• Aucun résultat trouvé

On the moments and the distribution of the Choquet integral

N/A
N/A
Protected

Academic year: 2021

Partager "On the moments and the distribution of the Choquet integral"

Copied!
10
0
0

Texte intégral

(1)

On the moments and the distribution of the Choquet integral

Ivan Kojadinovic Jean-Luc Marichal Kindly presented by Michel Grabisch

Department of Statistics, University of Auckland Applied Mathematics Unit, University of Luxembourg

EUSFLAT 2007

(2)

university-logo

The Choquet integral

The Choquet integralof x∈Rn w.r.t. a capacity ν onN is defined by Cν(x) :=

n

X

i=1

pν,σi xσ(i),

where σ is a permutation onN such that xσ(1) >· · ·>xσ(n), where piν,σ:=νiσ−νi−1σ , ∀i ∈N,

and where νiσ :=ν {σ(1), . . . , σ(i)}

for anyi = 0, . . . ,n. In particular, ν0σ := 0.

(3)

The Choquet integral

Let Sn denote the set of permutations on N.

The Choquet integral can therefore be regarded as a piecewise linear function that coincides with a weighted arithmetic mean on each n-dimensional region

Rσ :={x∈Rn|xσ(1)>· · ·>xσ(n)} (σ ∈Sn), whose union covers Rn.

We state hereafter immediate relationshipsbetween the moments (resp.

the c.d.f.) of the Choquet integraland the moments (resp. the c.d.f.) of linear combination of order statistics.

(4)

university-logo

Distributional relationships with linear combination of order statistics

Let X1, . . . ,Xn be a random sample from a continuous distribution with p.d.f.f.

Let X1:n6· · ·6Xn:n denote the corresponding order statistics.

Furthermore, let Yν :=Cν(X1, . . . ,Xn) and let h be any function.

By definition of the expectation, we have E[h(Yν)] =

Z

Rn

h(Cν(x1, . . . ,xn))

n

Y

i=1

f(xi)dxi

= X

σ∈Sn

Z

Rσ

h

n

X

i=1

piν,σxσ(i)

! n Y

i=1

f(xi)dxi

(5)

Distributional relationships with linear combination of order statistics

Using the well-known fact that the joint p.d.f. of Xn:n>· · ·>X1:n is n!

n

Y

i=1

f(xi), xn>· · ·>x1,

we obtain

E[h(Yν)] = 1 n!

X

σ∈Sn

E

"

h

n

X

i=1

pν,σi Xn−i+1:n

!#

= 1 n!

X

σ∈Sn

E[h(Yνσ)].

where

Yνσ :=

n

X

i=1

piν,σXn−i+1:n

are linear combinations of order statistics.

(6)

university-logo

Distributional relationships with linear combination of order statistics

The special cases

h(x) =xr, [x−E(Yν)]r, andetx

provide similar relationships, respectively, for raw moments,central moments, andmoment-generating functions.

Now, consider the minus(resp.plus) truncated power function xn (resp.x+n) defined to be xn ifx <0 (resp.x >0) and zero otherwise.

(7)

Distributional relationships with linear combination of order statistics

Given y ∈R, taking h(x) = (x−y)0 provides a relationship between the c.d.f. Fν ofYν and those of the random variablesPn

i=1piν,σXn−i+1:n. Indeed, we clearly have

Fν(y) :=Pr[Yν 6y] =E[(Yν−y)0], and, denoting byFνσ the c.d.f. ofYνσ,

Fνσ(y) :=Pr

" n X

i=1

piν,σXn−i+1:n6y

#

=E

n

X

i=1

piν,σXn−i+1:n−y

!0

.

(8)

university-logo

Distributional relationships with linear combination of order statistics

This immediately gives

Fν(y) = 1 n!

X

σ∈Sn

Fνσ(y).

As one could have expected from the definition of the Choquet integral, the determination of the moments and the distribution functions of the Choquet integral is closely related to the determination of the moments and the distribution functions of linear combinations of order statistics.

(9)

The uniform case: main results

Mathematical tools: divided differences, Hermite-Genocchi formula.

There holds

Fν(y) = 1 n!

X

σ∈Sn

∆[(·−y)n0σ, . . . , νnσ]

where ∆[(·−y)n0σ, . . . , νnσ] is the divided difference ofx 7→(x−y)n at pointsν0σ, . . . , νnσ.

From a practical perspective: there exists O(n2) algorithms for computing divided differences. Implemented in kappalab.

Also, a result giving the momemts of the Choquet integral in closed form.

(10)

university-logo

The non-uniform case: main results

Mathematical tools: approximations of moments of order statistics Main result: approximations of the 2 first moments of Choquet integral.

From a practical perspective: efficient algorithms for computing the moments in the normal caseand in theexponential case. Implemented in kappalab.

Thanks very much Michel!

Références

Documents relatifs

Predicting chaos with Lyapunov exponents: Zero plays no role in forecasting chaotic systems.. Dominique Guegan,

We report here on the qualification of an all-fiber counter propagating pumped amplifier in the nanosecond regime, tailored to be used for coherent beam combining of fiber

The home country increases its own tax rate compared to the Nash level, which will trigger a larger increase in the tax rate of the foreign country because of the

Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA

In this paper, we propose a model allowing to identify the weights of interacting criteria from a partial preorder over a reference set of alternatives, a partial preorder over the

Now, a suitable aggregation operator, which generalizes the weighted arithmetic mean, is the discrete Choquet integral (see Definition 2.2 below), whose use in multicriteria

In measurement theory, it is classical to split the overall evaluation model u into two parts [11]: the utility functions (that map the attributes onto a sin- gle satisfaction

Every n-variable Choquet integral is comonotonically additive.. Axiomatizations of the Choquet