fuzzy sets and applications introduction introduction fuzzy sets and operations fuzzy sets and...
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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”
Why fuzzy sets?Why fuzzy sets?
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
Why fuzzy sets?Why fuzzy sets?
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
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± 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
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° 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 ΰ £À ̳ ª ± â° è ¾ î À Ç Ñ ʿ ¡ ¸
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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 ΰ £° 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 ΰ £À ̳ ª ± â° è ¾ î À Ç Ñ ʿ ¡ ¸
À ÇÁ Ç Ò ¼ö ¾ø  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}.
Fuzzy SetsFuzzy Sets
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
Fuzzy SetsFuzzy Sets
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) }}
Fuzzy SetsFuzzy Sets
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
OperationsOperations
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
OperationsOperations
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)
OperationsOperations
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
OperationsOperations
(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([
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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
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Decomposition of Fuzzy Decomposition of Fuzzy SetsSets
Decomposition using Decomposition using - level set - level set
Ex: Ex:
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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
Additional Notions of OperatorsAdditional Notions of Operators
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.
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Additional Notions of OperatorsAdditional Notions of Operators
Characterization Theorem of Characterization Theorem of ComplementComplement By strictly increasing functionBy strictly increasing function
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Additional Notions of OperatorsAdditional Notions of Operators
Axiomatic Definition of t-norm Axiomatic Definition of t-norm ii Boundary ConditionBoundary Condition MonotonicityMonotonicity CommutativeCommutative AssociativeAssociative ContinuousContinuous SubidempotecySubidempotecy Strict MonotonicityStrict Monotonicity
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aaai ),(
),(),( and 22112121 baibaibbaa
Additional Notions of OperatorsAdditional Notions of Operators
Some intersection operatorsSome intersection operators Algebraic Product Algebraic Product Bounded DifferenceBounded Difference Drastic IntersectionDrastic Intersection
Yager’s t-normYager’s t-norm
babai ),(
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Additional Notions of OperatorsAdditional Notions of Operators
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
Additional Notions of OperatorsAdditional Notions of Operators
Axiomatic Definition of co-norm Axiomatic Definition of co-norm uu Boundary ConditionBoundary Condition MonotonicityMonotonicity CommutativeCommutative AssociativeAssociative ContinuousContinuous SubidempotecySubidempotecy Strict MonotonicityStrict Monotonicity
aau )0,(
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Additional Notions of OperatorsAdditional Notions of Operators
Some union operatorsSome union operators Algebraic SumAlgebraic Sum Bounded SumBounded Sum Drastic UnionDrastic Union
Yager’s conormYager’s conorm
bababau ),(
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otherwise. 1
0a when
0b when
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a
bau
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Additional Notions of OperatorsAdditional Notions of Operators
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
Additional Notions of OperatorsAdditional Notions of Operators
Dual Triples <intersec, union, comp>Dual Triples <intersec, union, comp> Generalized DeMorgan’s LawGeneralized DeMorgan’s Law
ExamplesExamples
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Additional Notions of OperatorsAdditional Notions of Operators
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
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Additional Notions of OperatorsAdditional Notions of Operators
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]
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