Kira Adaricheva Agata Pilitowska*pages.mini.pw.edu.pl/aab/PilitowskaAggregation.pdf · Kira...

Survey on aggregation theory

Kira Adaricheva Agata Pilitowska*

Yeshiva University

Warsaw University of Technology

Workshop ”Algebra across the borders”Yeshiva University, New York, August 11, 2011

*The work has been supported by the European

Union in the framework of European Social Fund through the Warsaw

University of Technology Development Programme

Adaricheva, Pilitowska Survey on aggregation theory

Survey on aggregation theory

PART I - INTRODUCTION

Definition, basic examples, elementary property

Some classes of aggregation functions

MeansAssociative aggregation functionst-norms, t-conorms, uninormsFuzzy integrals

Aggregation

The process of aggregation

An arbitrarily long vector of inputs x = (x1, . . . , xn)

A single output value A(x)

Example

Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.

In order to reach a consensus on these judgements, aggregationfunctions may be applied.

Aggregation

Example

Aggregation

Example

Aggregation

Example

Aggregation

Example

Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.In order to reach a consensus on these judgements, aggregationfunctions may be applied.

Aggregation functions

Aggregation functions are used in:

pure mathematics (theory of means and averages, measureand integration theory),

applied mathematics (probability, statistics),

computer and engineering sciences (artificial intelligence,information theory, automated reasoning),

economics and finance (game theory, voting theory, decisionmaking),

social sciences,

physics and natural sciences

I - a nonempty real interval, bounded or not

f : In → ℝ

Requirements:

boundary conditions

nondecreasing monotonicity

f : In → ℝ

Requirements:

boundary conditions

f : In → ℝ

Requirements:

boundary conditions

f : In → ℝ

Requirements:

boundary conditions

f : In → ℝ

Requirements:

boundary conditions

Definition

Let I ⊆ ℝ, n ∈ ℕ and let x := (x1, . . . , xn) ∈ In. An aggregationfunction with n variables is a function A : In → I such that

A is nondecreasing: for any x, x∗ ∈ In

x ≤ x∗ ⇒ A(x) ≤ A(x∗)

A fulfills the boundary conditions

infx∈In

A(x) = inf I and supx∈In

A(x) = sup I.

If I = [0, 1], the boundary conditions say:

A(0, . . . , 0) = 0 and A(1, . . . , 1) = 1.

Definition

Let I ⊆ ℝ, n ∈ ℕ and let x := (x1, . . . , xn) ∈ In. An aggregationfunction with n variables is a function A : In → I such that

A is nondecreasing: for any x, x∗ ∈ In

x ≤ x∗ ⇒ A(x) ≤ A(x∗)

A fulfills the boundary conditions

infx∈In

A(x) = inf I and supx∈In

A(x) = sup I.

If I = [0, 1], the boundary conditions say:

A(0, . . . , 0) = 0 and A(1, . . . , 1) = 1.

An extended aggregation function

The mapping

A :∪n∈ℕ

In → I

such thatA(n) := A∣In

is an n-ary aggregation function for any n ∈ ℕ.

A sequence of functions

(A(n) : In → I)n∈ℕ,

whose nth element is an n-ary function A(n) : In → I.

An extended aggregation function

The mapping

A :∪n∈ℕ

In → I

such thatA(n) := A∣In

is an n-ary aggregation function for any n ∈ ℕ.

A sequence of functions

(A(n) : In → I)n∈ℕ,

whose nth element is an n-ary function A(n) : In → I.

Aggregation functions - Basic examples

Let x := (x1, . . . , xn) ∈ In.

The arithmetic mean function

Aℳ(x) :=1

n∑i=1

The geometric mean function (for I ⊆ [0,∞])

Gℳ(x) := (n∏

xi )1n

The k-projection function, for any k ∈ {1, . . . , n}

Pk(x) := xk

Let � be a permutation on the set {1, . . . , n} such thatx�(1) ≤ . . . ≤ x�(n).

The k-order statistic function, for any k ∈ {1, . . . , n}

OSk(x) := x�(k)

The extreme order statistic functions

OS1(x) = min{x1, . . . , xn} = Min(x)

OSn(x) = max{x1, . . . , xn} = Max(x)

The median of values x1, . . . , xn

Med(x1, . . . , xn) =

{x�(k) if n = 2k − 1x�(k)+x�(k+1)

2 if n = 2k .

Let w := (w1, . . . ,wn) ∈ In be any (weight) vector such thatn∑

i=1wi = 1.

The weighted arithmetic mean function

WAℳw(x) :=n∑

The ordered weighted averaging function

OWAw(x) :=n∑

wix�(i)

Let S be a nonempty subset of {1, . . . , n}.The partial minimum function (associated with S)

MinS(x) := mini∈S

The partial maximum function (associated with S)

MaxS(x) := maxi∈S

The sum Σ : (0,∞)n → (0,∞)

Σ(x) :=n∑

The product Π : (0,∞)n → (0,∞)

Π(x) :=n∏

Aggregation functions - Examples

Assume I = [a, b] is a closed interval.

The smallest aggregation function

A⊥(x) :=

{b, if xi = b for all i ∈ {1, . . . , n}a, otherwise

The greatest aggregation function

A⊤(x) :=

{a, if xi = a for all i ∈ {1, . . . , n}b, otherwise

Non aggregation function

The constant function Kc : In → I, given by

Kc(x) := c ,

where c ∈ I is a fixed constant (unless I = {c}).

Aggregation functions - Examples

Assume I = [a, b] is a closed interval.

The smallest aggregation function

A⊥(x) :=

{b, if xi = b for all i ∈ {1, . . . , n}a, otherwise

The greatest aggregation function

A⊤(x) :=

{a, if xi = a for all i ∈ {1, . . . , n}b, otherwise

Non aggregation function

The constant function Kc : In → I, given by

Kc(x) := c ,

where c ∈ I is a fixed constant (unless I = {c}).

Elementary properties

IMPORTANT

The choice of aggregation function should be based uponproperties dedicated by the framework in which the aggregation isperformed.

Example

If we consider the aggregation of opinions in voting procedures,and as usual, the voters are anonymous, the aggregation functionmust be symmetric.

IMPORTANT

Example

IMPORTANT

Example

Continuity

Example

The function A : (0,∞)n → (0,∞)

A(x) :=

i=1xi )

2 −n∑

i=1x2i

n∑i=1

is continuous aggregation function.

Idempotency

Definition

A : In → ℝ is an idempotent function if, for all x ∈ I,

A(x , . . . , x) = x .

Aℳ, WAℳw, OWAw, Min, Max and Med are idempotentaggregation functions.

Idempotency

Definition

A : In → ℝ is an idempotent function if, for all x ∈ I,

A(x , . . . , x) = x .

Aℳ, WAℳw, OWAw, Min, Max and Med are idempotentaggregation functions.

Idempotency

Definition

An element e ∈ I is idempotent for A : [a, b]n → ℝ if,

A(e, . . . , e) = e.

Let c ∈ (0, 1) be an arbitrarily chosen element. The aggregationfunction Ac : [0, 1]n → [0, 1]

Ac(x) := Med(0, c +n∑

(xi − c), 1)

is not idempotent but has a nonextreme idempotent element c .

Idempotency

Definition

An element e ∈ I is idempotent for A : [a, b]n → ℝ if,

A(e, . . . , e) = e.

Let c ∈ (0, 1) be an arbitrarily chosen element. The aggregationfunction Ac : [0, 1]n → [0, 1]

Ac(x) := Med(0, c +n∑

(xi − c), 1)

is not idempotent but has a nonextreme idempotent element c .

Idempotency

Definition

A : [a, b]n → ℝ is weakly idempotent function if a and b areidempotent elements for A.

Every aggregation function on I = [a, b] is weakly idempotent.

Idempotency

Definition

A : [a, b]n → ℝ is weakly idempotent function if a and b areidempotent elements for A.

Every aggregation function on I = [a, b] is weakly idempotent.

Symmetry

Definition

A : In → ℝ is symmetric function if, for all x ∈ In and anypermutation � on the set {1, . . . , n}

A(x) = A(x�(1), . . . , x�(n)).

Functions: Aℳ, Gℳ and OWAw are symmetric.Nonsymmetric aggregation function is WAℳw.

Symmetry

Definition

A(x) = A(x�(1), . . . , x�(n)).

Functions: Aℳ, Gℳ and OWAw are symmetric.

Nonsymmetric aggregation function is WAℳw.

Symmetry

Definition

A(x) = A(x�(1), . . . , x�(n)).

Functions: Aℳ, Gℳ and OWAw are symmetric.Nonsymmetric aggregation function is WAℳw.

Symmetry

Proposition [J.J.Rotman, 1995]

A : In → ℝ is symmetric function if and only if, for all x ∈ In wehave

A(x2, x1, x3, . . . , xn) = A(x1, x2, x3, . . . , xn)

A(x2, x3, . . . , xn, x1) = A(x1, x2, x3, . . . , xn)

Conjunction

Definition

A function A : In → ℝ is conjunctive if, for all x ∈ In

inf I ≤ A(x) ≤ Min(x).

The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.Min is the greatest conjunctive aggregation function.Min is also the only idempotent conjunctive aggregation function.

Conjunction

Definition

The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.

Min is the greatest conjunctive aggregation function.Min is also the only idempotent conjunctive aggregation function.

Conjunction

Definition

The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.Min is the greatest conjunctive aggregation function.

Min is also the only idempotent conjunctive aggregation function.

Conjunction

Definition

The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.Min is the greatest conjunctive aggregation function.Min is also the only idempotent conjunctive aggregation function.

Disjunction

Definition

A function A : In → ℝ is disjunctive if, for all x ∈ In

Max(x) ≤ A(x) ≤ sup I.

The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.Max is the smallest conjunctive aggregation function.Max is the only idempotent disjunctive aggregation function.

Disjunction

Definition

The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.

Max is the smallest conjunctive aggregation function.Max is the only idempotent disjunctive aggregation function.

Disjunction

Definition

The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.Max is the smallest conjunctive aggregation function.

Max is the only idempotent disjunctive aggregation function.

Disjunction

Definition

The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.Max is the smallest conjunctive aggregation function.Max is the only idempotent disjunctive aggregation function.

Internality

Definition

A function A : In → ℝ is internal if, for all x ∈ In

Min(x) ≤ A(x) ≤ Max(x).

Proposition [B.de Finetti, 1931]

If A : In → ℝ is internal, then it is idempotent.

If A : In → ℝ is nondecreasing and idempotent then it is internal.

Internality

Definition

If A : In → ℝ is internal, then it is idempotent.

If A : In → ℝ is nondecreasing and idempotent then it is internal.

Internality

Definition

If A : In → ℝ is internal, then it is idempotent.If A : In → ℝ is nondecreasing and idempotent then it is internal.

Associativity

Min and Max are associative functions.

Functions Aℳ and Gℳ are not associative.

Extended associativity

A sequence of functions (A(n) : In → ℝ)n∈ℕ is associative if, forany x ∈ I, x1 ∈ Ip and x2 ∈ Ik with n = p + k ,

A(1)(x) = x , and

A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).

The aggregation procedure can be decomposed into partialaggregation and we can start with the aggregation procedurebefore knowing all inputs to be aggregated.

Associativity

Min and Max are associative functions.Functions Aℳ and Gℳ are not associative.

A(1)(x) = x , and

A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).

Associativity

A(1)(x) = x , and

A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).

Associativity

A(1)(x) = x , and

A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).

Definition

An n-ary internal aggregation function A : In → I is called a mean.

Quasi-arithmetic mean

Definition

Let f : I→ ℝ be a continuous and strictly monotonic function.The n-ary quasi-arithmetic mean (generated by f ) is the functionAf : I n → I defined as

Af (x) := f −1(1

n∑i=1

f (xi )).

Remark

Quasi-arithmetic means are symmetric functions.

Quasi-arithmetic mean

Definition

Let f : I→ ℝ be a continuous and strictly monotonic function.The n-ary quasi-arithmetic mean (generated by f ) is the functionAf : I n → I defined as

Af (x) := f −1(1

n∑i=1

f (xi )).

Remark

Quasi-arithmetic means are symmetric functions.

Quasi-arithmetic means - Examples

Arithmetic mean

Aℳ(x) :=1

n∑i=1

xi , with f (x) = x

Quadratic mean

Qℳ(x) := (1

n∑i=1

x2i )12 , with f (x) = x2

Geometric mean

Gℳ(x) := (n∏

xi )1n , with f (x) = log x

Harmonic mean

ℋℳ(x) :=1

n∑i=1

, with f (x) = x−1

Root mean power

ℳid�(x) := (1

n∑i=1

x�i )1� , with f (x) = x�, 0 ∕= � ∈ ℝ

Exponential mean

ℰℳ�(x) :=1

�ln(

n∑i=1

e�xi ), with f (x) = e�x , 0 ∕= � ∈ ℝ

A : (0, 1)n → (0, 1)

A(x) :=Gℳ(x)

Gℳ(x) + Gℳ(1− x),

where 1 := (1, . . . , 1).The function A is a quasi-arithmetic mean generated by theincreasing function f (x) = log x

1−x , whose inverse function is given

by f −1(x) = ex

1+ex .

Quasi-linear mean (weighted quasi-arithmetic mean)

Definition

Let f : I→ ℝ be a continuous, strictly monotonic function and let

w1, . . . ,wn > 0 be real numbers fulfillingn∑

i=1wi = 1. The n-ary

quasi-linear mean is the function Afw : I n → I defined as

Afw(x) := f −1(n∑

wi f (xi )).

Quasi-linear means - Examples

Weighted arithmetic mean

WAℳw(x) :=n∑

wixi , with f (x) = x

Weighted quadratic mean

WQℳw(x) := (n∑

wix2i )

12 , with f (x) = x2

Quasi-linear means - Examples

Weighted geometric mean

WGℳw(x) :=n∏

xwii , with f (x) = log x

Weighted root mean power

Wℳid�,w(x) := (n∑

wix�i )

1� , with f (x) = x�, 0 ∕= � ∈ ℝ

Associative aggregation functions

Theorem, [J.Aczel, 1948]

Let I be a real interval, which is open on one side. A : I2 → I iscontinuous, strictly increasing, and associative if and only if thereexists a continuous and strictly monotonic function f : I→ ℝ suchthat

A(x , y) = f −1(f (x) + f (y)).

Corollary

Every continuous, strictly increasing, and associative function issymmetric.

A(x , y) = f −1(f (x) + f (y)).

Corollary

A(x , y) = f −1(f (x) + f (y)).

Corollary

A(x , y) = f −1(f (x) + f (y))

Example

The sum Σ : (0,∞)2 → (0,∞)

Σ(x , y) = x + y , with f (x) = x

and the product Π : (0,∞)2 → (0,∞)

Π(x , y) = x ⋅ y = 10log(x)+log(y), with f (x) = log(x)

are continuous, strictly increasing, and associative functions.

A(x , y) = f −1(f (x) + f (y))

If x ∈ I then

A(x , x) = f −1(f (x) + f (x)) = f −1(2f (x)) = x ⇔ f (x) = 0.

Since f is strictly monotonic, A cannot be idempotent.

Corollary

There is no continuous, strictly increasing, idempotent, andassociative function.

A(x , y) = f −1(f (x) + f (y))

If x ∈ I then

A(x , x) = f −1(f (x) + f (x)) = f −1(2f (x)) = x ⇔ f (x) = 0.

Corollary

A(x , y) = f −1(f (x) + f (y))

If x ∈ I then

A(x , x) = f −1(f (x) + f (x)) = f −1(2f (x)) = x ⇔ f (x) = 0.

Corollary

Theorem [C.-H. Ling, 1965]

Let I = [a, b] be a real interval. A : I2 → I is continuous,nondecreasing, associative, and such that

A(b, x) = x , for x ∈ I (b is an identity for A)

A(x , x) < x , for x ∈ (a, b) (there is no idempotents for A in (a, b))

if and only if there exists a continuous and strictly decreasingfunction f : I→ [0,∞], with f (b) = 0, such that

A(x , y) = f −1(Min(f (x) + f (y), f (a))).

If A : I2 → I is an associative function then (I,A) is a semigroup.

Definition

Let {(Ik ,Ak) ∣ k ∈ K} be a collection of disjoint semigroupsindexed by a set K . The sum of {(Ik ,Ak) ∣ k ∈ K} is thesemigroup (

∪k∈K

Ik ,A) defined on the set-theoretic union∪

k∈KIk

under the following binary operation:

A(x , y) :=

{Ak(x , y), if exists k ∈ K such that x , y ∈ IkMin(x , y), if exist k1 ∕= k2 ∈ K with x ∈ Ik1 , y ∈ Ik2

If A : I2 → I is an associative function then (I,A) is a semigroup.

Definition

Let {(Ik ,Ak) ∣ k ∈ K} be a collection of disjoint semigroupsindexed by a set K . The sum of {(Ik ,Ak) ∣ k ∈ K} is thesemigroup (

∪k∈K

Ik ,A) defined on the set-theoretic union∪

k∈KIk

under the following binary operation:

A(x , y) :=

{Ak(x , y), if exists k ∈ K such that x , y ∈ IkMin(x , y), if exist k1 ∕= k2 ∈ K with x ∈ Ik1 , y ∈ Ik2

Definition

A semigroup (I = [a, b],A) is called Archimedean, if the functionA : I2 → I is continuous, nondecreasing, A(b, x) = x , for x ∈ I,and A(x , x) < x , for x ∈ (a, b) or A(a, x) = x , for x ∈ I, andA(x , x) < x , for x ∈ (a, b) (one endpoint of I is an identity for A,and there are no idempotents for A in (a, b)).

Theorem [P.S.Mostert and A.L.Shields, 1957]

A function A : [a, b]2 → [a, b] is continuous, associative, and suchthat A(a, x) = A(x , a) = a (a is a zero element) andA(b, x) = A(x , b) = x (b is an identity) if and only if

either A(x , y) = Min(x , y),

or ([a, b],A) is a conjunctive Archimedean semigroup,

or ([a, b],A) is the sum of conjunctive Archimedeansemigroups and one-point semigroups.

Theorem [J.C.Fodor, 1991]

A function A : I2 → I is continuous, nondecreasing, idempotent,and associative if and only if there exist �, � ∈ I such that for allx , y ∈ I

A(x , y) = Max(Min(�, x),Min(�, x),Min(x , y)).

Theorem [L.W.Fung and K.S.Fu, 1975]

A function A : I2 → I is symmetric, continuous, nondecreasing,idempotent, and associative if and only if there exists � ∈ I suchthat for all x , y ∈ I

A(x , y) = Med(x , y , �).

Theorem [J.C.Fodor, 1991]

A function A : I2 → I is continuous, nondecreasing, idempotent,and associative if and only if there exist �, � ∈ I such that for allx , y ∈ I

A(x , y) = Max(Min(�, x),Min(�, x),Min(x , y)).

Theorem [L.W.Fung and K.S.Fu, 1975]

A function A : I2 → I is symmetric, continuous, nondecreasing,idempotent, and associative if and only if there exists � ∈ I suchthat for all x , y ∈ I

A(x , y) = Med(x , y , �).

Theorem [E.Czogala and J.Drewniak, 1984]

If a function A : I2 → I is nondecreasing, idempotent, associative,and has an identity element e ∈ I, then there exists a decreasingfunction g : I→ I, with g(e) = e, such that for all x , y ∈ I

A(x , y) :=

⎧⎨⎩Min(x , y), if y < g(x),

Max(x , y), if y > g(x),

Max(x , y) or Min(x , y), if y = g(x).

If a function A : I2 → I is continuous, nondecreasing, idempotent,associative, and has an identity, then

A(x , y) = Min(x , y) or A(x , y) = Max(x , y).

Theorem [E.Czogala and J.Drewniak, 1984]

If a function A : I2 → I is nondecreasing, idempotent, associative,and has an identity element e ∈ I, then there exists a decreasingfunction g : I→ I, with g(e) = e, such that for all x , y ∈ I

A(x , y) :=

⎧⎨⎩Min(x , y), if y < g(x),

Max(x , y), if y > g(x),

Max(x , y) or Min(x , y), if y = g(x).

If a function A : I2 → I is continuous, nondecreasing, idempotent,associative, and has an identity, then

A(x , y) = Min(x , y) or A(x , y) = Max(x , y).

t-norms, t-conorms, uninorms

Definition

A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.

A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.

t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)

Definition

A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.

A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.

Definition

A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.

A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.

Definition

A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.

Definition

t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))

t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)

Definition

t-norms, t-conorms, uninorms - Examples

t-norms

N : [0, 1]2 → [0, 1], N(x , y) := xy ,

N : [0, 1]2 → [0, 1], N(x , y) := Max(x + y − 1, 0).

t-conorms

N : [0, 1]2 → [0, 1], N(x , y) := x + y − xy ,

N : [0, 1]2 → [0, 1], N(x , y) := Min(x + y , 1).

t-norms, t-conorms, uninorms - Examples

Uninorms with neutral element e = 12

N : [0, 1]2 → [0, 1], N(x , y) :=

{2xy , if (x , y) ∈ [0, 12 ]2

Max(x , y), otherwise

N : [0, 1]2 → [0, 1], N(x , y) :=

⎧⎨⎩

Max(x , y), if (x , y) ∈ [12 , 1]2

4xy , if (x , y) ∈ [0, 14 ]2

14(4x − 1)(4y − 1) + 1

4 , if

(x , y) ∈ [14 ,12 ]2

Min(x , y), otherwise.

Theorem [J.-L.Marichal, 2009]

A function A : [0, 1]2 → [0, 1] is continuous, nondecreasing, weaklyidempotent, associative and has an identity in [0, 1] if and only ifA is a continuous t-norm or a continuous t-conorm.

Fuzzy integrals

Definition

A (discrete) fuzzy measure on N := {1, . . . , n} is a set function

� : 2N → [0, 1]

that is

monotonic (S ⊆ T ⇒ �(S) ≤ �(T )),

fulfills the boundary conditions

�(∅) = 0 and �(N) = 1.

ℱN := the set of all fuzzy measures on N

Fuzzy integrals

Definition

A (discrete) fuzzy measure on N := {1, . . . , n} is a set function

� : 2N → [0, 1]

that is

monotonic (S ⊆ T ⇒ �(S) ≤ �(T )),

fulfills the boundary conditions

�(∅) = 0 and �(N) = 1.

ℱN := the set of all fuzzy measures on N

The Choquet integrals

Definition

Let � ∈ ℱN . The (discrete) Choquet integral of x ∈ ℝn withrespect to � is defined by

C�(x) :=n∑

x�(i)(�(A�(i))− �(A�(i+1))),

where A�(i) := {�(i), . . . , �(n)} and A�(n+1) := ∅.(x�(1) ≤ . . . ≤ x�(n))

Example

If x3 ≤ x1 ≤ x2, we have C�(x1, x2, x3) =

x3(�({3, 1, 2})− �({1, 2})) + x1(�({1, 2})− �({2})) + x2�({2}).

Definition

Let � ∈ ℱN . The (discrete) Choquet integral of x ∈ ℝn withrespect to � is defined by

C�(x) :=n∑

x�(i)(�(A�(i))− �(A�(i+1))),

where A�(i) := {�(i), . . . , �(n)} and A�(n+1) := ∅.(x�(1) ≤ . . . ≤ x�(n))

Example

If x3 ≤ x1 ≤ x2, we have C�(x1, x2, x3) =

x3(�({3, 1, 2})− �({1, 2})) + x1(�({1, 2})− �({2})) + x2�({2}).

The Choquet integral - properties

continuous

nondecreasing

idempotent

internal

comonotonic additive:

f (x1 + x ′1, . . . , xn + x ′n) = f (x1, . . . , xn) + f (x ′1, . . . , x′n),

for all vectors such that there is no i , j ∈ N such that xi > xjand x ′i < x ′jlinear with respect to the fuzzy measure:

C�1�1+�2�2 = �1C�1 + �2C�2 ,

for any �1, �2 ∈ ℱN and all �1, �2 ∈ ℝ such that�1�1 + �2�2 ∈ ℱN .

Proposition [D.Schmeidler, 1986]

A function f : ℝn → ℝ is nondecreasing, comonotonic additive,and fulfills f (1) = 1 if and only if there exists � ∈ ℱN such thatf = C�.

Proposition [T.Murofushi and M.Sugeno, 1993]

The Choquet integral C� : ℝn → ℝ is additive(f (x1 + x ′1, . . . , xn + x ′n) = f (x1, . . . , xn) + f (x ′1, . . . , x

′n), for all

vectors x, x′ ∈ ℝn) if and only if there exists w ∈ [0, 1]n withn∑

i=1wi = 1 such that C� = WAℳw =

n∑i=1

wixi .

Proposition [M.Grabisch, 1995]

The Choquet integral C� : ℝn → ℝ is a symmetric function if and

only if there exists a weight vector w ∈ [0, 1]n withn∑

i=1wi = 1 such

C� = OWAw =n∑

wix�(i).

The Sugeno integrals

The unit interval [0, 1] has naturally defined a structure of adistributive lattice (chain) with binary operations:

x ∧ y := min{x , y} and x ∨ y := max{x , y}.

Definition

Let � ∈ ℱN . The (discrete) Sugeno integral of x ∈ [0, 1]n withrespect to � is defined by

S�(x) :=n⋁

(x�(i) ∧ �(A�(i))),

where A�(i) := {�(i), . . . , �(n)} and A�(n+i) := ∅.(x�(1) ≤ . . . ≤ x�(n))

Example

If x3 ≤ x1 ≤ x2, then we have S�(x1, x2, x3) =

(x3 ∧ �({3, 1, 2})) ∨ (x1 ∧ �({1, 2})) ∨ (x2 ∧ �({2})).

Definition

Let � ∈ ℱN . The (discrete) Sugeno integral of x ∈ [0, 1]n withrespect to � is defined by

S�(x) :=n⋁

(x�(i) ∧ �(A�(i))),

where A�(i) := {�(i), . . . , �(n)} and A�(n+i) := ∅.(x�(1) ≤ . . . ≤ x�(n))

Example

If x3 ≤ x1 ≤ x2, then we have S�(x1, x2, x3) =

(x3 ∧ �({3, 1, 2})) ∨ (x1 ∧ �({1, 2})) ∨ (x2 ∧ �({2})).

S�(x) :=⋁T⊆N

(�(T ) ∧ (⋀i∈T

The Sugeno integrals - properties

continuous

nondecreasing

idempotent

internal

comonotonic minitivity:

f (x1 ∧ x ′1, . . . , xn ∧ x ′n) = f (x1, . . . , xn) ∧ f (x ′1, . . . , x′n),

for all vectors such that there is no i , j ∈ N such that xi > xjand x ′i < x ′jcomonotonic maxitivity:

f (x1 ∨ x ′1, . . . , xn ∨ x ′n) = f (x1, . . . , xn) ∨ f (x ′1, . . . , x′n),

for all vectors such that there is no i , j ∈ N such that xi > xjand x ′i < x ′j

Proposition [J.-L.Marichal, 1998]

A function f : [0, 1]n → [0, 1] is nondecreasing, idempotent,comonotonic minitive and maxitive if and only if there exists� ∈ ℱN such that f = S�.

The set of Sugeno integrals coincides with the set of idempotentlattice polynomial functions.

Lattice term functions are Sugeno integrals defined from fuzzymeasures � : 2N → {0, 1} having their value in the set {0, 1}.

The set of Sugeno integrals coincides with the set of idempotentlattice polynomial functions.

Lattice term functions are Sugeno integrals defined from fuzzymeasures � : 2N → {0, 1} having their value in the set {0, 1}.

References

M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap,AggregationFunctions, Encyclopedia of Mathematics and Its Applicationsvol. 127, Cambridge University Press, Cambrige, 2009.

J.-L. Marichal, Aggregation functions for decision making,arXiv:0901.4232v1 [math.ST] 27 Jan 2009.

Kira Adaricheva Agata Pilitowska*pages.mini.pw.edu.pl/aab/PilitowskaAggregation.pdf · Kira...

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