Pivotal decompositions of aggregation functions

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Pivotal decompositions of aggregation

functions

Jean-Luc Marichal Bruno Teheux

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Shannon decomposition

For f : {0, 1}n → {0, 1}, f (x) = xkf (x xk=1) + (1 − xk)f (x xk=0), ∀x, k .

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Shannon decomposition

For f : {0, 1}n → {0, 1}, f (x) = xkf (x xk=1) + (1 − xk)f (x xk=0), ∀x, k .

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Median decomposition

For a Sugeno integral

f (x) = _ S⊆[n] µ(S) ∧^ i∈S xi, f (x) = med(xk,f (x _x k=0),f (x _x k=1)), ∀x, k

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Median decomposition

For a Sugeno integral

f (x) = _ S⊆[n] µ(S) ∧^ i∈S xi, f (x) = med(xk,f (x _x k=0),f (x _x k=1)), ∀x, k

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Pivotal decomposition

General assumption :

f : [0, 1]n _{→ R}

Definition

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Median decomposition For a Sugeno integral

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Multilinear polynomial functions For a multilinear polynomial function

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Multilinear polynomial functions For a multilinear polynomial function

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Conjugate weighted means

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Conjugate weighted means

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Discrete Choquet integrals For discrete Choquet integrals

f (x) = X

S⊂[n]

a_S^

i∈S

xi, aS ∈ R,

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Discrete Choquet integrals For discrete Choquet integrals

f (x) = X

S⊂[n]

a_S^

i∈S

xi, aS ∈ R,

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Why pivotal decomposition ?

f (x) = Φxk,f (x xk=0),f (x xk=1) , ∀x, k .

I Uniformly isolate the marginal contribution of a factor.

I Ease computations.

I Allow proofs by induction.

I Repeated applications of the decomposition lead to canonical forms.

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Proposition

Let f be Φ-decomposable

I Uniqueness of Φ,

I f is determined by Φ and f

{0,1}n.

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Two characterizations of classes

A function f is a multilinear polynomial functionif and only if

f (x) = xkf (x xk=1) + (1 − xk)f (x xk=0), ∀x, lk

A function f is is a lattice polynomial functionif and only if

f (x) = med(xk,f (x xk=0),f (x xk=1)), ∀x, ∀k

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Two characterizations of classes

A function f is a multilinear polynomial functionif and only if

f (x) = xkf (x xk=1) + (1 − xk)f (x xk=0), ∀x, lk

A function f is is a lattice polynomial functionif and only if

f (x) = med(xk,f (x xk=0),f (x xk=1)), ∀x, ∀k

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We note f ≡ g if f and g are equal up to permutation of variables, addition or deletion of inessential variables.

Definition

Let D ⊆ R2and Φ : [0, 1] × D. Let ΓΦdefined by

f ∈ Γ_Φiff f is Φ-decomposable

(up to inessential variables).

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We note f ≡ g if f and g are equal up to permutation of variables, addition or deletion of inessential variables. Definition

Let D ⊆ R2and Φ : [0, 1] × D. Let ΓΦdefined by

f ∈ Γ_Φiff f is Φ-decomposable(up to inessential variables).

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Multilinear polynomial functions

The class of multilinear polynomial functions is ΓΦ for

Φ(x , y , z) = (1 − x ) u + y z

Lattice polynomial functions

The class of lattice polynomial functions is ΓΦfor

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Multilinear polynomial functions

The class of multilinear polynomial functions is ΓΦ for

Φ(x , y , z) = (1 − x ) u + y z

Lattice polynomial functions

The class of lattice polynomial functions is ΓΦfor

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Non-decreasing multilinear polynomial functions

The class of non-decreasing multilinear polynomial functions is ΓΦ for

Φ(x , y , z) = (1 − x ) (y ∧ z) + x (y ∨ z)

t-norms

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Non-decreasing multilinear polynomial functions

The class of non-decreasing multilinear polynomial functions is ΓΦ for

Φ(x , y , z) = (1 − x ) (y ∧ z) + x (y ∨ z)

t-norms

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Symmetric multilinear polynomials No pivotal characterization

Choquet integrals

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How to recognize pivotally characterized classes ?

Problem

Given a class C of functions, can you determine if it is pivotally characterized ?

Example

Is the class of Sugeno integrals pivotally characterized ?

x ∧ y x ∧ y ∧ 3

med-decomposable med-decomposable

Sugeno Non Sugeno

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How to recognize pivotally characterized classes ?

Problem

Given a class C of functions, can you determine if it is pivotally characterized ?

Example

Is the class of Sugeno integrals pivotally characterized ?

x ∧ y x ∧ y ∧ 3

med-decomposable med-decomposable

Sugeno Non Sugeno

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Definition

A class C of functions is UM-characterized if for any f : [0, 1]n→ R

f ∈ C if and only if so is everyessentialunary section of f .

Example

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Definition

A class C of functions is UM-characterized if for any f : [0, 1]n→ R

f ∈ C if and only if so is everyessentialunary section of f .

Example

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Theorem

Assume that C is a Φ-characterized and that C0⊆ C. Then, C0is UM-characterized

if and only if

C0 is pivotally characterized.

Sugeno integral

Sugeno integrals arenotpivotally characterized : x ∧ 3 is a unary section of a Sugeno integral that is not a Sugeno integral.

Non-decreasing multilinear polynomial functions These are those multilinear polynomials that have non-decreasing unary sections.

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Theorem

if and only if

Sugeno integral

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Theorem

if and only if

Sugeno integral

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Definition

A function f is componentwise pivotally decomposable if there are some Φ1, . . . , Φnsuch that

f (x) = Φk(xk,f (x xk=0),f (x xk=1)), ∀x, k . Fact

A function f is pivotally decomposable

if and only if

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Definition

A function f is componentwise pivotally decomposable if there are some Φ1, . . . , Φnsuch that

f (x) = Φk(xk,f (x xk=0),f (x xk=1)), ∀x, k . Fact

A function f is pivotally decomposable

if and only if

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Binary Choquet integrals

Every binary Choquet integral f (x) = a x1+b x2+c (x1∧ x2)is

componentwise pivotally decomposable.

Choquet integrals

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Binary Choquet integrals

Every binary Choquet integral f (x) = a x1+b x2+c (x1∧ x2)is

componentwise pivotally decomposable.

Choquet integrals

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Questions

I Find pivotal decompositions / characterizations.

I Classes characterized by componentwise pivotal decompositions.

I Two pivots decomposition f (x) = Φ(x_k,xj,f (x