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 fuzzy arithmetic Pedrycz and Gomide, FSE 2007

7.1 The extension principle Pedrycz and Gomide, FSE 2007

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

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

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

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

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

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

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

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

7.2 Compositions of fuzzy relations Pedrycz and Gomide, FSE 2007

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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