1. Fuzzy Sets and Applications Introduction Introduction Fuzzy Sets and Operations Fuzzy Sets and Operations

2. Why fuzzy sets? Types of Uncertainty 1. 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”

3. Why fuzzy sets? 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

4. Why fuzzy sets? Powerful tool for vagueness Description of vague linguistic terms and algorithms Description of vague linguistic terms and algorithms Operation on vague linguistic terms Operation on vague linguistic terms Reasoning with vague linguistic rules Reasoning with vague linguistic rules Representation of clusters with vague boundaries Representation of clusters with vague boundaries

5. History of Fuzzy Sets History of Fuzzy Sets and Applications 1965 Zadeh Fuzzy Sets 1965 Zadeh Fuzzy Sets 1972 Sugeno Fuzzy Integrals 1972 Sugeno Fuzzy Integrals 1975 Zadeh 1975 Zadeh Fuzzy Algorithm & Approximate Reasoning 1974 Mamdani Fuzzy Control 1974 Mamdani Fuzzy Control 1978 North Holland Fuzzy Sets and Systems 1978 North Holland Fuzzy Sets and Systems 1982 Bezdek Fuzzy C-Mean 1982 Bezdek Fuzzy C-Mean 1987 Korea Fuzzy Temperature Control 1987 Korea Fuzzy Temperature Control

6. Current Scope of Fuzzy Society Fuzzy Sets Applications Methods Fuzzy Measure Fuzzy Logic Fuzzy Integrals Fuzzy Measure Fuzzy Relation Fuzzy Numbers Extension Principle Fuzzy Optimization Linguistic Variable Fuzzy Algorithm Approximate Reasoning Foundation Clustering Statistics Pattern Recognition Data Processing Decision Making Evaluation Estimation Expert Systems Fuzzy Computer Fuzzy Control

7. Applications

8. Topic in the Class Theory on fuzzy sets 1) fuzzy set 2) fuzzy number 3) fuzzy logic 4) fuzzy relation Applications 1) fuzzy database 2) fuzzy control and expert system 3) robot 4) fuzzy computer 5) pattern recognition Rough Sets & Applications

9. Fuzzy Sets Definition) Fuzzy sunset F on U, the universe of discourse can be represented with the membership grade,  F (u) for all u  U, which is defined by  F : U  [0,1]. Note: 1) The membership function  F (u) represents the degree of belongedness of u to the set F. 1) The membership function  F (u) represents the degree of belongedness of u to the set F. 2) A crisp set is a special case of a fuzzy set, where 2) A crisp set is a special case of a fuzzy set, where  F : U  {0,1}.

10. Fuzzy Sets F = {(u i,  F (u i ) |u i  U } = {  F (u i ) / u i |u i  U } = {  F (u i ) / u i |u i  U } =   F (u i ) / u i if U is discrete =   F (u i ) / u i if U is discrete F =   F (u) / u if U is continuous ex) F = {(a, 0.5), (b, 0.7), (c, 0.1)} ex) F = Real numbers close to 0 F =   F (x) / x where  F (x) = 1/(1+x 2 ) F =   F (x) / x where  F (x) = 1/(1+x 2 ) ex) F = Real numbers very close to 0 F =   F (x) / x where  F (x) = {1/(1+x 2 )} 2 F =   F (x) / x where  F (x) = {1/(1+x 2 )} 2

11. Fuzzy Sets Definition) Support of set F is defined by supp(F) = { u  U|  F (u)  0} supp(F) = { u  U|  F (u)  0} Definition) Height of set F h(F) = Max{  F (u),  u  U} h(F) = Max{  F (u),  u  U} Definition) Normalized fuzzy set is the fuzzy set with h(F) = 1 Definition)  - level set,  - cut of F F  = {u  U|  F (u)   } F  = {u  U|  F (u)   }

12. Fuzzy Sets Definition) Convex fuzzy set F: The fuzzy set that satisfies  F (u)   F (u 1 )   F (u 2 ) (u 1 < u < u 2 )  u  F  F (u)   F (u 1 )   F (u 2 ) (u 1 < u < u 2 )  u  F u  u1u1 u2u2

13. Operations Suppose U is the universe of discourse and F, and G are fuzzy sets defined on U. Definition) F = G (Identity)   F (u) =  G (u) Definition) F  G (Subset)   F (u) <  G (u) Definition) Fuzzy union: F  G  F  G (u) = Max[  F (u),  G (u)]  F  G (u) = Max[  F (u),  G (u)] =  F (u)   G (u)  u  U =  F (u)   G (u)  u  U Definition) Fuzzy intersection: F  G  F  G (u) = Min[  F (u),  G (u)]  F  G (u) = Min[  F (u),  G (u)] =  F (u)   G (u)  u  U =  F (u)   G (u)  u  U Definition) Fuzzy complement) F C (~F)  F c(u) = 1-  F (u)  u  U  F c(u) = 1-  F (u)  u  U

14. Operations Properties of Standard Fuzzy Operators 1) Involution : (F c ) c = F 2) Commutative : F  G = G  F F  G = G  F F  G = G  F 3) Associativity : F  (G  H) = (F  G)  H F  (G  H) = (F  G)  H F  (G  H) = (F  G)  H 4) Distributivity : F  (G  H) = (F  G)  (F  H) F  (G  H) = (F  G)  (F  H) F  (G  H) = (F  G)  (F  H) 5) Idempotency : F  F = F F  F = F F  F = F

15. Operations 6) Absorption : F  (F  G) = FF  (F  G ) = F 7) Absorption by  and U : F   = , F  U = U 8) Identity : F   = FF  U = F 9) DeMorgan’s Law: (F  G) C = F C  G C (F  G) C = F C  G C (F  G) C = F C  G C (F  G) C = F C  G C 10) Equivalence : (F C  G)  (F  G C ) = (F C  G C )  (F  G) (F C  G)  (F  G C ) = (F C  G C )  (F  G) 11) Symmetrical difference: (F C  G)  (F  G C ) = (F C  G C )  (F  G)

16. Operations Note: The two conventional identity do not satisfy in standard operation; Law of contradiction : F  F C =  Law of excluded middle : F  F C = U Other fuzzy operations (1) Disjunctive Sum: F  G = (F  G C )  (F C  G) (2) Set Difference:  Simple Difference : F-G = F  G C  Simple Difference : F-G = F  G C  F -G (u) = Min[  F (u), 1-  G (u)]  u  U  F -G (u) = Min[  F (u), 1-  G (u)]  u  U  Bounded Difference: F  G  Bounded Difference: F  G  F  G (u) = Max[0,  F (u)-  G (u)]  u  U  F  G (u) = Max[0,  F (u)-  G (u)]  u  U

17. Operations (3) Bounded Sum: F  G  F  G (u) = Min[1,  F (u) +  G (u)]  u  U  F  G (u) = Min[1,  F (u) +  G (u)]  u  U (4) Bounded Product: F  G  F  G (u) = Max[0,  F (u) +  G (u)-1]  u  U  F  G (u) = Max[0,  F (u) +  G (u)-1]  u  U (5) Product of Fuzzy Set for Hedge F 2 :  F 2 (u) = [  F (u)] 2 F m :  F m (u) = [  F (u)] m F m :  F m (u) = [  F (u)] m (6) Cartesian Product of Fuzzy Sets F 1  F 2    F n  F 1  F 2    F n (u 1, u 2, ,u n ) = Min[  F 1 (u 1 ), ,,  F n (u n ) ]  F 1  F 2    F n (u 1, u 2, ,u n ) = Min[  F 1 (u 1 ), ,,  F n (u n ) ]  u i  F i  u i  F i

18. Generalized Fuzzy Sets Interval-Valued Fuzzy Set Fuzzy Set of Type 2 L-Fuzzy Set

19. Generalized Fuzzy Sets Level-2 Fuzzy Set Ex: “x is close to r” Ex: “x is close to r” If r is precisely specified, then it can be represented by an ordinary fuzzy set If r is precisely specified, then it can be represented by an ordinary fuzzy set If r is approximately specified, A(B), the fuzzy set A of a fuzzy set B can be used. If r is approximately specified, A(B), the fuzzy set A of a fuzzy set B can be used.

20. Additional Definitions Cardinality of A (Sigma Count of A) Ex: A =.1/1 +.5/2 + 1./3 +.5/4 +.1/6 Ex: A =.1/1 +.5/2 + 1./3 +.5/4 +.1/6 |A| = 2.2 |A| = 2.2 Degree of Subsethood S(A,B) Hamming Distance

21. Decomposition of Fuzzy Sets Decomposition using  - level set Ex: Ex:

22. Additional Notions of Operators Axiomatic Definition of Complement C Boundary Condition Boundary Condition Monotonicity Monotonicity Continuity Continuity Involutive Involutive

23. Additional Notions of Operators Some complement operators Sugeno Class Sugeno Class Yager Class Yager Class Note: Parameters can be adjusted to obtain some desired behavior. Note: Parameters can be adjusted to obtain some desired behavior.

24. Additional Notions of Operators Characterization Theorem of Complement By strictly increasing function By strictly increasing function By strictly decreasing function By strictly decreasing function

25. Additional Notions of Operators Axiomatic Definition of t-norm i Boundary Condition Boundary Condition Monotonicity Monotonicity Commutative Commutative Associative Associative Continuous Continuous Subidempotecy Subidempotecy Strict Monotonicity Strict Monotonicity

26. Additional Notions of Operators Some intersection operators Algebraic Product Algebraic Product Bounded Difference Bounded Difference Drastic Intersection Drastic Intersection Yager’s t-norm Yager’s t-norm

27. Additional Notions of Operators Notes: Boundary of t-norm Boundary of t-norm Characterization Theorem Characterization Theorem t-norm can be generated by a generating function. t-norm can be generated by a generating function.

28. Additional Notions of Operators Axiomatic Definition of co-norm u Boundary Condition Boundary Condition Monotonicity Monotonicity Commutative Commutative Associative Associative Continuous Continuous Subidempotecy Subidempotecy Strict Monotonicity Strict Monotonicity

29. Additional Notions of Operators Some union operators Algebraic Sum Algebraic Sum Bounded Sum Bounded Sum Drastic Union Drastic Union Yager’s conorm Yager’s conorm

30. Additional Notions of Operators Notes: Boundary of co-norm Boundary of co-norm Characterization Theorem Characterization Theorem Co-norm can be generated by a generating function. Co-norm can be generated by a generating function.

31. Additional Notions of Operators Dual Triples Dual Triples Generalized DeMorgan’s Law Generalized DeMorgan’s Law Examples Examples

32. Additional Notions of Operators Aggregation Operators for IF Compensation (Averaging) Operators h Compensation (Averaging) Operators h Example: Example: Gamma Model Gamma Model Averaging Operators: Averaging Operators: Mean, Geometric Mean, Harmonic Mean Mean, Geometric Mean, Harmonic Mean

33. Additional Notions of Operators Ordered Weighted Averaging Definition Definition 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]