# 7 Transformations of Fuzzy Sets Fuzzy Systems Engineering Toward - PowerPoint PPT Presentation

## 7 Transformations of Fuzzy Sets Fuzzy Systems Engineering Toward Human-Centric Computing Contents 7.1 The extension principle 7.2 Composition of fuzzy relations 7.3 Fuzzy relational equations 7.4 Associative memories 7.5 Fuzzy numbers and

1. 7 Transformations of Fuzzy Sets Fuzzy Systems Engineering Toward Human-Centric Computing

2. Contents 7.1 The extension principle 7.2 Composition of fuzzy relations 7.3 Fuzzy relational equations 7.4 Associative memories 7.5 Fuzzy numbers and fuzzy arithmetic Pedrycz and Gomide, FSE 2007

3. 7.1 The extension principle Pedrycz and Gomide, FSE 2007

4. Extension principle • Extends point transformations to operations involving – sets – fuzzy sets • Given a function f : X → Y and a set (or fuzzy set) A on X the extension principle allows to map A into a set (or fuzzy set) on Y through f Pedrycz and Gomide, FSE 2007

5. Pointwise transformation is a function f 10 y 9 8 f 7 f : X → Y 6 5 y o 4 y o = f ( x o ) 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 x x o Pedrycz and Gomide, FSE 2007

6. Set transformation f : X → Y, A ∈ P ( X ) B= f ( A ) = { y ∈ Y | y = f ( x ), ∀ x ∈ X } 10 10 8 8 f B 6 6 y B ∈ P ( Y ) 4 4 2 2 0 0 1 0.5 0 0 5 10 x B ( y ) = ( ) sup ( ) B y A x = 1 / ( ) x y f x A A ( x ) 0 0 2 4 6 8 10 x Pedrycz and Gomide, FSE 2007

7. Fuzzy set transformation f : X → Y, A ∈ F ( X )  B= f ( A ), B ∈ F ( Y ) − − + ≤ ≤ 2 0 . 2 ( 5 ) 5 if 0 5 x x =  ( ) f x − + < ≤ 2  0 . 2 ( 5 ) 5 if 5 10 x x 10 10 8 8 f B = 6 6 ( ) sup ( ) B y A x y 4 4 = / ( ) x y f x 2 2 0 0 1 B ( y ) 0.5 0 0 5 10 x 1 0.8 A 0.6 A ( x ) 0.4 0.2 0 A = A ( x , 3, 5, 8) 0 5 10 x Pedrycz and Gomide, FSE 2007

8. Example y= f ( x ) = x 2 A = A ( x , –2, 2, 3) 10 10 8 8 f B 6 6 y 4 4 2 2 0 0 1 B ( y ) 0.5 0 -4 -3 -2 -1 0 1 2 3 4 x 1 0.8 A 0.6 A ( x ) 0.4 0.2 0 -4 -3 -2 -1 0 1 2 3 4 x Pedrycz and Gomide, FSE 2007

9. Example y= f ( x ) = x 2 X = {–3, –2, – 1, 0, 1, 2, 3} Y = {0, 1, 4, 9} 10 10 8 8 f B 6 6 y 4 4 2 2 0 0 1 B ( y ) 0.5 0 -4 -3 -2 -1 0 1 2 3 4 x 1 0.8 A 0.6 A ( x ) 0.4 0.2 0 -4 -3 -2 -1 0 1 2 3 4 x B = {1/0, max(0.2,0.3)/1, max(0, 0.1)/4, 0/9} = {1/0, 0.3/1, 0.1/4, 0/9} Pedrycz and Gomide, FSE 2007

10. Generalization X = X 1 × X 2 × ... × X n A i ∈ F ( X i ), i = 1,…, n y = f ( x ), x = [ x 1 , x 2 , …, x n ] = � ( ) sup {min [ ( ), ( ), , ( )]} B y A x A x A x 1 1 2 2 n n x = | ( ) y f x B ∈ F ( Y ) Pedrycz and Gomide, FSE 2007

11. Properties = ∅ = ∅ 1 . B iff A i i ⊆ ⇒ ⊆ 2 . A A B B 1 2 1 2 n n n = = � � � 3 . ( ) ( ) f A f A B = i = i = i i 1 i 1 i 1 n n n ⊆ = � � � 4 . ( ) ( ) f A f A B = i = i = i 1 1 1 i i i ⊇ 5 . ( ) B f A α α α = {y ∈ Y | B ( y ) > α } B + + + = strong α –cut 6 . ( ) B f A α α Pedrycz and Gomide, FSE 2007

12. 7.2 Compositions of fuzzy relations Pedrycz and Gomide, FSE 2007

13. Sup-t composition Given the fuzzy relations G : X × Z → [0,1] W : Z × Y → [0,1] R = G ° W sup-t composition = ∀ ( x , y ) ∈ X × Y ( , ) sup {min [ ( , ) ( , )} R x y G x z t W z y z Z ∈ R : X × Y → [0,1] Pedrycz and Gomide, FSE 2007

14. Example t = product sup-product composition = − − 2 ( , ) exp[ ( ) ] G x z x z = − − 2 ( , ) exp[ ( ) ] W z y z y − − − − − − − − = 2 2 = 2 2 ( ) ( ) ( ) ( ) x z z x x z z x ( , ) sup { } max { } R x y e e e e Z ∈ Z z ∈ z = − − 2 ( , ) exp[ ( ) / 2 ] R x y x y Pedrycz and Gomide, FSE 2007

15. G : X × Z R = G ° W = − − = − − 2 2 ( , ) exp[ ( ) ] ( , ) exp[ ( ) / 2 ] G x z x z R x y x y Pedrycz and Gomide, FSE 2007

16. Sup-t composition for matrix relations procedure SUP-T-COMPOSITION ( G , W ) returns composition of fuzzy relations static : fuzzy relations: G = [ g ik ], W =[ w kj ] 0 nm : n × m matrix with all entries equal to zero t : a t-norm R = 0 nm for i = 1: n do for j = 1: m d o for k = 1: p do tope ← g ik t w kj ← max( r ij , tope) r ij return R Pedrycz and Gomide, FSE 2007

17. Example   0 . 6 0 . 1   1 . 0 0 . 6 0 . 5 0 . 5     0 . 5 0 . 7   t = min = ∧ = = 0 . 6 0 . 8 1 . 0 0 . 2 G W     0 . 7 0 . 8      0 . 8 0 . 3 0 . 4 0 . 3    0 . 3 0 . 6 r 11 =max(1.0 ∧ 0.6, 0.6 ∧ 0.5, 0.5 ∧ 0.7, 0.5 ∧ 0.3) = max (0.6, 0.5, 0.5, 0.3) = 0.6 ……………….…… r 32 =max(0.8 ∧ 0.1, 0.3 ∧ 0.7, 0.4 ∧ 0.8, 0.3 ∧ 0.6) = max (0.1, 0.3, 0.4, 0.3) = 0.4   0 . 6 0 . 6   = = � 0 . 7 0 . 8 R G W     0 . 6 0 . 4   Pedrycz and Gomide, FSE 2007

18. Properties associativity = � � � � 1 . ( ) ( ) P Q R P Q R ∪ = ∪ distributivity over union � � � 2 . ( ) ( ) ( ) P Q R P Q P R ∩ ⊆ ∩ weak distributivity over intersection � � � 3 . ( ) ( ) ( ) P Q R P Q P R ⊆ ⊆ � � monotonicity 4 . If then Q S P Q P S ∪ ∪ , ∩ ∩ are standard operations ∪ ∪ ∩ ∩ Pedrycz and Gomide, FSE 2007

19. Interpretations = 1 . ( ) sup [ ( ) ( )] B y A x tR x possibility y ∈ X x existential = ∃ 2 . ( ) truth [ | ( ) ( )] B y x A x and R x y quantifier X = = = projection 3 . ( ) sup [ ( ) ( , )] sup [ 1 ( , )] sup ( , ) B y x tR x y t R x y R x y X X X ∈ ∈ ∈ x x x Pedrycz and Gomide, FSE 2007

20. Inf-s composition Given the fuzzy relations G : X × Z → [0,1] W : Z × Y → [0,1] R = G • W inf-s composition = ∀ ( x , y ) ∈ X × Y ( , ) inf {min [ ( , ) ( , )} R x y G x z s W z y z Z ∈ R : X × Y → [0,1] Pedrycz and Gomide, FSE 2007

21. procedure INF-S-COMPOSITION( G , W ) returns composition of fuzzy relations static : fuzzy relations: G = [ g ik ], W = [ w kj ] 1 nm : n × m matrix with all entries equal to unity s : a s-norm R = 1 nm for i = 1: n do for j = 1: m do for k = 1: p do sope ← g ik s w k j ← min( r ij , sope) r ij return R Pedrycz and Gomide, FSE 2007

22. Example   0 . 6 0 . 1   1 . 0 0 . 6 0 . 5 0 . 5   0 . 5 0 . 7     = = 0 . 6 0 . 8 1 . 0 0 . 2 s = probabilistic sum G W     0 . 7 0 . 8     0 . 8 0 . 3 0 . 4 0 . 3    0 . 3 0 . 6  r 11 = min (1.0+0.6-0.6, 0.6+0.5-0.3, 0.5+0.7-0.35, 0.5+0.3-0.15) = min (1.0, 0.8, 0.85, 0.65) = 0.65 ……………….…… r 32 = min (0.8+0.1-0.08, 0.3+0.7-0.21, 0.4+0.8-0.32, 0.3+0.6-01.8) = min (0.82, 0.79, 0.88, 0.72) = 0.72   0 . 65 0 . 80   = • = 0 . 44 064 R G W     0 . 51 0 . 72   Pedrycz and Gomide, FSE 2007

23. Example = − − = − − 2 2 ( , ) exp[ ( ) ] , ( , ) exp[ ( ) ] G x z x z W z y z y G : X × Z R = G • W Pedrycz and Gomide, FSE 2007

24. Properties • • = • • 1 . ( ) ( ) P Q R P Q R associativity • ∪ ⊇ • ∪ • 2 . ( ) ( ) ( ) P Q R P Q P R weak distributivity over union • ∩ = • ∩ • 3 . ( ) ( ) ( ) distributivity over intersection P Q R P Q P R ⊆ • ⊇ • monotonicity 4 . f then I Q S P Q P S ∪ ∪ , ∩ ∩ are standard operations ∪ ∪ ∩ ∩ Pedrycz and Gomide, FSE 2007

25. Interpretations = = = necessity 1 . ( ) inf [ ( ) ( )] inf [ ( ) ( )] inf [ ( ) ( )] B y A x sR x R x sA x R x s A x y y y ∈ X ∈ X ∈ X x x x universal = ∀ 2 . ( ) truth [ | ( ) ( )] B y x A x or R x quantifier y Pedrycz and Gomide, FSE 2007

26. Inf- ϕ ϕ composition ϕ ϕ Given the fuzzy relations G : X × Z → [0,1] W : Z × Y → [0,1] R = G ϕ W inf- ϕ composition ϕ : [0,1] → [0,1 a ϕ = ∈ ≤ ∀ ∈ { [ 0 , 1 ] | }, , [ 0 , 1 ] b c atc b a b = ϕ ∀ ( x , y ) ∈ X × Y ( , ) inf { ( , ) ( , )} R x y G x z W z y Z ∈ z Pedrycz and Gomide, FSE 2007

27. Example   0 . 6 0 . 1   1 . 0 0 . 6 0 . 5 0 . 5   0 . 5 0 . 7     = = 0 . 6 0 . 8 1 . 0 0 . 2 G W     0 . 7 0 . 8      0 . 8 0 . 3 0 . 4 0 . 3   0 . 3 0 . 6  If t is the bounded difference: a t b = max (0, a + b – 1) then a ϕ b = min (1, 1 – a + b ) Lukasiewicz implication   0 . 6 0 . 1   = ϕ = 0 . 7 06 R G W     0 . 8 0 . 3   Pedrycz and Gomide, FSE 2007

28. Properties ϕ ϕ = ϕ � 1 . ( ) ( ) P Q R P Q R associative ϕ ∪ ⊇ ϕ ∪ ϕ 2 . ( ) ( ) ( ) P Q R P Q P R weak distributivity over union ϕ ∩ = ϕ ∩ ϕ 3 . ( ) ( ) ( ) distributivity over intersection P Q R P Q P R ⊆ ϕ ⊆ ϕ monotonicity 4 . If then Q S P Q P S ∪ ∪ , ∩ ∩ are standard operations ∪ ∪ ∩ ∩ Pedrycz and Gomide, FSE 2007