Example f = – log(1 – x ) f ( a ) + f ( b ) = – log(1 – a ) – log(1 – b ) – (log(1 – a ) + log(1 – b )) – log(1 – a )(1 – b ) f - –1 ( f ( a ) + f ( b )) = 1 – e log(1 – a )(1 – b ) = a + b – ab � a s f b = a + b – ab ( Archimedean t-conorm) Pedrycz and Gomide, FSE 2007
Multiplicative generators � g : [0,1] → [0, 1], g (0) = 1 – continuous – strictly decreasing � a s g b = g –1 ( g ( a )g( b )) ⇔ is a Archimedean t-norm Pedrycz and Gomide, FSE 2007
Multiplicative generators of t-conorms R + g ( a ) g ( x ) g ( b ) g ( a ) g ( b ) a a s g b x b 1.0 R + ≡ [0, ∞ ), Pedrycz and Gomide, FSE 2007
Example � g = 1 – x a s g b = a + b – ab ( Archimedean t-norm) � g = e – f ( x ) multiplicative and additive generators → same t-conorm Pedrycz and Gomide, FSE 2007
Ordinal sums denoted s o = ( <α k , β k , s k > , k ∈ K ) s o : [0,1] → [0, 1] − α − α a b k k α + β − α ∈ α β s , a , b , ( ) if [ ] k k k k k k σ = s a , b , I , β − α β − α ( ) o k k k k a , b max( ) otherwise I = {[ α k , β k ], k ∈ K } � nonempty, countable family � pairwise disjoint subintervals of [0,1] σ = { s k , k ∈ K } � family of t-conorms Pedrycz and Gomide, FSE 2007
Example + − + − ∈ . a . b . - a- . b- . a , b . , . 0 2 ( 0 2 ) ( 0 2 ) 5 ( 0 2 )( 0 2 ) if [ 0 2 0 4 ] σ = + − + − ∈ s a , b , I , . . a . b . , , a , b . , . ( ) 0 5 0 2 min(5( 0 2 ) 5 ( 0 2 0 ) 1 ) if [ 0 5 0 7 ] o a , b max( ) otherwise I = {[0.2, 0.4], [0.5, 0.7]} K ={1,2} σ = { s p , s l }, s 1 = s p , s 2 = s l Pedrycz and Gomide, FSE 2007
+ − + − ∈ . a . b . - a- . b- . a , b . , . 0 2 ( 0 2 ) ( 0 2 ) 5 ( 0 2 )( 0 2 ) if [ 0 2 0 4 ] σ = + − + − ∈ s a , b , I , . . a . b . , , a , b . , . ( ) 0 5 0 2 min(5( 0 2 ) 5 ( 0 2 0 ) 1 ) if [ 0 5 0 7 ] o a , b max( ) otherwise Pedrycz and Gomide, FSE 2007
+ − + − ∈ . a . b . - a- . b- . a , b . , . 0 2 ( 0 2 ) ( 0 2 ) 5 ( 0 2 )( 0 2 ) if [ 0 2 0 4 ] σ = + − + − ∈ s a , b , I , . . a . b . , , a , b . , . ( ) 0 5 0 2 min(5( 0 2 ) 5 ( 0 2 0 ) 1 ) if [ 0 5 0 7 ] o a , b max( ) otherwise Pedrycz and Gomide, FSE 2007
5.5 Triangular norms as general category of logical operators Pedrycz and Gomide, FSE 2007
Motivation � Fuzzy propositions involves linguistic statements: – temperature is low and humidity is high – velocity is high or noise level is low � Logical operations: – and ( ∧ ) – or ( ∨ ) Pedrycz and Gomide, FSE 2007
Truth value assignment L = { P, Q, .... } P , Q , ... atomic statements truth: L → [0, 1] p , q ,... ∈ [0, 1] truth ( P and Q ) = truth ( P ∧ Q ) → p ∧ q = p t q truth ( P or Q ) = truth ( P ∨ Q ) → p ∨ q = p s q Pedrycz and Gomide, FSE 2007
Examples p q min( p , q ) max( p , q ) pq p + q – pq 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0.2 0.5 0.2 0.5 0.1 0.6 0.5 0.8 0.5 0.8 0.4 0.9 0.8 0.7 0.7 0.8 0.56 0.94 Pedrycz and Gomide, FSE 2007
Implication induced by a t-norm a ϕ b ≡ a ⇒ b a ϕ b = sup { c ∈ [0, 1] | a t c ≤ b } ∀ a , b ∈ [0,1] residuation ϕ operator or Boolean values of its arguments a ⇒ b a ϕ b a b 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1 Pedrycz and Gomide, FSE 2007
5.6 Aggregation operations Pedrycz and Gomide, FSE 2007
Definition g : [0,1] n → [0,1] 1. Monotonicity: g ( x 1 , x 2 ,..., x n ) ≥ g ( y 1 , y 2 ,.., y n ) if x i ≥ y i , i = 1,.., n 2. Boundary conditions: g (0, 0,..., 0) = 0 g (1, 1,..., 1) = 1 Pedrycz and Gomide, FSE 2007
1. Neutral element ( e ) : g ( x 1 , x 2 ,..., x i-1 , e , x i +1 ,..., x n ) = g ( x 1 , x 2 ,..., x i-1 , e , x i +1 ,..., x n ) n ≥ 2 2. Annihilator ( l ): g ( x 1 , x 2 ,..., x i-1 , l , x i +1 ,..., x n ) = l Observation: Annihilator ≡ absorbing element Pedrycz and Gomide, FSE 2007
Averaging operations n 1 p = p ∈ R ≠ g x , x , , x ∑ x , p , p � ( ) ( ) 0 n i 1 2 n = i 1 n 1 = = p g x , x , , x ∑ x � 1 ( ) arithmetic mean n i 1 2 n = i 1 n → = p g x , x , , x x ∏ n � 0 ( ) geometric mean n i 1 2 = i 1 n = − = p g x , x , , x � 1 ( ) n harmonic mean 1 2 n ∑ / x 1 i = i 1 Pedrycz and Gomide, FSE 2007
p → – ∝ g ( x 1 , x 2 ,..., x n ) = min ( x 1 , x 2 ,..., x n ) p → ∝ g ( x 1 , x 2 ,..., x n ) = max ( x 1 , x 2 ,..., x n ) Bounds min ( x 1 , x 2 ,..., x n ) ≤ g ( x 1 , x 2 ,..., x n ) ≤ max ( x 1 , x 2 ,..., x n ) Pedrycz and Gomide, FSE 2007
Examples Arithmetic mean (b) Arithmetic mean of A and B 1.2 1 0.8 A B 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 x Pedrycz and Gomide, FSE 2007
Geometric mean (d) Geometric mean of A and B 1.2 1 0.8 A B 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 x Pedrycz and Gomide, FSE 2007
Harmonic mean (f) Harmonic mean of A and B 1.2 1 0.8 A B 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 x Pedrycz and Gomide, FSE 2007
Ordered Weighted Averaging (OWA) n w = A ∑ w A x OWA( , ) ( ) i i = i 1 Σ w i = 1, w i ∈ [0, 1] A ( x 1 ) ≤ A ( x 2 ) ≤ ... ≤ A ( x n ) Pedrycz and Gomide, FSE 2007
Examples 1. w = [1, 0,...,0] OWA( A , w ) = min ( A ( x 1 ), A ( x 2 ),..., A ( x n )) 2. w = [0, 0,...,1] OWA( A , w ) = max ( A ( x 1 ), A ( x 2 ),..., A ( x n )) 3. w = [1/ n , 1/ n ,...,1/ n ] OWA( A , w ) = arithmetic mean min ( A ( x 1 ), A ( x 2 ),..., A ( x n )) ≤ OWA( A , w ) ≤ max ( A ( x 1 ), A ( x 2 ),..., A ( x n )) Pedrycz and Gomide, FSE 2007
Examples w = [0.8, 0.2] (h) Owa of A and B, w1 = 0.8, w2 = 0.2 1.2 1 0.8 A B 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 x Pedrycz and Gomide, FSE 2007
w = [0.2, 0.8] (f) Owa of A and B, w1 = 0.2, w2 = 0.8 1.2 1 0.8 A B 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 x Pedrycz and Gomide, FSE 2007
Uninorms u : [0,1] × [0,1] → [0,1] a u b = b u a � Commutativity: a u ( b u c ) = ( a u b ) u c � Associativity: if b ≤ c then a u b ≤ a u c � Monotonicity: a u e = a ∀ a ∈ [0, 1] � Identity: e ∈ [0, 1], e = 1 u is a t-norm, e = 0 u is a t-conorm Pedrycz and Gomide, FSE 2007
Results on uninorms 1. t u , s u : [0, 1] × [0, 1] → [0, 1] such that ∀ e ∈ [0, 1] ea u eb ( ) ( ) = at b u e + − + − − e e a u e e b e ( ( 1 ) ) ( ( 1 ) ) = as b u − e 1 − − t s and are t norm and t conorm u u Pedrycz and Gomide, FSE 2007
2. If a ≤ e ≤ b or a ≥ e ≥ b then ≤ ≤ ≥ ≥ a e b a e b If or then ≤ ≤ a b aub a b min( , ) max( , ) Pedrycz and Gomide, FSE 2007
3. For any u with e ∈ [0, 1] ≤ ≤ au b aub au b w s ≤ ≤ a , b e 0 if 0 = ≤ ≤ au b a , b e a , b max( ) if 1 w a , b min( ) otherwise ≤ ≤ a , b a , b e min( ) if 0 = ≤ ≤ au b e a , b 1 if 1 s a , b max( ) otherwise Pedrycz and Gomide, FSE 2007
4. Conjunctive and disjunctive uninorm = u if ( 0 1 ) 0 then a b ≤ ≤ e t a , b e ( ) ( ) if 0 e e − − a e b e = + − ≤ ≤ au b e e ) s ) e a , b ( 1 )( ( if 1 c − − e e 1 1 a , b min( ) otherwise = u if ( 0 1 ) 1 then a b ≤ ≤ e t a , b e ( ) ( ) if 0 e e − − a e b e = + − ≤ ≤ au b e e ) s ) e a , b ( 1 )( ( if 1 d − − e e 1 1 a , b max( ) otherwise Pedrycz and Gomide, FSE 2007
Conjunctive uninorm e = 0.5 Pedrycz and Gomide, FSE 2007
Disjunctive uninorm e = 0.5 Pedrycz and Gomide, FSE 2007
5. Almost continuous Archimedian uninorms a u a < a for 0 < a < e a u a > a for e < a < 1 6. Additive and multiplicative generators of almost continuous uninorms a u f b = f –1 ( f ( a ) + f ( b )) a u g b = g –1 ( g ( a )g( b )) g ( x ) = e – f ( x ) f strictly increasing g strictly decreasing Pedrycz and Gomide, FSE 2007
7. Ordinal sum − α − α a b k k α + β − α ∈ ι t , a , b ( ) if [ ] k k k k 1 β − α β − α k k k k − α − α a b k k α + β − α ∈ ι s , a , b ( ) if [ ] k k k k τ σ = u a , b , , , ι 2 β − α β − α ( ) co k k k k ∉ a,b , β α if [ ] k k a , b max( ) ≥ a , b e and a , b min( ) otherwise l = {[ α k , β k ], k ∈ K } l 1 = {[ α k , β k ] ∈ l | β k ≤ e } l 2 = {[ α k , β k ] ∈ l | α k ≥ e } τ = { t k , k ∈ K }, σ = { s k , k ∈ K } Pedrycz and Gomide, FSE 2007
− α − α a b k k α + β − α ∈ ι t , a , b ( ) if [ ] k k k k 1 β − α β − α k k k k − α − α a b k k α + β − α ∈ ι s , a , b ( ) if [ ] k k k k τ σ = u a , b , , , ι 2 β − α β − α ( ) co k k k k ∉ a,b , β α if [ ] k k a , b min( ) ≤ a , b e and a , b max( ) otherwise Pedrycz and Gomide, FSE 2007
Nullnorms v : [0,1] × [0,1] → [0,1] a v b = b v a � Commutativity: a v ( b v c ) = ( a v b ) v c � Associativity: if b ≤ c then a v b ≤ a v c � Monotonicity: a v e = e ∀ a ∈ [0, 1] � Absorbing element: a v 0 = a ∀ a ∈ [0, e ] � Boundary conditions: a v 1 = a ∀ a ∈ [ e , 1] Pedrycz and Gomide, FSE 2007
ea v eb ( ) ( ) = as b v e + − + − − e e a v e e b e ( ( 1 ) ) ( ( 1 ) ) = at b v − e 1 e ∈ [0, 1] v behaves as a t-norm in [0, e ] × [0, e ] v behaves as a t-conorm in [ e , 1] × [ e , 1] v = e in the rest of the unit square Pedrycz and Gomide, FSE 2007
Example e = 0.5, t-norm = min, t-conorm = max Pedrycz and Gomide, FSE 2007
Symmetric sums σ s ( a 1 , a 2 ,...., a n ) = 1 – σ s (1 – a 1 , 1 – a 2 ,...., 1 – a n ) − 1 − − f a , a , , a � ( 1 1 ) n σ = + 1 2 a , a , , a � ( ) 1 s n 1 2 f a , a , , a � ( ) n 1 2 f increasing, continuous f (0,0,...,0) = 0 Pedrycz and Gomide, FSE 2007
Example Symmetric sum of A and B 1.2 1 0.8 A B 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 x f ( a , b ) = a 2 + b 2 Pedrycz and Gomide, FSE 2007
Compensatory operations a � b = ( a t b ) 1 – γ ( a s b ) γ compensatory product a � b = (1 – γ )( a t b ) + γ ( a t b ) compensatory sum Pedrycz and Gomide, FSE 2007
Example (b) (a) γ = 0.5, t-norm = min, t-conorm = max Pedrycz and Gomide, FSE 2007
5.7 Fuzzy measure and integral Pedrycz and Gomide, FSE 2007
Fuzzy measure g : Ω → [0,1] g( ∅ ) = 0 • Boundary conditions: g ( X ) = 1 if A ⊂ B then g ( A ) ≤ g ( B ) • Monotonicity: Pedrycz and Gomide, FSE 2007
λ λ λ λ –fuzzy measure g ( A ∪ B ) = g ( A ) + g ( B ) + λ g ( A ) g ( B ), λ > − 1 • λ = 0 g ( A ∪ B ) = g ( A ) + g ( B ) additive • λ > 0 g ( A ∪ B ) ≥ g ( A ) + g ( B ) super-additive • λ < 0 g ( A ∪ B ) ≤ g ( A ) + g ( B ) sub-additive Pedrycz and Gomide, FSE 2007
Fuzzy integral h : X → [0,1] Ω measurable fuzzy integral of h with respect to g over A = α ∩ h x g , g A H ∫ � ( ) ( ) sup {min[ ( )] α A α ∈ , [ 0 1 ] H α = { x | h ( x ) ≥ α } Pedrycz and Gomide, FSE 2007
X = { x 1 , x 2 ,...., x n } h ( x 1 ) ≥ h ( x 2 ) ≥ ..... ≥ h ( x n ) A 1 = { x 1 ), A 2 = { x 1 , x 2 ), ..., A n = { x 1 , x 2 ,..., x n } = X = h x g h x , g A ∫ � ( ) ( ) max {min[ ( ) ( )] i i A = i , ,n 1 � Pedrycz and Gomide, FSE 2007
Example Fuzzy integral 1 0.9 0.8 0.7 g(Ai) 0.6 0.5 0.4 0.3 0.2 h(xi) 0.1 0 1 1.5 2 2.5 3 3.5 4 4.5 5 xi Pedrycz and Gomide, FSE 2007
Choquet integral n = − Ch f g ∑ h x h x g A ∫ � ( ) [ ( ) ( )] ( ) + i i i 1 = i 1 h ( x n+ 1 ) = 0 Pedrycz and Gomide, FSE 2007
5.8 Negations Pedrycz and Gomide, FSE 2007
Definition N : [0,1] → [0,1] 1. Monotonicity: N is nonincreasing 2. Boundary conditions: N (0) = 1 N (1) = 0 Pedrycz and Gomide, FSE 2007
3. Continuity: N is a continuous function 4. Involution: ∀ x ∈ [0, 1] N ( N ( x )) = x Pedrycz and Gomide, FSE 2007
Examples 1.0 < x a 1 if = N x ( ) ≥ x a 0 if x a 1.0 Pedrycz and Gomide, FSE 2007
1.0 = x 1 if 0 = N x ( ) = x 0 if 1 x 1.0 Pedrycz and Gomide, FSE 2007
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