Barycentrically associative aggregation functions

Barycentrically associative aggregation functions

Jean-Luc Marichal Bruno Teheux

University of Luxembourg

B-associative functions

LetX be a nonempty set and let X^∗ = [

n∈N

Xⁿ

F:X^∗ →X is B-associative(Bemporad, 1926) if

F(x₁, . . . ,x_p, y₁, . . . ,y_q, z₁, . . . ,z_r)

= F(x1, . . . ,xp,F(y1, . . . ,yq). . . ,F(y1, . . . ,yq)

| {z }

q times

,z1, . . . ,zr)

Example: F(x1, . . . ,xn) = ^x1+···+x_n ⁿ onX =R

Notation

We regardn-tuplesx in Xⁿ asn-stringsover X 0-string: ε

1-strings: x,y,z, . . . n-strings: x,y,z, . . .

X^∗ is endowed with concatenation

Example: x∈Xⁿ,y ∈X,z∈X^m ⇒ xyz∈X^n+1+m xⁿ = x· · ·x (n times)

|x| = length ofx

Functions of multiple arities

Let

X^∗ = [

n∈N

Xⁿ

F:X^∗→X

Components ofF :

F_n:Xⁿ→X F_n = F|_Xⁿ

F is described by its components F₁,F₂,F₃, . . . ,F_n, . . .

B-associative functions

F:X^∗ →X is B-associativeif

F(xyz) = F(xF(y)^|y|z) ∀ xyz∈X^∗

Theorem

F:X^∗ →X is B-associative if and only if

F(xy) = F(F(x)^|x|F(y)^|y|) ∀ xy∈X^∗ Interpretation ?

B-associative functions

Theorem(Kolomogorov-Nagumo, 1930)

F:R^∗→Ris B-associative and everyF_n is symmetric, continuous, idempotent (i.e.,F_n(xⁿ) =x), and strictly increasing in each argu- ment if and only if there exists a continuous and strictly monotone functionϕ:R→R such that

Fn(x) = ϕ⁻¹ 1 n

n

X

i=1

ϕ(xi)

!

ϕ(x) Mean x arithmetic x² quadratic x^k root-power 1/x harmonic log(x) geometric exp(x) exponential

B-associative functions

Theorem(M., Mathonet, and Tousset, 1999)

F:R^∗ → R is B-associative and everyF_n is nondecreasing in each argument and satisfies

F_n(rx₁+s, . . . ,rx_n+s) = r F_n(x₁, . . . ,x_n) +s r,s ∈R, r >0, if and only if either

(i) Fn= min for everyn∈N, or (ii) Fn= max for everyn∈N, or (iii) Fn(x) =P

iwi(θ)xi for every n∈N, where wi(θ) = θⁱ⁻¹(1−θ)ⁿ⁻ⁱ

P

jθ^j−1(1−θ)^n−j

B-associative functions

F:X^∗ →X is B-associativeif

F(xyz) = F(xF(y)^|y|z) ∀ xyz∈X^∗

Theorem

We can assume that|xz|61 in the definition above That is,F:X^∗ →X is B-associative if and only if

F(y) = F(F(y)^|y|) F(xy) = F(xF(y)^|y|) F(yz) = F(F(y)^|y|z)

B-preassociative functions

LetY be a nonempty set

Definition. We say that F:X^∗ →Y isB-preassociative if

|y|=|y⁰| and F(y) = F(y⁰) ⇒ F(xyz) = F(xy⁰z)

Example: Fn(x) =x₁²+· · ·+x_n² (X =Y =R)

Proposition

F:X^∗ →Y is B-preassociative if and only if

|x|=|x⁰| and F(x) = F(x⁰)

|y|=|y⁰| and F(y) = F(y⁰)

⇒ F(xy) = F(x⁰y⁰)

B-preassociative functions

Remark. If F:X^∗ →X is B-associative, then it is B-preassociative Proof. Suppose |y|=|y⁰|andF(y) =F(y⁰)

ThenF(xyz) =F(xF(y)^|y|z) =F(xF(y⁰)^|y0|z) =F(xy⁰z)

Proposition

F:X^∗ →X is B-associative if and only if it is B-preassociative and F(F(x)^|x|) =F(x)

Proof. (Necessity) OK.

(Sufficiency) We haveF(y) =F(F(y)^|y|)

Hence, by B-preassociativity,F(xyz) =F(xF(y)^|y|z)

B-preassociative functions

Proposition

IfF:X^∗ →Y is B-preassociative, then so isF◦(g, . . . ,g) for every functiong:X →X, where

F ◦(g, . . . ,g) : x1· · ·xn 7→Fn(g(x1)· · ·g(xn)) Example: Fn(x) =x₁²+· · ·+x_n² (X =Y =R)

Proposition

IfF:X^∗ →Y is B-preassociative, then so isg◦F for every function g:Y →Y such thatg|_ran(F) is constant or one-to-one

Example: F_n(x) = exp(x₁²+· · ·+x_n²) (X =Y =R)

B-preassociative functions

Proposition

AssumeF:X^∗→Y is B-preassociative IfFn is constant, then so isFn+1

Proof. If Fn(y) =Fn(y⁰) for all y,y⁰∈Xⁿ, then

F_n+1(xy) =F_n+1(xy⁰) and henceF_n+1 depends only on its first argument...

Open question:

Find necessary and sufficient conditions on a B-preassociative functionF forF_n+1 to be completely determined byF_n

B-preassociative functions

We have seen thatF:X^∗→X is B-associative if and only if it is B-preassociative andF(F(x)^|x|) =F(x)

Remark. The latter condition can be rewritten as

Fn(Fn(x)ⁿ) = Fn(x) ∀n∈N or

δ_Fn(Fn(x)) = Fn(x) ∀n∈N Relaxation of δFn◦Fn=Fn:

ran(δ_Fn) = ran(F_n) ∀n∈N

B-preassociative functions

Theorem

LetF:X^∗ →Y be a function. The following assertions are equiv- alent:

(i) F is B-preassociative and satisfiesran(δ_Fn) =ran(F_n) (ii) F_n can be factorized intoF_n=f_n◦H_n,

whereH:X^∗ →X is B-associative andfn:ran(Hn)→Y is one-to-one.

In this case, we havef_n=δ_Fn|_ran(Hn) andF_n=δ_Fn◦H_n

Open problems

(1) Suppress the conditionran(δ_Fn) =ran(F_n) in this theorem (2) Find necessary and sufficient conditions on δ_Fn for a function

F of the form F_n=δ_Fn◦H_n, whereH is B-associative, to be B-preassociative.

Axiomatizations of function classes

Theorem

F:R^∗→Ris B-preassociative and

every F_n is symmetric, continuous, and strictly increasing in each argument

if and only if there exist continuous and strictly increasing function ϕ:R→R andψ_n:R→R(n∈N) such that

F_n(x) = ψ_n

n

X

i=1

ϕ(x_i)

!

Open problem

Find new axiomatizations of classes of B-preassociative functions from existing axiomatizations of classes of B-associative functions

Thank you for your attention !