5 Operations and Aggregations of Fuzzy Sets - UNICAMP · 5 Operations and Aggregations of Fuzzy Sets Fuzzy Systems Engineering Toward Human-Centric Computing . 5.1 Standard operations

5 Operations andAggregations ofFuzzy Sets

Fuzzy Systems EngineeringToward Human-Centric Computing

5.1 Standard operations on sets and fuzzy sets

5.2 Generic requirements for operations on fuzzy se ts

5.3 Triangular norms

5.4 Triangular conorms

5.5 Triangular norms as a general category of logic al 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


Intersection of sets


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 B1.0

x

A(x)

BA

A∩B

1 2 3 4 1 2 3 4


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 B1.0

x

A(x)

BA

A∪B

1 2 3 4 1 2 3 4

Complement of sets

A = x∈R| 1 ≤ x ≤ 3

1.0

x

A(x)

A1.0

x1 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



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


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


Complement of fuzzy sets

Standard complement

1.0

x

A(x)A

1 2 3 4

1.0

x

A

1 2 3 4

AA(x)

A(x) = 1 –A(x) ∀x∈X

1 CommutativityA ∪ B = B ∪ A

A ∩ B = B ∩ A

2 AssociativityA ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∩ C) = (A ∩ B) ∩ C

3 DistributivityA ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

4 IdempotencyA ∪ A = A

A ∩ A = A

5 Boundary ConditionsA ∪ 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


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


Geometric view of standard operations

1.0

1.0

x2

x1

A

F(X)

∅

x1, x2

A∪A

AA∩A


1.0

1.0

x2

x1

F(X)

∅

x1, x2

A=A

1.0

x2x1

0.5

X

A(x)

Example



5.2 Generic requirements foroperations on fuzzy sets

Set operation is a binary operator


[0,1] ×[0,1] → [0,1]

Requirements:

– commutativity– associativity– identity

Identity:

– its form depends on the operation


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 = aa t 0 = 0


Examples

(a) (b)

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

minimum product



a tl b = max(a+b – 1,0)

(c) (d)

Lukasiewicz drasticproduct

==

=otherwise0

1if

1if

d ab

ba

bat

),min( baatbbatd ≤≤

aata ≤

lower and upper bounds

Archimedean (if t is continuous)

0=na nilpotent (e.g. Lukasiewicz)

a t a = a2, … , an-1 t a = an


Constructors of t-norms

Monotonic function transformation

Additive and multiplicative generators

Ordinal sums



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)


(a)

(b)

Examples

h = x2, x∈[0,1]

minimum

product

t th


(c)

(d)

Lukasiewicz

drastic product

h = x2, x∈[0,1]

t th


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


1.0 x

R+

f(x)

f(a)

f(b)

f(a) + f(b)

a ba tf b

Additive generators of t-norms

R+ ≡ [0, ∞),


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)


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


Multiplicative generators of t-norms

1.0 x

g(x)

g(a)

g(b)

g(a)g(b)

a ba tg b

1.0


Example

g = x2

a tf b = ab (Archimedean t-norm)

g = e–f(x)

multiplicative and additive generators → same t-norm


Ordinal sums

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

βα∈

α−βα−

α−βα−α−β+α=τ

otherwise)min(

][if)()(o

b,a

,b,ab

,a

t,I,b,at kk

kk

k

kk

kkkkk

I = [αk,βk], k∈K

nonempty, countable family pairwise disjoint subintervals of [0,1]

τ = tk, k∈K

family of t-normsPedrycz and Gomide, FSE 2007

Example

∈−++∈−−+

=τotherwise)min(

]7050[if)021max(50

]4020[if)20)(20(520

)(o

b,a

.,.b,a,.ba.

.,.b,a.b.a.

,I,b,at

I = [0.2, 0.4], [0.5, 0.7] K =1,2

τ = tp, tl, t1 = tp, t2 = tl


∈−++∈−−+

=τotherwise)min(

]7050[if)021max(50

]4020[if)20)(20(520

)(o

b,a

.,.b,a,.ba.

.,.b,a.b.a.

,I,b,at


∈−++∈−−+

=τotherwise)min(

]7050[if)021max(50

]4020[if)20)(20(520

)(o

b,a

.,.b,a,.ba.

.,.b,a.b.a.

,I,b,at



5.4 Triangular conorms


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 = 1a s 0 = a


Examples

a sm b = max(a,b) a sp b = a + b – ab

maximum probabilisticsum

(b) (a)


a sl b = max(a+b, 1)

Lukasiewicz drasticsum

==

=otherwise1

0if

0if

d ab

ba

bas

(c) (d)


basasbb,a d)max( ≤≤

aasa ≥

lower and upper bounds

Archimedean (if t is continuous)

1=na nilpotent (e.g. Lukasiewicz)

a s a = a2, … , an-1 s a = an


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

BABA

BABA

∩=∪∪=∩

Dual triangular norms

De Morgan


Constructors of s-norms


Additive and multiplicative generators

Ordinal sums



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


Examples

h = x2, x∈[0,1]

maximum

probabilisticsum

s sh

(a)

(b)


Lukasiewicz

drastic sum

h = x2, x∈[0,1]

s sh

(c)

(d)


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


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)


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


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


Ordinal sums

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

βα∈

α−βα−

α−βα−α−β+α=σ

otherwise)max(

][if)()(o

b,a

,b,ab

,a

s,I,b,as kk

kk

k

kk

kkkkk

I = [αk,βk], k∈K

nonempty, countable family pairwise disjoint subintervals of [0,1]

σ = sk, k∈K

family of t-conormsPedrycz and Gomide, FSE 2007

Example

∈−+−+∈−+−+

=σotherwise)max(

]7050[if)1)020(5)20min(5(2050

]4020[if)20)(20(5)20()20(20

)(o

b,a

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

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

,I,b,as

I = [0.2, 0.4], [0.5, 0.7] K =1,2

σ = sp, sl, s1 = sp, s2 = sl


∈−+−+∈−+−+

=σotherwise)max(

]7050[if)1)020(5)20min(5(2050

]4020[if)20)(20(5)20()20(20

)(o

b,a

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

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

,I,b,as


∈−+−+∈−+−+

=σotherwise)max(

]7050[if)1)020(5)20min(5(2050

]4020[if)20)(20(5)20()20(20

)(o

b,a

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

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

,I,b,as



5.5 Triangular norms as generalcategory of logical operators


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 (∨)


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


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

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

0 0 1 1

0 1 1 1

1 0 0 0

1 1 1 1

ϕ operator or Boolean values of its arguments

Implication induced by a t-norm

residuation


5.6 Aggregation operations



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) = 0g (1, 1,..., 1) = 1


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


0)(1

)(1

21 ≠∈∑==

p,p,xn

x,,x,xg pn

i

pin RK

∑

=−=

∏=→

∑==

=

=

=

n

ii

n

nn

iin

n

iin

x/

nx,,x,xgp

xx,,x,xgp

xn

x,,x,xgp

1

21

121

121

1)(1

)(0

1)(1

K

K

K arithmetic mean

geometric mean

harmonic mean


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)


Examples

0 1 2 3 4 5 6 7 8 0

0.2

0.4

0.6

0.8

1

1.2

x

(b) Arithmetic mean of A and B

A B

Arithmetic mean

0 1 2 3 4 5 6 7 8 0

0.2

0.4

0.6

0.8

1

1.2

x

(d) Geometric mean of A and B

A B

Geometric mean


0 1 2 3 4 5 6 7 8 0

0.2

0.4

0.6

0.8

1

1.2

x

(f) Harmonic mean of A and B

A B

Harmonic mean


Ordered Weighted Averaging (OWA)

∑==

n

iii xAwA

1)(),OWA( w

Σwi = 1, wi ∈ [0, 1]

A (x1) ≤ A(x2) ≤... ≤ A( xn)


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


0 1 2 3 4 5 6 7 8 0

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


0 1 2 3 4 5 6 7 8 0

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]


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


conormtandnormtareand1

))1(())1((

)()(

−−−

−−+−+=

=

uu

u

u

ste

ebeeuaeebas

e

ebueabat

Results on uninorms

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


2. If a ≤ e ≤ b or a ≥ e ≥ b then

),max(),min(

thenor If

baaubba

beabea

≤≤≥≥≤≤


≤≤≤≤

=

≤≤≤≤

=

≤≤

otherwise)max(

1if1

0if)min(

otherwise)min(

1if)max(

0if0

b,a

b,ae

eb,ab,a

bau

b,a

b,aeb,a

eb,a

bau

bauaubbau

s

w

sw

3. For any u with e ∈ [0, 1]


≤≤−−

−−−+

≤≤

=

=

≤≤−−

−−−+

≤≤

=

=

otherwise)max(

1if1

(1

)(1(

0if)()(

then1)10(if

otherwise)min(

1if1

(1

)(1(

0if)()(

then0)10(if

b,a

b,ae)e

ebs)

e

eaee

eb,ae

bt

e

ae

bau

u

b,a

b,ae)e

ebs)

e

eaee

eb,ae

bt

e

ae

bau

u

d

c

4. Conjunctive and disjunctive uninorm


Conjunctive uninorm

e = 0.5


e = 0.5

Disjunctive uninorm


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


7. Ordinal sum

≥∉

ι∈

α−βα−

α−βα−α−β+α

ι∈

α−βα−


=στ

otherwise)min(and

][if)max(

][if)(

][if)(

)( 2

1

co

b,aeb,a

,βαa,bb,a

b,ab

,a

s

b,ab

,a

t

,,ι,b,au

kk

kk

k

kk

kkkkk

kk

k

kk

kkkkk

l = [ αk, βk], k∈Kl1 = [ αk, βk] ∈ l | βk ≤ el2 = [ αk, βk] ∈ l | αk ≥ eτ = tk, k∈K, σ = sk, k∈K


≤∉

ι∈

α−βα−


ι∈

α−βα−


=στ

otherwise)max(and

][if)min(

][if)(

][if)(

)( 2

1

co

b,aeb,a

,βαa,bb,a

b,ab

,a

s

b,ab

,a

t

,,ι,b,au

kk

kk

k

kk

kkkkk

kk

k

kk

kkkkk


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]


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

ebeevaeebat

e

ebveabas

v

v

−−−+−+=

=

1))1(())1((

)()(


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

Example


Symmetric sums

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

1

21

2121 )(

)11(1)(

−

−−+=σn

nns a,,a,af

a,,a,afa,,a,a

K

KK

f increasing, continuous

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


0 1 2 3 4 5 6 7 8 0

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


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


(a) (b)

Example

γ = 0.5, t-norm = min, t-conorm = max


5.7 Fuzzy measure andintegral



Fuzzy measure

g : Ω → [0,1]

• Boundary conditions: g(∅) = 0g(X) = 1

• Monotonicity: if A ⊂ B then g(A) ≤ g(B)


λλλλ–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,

HAg,gxh )](min[sup)()(]10[

o

Hα = x | h(x) ≥ α


X = x1, x2,....,xn

h(x1) ≥ h(x2) ≥ ..... ≥ h(xn)

A1 = x1), A2 = x1, x2), ..., An = x1, x2,..., xn = X

∫ ==A ii

,n,iAg,xhgxh )]()(min[max)()(

1Ko


1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

xi

Fuzzy integra l

h(xi)

g(Ai)

Example


Choquet integral

)()]()([)( 11

iin

ii AgxhxhgfCh +

=−∫ ∑=o

h(xn+1) = 0


5.8 Negations


Definition

N : [0,1] → [0,1]

1. Monotonicity:

N is nonincreasing

2. Boundary conditions:

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


3. Continuity:

N is a continuous function

4. Involution:

N (N(x)) = x ∀x ∈ [0, 1]


Examples

≥<

=ax

axxN

if0

if1 )(

1.0

x1.0a


1.0

x1.0

==

=1 if0

0 if1 )(

x

xxN


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 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(11

)( ∞−∈λλ+

−= ,x

xxN


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 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

)0(1 )( ∞∈−= ,wxxN w w


(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]


Examples

)1(1

1

)11(min

)1

10(max

∞−∈+−=

+−+=+

+−+=

,λλx

xN(x)

λxyy,xxsyλ

λxyyx,xty

xxN

yxxsy

yxxty

−===

1 )(

),max(

),min(


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