5 Operations and Aggregations of Fuzzy Sets Fuzzy Systems - - PowerPoint PPT Presentation

5 Operations and Aggregations of Fuzzy Sets

Fuzzy Systems Engineering Toward Human-Centric Computing

5.1 Standard operations on sets and fuzzy sets 5.2 Generic requirements for operations on fuzzy sets 5.3 Triangular norms 5.4 Triangular conorms 5.5 Triangular norms as a general category of logical operations 5.6 Aggregation operations 5.7 Fuzzy measure and integral 5.8 Negations

Contents

Pedrycz and Gomide, FSE 2007

5.1 Standard operations on sets and fuzzy sets

Pedrycz and Gomide, FSE 2007

Intersection of sets

Pedrycz and Gomide, FSE 2007

A = {x∈R| 1 ≤ x ≤ 3} B = {x∈R| 2 ≤ x ≤ 4} (A∩B)(x) = min [A(x), B(x)] ∀x∈X A∩B: {x∈R| 2 ≤ x ≤ 3}

1.0 x A(x)

A B

1.0 x A(x)

B A A∩B

1 2 3 4 1 2 3 4

Pedrycz and Gomide, FSE 2007

Union of sets

A = {x∈R| 1 ≤ x ≤ 3} B = {x∈R| 2 ≤ x ≤ 4} (A∪B)(x) = max [A(x), B(x)] ∀x∈X A∪B: {x∈R| 1≤ x ≤ 4}

1.0 x A(x)

A B

1.0 x A(x)

B A A∪B

1 2 3 4 1 2 3 4

Complement of sets

A = {x∈R| 1 ≤ x ≤ 3}

1.0 x A(x)

A

1.0 x

1 2 3 4 1 2 3 4

A(x) A(x) = {x∈R| x < 1 , x > 3} A(x) = 1 – A(x) ∀x∈X A

Pedrycz and Gomide, FSE 2007

Pedrycz and Gomide, FSE 2007

Intersection of fuzzy sets

(A∩B)(x) = min [A(x), B(x)] ∀x∈X

Standard intersection

1.0 x A(x)

A B

1 2 3 4

1.0 x A(x)

A B

1 2 3 4

A∩B

Pedrycz and Gomide, FSE 2007

Union of fuzzy sets

(A∪B)(x) = max [A(x), B(x)] ∀x∈X

Standard union

1.0 x A(x)

A B

1 2 3 4

1.0 x A(x)

A B

1 2 3 4

A∪B

Pedrycz and Gomide, FSE 2007

Complement of fuzzy sets

Standard complement

1.0 x A(x)

A

1 2 3 4

1.0 x

A

1 2 3 4

A A(x) A(x) = 1 – A(x) ∀x∈X

1 Commutativity A ∪ B = B ∪ A A ∩ B = B ∩ A 2 Associativity A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C 3 Distributivity A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) 4 Idempotency A ∪ A = A A ∩ A = A 5 Boundary Conditions A ∪ φ = A and A ∪ X = X A ∩ φ = φ and A ∩ X = A 6 Involution 7 Transitivity if A ⊂ B and B ⊂ C then A ⊂ C A = A

Basic properties of sets and fuzzy sets

Pedrycz and Gomide, FSE 2007

Sets Fuzzy sets 8-Noncontradiction 9-Excluded middle

A∪A = X A∪A ≠ X A∩A = φ A∩A ≠ φ

Noncontradiction and excluded middle for standard operations

Pedrycz and Gomide, FSE 2007

Geometric view of standard operations

1.0 1.0 x2 x1 A F(X) ∅ { x1, x2} A∪A A A∩A

Pedrycz and Gomide, FSE 2007

1.0 1.0 x2 x1 F(X) ∅ { x1, x2} A=A 1.0 x2 x1 0.5 X A(x)

Example

Pedrycz and Gomide, FSE 2007

Pedrycz and Gomide, FSE 2007

5.2 Generic requirements for

• perations on fuzzy sets

Set operation is a binary operator

Pedrycz and Gomide, FSE 2007

[0,1] ×[0,1] → [0,1]

Requirements: – commutativity – associativity – identity Identity: – its form depends on the operation

Pedrycz and Gomide, FSE 2007

5.3 Triangular norms

Definition

t : [0,1] ×[0,1] → [0,1]

• Commutativity:

a t b = b t a

• Associativity:

a t (b t c) = (a t b) t c

• Monotonicity:

if b ≤ c then a t b ≤ a t c

• Boundary conditions:

a t 1 = a a t 0 = 0

Pedrycz and Gomide, FSE 2007

Examples

(a) (b)

a tm b = min(a,b) a tp b = ab minimum product

Pedrycz and Gomide, FSE 2007

Pedrycz and Gomide, FSE 2007

a tl b = max(a+b – 1,0)

(c) (d)

Lukasiewicz drastic product      = = =

• therwise

1 if 1 if

d

a b b a b at

) , min( b a atb b atd ≤ ≤ a ata ≤

lower and upper bounds Archimedean (if t is continuous)

=

n

a

nilpotent (e.g. Lukasiewicz) a t a = a2, … , an-1 t a = an

Pedrycz and Gomide, FSE 2007

Constructors of t-norms

Monotonic function transformation Additive and multiplicative generators Ordinal sums

Pedrycz and Gomide, FSE 2007

Monotonic function transformation

If h: [0,1] → [0,1] is a strictly increasing bijection then th(a,b) = h-1(t(h(a),h(b))) is a t-norm h is a scaling transformation Obs: h bijective means both, h injective (one-to-one) and h surjective (onto)

Pedrycz and Gomide, FSE 2007

(a) (b)

Examples

h = x2, x∈[0,1] minimum product

t th

Pedrycz and Gomide, FSE 2007

(c) (d)

Lukasiewicz drastic product h = x2, x∈[0,1]

t th

Pedrycz and Gomide, FSE 2007

Additive generators

f: [0,1] → [0, ∞), f(1) = 0 – continuous – strictly decreasing a tf b = f –1(f(a) + f(b)) ⇔ is a Archimedean t-norm

Pedrycz and Gomide, FSE 2007

1.0 x R+ f(x) f(a) f(b) f(a) + f(b) a b a tf b

Additive generators of t-norms

R+ ≡ [0, ∞),

Pedrycz and Gomide, FSE 2007

Example

f = – log(x) f(a) + f(b) = – log(a) – log(b) – (log(a) + log(b)) – log(ab) f- –1(f(a) + f(b)) = elog(ab) = ab a tf b = ab (Archimedean t-norm)

Pedrycz and Gomide, FSE 2007

Multiplicative generators

g: [0,1] → [0, 1], g(1) = 1 – continuous – strictly increasing a tg b = g –1(g(a)g(b)) ⇔ is a Archimedean t-norm

Pedrycz and Gomide, FSE 2007

Multiplicative generators of t-norms

1.0 x g(x) g(a) g(b) g(a)g(b) a b a tg b 1.0

Pedrycz and Gomide, FSE 2007

Example

g = x2 a tf b = ab (Archimedean t-norm) g = e–f(x) multiplicative and additive generators → same t-norm

Pedrycz and Gomide, FSE 2007

Ordinal sums

to: [0,1] → [0, 1] denoted to = (<αk,βk, tk>, k∈K)

     β α ∈         α − β α − α − β α − α − β + α = τ

• therwise

) min( ] [ if ) ( ) (

• b

, a , b , a b , a t , I , b , a t

k k k k k k k k k k k k

I = {[αk,βk], k∈K } nonempty, countable family pairwise disjoint subintervals of [0,1] τ = {tk, k∈K} family of t-norms

Pedrycz and Gomide, FSE 2007

Example

     ∈ − + + ∈ − − + = τ

• therwise

) min( ] 7 5 [ if ) 2 1 max( 5 ] 4 2 [ if ) 2 )( 2 ( 5 2 ) (

• b

, a . , . b , a , . b a . . , . b , a . b . a . , I , b , a t

I = {[0.2, 0.4], [0.5, 0.7]} K ={1,2} τ = {tp, tl}, t1 = tp, t2 = tl

Pedrycz and Gomide, FSE 2007

     ∈ − + + ∈ − − + = τ

• therwise

) min( ] 7 5 [ if ) 2 1 max( 5 ] 4 2 [ if ) 2 )( 2 ( 5 2 ) (

• b

, a . , . b , a , . b a . . , . b , a . b . a . , I , b , a t

Pedrycz and Gomide, FSE 2007

     ∈ − + + ∈ − − + = τ

• therwise

) min( ] 7 5 [ if ) 2 1 max( 5 ] 4 2 [ if ) 2 )( 2 ( 5 2 ) (

• b

, a . , . b , a , . b a . . , . b , a . b . a . , I , b , a t

Pedrycz and Gomide, FSE 2007

Pedrycz and Gomide, FSE 2007

5.4 Triangular conorms

Pedrycz and Gomide, FSE 2007

Definition

s : [0,1] ×[0,1] → [0,1]

• Commutativity:

a s b = b s a

• Associativity:

a s (b s c) = (a s b) s c

• Monotonicity:

if b ≤ c then a s b ≤ a s c

• Boundary conditions:

a s 1 = 1 a s 0 = a

Pedrycz and Gomide, FSE 2007

Examples

a sm b = max(a,b) a sp b = a + b – ab maximum probabilistic sum

(b) (a)

Pedrycz and Gomide, FSE 2007

a sl b = max(a+b, 1) Lukasiewicz drastic sum      = = =

• therwise

1 if if

d

a b b a b as

(c) (d)

Pedrycz and Gomide, FSE 2007

b as asb b , a

d

) max( ≤ ≤ a asa ≥

lower and upper bounds Archimedean (if t is continuous)

1 =

n

a

nilpotent (e.g. Lukasiewicz) a s a = a2, … , an-1 s a = an

Pedrycz and Gomide, FSE 2007

Dual norms and De Morgan laws

a s b = 1 – (1 – a) t (1 – b) a t b = 1 – (1 – a) s (1 – b) (1 – a) t (1 – b) = 1 – a s b (1 – a) s (1 – b) = 1 – a t b

B A B A B A B A ∩ = ∪ ∪ = ∩

Dual triangular norms De Morgan

Pedrycz and Gomide, FSE 2007

Constructors of s-norms

Monotonic function transformation Additive and multiplicative generators Ordinal sums

Pedrycz and Gomide, FSE 2007

Monotonic function transformation

If h: [0,1] → [0,1] is a strictly increasing bijection then sh(a,b) = h-1(s(h(a),s(b))) is a t-norm h is a scaling transformation

Pedrycz and Gomide, FSE 2007

Examples

h = x2, x∈[0,1] maximum probabilistic sum

s sh

(a) (b)

Pedrycz and Gomide, FSE 2007

Lukasiewicz drastic sum h = x2, x∈[0,1]

s sh

(c) (d)

Pedrycz and Gomide, FSE 2007

Additive generators

f: [0,1] → [0, ∞), f(0) = 0 – continuous – strictly increasing a sf b = f –1(f(a) + f(b)) ⇔ is a Archimedean t-conorm

Additive generators of t-conorms

1.0 x R+ f(x) f(a) f(b) f(a) + f(b) a b a sf b

Pedrycz and Gomide, FSE 2007

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 – elog(1 – a)(1 – b) = a + b – ab a sf 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 sg b = g –1(g(a)g(b)) ⇔ is a Archimedean t-norm

Pedrycz and Gomide, FSE 2007

1.0 x R+ g(x) g(a) g(b) g(a)g(b) a b a sg b

Multiplicative generators of t-conorms

R+ ≡ [0, ∞),

Pedrycz and Gomide, FSE 2007

Example

g = 1 – x a sg 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

so: [0,1] → [0, 1] denoted so = (<αk,βk, sk>, k∈K)

     β α ∈         α − β α − α − β α − α − β + α = σ

• therwise

) max( ] [ if ) ( ) (

• b

, a , b , a b , a s , I , b , a s

k k k k k k k k k k k k

I = {[αk,βk], k∈K } nonempty, countable family pairwise disjoint subintervals of [0,1] σ = {sk, k∈K} family of t-conorms

Pedrycz and Gomide, FSE 2007

Example

     ∈ − + − + ∈ − + − + = σ

• therwise

) max( ] 7 5 [ if ) 1 ) 2 ( 5 ) 2 min(5( 2 5 ] 4 2 [ if ) 2 )( 2 ( 5 ) 2 ( ) 2 ( 2 ) (

• b

, a . , . b , a , , . b . a . . . , . b , a . b- . a-

• .

b . a . , I , b , a s

I = {[0.2, 0.4], [0.5, 0.7]} K ={1,2} σ = {sp, sl}, s1 = sp, s2 = sl

Pedrycz and Gomide, FSE 2007

     ∈ − + − + ∈ − + − + = σ

• therwise

) max( ] 7 5 [ if ) 1 ) 2 ( 5 ) 2 min(5( 2 5 ] 4 2 [ if ) 2 )( 2 ( 5 ) 2 ( ) 2 ( 2 ) (

• b

, a . , . b , a , , . b . a . . . , . b , a . b- . a-

• .

b . a . , I , b , a s

Pedrycz and Gomide, FSE 2007

     ∈ − + − + ∈ − + − + = σ

• therwise

) max( ] 7 5 [ if ) 1 ) 2 ( 5 ) 2 min(5( 2 5 ] 4 2 [ if ) 2 )( 2 ( 5 ) 2 ( ) 2 ( 2 ) (

• b

, a . , . b , a , , . b . a . . . , . b , a . b- . a-

• .

b . a . , I , b , a s

Pedrycz and Gomide, FSE 2007

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

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

Truth value assignment

Pedrycz and Gomide, FSE 2007

p q min(p, q) max(p, q) pq p + q – pq 1 1 1 1 1 1 1 1 1 1 1 1 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

Examples

a ϕ b ≡ a ⇒ b a ϕ b = sup { c ∈ [0, 1] | a t c ≤ b} ∀a,b ∈ [0,1]

a b a ⇒ b a ϕ b 1 1 1 1 1 1 1 1 1 1

ϕ

• perator or Boolean values of its arguments

Implication induced by a t-norm

residuation

Pedrycz and Gomide, FSE 2007

5.6 Aggregation operations

Pedrycz and Gomide, FSE 2007

Pedrycz and Gomide, FSE 2007

Definition

g : [0,1]n → [0,1]

• 1. Monotonicity:

g (x1, x2,..., xn) ≥ g (y1, y2,.., yn) if xi ≥ yi, 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 (x1, x2,..., xi-1, e, xi+1,..., xn) = g (x1, x2,..., xi-1, e, xi+1,..., xn) n ≥ 2

• 2. Annihilator (l ):

g (x1, x2,..., xi-1, l, xi+1,..., xn) = l

Observation: Annihilator ≡ absorbing element

Averaging operations

Pedrycz and Gomide, FSE 2007

) ( 1 ) (

1 2 1

≠ ∈ ∑ =

=

p , p , x n x , , x , x g

p n i p i n

R

• ∑

= − = ∏ = → ∑ = =

= = = n i i n n n i i n n i i n

x / n x , , x , x g p x x , , x , x g p x n x , , x , x g p

1 2 1 1 2 1 1 2 1

1 ) ( 1 ) ( 1 ) ( 1

• arithmetic mean

geometric mean harmonic mean

Pedrycz and Gomide, FSE 2007

p → – ∝ g (x1, x2,..., xn) = min (x1, x2,..., xn) p → ∝ g (x1, x2,..., xn) = max (x1, x2,..., xn) Bounds min (x1, x2,..., xn) ≤ g (x1, x2,..., xn) ≤ max (x1, x2,..., xn)

Pedrycz and Gomide, FSE 2007

Examples

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 x (b) Arithmetic mean of A and B A B

Arithmetic mean

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 x (d) Geometric mean of A and B A B

Geometric mean

Pedrycz and Gomide, FSE 2007

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 x (f) Harmonic mean of A and B A B

Harmonic mean

Pedrycz and Gomide, FSE 2007

Ordered Weighted Averaging (OWA)

∑ =

= n i i i

x A w A

1

) ( ) , OWA( w

Σwi = 1, wi ∈ [0, 1] A (x1) ≤ A(x2) ≤... ≤ A( xn)

Pedrycz and Gomide, FSE 2007

1. w = [1, 0,...,0] OWA(A,w) = min (A(x1), A(x2),..., A(xn)) 2. w = [0, 0,...,1] OWA(A,w) = max (A(x1), A(x2),..., A(xn)) 3. w = [1/n, 1/n,...,1/n] OWA(A,w) = arithmetic mean min (A(x1), A(x2),..., A(xn)) ≤ OWA(A,w) ≤ max (A(x1), A(x2),..., A(xn))

Examples

Pedrycz and Gomide, FSE 2007

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 x (h) Owa of A and B, w1 = 0.8, w2 = 0.2 A B

w = [0.8, 0.2]

Examples

Pedrycz and Gomide, FSE 2007

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 x (f) Owa of A and B, w1 = 0.2, w2 = 0.8 A B

w = [0.2, 0.8]

Pedrycz and Gomide, FSE 2007

Uninorms

u : [0,1] ×[0,1] → [0,1] Commutativity: a u b = b u a Associativity: a u (b u c) = (a u b) u c Monotonicity: if b ≤ c then a u b ≤ a u c Identity: a u e = a ∀a ∈ [0, 1] e ∈ [0, 1], e = 1 u is a t-norm, e = 0 u is a t-conorm

Pedrycz and Gomide, FSE 2007

conorm t and norm t are and 1 ) ) 1 ( ( ) ) 1 ( ( ) ( ) ( − − − − − + − + = =

u u u u

s t e e b e e u a e e b as e eb u ea b at

Results on uninorms

• 1. tu, su: [0, 1] × [0, 1] → [0, 1] such that ∀e ∈ [0, 1]

Pedrycz and Gomide, FSE 2007

• 2. If a ≤ e ≤ b or a ≥ e ≥ b then

) , max( ) , min( then

• r

If b a aub b a b e a b e a ≤ ≤ ≥ ≥ ≤ ≤

Pedrycz and Gomide, FSE 2007

     ≤ ≤ ≤ ≤ =      ≤ ≤ ≤ ≤ = ≤ ≤

• therwise

) max( 1 if 1 if ) min(

• therwise

) min( 1 if ) max( if b , a b , a e e b , a b , a b au b , a b , a e b , a e b , a b au b au aub b au

s w s w

• 3. For any u with e ∈ [0, 1]

Pedrycz and Gomide, FSE 2007

         ≤ ≤ − − − − − + ≤ ≤ = =          ≤ ≤ − − − − − + ≤ ≤ = =

• therwise

) max( 1 if 1 ( 1 )( 1 ( if ) ( ) ( then 1 ) 1 ( if

• therwise

) min( 1 if 1 ( 1 )( 1 ( if ) ( ) ( then ) 1 ( if b , a b , a e ) e e b s ) e e a e e e b , a e b t e a e b au u b , a b , a e ) e e b s ) e e a e e e b , a e b t e a e b au u

d c

• 4. Conjunctive and disjunctive uninorm

Pedrycz and Gomide, FSE 2007

Conjunctive uninorm e = 0.5

Pedrycz and Gomide, FSE 2007

e = 0.5 Disjunctive uninorm

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 uf b = f –1(f(a) + f(b)) a ug 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

           ≥ ∉ ι ∈         α − β α − α − β α − α − β + α ι ∈         α − β α − α − β α − α − β + α = σ τ

• therwise

) min( and ] [ if ) max( ] [ if ) ( ] [ if ) ( ) (

2 1 co

b , a e b , a ,β α a,b b , a b , a b , a s b , a b , a t , , ι , b , a u

k k k k k k k k k k k k k k k k k k k k k k

l = {[αk, βk], k∈K} l1 = {[αk, βk] ∈ l | βk ≤ e} l2 = {[αk, βk] ∈ l | αk ≥ e} τ = {tk, k∈K}, σ = {sk, k∈K}

Pedrycz and Gomide, FSE 2007

           ≤ ∉ ι ∈         α − β α − α − β α − α − β + α ι ∈         α − β α − α − β α − α − β + α = σ τ

• therwise

) max( and ] [ if ) min( ] [ if ) ( ] [ if ) ( ) (

2 1 co

b , a e b , a ,β α a,b b , a b , a b , a s b , a b , a t , , ι , b , a u

k k k k k k k k k k k k k k k k k k k k k k

Pedrycz and Gomide, FSE 2007

Nullnorms

v : [0,1] ×[0,1] → [0,1] Commutativity: a v b = b v a Associativity: a v (b v c) = (a v b) v c Monotonicity: if b ≤ c then a v b ≤ a v c Absorbing element: a v e = e ∀a ∈ [0, 1] Boundary conditions: a v 0 = a ∀a ∈ [0, e] a v 1 = a ∀a ∈ [e, 1]

Pedrycz and Gomide, FSE 2007

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

e e b e e v a e e b at e eb v ea b as

v v

− − − + − + = = 1 ) ) 1 ( ( ) ) 1 ( ( ) ( ) (

Pedrycz and Gomide, FSE 2007

e = 0.5, t-norm = min, t-conorm = max

Example

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Symmetric sums

σs(a1, a2,...., an) = 1 – σs(1 – a1, 1 – a2,...., 1 – an)

1 2 1 2 1 2 1

) ( ) 1 1 ( 1 ) (

−

      − − + = σ

n n n s

a , , a , a f a , , a , a f a , , a , a

• f increasing, continuous

f (0,0,...,0) = 0

Pedrycz and Gomide, FSE 2007

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 x Symmetric sum of A and B A B

Example

f (a,b) = a2 + b2

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Compensatory operations

a b = (a t b)1 – γ (a s b)γ a b = (1 – γ)(a t b) + γ(a t b) compensatory product compensatory sum

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(a) (b)

Example

γ = 0.5, t-norm = min, t-conorm = max

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5.7 Fuzzy measure and integral

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Fuzzy measure

g : Ω → [0,1]

• Boundary conditions:

g(∅) = 0 g(X) = 1

• Monotonicity:

if A ⊂ B then g(A) ≤ g(B)

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λ λ λ λ–fuzzy measure

g(A∪B) = g(A) + g(B) + λg(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

Fuzzy integral

h : X → [0,1] Ω measurable fuzzy integral of h with respect to g over A ∫ ∩ α =

α ∈ α A ,

H A g , g x h )] ( {min[ sup ) ( ) (

] 1 [

• Hα = {x | h(x) ≥ α}

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X = {x1, x2,....,xn} h(x1) ≥ h(x2) ≥ ..... ≥ h(xn) A1 = {x1), A2 = {x1, x2), ..., An = {x1, x2,..., xn} = X ∫ =

= A i i ,n , i

A g , x h g x h )] ( ) ( {min[ max ) ( ) (

1

• Pedrycz and Gomide, FSE 2007

1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xi Fuzzy integral h(xi) g(Ai)

Example

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Choquet integral

) ( )] ( ) ( [ ) (

1 1 i i n i i

A g x h x h g f Ch

+ =

− ∫ ∑ =

• h(xn+1) = 0

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5.8 Negations

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Definition

N : [0,1] → [0,1]

• 1. Monotonicity:

N is nonincreasing

• 2. Boundary conditions:

N (0) = 1 N (1) = 0

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• 3. Continuity:

N is a continuous function

• 4. Involution:

N (N(x)) = x ∀x ∈ [0, 1]

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Examples

   ≥ < = a x a x x N if if 1 ) (

1.0 x 1.0 a

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1.0 x 1.0

   = = = 1 if if 1 ) ( x x x N

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sugeno Negation λ=20 λ=5 λ=0 λ=−0.9 N x

) 1 ( 1 1 ) ( ∞ − ∈ λ λ + − = , x x x N

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Yager Negation w=1 w=2 w=3 w=4 N x

) ( 1 ) ( ∞ ∈ − = , w x x N

w w

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(t, s, N) system

x s y = N (N(x) t N(y)) x t y = N (N(x) s N(y)) ∀x, y ∈ [0, 1]

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Examples

) 1 ( 1 1 ) 1 1 ( min ) 1 1 ( max ∞ − ∈ + − = + − + = + + − + = , λ λx x N(x) λxy y ,x xsy λ λxy y x , xty x x N y x xsy y x xty − = = = 1 ) ( ) , max( ) , min(

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