Fuzzy Sets and Applications Introduction Introduction Fuzzy Sets and Operations Fuzzy Sets and Operations

Fuzzy Sets and Fuzzy Sets and ApplicationsApplications

IntroductionIntroduction Fuzzy Sets and OperationsFuzzy Sets and Operations

Why fuzzy sets?Why fuzzy sets?

Types of UncertaintyTypes of Uncertainty1. Randomness : Probability

Knowledge about the relative frequency of each event in some domain

Lack of knowledge which event will be in next time

2. Incompleteness : Imputation by EM

Lack of knowledge or insufficient data

3. Ambiguity : Dempster-Shafer’s Belief Theory

=> Evidential Reasoning

Uncertainty due to the lack of evidence

ex) “The criminal is left-handed or not”


Types of Uncertainty (continued)Types of Uncertainty (continued)

4. Imprecision :

Ambiguity due to the lack of accuracy of observed data

ex) Character Recognition

5. Fuzziness (vagueness) : Uncertainty due to the vagueness of boundary

ex) Beautiful woman, Tall man


Powerful tool for vaguenessPowerful tool for vagueness Description of vague linguistic terms Description of vague linguistic terms

and algorithmsand algorithms Operation on vague linguistic termsOperation on vague linguistic terms Reasoning with vague linguistic rulesReasoning with vague linguistic rules Representation of clusters with vague Representation of clusters with vague

boundariesboundaries

History of Fuzzy SetsHistory of Fuzzy Sets

History of Fuzzy Sets and ApplicationsHistory of Fuzzy Sets and Applications 1965 Zadeh 1965 Zadeh Fuzzy SetsFuzzy Sets 1972 Sugeno 1972 Sugeno Fuzzy IntegralsFuzzy Integrals 1975 Zadeh1975 Zadeh

Fuzzy Algorithm & Approximate ReasoningFuzzy Algorithm & Approximate Reasoning 1974 Mamdani 1974 Mamdani Fuzzy ControlFuzzy Control 1978 North Holland Fuzzy Sets and Systems1978 North Holland Fuzzy Sets and Systems 1982 Bezdek 1982 Bezdek Fuzzy C-MeanFuzzy C-Mean 1987 1987 Korea Korea Fuzzy Temperature ControlFuzzy Temperature Control

Current Scope of Fuzzy SocietyCurrent Scope of Fuzzy Society

Fuzzy Sets

ApplicationsApplications

MethodsMethods

Fuzzy Measure Fuzzy Logic

Fuzzy Integrals Fuzzy Measure

Fuzzy RelationFuzzy Numbers

Extension Principle Fuzzy Optimization

Linguistic VariableFuzzy Algorithm

Approximate Reasoning

Foundation

ClusteringStatisticsPattern RecognitionData Processing

Decision MakingEvaluationEstimationExpert Systems

Fuzzy ComputerFuzzy Control

ApplicationsApplications

지 식인 식 추 론판 단 추 론평 가 추 론

자 동 기 능로 봇

인 공 지 능인 공 생 명

< >기 계 시 스 템° íµ µÀ Ç Á ö½ÄÀ »

± â° è¿ ¡ À Ô· ÂÇ Ï Â ¹ ®Á ¦

모 델 인 간 신 뢰 도 모 델

/ 사 고 행 동 모 델 수 요 경 향 모 델

분 석대 중 인 식 분 석

에 너 지 분 석분 류 분 석

평 가위 험 평 가환 경 평 가

< >인 간 시 스 템À Î° £° ú » çÈ À Ç ¹ ®Á ¦µ éÀ »

° úÇ ÐÀ û ¹ æ¹ ýÀ · Î ½Ãµ µÇ Ï Â ½Ã½ºÅ Û

시 스 템전 문 가 시 스 템

보 험 시 스 템C AD/ C AI

/진 단 결 정의 료 진 단장 비 진 단경 영 결 정

< >인 간 과 기 계 시 스 템À Î° £À Ì³ ª ± â° è ¾ î À Ç ÑÂ Ê¿ ¡ ¸

À ÇÁ Ç Ò ¼ö ¾ø Â ½Ã½ºÅ Û

인 간 과 정 보 시 스 템

지 식인 식 추 론판 단 추 론평 가 추 론

자 동 기 능로 봇

인 공 지 능인 공 생 명

< >기 계 시 스 템° íµ µÀ Ç Á ö½ÄÀ »

± â° è¿ ¡ À Ô· ÂÇ Ï Â ¹ ®Á ¦

모 델 인 간 신 뢰 도 모 델

/ 사 고 행 동 모 델 수 요 경 향 모 델

분 석대 중 인 식 분 석

에 너 지 분 석분 류 분 석

평 가위 험 평 가환 경 평 가

< >인 간 시 스 템À Î° £° ú » çÈ À Ç ¹ ®Á ¦µ éÀ »

° úÇ ÐÀ û ¹ æ¹ ýÀ · Î ½Ãµ µÇ Ï Â ½Ã½ºÅ Û

시 스 템전 문 가 시 스 템

보 험 시 스 템C AD/ C AI

/진 단 결 정의 료 진 단장 비 진 단경 영 결 정

< >인 간 과 기 계 시 스 템À Î° £À Ì³ ª ± â° è ¾ î À Ç ÑÂ Ê¿ ¡ ¸

À ÇÁ Ç Ò ¼ö ¾ø Â ½Ã½ºÅ Û

인 간 과 정 보 시 스 템

Topic in the ClassTopic in the Class

• Theory on fuzzy sets1) fuzzy set

2) fuzzy number

3) fuzzy logic

4) fuzzy relation

• Applications1) fuzzy database

2) fuzzy control and expert system

3) robot

4) fuzzy computer

5) pattern recognition

• Rough Sets & Applications

Fuzzy SetsFuzzy Sets

Definition) Fuzzy sunset F on U, the universe of discourse caDefinition) Fuzzy sunset F on U, the universe of discourse can be represented with the membership grade, n be represented with the membership grade, FF(u) for all (u) for all u u U, which is defined byU, which is defined by

F F : U : U [0,1]. [0,1].

Note: Note: 1) The membership function 1) The membership function FF(u) represents the degree o(u) represents the degree o

f belongedness of u to the set F.f belongedness of u to the set F. 2) A crisp set is a special case of a fuzzy set, where2) A crisp set is a special case of a fuzzy set, where

F F : U : U {0,1}. {0,1}.


F = {(uF = {(uii, , FF(u(uii) |u ) |u iiU }U }

= {= {FF(u(uii) / u) / uii |u |u iiU }U }

= = FF(u(uii) / u) / ui i if U is discreteif U is discrete

F = F = FF(u) / u(u) / u if U is continuousif U is continuous

ex) F = {(a, 0.5), (b, 0.7), (c, 0.1)} ex) F = {(a, 0.5), (b, 0.7), (c, 0.1)} ex) F = Real numbers close to 0ex) F = Real numbers close to 0 F = F = FF(x) / x where (x) / x where FF(x) = 1/(1+x(x) = 1/(1+x22))

ex) F = Real numbers very close to 0ex) F = Real numbers very close to 0 F = F = FF(x) / x where (x) / x where FF(x) = {1/(1+x(x) = {1/(1+x22)})}22


Definition) Support of set F is defined byDefinition) Support of set F is defined by supp(F) = { u supp(F) = { u U| U| FF(u) (u) 0}0}

Definition) Height of set FDefinition) Height of set F h(F) = Max{ h(F) = Max{ FF(u), (u), u u U} U}

Definition) Normalized fuzzy set is the fuzzy set with Definition) Normalized fuzzy set is the fuzzy set with h(F) = 1h(F) = 1

Definition) Definition) - level set, - level set, - cut of F - cut of F FF = {u = {u U| U| FF(u) (u) }}


Definition) Convex fuzzy set F: The fuzzy set that satisfies Definition) Convex fuzzy set F: The fuzzy set that satisfies FF(u) (u) FF(u(u11) ) FF(u(u22) (u) (u1 1 << uu < < uu22) ) u u F F

u

u1 u2

OperationsOperations

Suppose U is the universe of discourse and Suppose U is the universe of discourse and F, and G are fuzzy sets defined on U.F, and G are fuzzy sets defined on U.

Definition) F = G (Identity) Definition) F = G (Identity) FF(u) = (u) = GG(u) (u) Definition) F Definition) F G (Subset) G (Subset) FF(u) < (u) < GG(u) (u)

Definition) FDefinition) Fuzzy union:uzzy union: F F G G F F GG(u) = Max[(u) = Max[FF(u), (u), GG(u)](u)] = = FF(u) (u) GG(u) (u) u u UUDefinition) FDefinition) Fuzzy intersection: uzzy intersection: F F G G F F GG(u) = Min[(u) = Min[FF(u), (u), GG(u)](u)] = = FF(u) (u) GG(u) (u) u u UUDefinition) FDefinition) Fuzzy complement)uzzy complement) F FCC(~F)(~F) FFc(u) = 1- c(u) = 1- FF(u) (u) u u UU


Properties of Standard Fuzzy OperatorsProperties of Standard Fuzzy Operators

1) Involution : (F1) Involution : (Fcc))cc = F = F2) Commutative : F 2) Commutative : F G = G G = G F F F F G = G G = G F F3) Associativity : F 3) Associativity : F (G (G H) = (F H) = (F G) G) H H F F (G (G H) = (F H) = (F G) G) H H4) Distributivity : F 4) Distributivity : F (G (G H) = (F H) = (F G) G) (F (F H) H) F F (G (G H) = (F H) = (F G) G) (F (F H) H)5) Idempotency : F 5) Idempotency : F F = F F = F F F F = F F = F


6) Absorption : 6) Absorption : F F (F (F G) = F G) = F F F (F (F G ) = F G ) = F

7) Absorption by 7) Absorption by and U : and U : F F = = , , F F U = U U = U

8) Identity : 8) Identity : F F = F = F F F U = F U = F

9) DeMorgan’s Law:9) DeMorgan’s Law: (F (F G) G) CC= F= FCC G GCC (F (F G) G) CC= F= FCC G GCC 10) Equivalence : 10) Equivalence : (F(FCC G) G) (F (F G GCC) = (F) = (FCC G GCC) ) (F (F G) G)11) Symmetrical difference: 11) Symmetrical difference:

(F(FC C G) G) (F (F G GCC) = (F) = (FCC G GCC) ) (F (F G) G)


Note: The two conventional identity do not satisfy in standard operatNote: The two conventional identity do not satisfy in standard operation;ion;

Law of contradiction : F Law of contradiction : F F FCC = = Law of excluded middle : F Law of excluded middle : F F FCC = U = U

Other fuzzy operationsOther fuzzy operations(1) Disjunctive Sum: F(1) Disjunctive Sum: F G = (F G = (F G GCC) ) (F (FCC G) G)(2) Set Difference: (2) Set Difference: Simple Difference : F-G = F Simple Difference : F-G = F G GCC

F -GF -G(u) = Min[(u) = Min[FF(u), 1-(u), 1-GG(u)] (u)] u u U U Bounded Difference: FBounded Difference: F G G FFGG(u) = Max[0, (u) = Max[0, FF(u)-(u)-GG(u)] (u)] u u U U


(3) Bounded Sum: F (3) Bounded Sum: F G G F F G G(u) = Min[1, (u) = Min[1, FF(u) + (u) + GG(u)] (u)] u u U U

(4) Bounded Product: F (4) Bounded Product: F G G F F G G(u) = Max[0, (u) = Max[0, FF(u) + (u) + GG(u)-1] (u)-1] u u U U

(5) Product of Fuzzy Set for Hedge(5) Product of Fuzzy Set for HedgeFF2 2 : : FF

22 (u)(u) = [= [F F (u)](u)]2 2

FFmm : : FFmm

(u)(u) = [= [F F (u)](u)]mm

(6) Cartesian Product of Fuzzy Sets F(6) Cartesian Product of Fuzzy Sets F11 F F22 F Fn n

FF11 F F22 F Fnn(u(u11 , u , u22, , ,u ,unn) = Min[) = Min[FF11

(u(u11), ), ,,,,FFnn(u(unn) ]) ]

uui i F Fii

Generalized Fuzzy SetsGeneralized Fuzzy Sets

Interval-Valued Fuzzy SetInterval-Valued Fuzzy Set

Fuzzy Set of Type 2Fuzzy Set of Type 2

L-Fuzzy SetL-Fuzzy Set

[0,1]in intervals all ofset thedenotes ])1,0([

])1,0([:

XA

[0,1]in defined setsfuzzy all offamily thedenotes ])1,0([

])1,0([:

F

FXA

ordered.partially least at set thelattice, a denotes

:

L

LXA

Generalized Fuzzy SetsGeneralized Fuzzy Sets

Level-2 Fuzzy SetLevel-2 Fuzzy Set

Ex: “x is close to r”Ex: “x is close to r”

If r is precisely specified, then it can be If r is precisely specified, then it can be represented by an ordinary fuzzy setrepresented by an ordinary fuzzy set

If r is approximately specified, A(B), the If r is approximately specified, A(B), the fuzzy set A of a fuzzy set B can be used.fuzzy set A of a fuzzy set B can be used.

Xon defined setsfuzzy all thedenotes )(

]1,0[)(:

XF

XFA

Additional DefinitionsAdditional Definitions

Cardinality of A (Sigma Count of A)Cardinality of A (Sigma Count of A)

Ex: A = .1/1 + .5/2 + 1./3 + .5/4 + .1/6Ex: A = .1/1 + .5/2 + 1./3 + .5/4 + .1/6 |A| = 2.2|A| = 2.2

Degree of Subsethood S(A,B)Degree of Subsethood S(A,B)

Hamming DistanceHamming Distance

Xx

xAA )(

A

BABAS

xBxAAA

BASXx

),(

)])()(,0max[(1

),(

),( BAS

Xx

xBxABAd )()(),(

Decomposition of Fuzzy Decomposition of Fuzzy SetsSets

Decomposition using Decomposition using - level set - level set

Ex: Ex:

.

where

),(every For

]1,0[

AofsetlevelAA

AA

XFA

54321 /1/8./6./4./2. xxxxxA

Additional Notions of OperatorsAdditional Notions of Operators

Axiomatic Definition of Complement CAxiomatic Definition of Complement C Boundary ConditionBoundary Condition

MonotonicityMonotonicity

ContinuityContinuity

InvolutiveInvolutive

.0(1) and 1)0( CC

).( )( then ba if [0,1], allFor bCaCa, b

function. continuous a is C

]1,0[ allfor ))(( aaaCC


Some complement operatorsSome complement operators Sugeno ClassSugeno Class

Yager ClassYager Class

Note: Parameters can be adjusted to obtain soNote: Parameters can be adjusted to obtain some desired behavior. me desired behavior.

),1( 1

1)(

a

aac

),0( )1()( /1 waac www


Characterization Theorem of Characterization Theorem of ComplementComplement By strictly increasing functionBy strictly increasing function

By strictly decreasing functionBy strictly decreasing function

.0)0(such that to[0,1] from

function contiuous increasingstrictly a is where

))()1(()( 1

g

g

agggaC

R

.0)1(such that to[0,1] from

function contiuous decreasingstrictly a is where

))()0(()( 1

f

f

afffaC

R


Axiomatic Definition of t-norm Axiomatic Definition of t-norm ii Boundary ConditionBoundary Condition MonotonicityMonotonicity CommutativeCommutative AssociativeAssociative ContinuousContinuous SubidempotecySubidempotecy Strict MonotonicityStrict Monotonicity

aai )1,(

),(),( daibaidb

),(),( abibai

)),,(()),(,( dbaiidbiai

function. continuous a is i

aaai ),(

),(),( and 22112121 baibaibbaa


Some intersection operatorsSome intersection operators Algebraic Product Algebraic Product Bounded DifferenceBounded Difference Drastic IntersectionDrastic Intersection

Yager’s t-normYager’s t-norm

babai ),(

)1,0max(),( babai

otherwise. 0

1a when

1b when

),(min b

a

bai

).,0( where)])1()1[(,1min(1),( /1 wbabai wwww


Notes:Notes: Boundary of t-normBoundary of t-norm

Characterization TheoremCharacterization Theorem

t-norm can be generated by a t-norm can be generated by a generating function.generating function.

),min(),(),(min babaibai


Axiomatic Definition of co-norm Axiomatic Definition of co-norm uu Boundary ConditionBoundary Condition MonotonicityMonotonicity CommutativeCommutative AssociativeAssociative ContinuousContinuous SubidempotecySubidempotecy Strict MonotonicityStrict Monotonicity

aau )0,(

),(),( daubaudb

),(),( abubau

)),,(()),(,( dbauudbuau

function. continuous a is u

aaau ),(

),(),( and 22112121 baubaubbaa


Some union operatorsSome union operators Algebraic SumAlgebraic Sum Bounded SumBounded Sum Drastic UnionDrastic Union

Yager’s conormYager’s conorm

bababau ),(

),1min(),( babai

otherwise. 1

0a when

0b when

),(max b

a

bau

).,0( where))(,1min(),( /1 wbabau wwww


Notes:Notes: Boundary of co-normBoundary of co-norm

Characterization TheoremCharacterization Theorem

Co-norm can be generated by a Co-norm can be generated by a generating function.generating function.

),(),(),max( max baubauba


Dual Triples <intersec, union, comp>Dual Triples <intersec, union, comp> Generalized DeMorgan’s LawGeneralized DeMorgan’s Law

ExamplesExamples

))(),(()),((

))(),(()),((

bcacibauc

bcacubaic

abaubai

ababa

aabbaab

ababa

1),,(),,(

1),,1min(),1,0max(

1,,

1),,max(),,min(

maxmin


Aggregation Operators for IFAggregation Operators for IF Compensation (Averaging) Operators Compensation (Averaging) Operators hh

Example:Example: Gamma Model Gamma Model

Averaging Operators: Averaging Operators:

Mean, Geometric Mean, Harmonic MeanMean, Geometric Mean, Harmonic Mean

),(),(),(

),max(),(),min(

baubahbai

babahba

].1,0[ and where

))1(1()(),...,,(

1

1

1

121

n

ii

n

ii

n

iin

n

aaaaa ii


Ordered Weighted AveragingOrdered Weighted Averaging DefinitionDefinition

Note: Different operations Note: Different operations Min -> [1,0,…,0], Max -> [0, 0, …, 1] Min -> [1,0,…,0], Max -> [0, 0, …, 1] Median -> [0, 0, ..1, 0, ..0]Median -> [0, 0, ..1, 0, ..0] Mean -> [1/n, …. 1/n]Mean -> [1/n, …. 1/n]

.order. ascendingin

data sorted ofcomponent th -i theare and 1 where

),...,,(

1

121

sabw

bwaaaOWA

ji

n

ii

i

n

iin

Documents

Fuzzy Sets and Applications Introduction Introduction Fuzzy Sets and Operations Fuzzy Sets and Operations