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
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