = A′ ο R “ο“ is the composition operator. The inference

procedure is called “compositional rule of inference”.The inference mechanism is determined by two factors:

1. Implication operators: Mamdani: min Larsen: algebraic product

2. Composition operators: Mamdani: max-min Larsen: max-product



Compositional rule of inference can be represented graphically as a combination of cylindrical extension, intersection and projection of fuzzy sets:

1. Build a cylindrical extension of A, A(x,y)2. Determine intersection of R(x,y) and A(x,y)3. Build projection of R(x,y)∧A(x,y)

Compositional rule of inference can be represented graphically as a combination of cylindrical extension, intersection and projection of fuzzy sets:

1. Build a cylindrical extension of A, A(x,y)2. Determine intersection of R(x,y) and A(x,y)3. Build projection of R(x,y)∧A(x,y)

INFERENCE METHODSINFERENCE METHODS

There are many methods to perform fuzzy inference. Consider a fuzzy rule:

R1: if u is A1 and v is B1 then w is C1

Inputs u and v can be: crisp inputs. Crisp inputs can be treated as fuzzy singletons fuzzy sets A’ and B’

There are many methods to perform fuzzy inference. Consider a fuzzy rule:

R1: if u is A1 and v is B1 then w is C1

Inputs u and v can be: crisp inputs. Crisp inputs can be treated as fuzzy singletons fuzzy sets A’ and B’


MAMDANI METHODMAMDANI METHODThis method uses the minimum operation RC as a

fuzzy implication and the max-min operator for the composition.

Suppose a rule base is given in the following form: Ri: if u is Ai and v is Bi then w is Ci, i = 1, 2, …, n

for u ∈ U, v ∈ V, and w ∈ W.Then, Ri = (Ai and Bi) → Ci is defined by

This method uses the minimum operation RC as a fuzzy implication and the max-min operator for the composition.


for u ∈ U, v ∈ V, and w ∈ W.Then, Ri = (Ai and Bi) → Ci is defined by


),,()( wvuiiii CBandAR →= µµ ),,()( wvuiiii CBandAR →= µµ

MAMDANI METHODMAMDANI METHODCase 1: Inputs are crisp and treated as fuzzy singletons.

u = u0, v = v0

Inference if … then …Result

Example:

if temperature is high and humidity is high then fan speed is high

How to determine the fan speed for temperature 85ºF and humidity 93%?

Case 1: Inputs are crisp and treated as fuzzy singletons.uu = = uu00, , vv = = vv00

Inference if … then …Result

Example:

if temperature is high and humidity is high then fan speed is high

How to determine the fan speed for temperature 85ºF and humidity 93%?


)()]()([)( 00 wvanduwiiii CBAC µµµµ →=′ )()]()([)( 00 wvanduwiiii CBAC µµµµ →=′

MAMDANI METHODMAMDANI METHOD

Mamdani method uses min operator (∧) as fuzzy implication function (→):

αi is called “firing strength”, “matching degree”, “satisfaction degree”

Mamdani method uses min operator (∧) as fuzzy implication function (→):

ααii is called is called ““firing strengthfiring strength””, , ““matching degreematching degree””, , ““satisfaction degreesatisfaction degree””


0 00

1 1 1

u v wu0 v0

α1C1′

MIN Because of the “AND”connective

MIN of αi and Ci Because of the implication function

MAMDANI METHODMAMDANI METHODFor multiple rules (for example, two rules R1 and R2):For multiple rules (for example, two rules R1 and R2):


0 0

0

0 0 0

01

1 1

1 1 1

1

u

u

v

v

w

wminu0 v0

α1

α2

C1′

C2′

C1′ C2′

MAMDANI METHODMAMDANI METHODIn general:

max min

In general:

max min


0 0

0

0 0 0

0 1

1 1

1 1 1

1

u

u

v

v

w

wminu

0

v0

α1

α2

C1′

C2′

C1′ C2′


Case 2: Inputs are fuzzy sets A’, B’Case 2: Inputs are fuzzy sets AA’’, B, B’’


))]()((max)),()((maxmin[

)()(

vvuuwhere

ww

ii

ii

BBvAAui

CiC

µµµµα

µαµ

∧∧=

∧=

′′

′

))]()((max)),()((maxmin[

)()(

vvuuwhere

ww

ii

ii

BBvAAui

CiC

µµµµα

µαµ

∧∧=

∧=

′′

′

0 00

1 1 1

u v w

α1

A1 A′ B′ B1 C1


For multiple rules, For multiple rules,


0 0

0

0 0 0

01

1 1

1 1 1

1

u

u

v

v

w

wmin

α1

α2

A1 A′ B′

A′ B′

B1 C1

A2 B2 C2

EXAMPLE OF MAMDANI EXAMPLE OF MAMDANI METHODMETHOD

Let the fuzzy rule base consist of one rule:R: If u is A then v is B

where A=(0, 2, 4) and B=(3, 4, 5) are triangular fuzzy sets

Question 1: What is the output B’ if the input is a crisp value u0=3?

Question 2: What is the output B’ if the input is a fuzzy set A′=(0, 1, 2)?

Let the fuzzy rule base consist of one rule:R: If R: If uu is A then is A then vv is B is B

where where A=(0, 2, 4)A=(0, 2, 4) and and B=(3, 4, 5)B=(3, 4, 5) are are triangular fuzzy setstriangular fuzzy sets

Question 1: What is the output Question 1: What is the output BB’’ if the input is a if the input is a crisp value crisp value uu00=3=3??

Question 2: What is the output Question 2: What is the output BB’’ if the input is a if the input is a fuzzy set fuzzy set AA′=(0, 1, 2)?


EXAMPLE OF MAMDANI EXAMPLE OF MAMDANI METHODMETHOD


0 2 3 4

u0

1

1

0.5

B

B′

3 4 5

A

0 2 3 4

1

1

0.5

B

B′

3 4 5

AA’

Fuzzy inference with input u0=3 Fuzzy inference with input A′=(0, 1, 2).

LARSEN METHODLARSEN METHOD

This method uses the product operation RP as a fuzzy implication and the max-product operator for the composition.


for u ∈ U, v ∈ V, and w ∈ W. Then, Ri = (Ai and Bi) → Ci is defined by

This method uses the product operation RP as a fuzzy implication and the max-product operator for the composition.


for u ∈ U, v ∈ V, and w ∈ W. Then, Ri = (Ai and Bi) → Ci is defined by


),,()( wvuiiii CBandAR →= µµ ),,()( wvuiiii CBandAR →= µµ

LARSEN METHODLARSEN METHODCase 1: Inputs are crisp and treated as fuzzy singletons.

u = u0, v = v0

` Inference if … then …result

For multiple rules:

Case 1: Inputs are crisp and treated as fuzzy singletons.uu = = uu00, , vv = = vv00

` ` Inference if … then …result

For multiple rules:

)()()(

)()]()([

)()]()([)(

00

00

00

vuwherew

wvu

wvanduw

iii

iii

iiii

BAiCi

CBA

CBAC

µµαµα

µµµ

µµµµ

∧==

∧=

→=

•

•

′

)()()(

)()]()([

)()]()([)(

00

00

00

vuwherew

wvu

wvanduw

iii

iii

iiii

BAiCi

CBA

CBAC

µµαµα

µµµ

µµµµ

∧==

∧=

→=

•

•

′


i

n

i

C

n

iCi

n

iC

CUC

wwwii

′=′

==

=

′=

•=

′

1

11)(V)]([V)( µµαµ

i

n

i

C

n

iCi

n

iC

CUC

wwwii

′=′

==

=

′=

•=

′

1

11)(V)]([V)( µµαµ



1 1 1

1 1 1

1

0 0 0

0

000

u v w

u v w

min

w

α1

α2

Graphical representation of Larsen method with singleton input


Case 2: Inputs are fuzzy sets A’, B’

For multiple rules:

Case 2: Inputs are fuzzy sets A’, B’

For multiple rules:

))()((max)),()((maxmin[

)()(

vvuuwhere

ww

ii

ii

BBvAAui

CiC

µµµµα

µαµ

∧∧=

=

′′

•′

))()((max)),()((maxmin[

)()(

vvuuwhere

ww

ii

ii

BBvAAui

CiC

µµµµα

µαµ

∧∧=

=

′′

•′


i

n

i

C

n

iCi

n

iC

CUC

wwwii

′=′

==

=

′=

•=

′

1

11)(V)]([V)( µµαµ

i

n

i

C

n

iCi

n

iC

CUC

wwwii

′=′

==

=

′=

•=

′

1

11)(V)]([V)( µµαµ


Graphical representation of Larsen method with fuzzy set inputs

0 0

0

0 0 0

01

1 1

1 1 1

1

u

u

v

v

w

wmin

α1

α2

EXAMPLE OF LARSEN METHODEXAMPLE OF LARSEN METHOD

the fuzzy rule base consist of one rule:R: If u is A and v is B then w is C

ere A=(0, 2, 4), B=(3, 4, 5) and C=(3,4,5)re triangular fuzzy sets

estion 1: What is the output C’ if the inputs are risp values u0=3, v0=4?

estion 2: What is the output C’ if the inputs are zzy sets A′=(0, 1, 2) and B’=(2,3,4)?

the fuzzy rule base consist of one rule:R: If R: If uu is A and is A and vv is B then is B then ww is C is C

ere ere A=(0, 2, 4),A=(0, 2, 4), B=(3, 4, 5)B=(3, 4, 5) and and C=(3,4,5)C=(3,4,5)re triangular fuzzy setsre triangular fuzzy sets

estion 1: What is the output estion 1: What is the output CC’’ if the inputs are if the inputs are risp values risp values uu00=3, v=3, v00=4=4??

estion 2: What is the output estion 2: What is the output CC’’ if the inputs are if the inputs are zzy sets zzy sets AA′=(0, 1, 2) and B’=(2,3,4)?

XAMPLE OF LARSEN METHODXAMPLE OF LARSEN METHOD

1

0.5

2 3 4

u0

A 1

5

v0

3 4

B

1

3 4

C

5

Larsen method with input u0 =3, v0=4

2 40

1 1

0.5

AA′

1

3 4

C

51

BB′

3 4 52

2/3

Larsen method with input A′=(0, 1, 2), B′=(2, 3, 4).

DEFUZZIFICATIONDEFUZZIFICATION

The output of Mamdani and Larsen nference methods is a fuzzy set!

For practical applications a crisp value is often needed

The process of converting a fuzzy answer nto a crisp value is called defuzzification

The output of Mamdani and Larsen nference methods is a fuzzy set!

For practical applications a crisp value is often needed

The process of converting a fuzzy answer nto a crisp value is called defuzzification

SUMMARYSUMMARY

nference - the logical process by which new facts are derived from known facts by he application of inference rules.

Fuzzy rules – a convenient way to epresent knowledge

A fuzzy rule can be represented as a fuzzy elation connected by a fuzzy implication unction The fuzzy inference procedure is called thecompositional rules of inference

nference - the logical process by which new facts are derived from known facts by he application of inference rules.

Fuzzy rules – a convenient way to epresent knowledge

A fuzzy rule can be represented as a fuzzy elation connected by a fuzzy implication unction The fuzzy inference procedure is called thecompositional rules of inference

SUMMARYSUMMARY

Mamdani and Larsen methods are two very popular methods of fuzzy inference. There are many more inference methodsDefuzzification is needed for the results obtained through fuzzy inference.

Mamdani and Larsen methods are two very popular methods of fuzzy inference. There are many more inference methodsDefuzzification is needed for the results obtained through fuzzy inference.

IF-THEN RULES AND FUZZY INFERENCEsite.iugaza.edu.ps › mahir › files › 2010 › 02 ›...

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