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Chapter 8 Fuzzy Inference 模糊推論

Chapter 8 Fuzzy Inference 模糊推論. Fuzzy set and Fuzzy Logic why “Fuzzy Subset” ? Ordinary set -- the foundation of present day mathematics.(S) S : a set e 5 : an element But in real world , the relation is usually “fuzzy” ! John is 170 cm John : an element. S = {x|x is tall}

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Chapter 8 Fuzzy Inference 模糊推論

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  1. Chapter 8Fuzzy Inference模糊推論

  2. Fuzzy set and Fuzzy Logic • why “Fuzzy Subset” ? Ordinary set -- the foundation of present day mathematics.(S) S : a set e5 : an element But in real world , the relation is usually “fuzzy” ! John is 170 cm John : an element G.J. Hwang

  3. S = {x|x is tall} 180cm 高的人 S ? Yes 179cm 高的人 S ? Yes (179 和180只差 1cm) 178cm 高的人 S ? Yes (178 和179只差 1cm) • • • 170cm 高的人 S ? Yes (170 和171只差 1cm) 169cm 高的人 S ? Yes No (169 和170只差 1cm) • • • Why? 既然170是, 為何169不是? 120cm 高的人 S ? Yes (120 和121只差 1cm) G.J. Hwang

  4. S ={x|x is tall} 假如找100個人投票,互相推選屬於S和不屬於S的人 150cm 160cm 170cm 180cm 1 0.5 0 0 John is 180cm  John  S with degree 1.0 John is 165cm  John  S with degree 0.5 John is 150cm  John  S with degree 0 JohnS G.J. Hwang

  5. ordinary set is a particular case of the theory of fuzzy subset. let E be a set and A be a subset of E A E Characteristic function(x) x)= 1 if x  A (yes) x)= 0 if x  A (no) e.g. E={x1,x2,x3,x4,x5} let A = {x2,x3,x5} x1) = 0, x2 ) = 1, x3)= 1 x4) = 0, x5) = 1 G.J. Hwang

  6. A different representation A = {(x1,0),(x2,1),(x3,1),(x4,0),(x5,1)}  A A = 0 A A = E  IF x A , x A (x)= 1, A(x)= 0 consider A ={x2,x3,x5} A(x1) = 1, A(x2) = 0, A(x3) = 0 A(x4) = 1, A(x5) = 0 A = {(x1,1),(x2,0),(x3,0),(x4,1),(x5,0)} G.J. Hwang

  7. 0 1 0 1 0 0 0 1 Given two subsets A and B (x) = 1, if x  A = 0, if x A (x) = 1, if x  B = 0, if x  AB(x)= 1, if x  A B = 0, if x A B  AB(x)= (x) • (x)  Boolean product G.J. Hwang

  8. Union AB (x)= 1, if x  A B = 0, if x A B AB (x)= (x) (x)  + 0 1  + Boolean sum 0 1 0 1 1 1 • e.g. E = {x1,x2,x3,x4,x5} • two subsets A and B • A={x2,x3,x5}, B={x1,x3,x5} • AB = {(x1,0 1),(x2,1 0), (x3,1 1),(x4,0 0),(x5,1 1)} • = {(x1,1),(x2,1),(x3,1),(x4,0),(x5,1)}  +  +  +  +  + G.J. Hwang

  9. The concept of Fuzzy Subset • xi of E 或多或少  是A的元素 • A = {(x1|0.2),(x2|0),(x3|0.3),(x4|1),(x5|0.8)} Fuzzy Subset x1屬於A的 程度 (可能由0~1.0) 通常是主觀的認定,但至少 表達了Xis之間的相對程度 AE A is a Fuzzy Subset of E  AE x1 , x2 , x3 0.2 0 0.3   membership x2A G.J. Hwang

  10. Zadehs definition of Fuzzy subset Let E be a set, denumerable or not, let x be an element of E. Then Fuzzy subset A of E is a set of ordered pairs {(x|(x)}, xE. Where (x) : grade of membership of x in A (x) takes its values in a set M (membership set) x M IF M={0,1} fuzzy subset of A will be a nonfuzzy subset  (or ordinary set) mapping (x) G.J. Hwang

  11. E.g. Let N be the set of natural numbers N = {0,1,2,3,4,5,6,...} consider the fuzzy set A of smallnatural numbers  A = {(0/1),(1/0.8),(2/0.6),(3/0.4),(4/0.2),(5/0),(6/0),...}  用傳統的ordinary set很難表達 A = {(0,1),(1,1),(2,1),(3,1),(4,1),(5,0),(6,0),...} ? G.J. Hwang

  12. 0 for x  2[(x- )/()]2 for x 1- 2[(x- )/()]2 for x 1 for x S (x; ) = S- Function 1 0.5 0    G.J. Hwang

  13. 0 for x  2[(x- 5)/(7)]2 = [(x- 5) 2/2] for 5x 1- 2[(x- 7)/(7)]2 =1-[(x- 7) 2 /2] for 6x 1 for x S (x; ) = 1.0 0.9 TALL Membership Function 0.5 6 6.5 7 Height in Feet A membership Function for the Fuzzy Set TALL G.J. Hwang

  14. Close- Function close(x; ,)= with crossover points x =  1.0 close(x; ,) 0.5 x    G.J. Hwang

  15. E = { x|x= 價格合理的牛排 ?} 220NT  120 =220NT =120NT 1.0 close(x; 220,120) 0.5 100 340 220NT G.J. Hwang

  16. 1  0.5  x       function G.J. Hwang

  17.    價格合理的牛排 1  0.5    120 220NT 320   G.J. Hwang

  18. d1 d3 d2 d4 d5 Fuzzy Database systems 找一個停車容易,且價格合理的餐廳 以停車為優先考慮 E = {x|x = 離火車站近的餐館 }  Km G.J. Hwang

  19. dialogue Laws of thought are Fuzzy Fuzzy Logic Binary Logic: The logic associated with the Boolean theory of set Fuzzy Logic : The Logic associated with the same manner with the theory of fuzzy subsets G.J. Hwang

  20. A(x) : membership function of the element x in the fuzzy subset A M = [0,1] Let A, B be two fuzzy subsets of E and x is an element of E a = A(x) , b = A(x) a,b,...M = [0,1] G.J. Hwang

  21. Distributivity Commutativity Associativity G.J. Hwang

  22. DeMorgan s Law are true, but not trivial G.J. Hwang

  23. Tall Not Short 5 0 0.00 5 4 0.08 5 8 0.326 0 0.50 6 4 0.82 6 8 0.98 7 0 1.00 5 0 0.00 5 4 0.08 5 8 0.326 0 0.50 6 4 0.82 6 8 0.98 7 0 1.00 IF tall THEN not short G.J. Hwang

  24. Complementation Tall Not Tall 5 0 0.00 5 4 0.08 5 8 0.326 0 0.50 6 4 0.82 6 8 0.98 7 0 1.00 5 0 1.00 5 4 0.92 5 8 0.686 0 0.50 6 4 0.18 6 8 0.02 7 0 0.00 G.J. Hwang

  25. 5 0 1.00 5 4 0.92 5 8 0.686 0 0.50 AND 6 4 0.18 6 8 0.02 7 0 0.00 5 0 0.00 5 4 0.08 5 8 0.326 0 0.50 6 4 0.82 6 8 0.98 7 0 1.00 5 0 0.00 5 4 0.08 5 8 0.326 0 0.50 6 4 0.18 6 8 0.02 7 0 0.00 Not Tall Not Short Middle-Sized G.J. Hwang

  26. r=0.7 r=0.5 r=0.3 Linguistic Hedge Operation- Scalar Ura(x) = rUa(x) G.J. Hwang

  27. r = 0.5 r = 2 r = 4 Linguistic Hedge Operation- Power Uar(x) = ((Ua(x))r G.J. Hwang

  28. NORM(A) A Linguistic Hedge Operation- Normalization UA =supUA(X) G.J. Hwang

  29. A CON(A) Linguistic Hedge Operation- Concentration Ucon(A) = UA2(X) G.J. Hwang

  30. Linguistic Hedge Operation- Dilation UDIL(A)(X) = UA0.5(X)  DIL A .5  G.J. Hwang

  31. A  INT(A) 0.5  2(UA(X))2 0 UA(X) 0.5 1-2(1-UA(X))2 0.5 UA(X) 1.0  UINT(A)(X) = Linguistic Hedge Operation- Intensification G.J. Hwang

  32. Usage of Linguistic Hedge Operations Very A = CON(A) More Or less A = DIL(A) Slightly A = NORM(A and not (very A)) Sort of A = NORM(not (CON(A)2and DIL(A)) Pretty A = NORM(INT(A) and not INT(CON(A))) Rather A = NORM(INT(A)) G.J. Hwang

  33. Linguistic truth value True Very true More or less true Completely true False Very False More or less false Completely false Unknown Undefined G.J. Hwang

  34. Fuzzy Proposition “Mr.Wang is young.” is true. “Mr.Wang is young.” is very true. “Mr.Wang is young.”is more or less true. G.J. Hwang

  35. Tall Height Degree of membership 5 0 0.0 5 4 0.1 5 8 0.3 6 0 0.5 6 4 0.8 6 8 0.9 7 0 1.00 VERY Tall Height Degree of membership 5 0 0.0 5 4 0.01 5 8 0.09 6 0 0.25 6 4 0.64 6 8 0.81 7 0 1.00 G.J. Hwang

  36. ~ A  A E ~   ~ A  A  ~   ~ A  A = min ( A(X),A(X))  0.5 ~  ~ ~  ~ A  A = max ( A(X), A (X)0.5 ~  ~ ~  ~ A ~ A 1 0.5 0 X Figure 5-12 Fuzzy Complement G.J. Hwang

  37. Fuzzy Relation A crisp relation represents the presence or absence of association, interaction, or interconnectedness between the elements of two or more sets. G.J. Hwang

  38. Heavy(100) (140) (160) (200) (240) (280) (300) Tall 0.00 0.00 0.18 0.50 0.98 1.00 1.00 (5 0 ) 0.00 .00 .00 .00 .00 .00 .00 .00 (5 4 ) 0.08 .00 .00 .08 .08 .08 .08 .08 (5 8 ) 0.32 .00 .00 .18 .32 .32 .32 .32 (6 0 ) 0.50 .00 .00 .18 .50 .50 .50 .50 (6 4 ) 0.82 .00 .00 .18 .50 .82 .82 .82 (6 8 ) 0.98 .00 .00 .18 .50 .98 .98 .98 (7 0 ) 1.00 .00 .00 .18 .50 .98 1.00 1.00 Binary Relation • Any relation between two sets X and Y is known as a binary relation. It is usually denoted by R(X,Y). G.J. Hwang

  39. Y1 Y2 Y3 Y4 X1 X2 X3 X4 .9 0 .5 .3 .4 .2 .1 .9 0 0 .5 .6 0 .2 0 .4 Representation of binary relations Membership matrices G.J. Hwang

  40. y x 120 130 140 150 160 120 1.0 0.7 0.4 0.2 0.0 130 0.7 1.0 0.6 0.5 0.2 140 0.4 0.6 1.0 0.8 0.5 150 0.2 0.5 0.8 1.0 0.8 160 0.0 0.2 0.5 0.8 1.0 R3(y) = R1(x) R2(x,y) max min(u1(x),u2(x,y)) x Max_Min Composition R1(x) = 0.6/140 + 0.8/150 + 1.0/160 R2: The Relation APPROXIMATELY EQUAL Defined on Weights G.J. Hwang

  41. 1.0 0.7 0.4 0.2 0.0 0.7 1.0 0.6 0.5 0.2 0.4 0.6 1.0 0.8 0.5 0.2 0.5 0.8 1.0 0.8 0.0 0.2 0.5 0.8 1.0 R1(x) R2(x,y) 0.0/x1 x1 1.0 y1 + 0.7 0.0/x2 x2 y2 + 0.4 0.6/x3 x3 y3 + 0.2 0.8/x4 x4 y4 + 0.0 1.0/x5 x5 y5 0.4 R3(y) = [0.0 0.0 0.6 0.8 1.0] G.J. Hwang

  42. R3(120) = max min[(.6,.4),(.8,.2)]= max (.4,.2) = 0.4 R3(130) = max min[(.6,.6),(.8,.5),(1,.2)]= max (.6,.5,.2) = 0.6 R3(140) = max min[(.6,.1),(.8,.8),(1,.5)]= max (.6,.8,.5) = 0.8 R3(150) = max min[(.6,.8),(.8,.1),(1,.8)] = max (.6,.8,.8) = 0.8 R3(160) = max min[(.6,.5),(.8,.8),(1,1)] = max (.5,.8,1) = 1 G.J. Hwang

  43. y1 y2 y3 y4 y5 x1 x2 x3 0.1 0.2 0 1 0.7 0.3 0.5 0 0.2 1 0.8 0 1 0.4 0.3 z1 z2 z3 z4 y1 y2 y3 y4 y5 0.9 0 0.3 0.4 0.2 1 0.8 0 0.8 0 0.7 1 0.4 0.2 0.3 0 0 1 0 0.8 Composition of Two Fuzzy Relations R1(x,y) R2(y,z) R3(x,z) = ? G.J. Hwang

  44. R(x) = [0.5 0.2 0.6] R(z) = ? R(z) = R(x) R1(x,y) R2(y,z) = R(x) R3(x,z) 0.10.9 x1 y1 z1 0.4 0.2 0.2 Max 0 y2 0.8 1 y3 0.70.4 y4 0 y5 Mix G.J. Hwang

  45. Membership Grade Image Missile Fighter Airliner 1 1.0 0.0 0.0 2 0.9 0.0 0.1 3 0.4 0.3 0.2 4 0.2 0.3 0.5 5 0.1 0.2 0.7 6 0.1 0.6 0.4 7 0.0 0.7 0.2 8 0.0 0.0 1.0 9 0.0 0.8 0.2 10 0.0 1.0 0.0 Membership Grades for Images Fuzzy Rules G.J. Hwang

  46. 1/M 1 . 9/M + .1/A 2 .4/M + .3/F + .2/A 3 .2/M + .3/F + .5/A 4 .1/M + .2/F + .7/A 5 .1/M + .6/F + .4/A 6 .7/M + .2/A 7 1/A 8 .8/M + .2/A 9 1/F 10 G.J. Hwang

  47. IF IMAGE4 THEN TARGET4 = 0.2/M + 0.3/F + 0.5/A IF IMAGE6 THEN TARGET6 = 0.1/M + 0.6/F + 0.4/A + : set union 假設現由二個不同觀測點得到IMAGE4及IMAGE6 TARGET = TARGET 4 + TARGET 6 = 0.2/M + 0.3/F + 0.5/A + 0.1/M + 0.6/F + 0.4/A = 0.2/M + 0.6/F + 0.5/A G.J. Hwang

  48. removed cement water sand . Maximum and Moments Methods R1: IF MIX is too-wet THEN Add sand and coarse aggregate R2: IF MIX is Workable THEN Leave alone R3: IF MIX is too-stiff THEN Decrease sand and coarse aggregate G.J. Hwang

  49. TOO STIFF WORKABLE TOO WET 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 3 4 5 6 7 8 9 Concrete Slump (inches) Membership Grade Fuzzy Production Rule Antecedents for Concrete Mixture Process G.J. Hwang

  50. IF Concrete-slump = 6 THEN MIX = 0.0/Too-stiff + 1.0/workable + 0.0/Too-wet IF Concrete-slump = 7 THEN MIX = 0.0/Too-stiff + 0.3/workable + 0.0/Too-wet . . . IF Concrete-slump = 4.8 THEN MIX = 0.05/Too-stiff + 0.2/workable +0.0/Too-wet G.J. Hwang

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