Sophie Dabo-Niang
email: sophie.dabo@univ-lille3.fr

Baba Thiam
email: baba.thiam@univ-lille3.fr

Robust quantile estimation and prediction for spatial processes

Keywords: Spatial processes, Kernel estimate, Conditional quantile, Spatial prediction

In this paper, we present a statistical framework for modeling conditional quantiles of spatial processes assumed to be strongly mixing in space. We establish the L1 consistency and the asymptotic normality of the kernel conditional quantile estimator in the case of random fields. We also define a nonparametric spatial predictor and illustrate the methodology used with some simulations.

Introduction

Let (X, Y ) be a pair of random variables with values in R d × R and defined on a probability space (Ω, A, P ). Assume that the joint density of (X, Y ) and the marginal density of X exist and are denoted respectively by f (x, y) and g(x). In the following, we suppose that F (•|x), the conditional distribution function of Y given X = x exists and we denote by f (•|x) the density of Y given X = x. For p ∈]0, 1[ and for fixed x ∈ R d , let µ p (x) be the conditional quantile of order p of F (•|x), that can be seen as a solution of the equation F (y|x) = p. Another alternative characterization of the pth conditional quantile (see for example Gannoun et al. [START_REF] Gannoun | Nonparametric prediction by conditional median and quantiles[END_REF]) is

µ p (x) = argmin θ∈R E [(2p -1)(Y -θ) + |Y -θ| |X = x].
We are interested to the non-parametric estimation of µ p (x) in the case of spatial dependent observations. Nonparametric conditional quantile estimation technics have already been developed for non spatial (independent or mixing) real valued processes. Such results have provided useful tools for solving for example some prediction problems of strictly stationary processes satisfying the α-mixing condition. The existing results in the non-spatial case include the works of Matzner-Løber [START_REF] Matzner-Løber | Prévison nonparamétrique des processus stochastiques[END_REF], Collomb [START_REF] Collomb | Estimation non paramétrique de probabilités conditionnelles[END_REF], Gannoun et al. [START_REF] Gannoun | Nonparametric prediction by conditional median and quantiles[END_REF], Laksaci et al. [START_REF] Laksaci | A generalized L 1approach for a kernel estimator of conditional quantile with functionnal regressors: Consistency and asymptotic normality[END_REF]. In nonparametric spatial estimation, the existing works concern mainly the estimation of a probability density and regression functions, see the key references: Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF], Biau and Cadre [START_REF] Biau | Nonparametric spatial regression[END_REF], Carbon et al. [START_REF] Carbon | Kernel density estimation for random fields[END_REF]. For the spatial quantile conditional estimation case, there exist only few results in our knowledge. Abdi et al. [START_REF] Abdi | Consistency of a nonparametric conditional quantile estimator for random fields[END_REF] considered the pointwise p-mean and almost complete consistencies of a double kernel quantile estimator for real-valued random fields. Hallin et al. [START_REF] Hallin | Local linear spatial quantile regression[END_REF] give a Bahadur representation and asymptotic normality results of the local linear quantile estimator. Laksaci and Fouzia [START_REF] Laksaci | Estimation non paramétrique de quantiles conditionnels pour des variables fonctionnelles spatialement dépendantes[END_REF] consider the case where the regressor take their values in a semi-metric space and show the strong and weak consistency of the conditional quantile. In this paper, we will go beyond all these last spatial works and provide the L 1 consistency and an asymptotic normality of a kernel conditional quantile estimate of a strictly stationary spatial process satisfying the α-mixing condition. In addition, we employe our results to solve some nonparametric prediction problems. The organization of this paper is as follows. The estimation procedure is presented in Section 2. Section 3 gives some necessary conditions and then establishes the main asymptotic results. Section 4 is devoted to simulations results. Technical proofs are given in Section 5.

Nonparametric estimator of the conditional quantile

Let us consider a strictly stationary process ((X i , Y i ), i ∈ I n ) with values in R d × R where (X i , Y i ) has the same distribution as (X, Y ). For n ∈ (N * ) N , we define a rectangular region I n by

I n = {i = (i 1 , . . . , i N ) ∈ (N * ) N , 1 ≤ i k ≤ n k , k = 1, . . . , N }. We set n = n 1 . . . n N , and we write n → ∞ if min k=1,...,N n k → ∞.
The well known kernel estimates of f and g are defined by

f n (x, y) = 1 nh d+1 i∈In K x -X i h w y -Y i h and g n (x) = 1 nh d i∈In K x -X i h
, where K and w are two probability density functions, and the bandwidths h = h(n) is a sequence of positive real numbers such that h → 0 as n → ∞. The kernel estimate of the conditional density f n (y|x) is naturally defined by the ratio f n (x, y) over g n (x) while the estimator of the conditional distribution function (see the one introduced by Roussas [START_REF] Roussas | Nonparametric estimation of the transition distribution function of a Markov process[END_REF]) is defined by

F n (y|x) = ψ n (x, y) g n (x) 1I {gn(x) =0} , ψ n (x, y) = 1 nh d+1 i∈In K x -X i h y -∞ w z -Y i h dz.
For a fixed x, the estimator of the pth conditional quantile noted µ p,n (x) can be defined as the root of the equation F n (z|x) = p. Alternatively, one can consider the local constant estimator defined by

ν p,n (x) = argmin θ∈R i∈In (|Y i -θ| + (2p -1)(Y i -θ)) K x -X i h .
In this paper, we will focus on the study of the asymptotic behavior of µ p,n . For the study of ν p,n , one can adapt the technics developed in Zhou [START_REF] Zhou | Asymptotic Normality for L 1 Norm Kernel estimator of Conditional Median under α-Mixing Dependence[END_REF].

Main results

To establish the asymptotic results, we will suppose that the sequence (X i , Y i )) i∈(N * ) N satisfies the following mixing condition: there exists a function χ : R + → R + with χ(t) ↓ 0 as t → ∞, such that whenever E, E ′ ⊂ (N * ) N with finite cardinals,

α (B(E), B(E ′ )) := sup {|P (A ∩ B) -P (A)P (B)|; A ∈ B(E), B ∈ B(E ′ )} ≤ φ(CardE, CardE ′ )χ(dist(E, E ′ )),
where B(E) (resp. B(E ′ )) denotes the Borel σ-fields generated by

(X i , Y i ) i∈E (resp. (X i , Y i ) i∈E ′ ), Card E (resp. Card E ′ ) the cardinality of E (resp. E ′ ), dist(E, E ′
) the Euclidean distance between E and E ′ , and φ : N 2 → R + is a symmetric positive function which is non decreasing in each variable. Throughout this paper, we will assume that φ satisfies

φ (n, m) ≤ C min (n, m) , ∀n, m ∈ N (1) or φ (n, m) ≤ C (n + m + 1) κ , ∀n, m ∈ N (2)
for some κ ≥ 1 and some C > 0. If φ ≡ 1, then the field (X i , Y i ) i∈(N * ) N is called strongly mixing. In this paper, we consider the case where χ(i) tends to zero at a polynomial rate, that is,

χ(i) = O(i -β ), (3)
with β > 0. We fix a compact subset S of R d . Denote a = inf{y : F (y|x) > 0} and b = sup{y : F (y|x) < 1}, we will suppose that V ⊆ [a, b] is a compact neighborhood of the unknown quantile µ(x). For mixing coefficients with polynomial decreasing rate (3), the constraints on the bandwidth will be related to β by means of

θ 1 = N (d + 1)(d + 2) + (d + 1)β β -N (d + 5) , θ 2 = N (d + 2) -β β -N (d + 5) , θ 3 = N (d 2 + 4d + 2) + (d + 1)β β -N (d + 4 + 2κ) , θ 4 = N (d + 1) -β β -N (d + 4 + 2κ)
.

Denote Ω n = log n nh d . Let ε be an arbitrary small positive number and set u(n)

= N i=1 (log n i )(log log n i ) 1+ε . It is clear that n∈Z N 1/( nu(n)) < ∞.
In the sequel, we use the following hypotheses.

(A1) f and g are respectively continuous on R d+1 and R d , g satisfies a Lipschitz condition, g(x) > 0, ∀x ∈ S.

(A2) There exists D ≥ 0 such that the pairs (X i , X j ) and ((X i , Y i ), (X j , Y j )) admit a density, say g i,j and f i,j , as soon as dist(i, j) > D. Moreover, for some constant c ≥ 0,

|f i,j (s, t) -f (s)f (t)| ≤ c, ∀s, t ∈ R d+1
and

|g i,j (u, v) -g(u)g(v)| ≤ c, ∀u, v ∈ R d . (A3) i) f (i,j) (•, •) = ∂ i+j f ∂x i l ∂y j (•,
•) exists, is bounded and integrable for 0 ≤ i + j ≤ 2, and 0 ≤ l ≤ d. ii) F (y|x) has continuous second partial derivatives with respect to x.

(A4) F (y|x) has continuous second derivative with respect to y.

(A5) The kernel K is integrable, symmetric and is a lipschitzian density function on R d with compact support. Moreover R d s 2 K(s)ds < ∞.

(A6) The kernel w is a symmetric and lipschitzian density function on R and has compact support.

(A7) lim n→∞ nh d+2 (log n) -1 = 0.

(A8) The function µ p (x) satisfies a uniform uniqueness property on S:

∀ε > 0, ∃η > 0, ∀r : S → R, sup x∈S |µ p (x) -r(x)| ≥ ε ⇒ sup x∈S |F (µ p (x)|x) -F (r(x)|x)| ≥ η. (A9) nh θ1 (log n) θ2 (u(n)) -2N β-N (d+5) → ∞. (A10) nh θ3 (log n) θ4 (u(n)) -2N β-N (d+4+2κ) → ∞.
Comments on the hypotheses: Assumptions (A9) and (A10) imply conditions (3.7) and (3.8) of Theorem 3.3 in Carbon et al. [START_REF] Carbon | Kernel density estimation for random fields[END_REF] and they also imply the classical condition nh d+1 / log n → ∞. Assumption (A8) is introduced for getting consistency results on the quantile from those of the conditional distribution.

In order to state the asymptotic results, we will suppose that (A9) and ( 1) or (A10) and ( 2) are satisfied.

The following two theorems give uniform almost sure convergence results of respectively F n (y|x) and µ p,n (x) and permit to establish the L 1 consistency of µ p,n (x) (see Corollary 1).

Theorem 1 Assume (A1)-(A7) hold, then

sup y∈V sup x∈S |F n (y|x) -F (y|x)| = O (Ω n ) a.s.
Theorem 2 If (A1)-(A8) are satisfied, then we have

sup x∈S |µ p,n (x) -µ p (x)| a.s. → 0.
Corollary 1 Assume (A1)-(A8) hold, then

E {µ p,n (X n ) -µ p (X n )}1I {Xn∈S} a.s.
→ 0.

To establish the following asymptotic normality of µ p,n (x) (Theorem 3), we will suppose that for any (x, y) ∈ S × V, there exists c > 0 such that f (y|x) > c. Moreover, we will assume that the following additional conditions on the bandwidth hold for some 0 < γ < 1.

(C1) nh d(1+2N (1-γ)) → ∞.
(C2) There exists a sequence of positive integers

q = q n → ∞ with q = o ( nh d(1+2N (1-γ)) ) 1/2N such that n ∞ i=1 i N -1 χ(iq) → 0 and h -d(1-γ) ∞ i=q i N -1 (χ(i)) 1-γ → 0.
Theorem 3 Assume that (A1)-(A8), (C1) and (C2) hold. If there exists c ≥ 0 such that nh d+4 → c, then

√ nh d (µ p,n (x) -µ p (x)) L → N c B(x, µ p (x)) f (µ p (x)|x) , σ 2 (x, µ p (x)) (f (µ p (x)|x)) 2 ,
where

B(x, y) = 1 2 d i,j=1 ∂ 2 F (y|x) ∂x i ∂x j + 2 g(x) ∂g(x) ∂x i ∂F (y|x) ∂x j R d s 2 K(s)ds + ∂ 2 F (y|x) ∂y 2 R t 2 w(t)dt . ( 4
)
σ 2 (x, y) = F (y|x) [1 -F (y|x)] g(x) R d K 2 (z)dz. ( 5
)

Prediction

Let (ξ i , i ∈ I n ) be a R-valued strictly stationary random spatial process, assumed to be bounded, observable over a region I n ⊂ N N and observed over a subset O n of I n , n = (n 1 , ..., n N ) ∈ N N . The aim of this section is to predict ξ i0 , at a given fixed point i 0 not in O n ⊂ N N . In practice (e.g. for simplicity), we expect that ξ i0 depends only on the values of the process on a bounded neighborhood V i0 ⊂ O n . In other words, we expect that the process (ξ i ) satisfies a Markov property, see for example Biau and Cadre [START_REF] Biau | Nonparametric spatial regression[END_REF], Dabo-Niang and Yao [START_REF] Dabo-Niang | Spatial kernel regression estimation[END_REF]. Moreover, we assume that V i0 = V + i 0 , where V is a fixed bounded set of sites that does not contain 0. It is well known that the best predictor of ξ i0 given the data in V i0 in the sense of mean-square error is

E(ξ i0 |ξ i , i ∈ V i0 ). Let V i = V + i = {u + i, u ∈ V} for each i ∈ N N , and d be the cardinal of V (d is also the cardinal of each V i ). To define a predictor of ξ i0 , let us consider the R d -valued random variables ξi = {ξ u , u ∈ V i ⊂ O n }.
The notation of the previous sections are used by setting

X i = ξi , Y i = ξ i , i ∈ N N .
As a predictor of ξ i0 , we take the conditional quantile estimate ξ i0 = µ p,n ( ξi0 ) of order p, particularly the conditional median p = 0.5. We deduce from the previous consistency results, the following corollary that gives the convergence of the predictor ξ i0 .

Corollary 2 i) Under the conditions of Corollary 1, we have

E {µ p,n ( ξi0 ) -µ p ( ξi0 )}1I { ξi 0 ∈S} a.s. → 0.
ii) Under the conditions of Theorem 3, and if nh d+4 → 0, then

√ nh d µ p,n ( ξi0 ) -µ p ( ξi0 ) L → N 0, σ 2 ( ξi0 , µ p ( ξi0 )) (f (µ p ( ξi0 )| ξi0 )) 2 ,
where

σ 2 (x, y) = F n (y|x) [1 -F n (y|x)] g n (x) R d K 2 (z)dz.
These consistency results permit to have an approximation of an 1α confidence interval of ξ i0 given by I α = [a -( ξi0 ), a + ( ξi0 )], where

a ± ( ξi0 ) = µ p,n ( ξi0 ) ± Q 1-α 2 σ( ξi0 , µ p,n ( ξi0 )) √ nh d f n (µ p,n ( ξi0 )| ξi0 ) , (6)
where Q ζ denotes the ζ-quantile of the standard normal distribution, and the unknown parameters (of the asymptotic variance in Corollary 2) are replaced by kernel estimates. Note also that the quantiles of order p 1 and p 2 (p 1 < p 2 ) can be used to construct a predictive interval that consists of the (p 2p 1 )100% confidence interval with bounds µ p1,n ( ξi0 ) and µ p2,n ( ξi0 ).

A simulation study

In this section, we study the performance of the conditional quantile predictor introduced in the previous section towards some simulations. Let us denoted by GRF (m, σ 2 , s) a Gaussian random field with mean m and covariance function defined by

ϑ(h) = σ 2 exp - h s 2 , h ∈ R 2 .
Set

I n = {i = (i, j) ∈ (N * ) 2 , 1 ≤ i ≤ 61, 1 ≤ j ≤ 61} (7) O n = {i = (i, j) ∈ (N * ) 2 , 1 ≤ i ≤ 21, 1 ≤ j ≤ 21} ∪ {i = (22, j), 1 ≤ j ≤ 15}. ( 8
)
We consider a random field (ξ i ) i∈In from the following model

ξ i = U i * sin(2X i ) + 2 exp{-(16X i ) 2 } + Z i , i ∈ N 2 (9)
where

X = (X i ) i∈In is a GRF (0, 5, 3), Z = (Z i ) i∈In is a GRF (0, 0.1, 5) independent of X and U i = 1 n j∈In exp -i-j 2 .
The choice of U i in the model ( 9) is motivated by a reinforcement of the spatial local dependency. The field (ξ i , i ∈ I n ) is observable over the rectangular region I n and observed over the subset O n defined in [START_REF] Hallin | Local linear spatial quantile regression[END_REF] and [START_REF] Laksaci | A generalized L 1approach for a kernel estimator of conditional quantile with functionnal regressors: Consistency and asymptotic normality[END_REF]. We want to predict the values ξ i1 , . . . , ξ im at given fixed sites i 1 , . . . , i m not in O n , with m = 10. The sample obtained from model [START_REF] Laksaci | Estimation non paramétrique de quantiles conditionnels pour des variables fonctionnelles spatialement dépendantes[END_REF], observed in O n is plotted in Figure 1 below with the 10 non observable values of the field at i 1 , . . . , i m . As explained in Section 3.1, for any k ∈ {1, . . . , m}, we take the conditional quantile estimate ξ i k = µ p,n ( ξi k ) as a predictor of ξ i k , where ξi k are observed on O n and the vicinity

V = {-1, 1} × {-1, 1} or {-2, -1, 1, 2} × {-2, -1, 1, 2}.
To compute µ p,n , we select the standard normal density as kernel K and the Epanechnikov kernel as w. For the bandwidth selection, we use the rule developed in Yu and Jones [START_REF] Yu | Local linear quantile regression[END_REF],

h n = h mean p(1 -p) φ(Φ -1 (p)) 2 1/5
, where h mean is the bandwidth for kernel smoothing estimation of the regression mean, φ and Φ, are respectively, the standard normal density and distribution function.

To evaluate the performance of the predictor ξ i k , we compute the mean absolute error (MAE):

M AE = 1 m m k=1 | ξ i k -ξ i k |.
The following Table gives the predictors of ξ i k , k = 1, ..., m for p ∈ {0.05, 0.5, 0.95}, V = {-1, 1} × {-1, 1} on the left, V = {-2, -1, 1, 2} × {-2, -1, 1, 2} on the right and the prediction error. We derive from the results of Tables 1 a 90% predictive interval where the extremities are the 5% and 95% quantiles estimates, for each of the 10 prediction sites (see Section 3.1 for more details). Note that these 90% predictive intervals contain the true values. The average length for the 10 intervals is 0.05604 for V = {-1, 1} × {-1, 1} and 0.04455 for V = {-2, -1, 1, 2} × {-2, -1, 1, 2}. The numerical results show that our proposed predictor gives good results on the above simulated field. A next step would be to apply the predictor to a spatial real data and deserves futur investigations.

Appendix

The letter C will be used to denote constants whose values are unimportant. Before proving the main results, let us give the following notations:

ψ(x, y) = y -∞ f (x, z)dz, K 1 (y) = y -∞ w(z)dz.
Next, we need the following results.

Lemma 1 Assume that (A3)-(A7) hold, then,

sup y∈V sup x∈S |Eψ n (x, y) -ψ(x, y)| = O (Ω n ) .
The proof of Lemma 1 is classical (see for example Matzner and Løber [START_REF] Matzner-Løber | Prévison nonparamétrique des processus stochastiques[END_REF]).

Lemma 2 Let Assumptions (A1)-(A6) hold. If (A9) and ( 1) or (A10) and ( 2) are satisfied, then,

sup y∈V sup x∈S |ψ n (x, y) -Eψ n (x, y)| = O (Ω n ) a.s. Proof of Lemma 2. Define Z i (x, y) = 1 nh d+1 K x -X i h y -∞ w t -Y i h dt = 1 nh d K x -X i h K 1 y -Y i h , ∆ i (x, y) = Z i (x, y) -EZ i (x, y), then S n (x, y) = ψ n (x, y) -Eψ n (x, y) = i∈In ∆ i (x, y).
Lemma 3 Under Assumptions (A1), (A2), (A5) and (A6), we have

i∈In E(∆ i (x, y)) 2 + i =j i,j∈In E∆ i (x, y)∆ j (x, y) = O 1 nh d .
The proof of Lemma 3 is similar to the proof of Lemma 2.2 in Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF].

Let us introduce a spatial block decomposition that has been used by Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF] and Carbon et al. [START_REF] Carbon | Kernel density estimation for random fields[END_REF]. Without loss of generality, assume that

n i = 2pq i for 1 ≤ i ≤ N . The random variables ∆ i (x, y) can be grouped into 2 N q 1 × q 2 × • • • × q N cubic blocks of side p. Denote U x (1, n, j, y) = (2j k +1)p i k =2j k p+1 k=1,...,N ∆ i (x, y), U x (2, n, j, y) = (2j k +1)p i k =2j k p+1 k=1,...,N -1 2(jN +1)p iN =(2jN +1)p+1 ∆ i (x, y), U x (3, n, j, y) = (2j k +1)p i k =2j k p+1 k=1,...,N -2 2(jN-1+1)p iN-1=(2jN-1+1)p+1 (2jN +1)p iN =2jN p+1 ∆ i (x, y), U x (4, n, j, y) = (2j k +1)p i k =2j k p+1 k=1,...,N -2 2(jN-1+1)p iN-1=(2jN-1+1)p+1 2(jN +1)p iN =(2jN +1)p+1 ∆ i (x, y),
and so on. Finally, note that

U x (2 N -1 , n, j, y) = (2j k +1)p i k =2j k p+1 (2jN +1)p iN =2jN p+1 ∆ i (x, y), U x (2 N , n, j, y) = (2j k +1)p i k =2j k p+1 k=1,...,N ∆ i (x, y). For each integer 1 ≤ i ≤ 2 N , let T x (n, i, y) = q k -1 j k =0 k=1,...,N U x (i, n, j, y), S n (x, y) = 2 N i=1 T x (n, i, y).
Observe that, for any ε > 0

P (|S n (x, y)| > ε) = P   2 N i=1 T x (n, i, y) > ε   ≤ 2 N P |T x (n, i, y)| > ε/2 N .
Without loss of generality, we consider just the case where i = 1 and we enumerate in an arbitrary way the q = q 1 . . . q N terms U x (1, n, j, y) of the sum T x (n, 1, y) that we call W 1 , . . . , W q . Note that U x (i, n, j, y) is measurable with respect to the σ-field generated by Z i , with i such that 2j k p + 1 ≤ i k ≤ (2j k + 1)p, k = 1, . . . , N . These sets of sites are separeted by a distance at least p and since the Z i are bounded, then we have for all i = 1, . . . , q, |W i | ≤ C( nh d ) -1 p N K ∞ . Lemma 4.4 in Carbon et al. [START_REF] Carbon | Kernel density estimation for random fields[END_REF] ensures that there exist independent random variables W * 1 , . . . , W * q such that for all i = 1, . . . , q,

E|W i -W * i | ≤ C( nh d ) -1 p N K ∞ φ( n, p N )χ(p).
Markov's inequality leads to

P q i=1 |W i -W * i | > ε/2 N +1 ≤ C2 N +1 ( nh d ) -1 p N q K ∞ φ( n, p N )ε -1 χ(p). (10)
By Bernstein's inequality, we have

P | q i=1 W * i | > ε/2 N +1 ≤ 2 exp -ε 2 /(2 N +1 ) 2 4 q i=1 EW * 2 i + 2C( nh d ) -1 p N K ∞ ε/2 N +1 . (11)
Combining ( 10) and ( 11), we have

P (|S n (x, y)| > ε) ≤ 2 N P q i=1 |W i -W * i | > ε/2 N +1 + 2 N P | q i=1 W * i | > ε/2 N +1 ≤ 2 N +1 exp -ε 2 /(2 N +1 ) 2 4 q i=1 EW * 2 i + 2C( nh d ) -1 p N K ∞ ε/2 N +1 +C2 2N +1 φ( n, p N )( nh d ) -1 p N q K ∞ ε -1 χ(p).
By Lemma 3, one has q i=1 EW * 2 i = O( 1 nh d ) and since n = 2 N p N q, we have

P (|S n (x, y)| > ε) ≤ 2 N +1 exp -ε 2 nh d 2 2N +4 C + 2 N +2 Cp N ε + C2 N +1 φ( n, p N )h -d ε -1 χ(p).
Let λ > 0 and set ε = λΩ n , p = Ω -1/N n

. For the first part of Lemma 2, a simple computation shows that for sufficiently large n,

P (|S n (x, y)| > λΩ n ) ≤ 2 N +1 exp -λ 2 log n 2 2N +4 C + 2 N +2 Cλ + C2 N +1 p N h -d λ -1 Ω -1 n χ(p) ≤ C n -b + Cλ -1 h -d Ω β-2N N n , (12)
with b > 0. Analogously, for the second part, as above we have

P (|S n (x, y)| > λΩ n ) ≤ 2 N +1 exp -λ 2 log n 2 2N +4 C + 2 N +2 Cλ + C2 N +1 n κ h -d λ -1 Ω -1 n χ(p) ≤ C n -b + Cλ -1 n κ h -d Ω β-N N n . (13)
Now, set R n = h d+1 Ω n and r n = h d+2 n 1/2

. Since S is a compact, it can be covered with d n cubes B k having sides of length R n and center at x k with d n ≤ CR -d n . The compact set V can be covered with l n intervalls I l having length r n and center at y l , with l n ≤ Cr -1 n . We have Lemma 5 Assume that (A1), (A2), (A5) and (A6) hold. If (A9) and (1) or A( 10) and ( 2) are satisfied, then

d n l n ≤ Cn 1/2 h -d+2 2 h d+1 Ω n -d-1 . (14)
S 5n = O (Ω n ) a.s.
Proof of Lemma 4. On one hand, since the kernel K satisfies the Lipschitz condition, we have clearly

|ψ n (x, y) -ψ n (x k , y)| ≤ C n -1 h -d-1 i∈In x -x k ≤ Ch -d-1 R n = O (Ω n ) .
On the other hand, observe that h

-d-1 r n = nh d -1/2 = O (Ω n ). Since w satisfies the Lipschitz condition, |ψ n (x k , y) -ψ n (x k , y l )| ≤ C n -1 h -d-1 i∈In |y -y l | ≤ Ch -d-1 r n = O (Ω n ) ,
which gives the proof of Lemma 4.

Proof of Lemma 5. For ε > 0, we have

P max l max k |ψ n (x k , y l ) -Eψ n (x k , y l )| > ε ≤ l n d n P (|ψ n (x k , y l ) -Eψ n (x k , y l )| > ε) .
Setting ε = λΩ n with λ > 0 and taking into account ( 12) and ( 13), it suffices to show that

l n d n n -b nu(n) → 0 and l n d n h -d Ω β-2N N n nu(n) → 0, or l n d n n κ h -d Ω β-N N n nu(n) → 0. First, observe that condition nh d → ∞ implies that n > Ch -d , so that n (d+1)(d+2)/2d > Ch -(d+1)(d+2)/2 . (15)
Using ( 14) and (15), we have

l n d n n -b nu(n) ≤ Ch -(d+1)(d+2)/2 (log n) -d/2 n 1/2 n -b n d/2 nu(n) ≤ C n (d 2 +3d+1)/d-b (log n) -d/2 u(n),
which goes to 0 if b > (d 2 + 3d + 1)/d. Next, again [START_REF] Zhou | Asymptotic Normality for L 1 Norm Kernel estimator of Conditional Median under α-Mixing Dependence[END_REF] and a computation show that

l n d n h -d Ω β-2N N n nu(n) ≤ C n(log n) β-N (d+2) N (d+5)-β u(n) 2N N (d+5)-β h -N (d+1)(d+2)-dβ) N (d+5)-β N (d+5)-β 2N
, which goes to 0 by Assumption (A9) and β > N (d + 5). Analogously, ( 14) and a computation show that Proof of Corollary 1. First, by Lyapounov's inequality, we have

l n d n n κ h -d Ω β-N N n nu(n) ≤ C n(log n) β-N (d+1) N (d+4+2κ)-β u(n) 2N N (d+4+2κ)-β h -N (d 2 +4d+2)-dβ) N (
E [µ p,n (X n ) -µ p (X n )] ≤ E [|µ p,n (X n ) -µ p (X n )|] ≤ E (µ p,n (X n ) -µ p (X n )) 2 1/2 ,
so, we can write

E (µ p,n (X n ) -µ p (X n )) 1I {Xn∈S} ≤ E (µ p,n (X n ) -µ p (X n )) 2 1I {Xn∈S} 1/2 .
Then we also have

E (µ p,n (X n ) -µ p (X n )) 2 1I {Xn∈S} ≤ E sup x∈S (µ p,n (x) -µ p (x)) 2 .
An integration by parts gives where µ * p,n (x) lies between µ p,n (x) and µ p (x). To prove the asymptotic normality, it suffices to show that the numerator is normally distributed, and the denominator converges to f (µ p (x)|x) in probability. We have the following propositions. where B(x, y) and σ 2 (x, y) are defined in (4) and (5).

Figure 1 :

 1 Figure 1: The random field (ξ i ) with non observed values ξ i1 , . . . , ξ i10 in the white rectangular cases.

DefineLemma 4

 4 S 1n = sup y∈V max k sup x∈B k |ψ n (x, y)ψ n (x k , y)| S 2n = max k max l sup y∈I l |ψ n (x k , y)ψ n (x k , y l )| , S 3n = max k max l sup y∈I l |Eψ n (x k , y l ) -Eψ n (x k , y)| S 4n = sup y∈V max k sup x∈B k |Eψ n (x k , y) -Eψ n (x, y)| , S 5n = max k max l |ψ n (x k , y l ) -Eψ n (x k , y l )| .Then, we can write supy∈V sup x∈S |ψ n (x, y) -Eψ n (x, y)| ≤ S 1n + S 2n + S 3n + S 4n + S 5n . The proof of Lemma 2 follows easily from the combination of the two following lemmas. Under Assumptions (A5) and (A6), S in = O (Ω n ) a.s., i = 1, 2, 3, 4.

  d+4+2κ)-β N (d+4+2κ)-β 2N, which goes to 0 by Assumption (A10) and β > N (d + 4 + 2κ). The conclusion of Lemma 5 follows from the Borel Cantelli's Lemma.Proof of Theorems 1 and 2. First, from Carbon et al.[START_REF] Carbon | Kernel density estimation for random fields[END_REF], we have supx∈S |g n (x)g(x)| = O (Ω n ) a.s. Now, a standard decomposition gives sup y∈V sup x∈S |F n (y|x) -F (y|x)| ≤ 1 inf x∈S g n (x) sup y∈V sup x∈S |ψ n (x, y)ψ(x, y)| + sup y∈V sup x∈S (F (y|x)|g(x)g n (x)|) ≤ 1 inf x∈S g n (x) sup y∈V sup x∈S |ψ n (x, y) -Eψ n (x, y)| + sup y∈V sup x∈S |Eψ n (x, y)ψ(x, y)| + sup x∈S |g(x)g n (x)| .Since by (A1), g n (x) is bounded away from 0, Theorem 1 follows from the preceding inequality, Lemmas 1 and 2. Next, from (A8), to prove Theorem 2, it suffices to show that sup x∈S |F (µ p,n (x)|x) -F (µ p (x)|x)| → 0. We have|F (µ p,n (x)|x) -F (µ p (x)|x)| ≤ |F (µ p,n (x)|x) -F n (µ p,n (x)|x)| + |F n (µ p,n (x)|x) -F (µ p (x)|x)| ≤ |F (µ p,n (x)|x) -F n (µ p,n(x)|x)| ≤ sup y∈V |F n (y|x) -F (y|x)| . Thus, sup x∈S |F (µ p,n (x)|x) -F (µ p (x)|x)| ≤ sup x∈S sup y∈V |F n (y|x) -F (y|x)| , so that Theorem 2 follows from an application of Theorem 1.

  ,n (x)µ p (x)) n (x)µ p (x)| > v dv.Using Assumption (A8), we have that for n large enough,∞ 0 vP sup x∈S |µ p,n (x)µ p (x)| > v dv ≤ 2 ∞ 0 tP sup x∈S |F (µ p (x)|x) -F n (µ p (x)|x)| > t dt |x) -F n (y|x)| > t dt,and Corollary 1 follows from Theorem 1.Proof of Theorem 3. SinceF (µ p (x)|x) = p = F n (µ p,n (x)|x),by Taylor's expansion, we haveµ p,n (x)µ p (x) = 1 f n (µ * p,n (x)|x) [F (µ p (x)|x) -F n (µ p (x)|x)] ,

Proposition 1

 1 Under (A1)-(A6), (C1) and (C2), if there exists c ≥ 0 such that nh d+4 → c, then√ nh d (F n (y|x) -F (y|x)) L → N cB(x, y), σ 2 (x, y) ,

Table 1 :

 1

	0.95

Predictive data for V = {-1, 1} × {-1, 1} on the left and V = {-2, -1, 1, 2} × {-2, -1, 1, 2} on the right. p = 0.05 True data p = 0.5 p = 0.95 p = 0.05 True data p = 0.5 p =

→ 0.

Proof of Proposition 1. Proposition 1 is a consequence of the two following lemmas.

Lemma 6 If Assumptions (A1) and (A3)-(A6) are satisfied, then

where B(x, y) is defined in (4).

The proof of Lemma 6 is classical and therefore is omitted (see Matzner-Løber [START_REF] Matzner-Løber | Prévison nonparamétrique des processus stochastiques[END_REF]).

Lemma 7 Assume that Assumptions (A1)-(A6), (C1) and (C2) hold, then

where σ 2 (x, y) is defined in [START_REF] Dabo-Niang | Spatial kernel regression estimation[END_REF].

Proof of Lemma 7. Assume for the moment that for any pair (c

where

On one hand, according to (16), we have

→ N 0, σ 2 , with σ 2 being defined in [START_REF] Dabo-Niang | Spatial kernel regression estimation[END_REF]. On the other hand,

which goes to 0 in probability. Finally, we conclude with the decomposition 16) is proved following the same lines as the proof of Theorem 3.1 in Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF].

Proof of Proposition 2. We have

Hence to prove Proposition 2, it suffices to show that sup


Escrts Anne Petiot

Rémy Sadoul
email: remy.sadoul@ujf-grenoble.fr

Autophagy discriminates between

Keywords: Alix, Autophagy, ESCRT, endocytosis, MVB

Alix and ESCRT proteins are required for membrane fission during viral budding and egress and during the abscission stage of cytokinesis. These common roles have suggested that Alix functions as an ESCRT protein, a conclusion challenged by the finding that unlike ESCRTs, which control the formation of multivesicular endosomes, Alix does not influence the degradation of the EGF receptor. We previously showed that Alix controls neuronal death by an unknown mechanism, but dependent on its interaction with ESCRT proteins. Since then, numerous reports have shown that ESCRTs participate in macroautophagy. Given the direct interaction between ESCRTs and Alix, together with the known contribution of autophagy to cell death, it was hypothesized that Alix controls autophagy and thereby cell death. Our recent published results show that this is not the case. ESCRT protein activity therefore needs Alix for viral budding and cytokinesis but not for autophagy. The function of ESCRT can thus be clearly be disconnected from that of Alix.

endosomes to late endosomes and then into lysosomes, whereas phagocytosis and autophagy provide alternative entry points. Endocytosed membrane proteins meant for degradation are selectively sequestered within vesicles budding from the membrane into the lumen of endosomes. This process leads to the formation of endosome intermediates called multivesicular endosomes (MVEs), filled with intralumenal vesicles, which are hydrolyzed after fusion of the MVEs with lysosomes. [START_REF] Van Der Goot | Intra-endosomal membrane traffic[END_REF] Intracellular protein and organelle degradation occurs via autophagy after MVEs or endosomes fuse with autophagosomes to form amphisomes. [START_REF] Liou | The autophagic and endocytic pathways converge at the nascent autophagic vacuoles[END_REF] The autophagic process ends with the fusion of amphisomes with lysosomes.

After endocytosis, the cytoplasmic domains of receptors pointing out into the cytosol trigger the sequential building onto the endosomal membrane, of protein complexes called the endosomal sorting complex required for transport (ESCRT I, ESCRT II, ESCRT III). ESCRT III complexes, made of CHMP proteins in mammalian cells, entrap transmembrane proteins and contribute to the deforming of the endosomal membrane into vesicles. [START_REF] Williams | The emerging shape of the ESCRT machinery[END_REF] The role played by ESCRTs is central to the replication of enveloped viruses, which recruit them via their GAGs in order to bud off membranes. [START_REF] Morita | Retrovirus budding[END_REF] Both retroviruses and vesicles of MVEs bud away from the cytoplasm, and CHMP proteins of ESCRT-III seem instrumental for the fission of vesicles/virus from membranes. More recently, ESCRT proteins were shown to function in the last step of cytokinesis, which allows separation of the daughter cells after cell division. [START_REF] Morita | Human ESCRT and ALIX proteins interact with proteins of the midbody and function in cytokinesis[END_REF][START_REF] Carlton | Parallels between cytokinesis and retroviral budding: a role for the ESCRT machinery[END_REF] These events all require fission of a thin membrane tubule from within. CHMP proteins make helical tubular structures possibly inside deforming membranes. Fission would occur upon disassembly of the helical structures by the AAA-type ATPase Vps4/SKD1, which is known to be required for MVE genesis and viral abscission. [START_REF] Lata | Helical structures of ESCRT-III are disassembled by VPS4[END_REF] Alix is another protein that might be involved in membrane deformation/fission in tight relation with ESCRT proteins. This cytoplasmic protein binds to both Tsg101 and SNF7/CHMP4B, members of the ESCRT I and III, respectively, [START_REF] Strack | AIP1/ALIX is a binding partner for HIV-1 p6 and EIAV p9 functioning in virus budding[END_REF][START_REF] Von Schwedler | The protein network of HIV budding[END_REF] and to lysobisphosphatidic acid (LBPA), a phospholipid which is highly concentrated inside MVEs and may help the intralumenal vesiculation of endosomes. [START_REF] Matsuo | Role of LBPA and Alix in multivesicular liposome formation and endosome organization[END_REF] In both cases of cytokinesis and virus budding, Alix represents a way to recruit the ESCRT machinery necessary for membrane fission. The situation is far less clear regarding a role of Alix in MVE genesis. A first hint that it might play a role in this process stems from its homology with a yeast protein, Bro1, required for the sorting of mono-ubiquitinated transmembrane proteins into intralumenal vesicles of MVEs. [START_REF] Odorizzi | Bro1 is an endosome-associated protein that functions in the MVB pathway in Saccharomyces cerevisiae[END_REF] Furthermore, in HeLa cells knockdown of Alix results in a loss of intralumenal vesicles inside endosomes. [START_REF] Matsuo | Role of LBPA and Alix in multivesicular liposome formation and endosome organization[END_REF] Surprisingly, however, neither we nor other labs have found any significant effect of Alix on endocytosis and degradation of the EGF receptor, a process known to require ESCRT proteins. [START_REF] Cabezas | Alix regulates cortical actin and the spatial distribution of endosomes[END_REF][START_REF] Doyotte | The Bro1-related protein HD-PTP/PTPN23 is required for endosomal cargo sorting and multivesicular body morphogenesis[END_REF] Thus, the final consequence of Alix activity in MVEs remains largely unclear.

Originally, Alix was characterized in a screen for proteins capable of binding the calcium binding protein ALG-2 (Apoptosis Linked Gene-2), [START_REF] Missotten | Alix, a novel mouse protein undergoing calcium-dependent interaction with the apoptosis-linked-gene 2 (ALG-2) protein[END_REF] first reported to be required for T cell apoptosis. [START_REF] Vito | Interfering with apoptosis: Ca 2+ -binding protein ALG-2 and Alzheimer's disease gene ALG-3[END_REF] Alix expression is up-regulated in degenerating neurons [START_REF] Hemming | Early increase of apoptosislinked gene-2 interacting protein X in areas of kainate-induced neurodegeneration[END_REF] and overexpression is sufficient to activate caspases and thereby neuronal death. [START_REF] Trioulier | Alix, a protein regulating endosomal trafficking, is involved in neuronal death[END_REF][START_REF] Mahul-Mellier | Alix, making a link between apoptosis-linked gene-2, the endosomal sorting complexes required for transport, and neuronal death in vivo[END_REF] In contrast, dominant negative mutants of the protein block neuronal death. In particular, we found that several Alix mutants electroporated into chick embryo motoneurons in ovo rescue them from cell death occurring naturally during development. The death inhibiting activity of the mutants was tightly dependent on their capacity to bind to ESCRT proteins. [START_REF] Mahul-Mellier | Alix, making a link between apoptosis-linked gene-2, the endosomal sorting complexes required for transport, and neuronal death in vivo[END_REF] Programmed cell death of motoneurons is known to involve caspases, even though inhibiting the proteases only blocks nuclear destruction without allowing motoneuron survival. [START_REF] Mahul-Mellier | Alix, making a link between apoptosis-linked gene-2, the endosomal sorting complexes required for transport, and neuronal death in vivo[END_REF][START_REF] Oppenheim | Programmed cell death of developing mammalian neurons after genetic deletion of caspases[END_REF] Death might thus occur through a caspase-independent mechanism, which is inhibited by Alix mutants.

Caspase-independent cell death remains ill defined, although autophagy has often been designated as responsible for cytoplasmic destruction. [START_REF] Clarke | Developmental cell death: morphological diversity and multiple mechanisms[END_REF][START_REF] Bursch | The autophagosomal-lysosomal compartment in programmed cell death[END_REF] Because of the tight relationship between Alix and the endo-lysosomal system we reasoned that its capacity to block caspase-independent neuronal death might relate to a possible role in the process of autophagy. In a first set of experiments we found that autophagosomes decorated with the microtubule-associated protein 1 light chain 3 LC3 accumulated in cultured post-mitotic neurons expressing Alix or deprived of trophic support. Furthermore, an Alix dominant negative mutant inhibited accumulation of such vacuoles in neurons deprived of trophic support (Strappazzon et al., unpublished observation). These results could suggest that Alix drives autophagy and thereby neuronal death, or simply reflect the fact that autophagy accompanies neuronal death, which is itself controlled by Alix through an unrelated mechanism. In our recent paper [START_REF] Petiot | Alix differs from ESCRT proteins in the control of autophagy[END_REF] we used BHK-21 cells to discriminate between these interpretations by uncoupling autophagy from cell death. In these cells Alix overexpression has no apparent effect on cell survival, and autophagy can be induced independently of cell death by depletion of essential amino acids. Using this paradigm, we found that Alix depletion or expression of Alix mutants did not affect LC3 localization or bulk protein degradation. These results were unexpected in view of recent reports implicating ESCRT proteins in autophagy. In particular depletion of Tsg-101 of ESCRT-I and SNF7/CHMP4B of ESCRT-III, which both interact with Alix, were shown to lead to accumulation of autophagosomes in mammalian and insect cells, respectively. [START_REF] Rusten | ESCRT functions in autophagy and associated disease[END_REF] Thus, our findings demonstrate that Alix control of neuronal death is unrelated to autophagy. They also clearly discriminate between the functions of Alix and ESCRT proteins and suggest that during autophagy Alix does not intervene in the recruitment/activation of ESCRTs as in the case of viral budding or cytokinesis.

The authors would like to thank Fiona Hemming for critical reading of the manuscript. This work was supported by Inserm, the University Joseph Fourier, and grants from the Association Française contre les Myopathies, (AFM), the Association pour la Recherche sur la Sclérose Latérale Amyotrophique (ARS), the Association pour la recherche sur le cancer (ARC), the Fondation pour la Recherche Médicale (FRM) and region Rhone Alpes.


Claire Mantel

Patricia Ladret

Thomas Kunlin

A TEMPORAL MOSQUITO NOISE CORRECTOR

Keywords: Mosquito Noise, Video, Correction, Temporal Filtering, Temporal Artifact

Due to the required memory space and computing time, temporal processing is rarely used even in video processing. However, when the temporal aspect of a compression noise is major, it cannot be treated spatially only. The following article presents an example of such a noise: the mosquito noise. The equally spatial and temporal nature of the mosquito artifact is first explained through its definition. It is followed by the different ways already used to treat it. A spatio-temporal and compression-independent mosquito noise corrector is then introduced. Finally, the interest of a temporal corrector over purely spatial ones is shown through tests results.

INTRODUCTION

Temporal processing is seldom used for denoising a decompressed video because of the complexity involved. However, in [START_REF] Anderson | The myth of persistence of vision revisited[END_REF] Anderson and Anderson show that images are not perceived by the Human Visual System (HVS) as what they are: a succession of static images, but that their temporal facet is interpreted as a real one. In other words, video motion goes through the same treatment as natural motion. As a result a video is not just series of images to our eyes, and considering its temporal aspect is simply emulating the HVS. Temporal variations have been taken into account in video compression since its very beginning. To gain bandwidth temporal redundancy of a video was profited from by detecting still zones that did not need to be coded anew in H.120, the first video compression standard released in 1984, and by compensating motion since the MPEG1 standard in 1988. As the main principles of the different video standards have not changed for a while (motion compensation and DCTbased transform are present in MPEG2, MPEG4 and H.264), the type of defects resulting from video compression remains the same. However, with the expansion of LCD displays, High Definition sources and monitors, the perception and expectations of video quality altered. Precisely, it is the respective annoyance of the different noises compared to one another that has completely changed: for example high frequency noises, who were faded by the low pass answer of cathodic monitors, are emphasized by the increase in displays size and the sharpness of the pixel grid in LCD. This paper deals with such an artifact: the mosquito noise. Whereas other purely spatial defects (such as blocking or ringing) have been thoroughly studied, this one has been less addressed in literature. The temporal aspect is especially major for this noise since it would not be that annoying in a static image but is mainly noticeable due to its variation through time. Our purpose was to build an efficient corrector that takes into account the temporal feature of this noise so it contains both a spatial and a temporal part. To extend the scope of our corrector as much as possible, we set it up to be totally independent from the compression parameters or information. The interest of treating mosquito noise temporally will be explained in the first part of this article, describing its both spatial and temporal nature. A second paragraph will sum up the different denoising processes found in the literature review. Then we will describe our Temporal Variation-Inverse Filter and finally compare it with other algorithms.

MOSQUITO NOISE: A COMPLICATED DEFINITION

The first difficulty when studying mosquito noise is to establish what we actually deal with: there are as many definitions as there are publications on that topic. In its recommendation for assessing subjective video quality [START_REF] Itu | Principles of a reference impairment system for video[END_REF], the ITU-T defines mosquito noise as: "Form of edge busyness distortion sometimes associated with movement, characterized by moving artifacts and/or blotchy noise pattern superimposed over the objects (resembling a mosquito flying around a person's head and shoulders)." and the edge busyness by "Distortion concentrated at the edge of objects, and further characterized by its temporal and spatial characteristics." . Though those definitions should be a reference, they are somehow open and not sufficient.

Resemblance to other artifacts

The ITU characterizes the mosquito effect with a number of specificities which are unfortunately also valid to other kinds of artifacts. By extension, we can see that it is difficult to identify, and easy to mistake for other artifacts. Mosquito noise can typically be assimilated to ringing, since they are both located next to the edges. Some, as Segall et al. in [START_REF] Segall | Pre-and postprocessing algorithms for compressed video enhancement[END_REF], clearly integrate the temporal variations of ringing as mosquito artifact; while others, such as Coudoux et al. in [START_REF] Coudoux | A postprocessor for reducing temporal busyness in low-bit-rate video applications[END_REF], make even no difference between those two noises. There is actually no definition of "ringing" in the ITU documentation [START_REF] Itu | Principles of a reference impairment system for video[END_REF]. This artifact is in fact purely spatial and has common properties with the mosquito noise: they both appear on unified zones near the border of objects.

Another difficulty to consider with mosquito noise is that its cause is not clearly defined. As quantization is the only source of information loss in compression, it is inevitably linked to mosquito noise, but that is also true for every other compression noise.

Spatial aspects

Almost all papers on mosquito noise agree on the fact that it appears near the border of objects. Several articles put an additional indication concerning this kind of zone: it must be near a portion of intense activity. Some authors, such as Coudoux et al. in [START_REF] Coudoux | A postprocessor for reducing temporal busyness in low-bit-rate video applications[END_REF], specify that mosquito effects can commonly be seen at the border of uniform or nearly uniform zones (denoted as "flat"). Others, as Westen et al. in [START_REF] Westen | Adaptive spatial noise shaping for dct based image compression[END_REF], also indicate that the blocks presenting mosquito artifact must contain both flat and textured areas. DelCorso et al. assert in [START_REF] Del Corso | Mnr: a novel approach to correct mpeg temporal distortions[END_REF] that mosquito noise is located near borders of objects in motion.

The second major feature of mosquito noise is its really low intensity. Every scientific paper concurs on this point: in a region where mosquito is present, the oscillations around the right gray level due to noise are quite small [START_REF] Coudoux | A postprocessor for reducing temporal busyness in low-bit-rate video applications[END_REF][START_REF] Del Corso | Mnr: a novel approach to correct mpeg temporal distortions[END_REF][START_REF] Abbas | Suppression of mosquito noise by recursive epsilon-filters[END_REF]. This explains why the artifact is invisible in portions including many details, or textured zones.

To sum up, if the various articles on the subject sometimes diverge about the surroundings of mosquito noise, a characteristic that is globally accepted is that the noise level is very low compared to the intensity variation of the image (edges or textures) neighboring the noise.

Temporal aspects

There are two different ways to consider the temporal aspect of mosquito noise. The first is a perceptual one: the reason this small-amplitude noise is a nuisance is that it varies through time as stated in [START_REF] Segall | Pre-and postprocessing algorithms for compressed video enhancement[END_REF]. The fact that the temporal oscillations are not a consistent motion and that their frequency is the video framerate, prevents the HVS from stabilizing on the noise. It is this "discontinuity along the temporal axis", as named in [START_REF] Coudoux | A postprocessor for reducing temporal busyness in low-bit-rate video applications[END_REF], that is perceived as annoying. Since peripheral vision is very sensitive to movement, it can be noticed everywhere in the frame and not only in salient regions.

The second way is its relationship to the temporal part of video compression. In the most common video standards, some blocks are coded using inter-coding (or predictive coding). They are first motion-compensated regarding a reference frame and then only the remaining of the difference between the current block and its motion compensation are coded. That is what we call "temporal aspect" of video compression. Opinions are divided on whether mosquito effect can appear in immobile sequences, as asserted by Fenimore et al. in [START_REF] Fenimore | Mosquito noise in mpeg-compressed video: test patterns and metrics[END_REF] whereas DelCorso et al. insist in [START_REF] Del Corso | Mnr: a novel approach to correct mpeg temporal distortions[END_REF] on the fact that the presence of a moving object and a motionless background are required for mosquito noise to occur. But it is clear from all papers on the subject that the coding of residuals after motion compensation increases the presence of mosquito noise. We can also deduce that there is a correlation between the fact that a sequence includes a wide number of movements and the presence of mosquito noise.

Spiral videos

A completely different approach has been set up by Fenimore et al. in [START_REF] Fenimore | Mosquito noise in mpeg-compressed video: test patterns and metrics[END_REF], where they display a mosquito noise reference metric. When they confront themselves with the problem of separating mosquito noise from others to measure it exclusively, they adapt the videos they are using and not their metric. They describe video sequences in which compression noises is strictly mosquito noise. Their videos are made of black spirals moving on a white background.

Creating such videos allowed us to make sure that their compression produced mosquito noise as the most annoying artifact in MPEG2 (figure 1). For the H.264 standard results were less distinct, so we produced new sequences were the spirals were turning and shifted throughout the framework. The motion compensator of the encoder does not handle rotations, therefore it could not correctly estimate the current motion. It clearly appeared that by putting it in difficulty, we greatly increased the presence of mosquito noise (figure 1). As modifying motion in videos can only affect inter coded macroblocks, the link between mosquito noise and the type of encoding of a macroblock is thus proved by this experiment.

The most probable explanation of the presence of mosquito effects in image sequences is given by Abbas in [START_REF] Abbas | Suppression of mosquito noise by recursive epsilon-filters[END_REF]: the noise is not due to a unique cause, but the conjunction of several criteria. This approach allows understanding the variety of interpretations and more specifically to clarify the spatial/temporal ambivalence: the origins and the perception of the mosquito noise are both spatial and temporal.

MOSQUITO DENOISING METHODS

Since the focus of this paper is correcting mosquito noise, the state-of-the-art will not address pre-processing methods (as [START_REF] Westen | Adaptive spatial noise shaping for dct based image compression[END_REF]) but only post-processing ones that are totally independent from compression. Practically, proximity of mosquito noise to contours means that correctors must be precise enough to suppress it without damaging edges. To meet this constraint, various strategies have been adopted. In [START_REF] Coudoux | A postprocessor for reducing temporal busyness in low-bit-rate video applications[END_REF] Coudoux et al. first mark as edge pixels those which gradient is higher than 80 and afterward use that to tag each 8x8 block as including or not borders. They filter only the non-contour pixels in the blocks labeled as "containing borders", using a 2-D Wiener filter to remove the noise. We agree with Abbas and Karam ( [START_REF] Abbas | Suppression of mosquito noise by recursive epsilon-filters[END_REF]) that it raises two issues: the non-correlation hypothesis of the Wiener filter is not respected since mosquito noise is not decorrelated from the signal (2.2) and a pixel can end-up being filtered with pixels belonging to another object, thus blurring the contours. In [START_REF] Abbas | Suppression of mosquito noise by recursive epsilon-filters[END_REF] Abbas and Karam employ a Canny edge detector to find out where contours are precisely and then apply their filter to the whole image except to edge pixels as Coudoux et al.. To meet with spatial characteristics of mosquito effect, chose -filter: a simple averaging filter which restrains its support to pixels that are within an from the current pixel. They state their Recursive -filter (REF) in which an -filter is recursively applied to correct mosquito noise, using the standard deviation of a block to approximate the one of the noise. Same as Coudoux et al., the edge pixels are not altered but may serve to filter pixels belonging to other objects. Indeed, in some cases the standard deviation might be high enough to introduce pixel of another object in the filter support. That might happen if a block is cut in half by the frontier between two objects, or when the edges are not sharp. It results in introducing blur.

As it has been said in 2.3, DelCorso et al. focus their corrector on moving boundaries only ( [START_REF] Del Corso | Mnr: a novel approach to correct mpeg temporal distortions[END_REF]). Thus, their solution is applied on the entire image and uses a motion detector, skipping the edge detector part. But the surprising part is that, in order not to degrade moving zones, they do not filter them. They consider that mosquito noise is more visible on the background close to moving objects than on the ob-jects themselves. Anyway since they partition the frame into 8x8 blocks, they miss potentially noisy pixels contained in moving blocks. They utilize the small amplitude of mosquito noise by asserting that correcting the average value of an 8x8 block is enough. To do so, they first apply a median filter on DC values of the current, the upper and lower blocks. The result is once again filtered by a median filter with the DC coefficient of the equivalent block in the previous and following frames. In the end, the temporal feature of mosquito noise has been either not addressed at all, as in [START_REF] Abbas | Suppression of mosquito noise by recursive epsilon-filters[END_REF] or [START_REF] Coudoux | A postprocessor for reducing temporal busyness in low-bit-rate video applications[END_REF] or only for motionless areas through a motion-adapted filter in [START_REF] Del Corso | Mnr: a novel approach to correct mpeg temporal distortions[END_REF].

THE TEMPORAL VARIATION-INVERSE FILTER

The corrector we are to present here, the Temporal Variation-Inverse Filter (T-VIF), covers all three aspects of the mosquito noise definition. It is composed of a filter, the Variation-Inverse Filter (VIF) that will be described in 4.1, which support will be adjusted according to the two other major features of mosquito noise: its proximity to edges (4.2) and its temporal variation (4.3). Finally a fast motion-adapted version of our filter, especially accommodated for mosquito, is explained in 4.4.

Small Gray Level Fluctuation

The Variation Inverse Filter is a 1-D filter set up by Crété-Roffet in [START_REF] Crété-Roffet | mesurer et corriger les artefacts de compression pour la télévision numérique[END_REF] to suppress ringing artifact. We will describe here the vertical filtering process. It is an answer to one of the main issues in treating ringing artifact: to distinguish correctly what is noise and what is an edge. As noise amplitude depends directly on the edge strength, the dynamic of a ringing effect due to a strong edge can be higher than the one of a weak edge present in the same image. The first step of the VIF is thus to fit the strength of the filter accordingly to the one of the edge. As it was designed for MPEG2, the normalization N orm is made according to the following formulas, for (i,j) covering a whole 8x8 DCT block in the current image Im:

D i,j = |Im i,j+1 -Im i,j | M ax = max(D i,j ) and M in = min(D i,j ) If M ax = M in, N orm i,j = T hresh max • D i,j -M in M ax -M in
If M ax and M in are equal then variations are constant and the noise cannot be separated from the edge so we will not filter. T hresh max represents the number of possible levels.

The VIF applies then an adaptive filtering. It relies on the idea that the more a neighboring pixel is distinct from the pixel being filtered, the more likely they do not belong to the same object. The weight given to each pixel is therefore based on the inverse of its difference with the current pixel. The coefficients W i,j±1 of pixels (i,j-1) and (i,j+1), which are used to filter vertically the pixel (i,j), are calculated according to the subsequent equations, where F ilter N orm is the norm of the filter:

W i,j±1 = 1 F ilter N orm • 1 N orm i,j±1 + 1
As explained in 2.2, the amplitude of mosquito noise is low regarding the neighboring edge amplitude. This is why the VIF is not only adapted for ringing artifact but also suited for the spatial characteristics of mosquito noise. Indeed, it was set up to correct a noise of small-amplitude measured up to an edge variation. Anyhow, the VIF needed to be adapted to 3-D filtering (2-D plus temporal) and to other partitions of image than MPEG2's 8x8 blocks. As gradient calculation is a stage of border detection we already have them available. We used the highest gradient value in the 15x15 neighboring of the currently filtered pixel as M ax and the lowest as M in. The result F i,j is then calculated the following way:

F i,j = 1 F ilter N orm •   1 + (k,l)∈{-1,1} 2 1 N orm i+k,j+1 + 1  
The paragraphs 4.2, 4.3 and 4.4 modify the support of the VIF filter. Figure 2 shows the global structure of the T-VIF.

Close to Contours

As stated in 2.2, mosquito noise is present beside object edges. Thus, we decided to use an edge detector for two distinct reasons: the first is to determine for each pixel whether it presents a risk of mosquito. If a pixel could be within the same quantization block as a contour pixel during encoding, practically if it is in an 8x8 neighborhood of a contour pixel, it has to be filtered, otherwise no. Filtering uniquely where there may be ringing or mosquito noise allows time saving because the filter is only applied on "potentially noisy" pixels and also eliminates risk of degrading healthy zones that do not need correcting. The other advantage to detect edges is the possibility to adapt the filter support to the frame content. To avoid degrading or blurring edges, we decided to restrain the filter support to pixels with the same tag from the edge detector (i.e. border/no border). Our aim is not to filter edges with edges and objects with objects. Indeed, it would not match reality because, except for synthetic videos, there is never a one-pixel border surrounding each object in videos: the "border" is part of an object. The benefit of restricting the filter support that way is to make sure we only use pixels belonging to the same object to filter. We chose Canny's edge detector because it provides exclusively one-pixel width edges. As we filter on a 3x3 neighborhood, the only case in which two pixels of the neighborhood can be part of different items is if there is an edge pixel between them. Since edges obtained through Canny's detector are connected, it means that the central pixel is an edge one. And in our algorithm it will be filtered using only other edge pixels. If the edge detector was to make a mistake, the VIF also offers a security: if by mistake an outsider pixel was to be added to the filter support, its influence would be minimal due to its difference from the filtered pixel.

Temporal Variation

The major specificity of the T-VIF is its temporal aspect since, as shown in 3, there is currently few temporal corrector for mosquito noise and none addressing both the edge issue and the temporal one. As mosquito noise varies from one frame to another, our idea is that using the previous frame in the support of the filter will help converging toward the right value. Another gain is that using the time-preceding pixels to filter will diminish the temporal discontinuities. Thus we decided to modify our VIF into a Motion-Compensated spatiotemporal filter to improve the temporal smoothness of the processed sequence. We used the Robust Optic Flow estimation from Black and Anandan stated in [START_REF] Black | The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields[END_REF] and whose code is available on their personal webpage. Applied between two images, it provides motion vectors that optimize the match between the two images with pixel granularity. A motion-compensated frame is then created. The 3x3 motion-compensated neighborhood with identical coordinates as the current pixel is then adjoined to the filter support. This add-on follows the same sorting (by membership to border or non-border class) as the original neighboring pixels.

Fast Temporal Filtering

Motion compensation is quite time and resource consuming. We studied another way to add pixels from a prior image in the filter support: turning our T-VIF into a Motion-Adaptive Filter. This version will be called Fast-T-VIF in the paper. The advantage with mosquito effect is that its small amplitude and its alteration from frame to frame enable using a simple but efficient approach. The concept is to classify pixels not accordingly to their stillness but to their belonging to the same object or not. To guarantee that, the highest gradient amplitude in the 8x8 neighboring, which is identified as a contour one, serves once again as a standard besides which other pixels will be evaluated. A difference frame is calculated between the current and preceding images (figure 3). Each neighboring pixel in the previous frame is then appended to the filter support if this difference is relatively smaller than the gradient maximum (i.e. if they are part of the same object). Like for other VIF methods, pixels are then sorted by their belonging to a contour or not before being used.

SIMULATION RESULTS

Video used

In this section the versions of VIF will be compared to the REF. The video sequences used are Aspen, InToTree (from the VQEG website: www.its.bldrdoc.gov/vqeg/) and Life (from www.hdgreetings.com/other/ecards-video/video-1080p.aspx). Aspen and IntoTree are natural videos with respectively wide and small motion, while Life is a synthetic video with average motion. All video sequences were retrieved in "perfect" quality, cropped to SD and then encoded using the JM13 from the JVT, which favorably allows canceling the deblocking filter through its parameters. The chosen bitrates were respectively 1Mbps for Aspen and 600Kbps for the others. The three sequences were cropped to 500x500 format.

Spatial Analysis

The spacial results are evaluated through SSIM and PSNR computing between the corrected frames and the original ones. The corresponding graph is shown for Life in figure 4

Temporal Analysis

The main advantage of the T-VIF is its temporal aspect. The idea is that the correction will gain in smoothness on a temporal level. To demonstrate its effect, we calculated PSNR and SSIM on the difference between two successive images. Thus we study the effects of the different filters to the variation through time. For the three tested videos, the variation from one frame to another in the T-VIF and Fast-T-VIF are better. For Life and Aspen this enhancement is significant, while for InToTree it is too slight to be meaningful. The PSNR values for the Aspen sequence are shown in figure 6.

CONCLUSION

On a spatial level, our Fast-T-VIF produces a significant enhancement for one of the test videos, a slight one for another and none for the third in terms of fidelity metric. On the temporal point of view the results indicate a moderate improvement in the sequence continuity. The tests should be extended to a wider video database for more significant tendencies. But the main issue while assessing the efficiency of mosquito noise correctors is how to evaluate the quality of the correction. Indeed, the precise localization and small amplitude of this noise both tend to level the usual global metrics and make its temporal aspect difficult to deal with. The next step for mosquito correction is thus the making of a dedicated metric, which exactly what we are currently working.

Fig. 1 .

 1 Fig. 1. Mosquito noise on spiral videos created by compression using MPEG2 (left) and H-264 (right)

Fig. 2 .

 2 Fig. 2. The T-VIF structure.

Fig. 3 .

 3 Fig. 3. Differences between image 17 and 16 of Aspen.

  . For this video, the results are clearly in favor of the T-VIF and Fast-T-VIF. The results for Aspen are slightly better for the T-VIF and Fast-T-VIF than for REF. Regarding the InToTree video, the SSIM values are equivalent, but they are also similar to those of the compressed video although the corrected sequences are visually better. In fact on figure 5, the corrections due to T-VIF, Fast-T-VIF or REF are conspicuous. It is also visible that the temporal versions of VIF suppress as much noise as the REF while keeping a better dynamic for gray levels (in the windows for example).

Fig. 4 .

 4 Fig. 4. SSIM results for Life.


Quang Anh

Nguyen Vu

Benoit Gaudou
email: benoit.gaudou@alumni.enseeiht.fr

Richard Canal
email: richard.canal@auf.org

Salima Hassas
email: hassas@bat710.univ-lyon1.fr

Coherence and robustness in a disturbed MAS

Keywords: Multi-Agents System, trust, coherence, robustness, TrustNet, mapping

This paper presents a study on how to ensure the coherence of a distributed information system in which information is collected by a multi-agent system with the hypothesis that some agents of the system are dissonant, i.e. can produce or incorrect information.

I. INTRODUCTION

We consider a multi-agents system (MAS) in which each mobile agent aims at obtaining the most precise representation of its environment by collecting information directly via sensors or indirectly via communication with other agents. We assume that some agents disturb the system by transmitting false or inaccurate information about the environment either because their perception is flawed (i.e. their sensors are awry or inoperative) or because their interest goes against the community's one and encourages them to lie.

In this article, we study ways to ensure the coherence (i.e. the adequation between the agents' environment representation and the actual environment) and the robustness (i.e. the agents' capacity to adopt strategies allowing them to obtain this coherence despite the perturbed communication) of such a disturbed distributed information system. To limit the influence of agents transmitting incorrect information, we propose to use the social concepts of trust and reputation. TrustNet [START_REF] Schillo | Using trust for detecting deceitful agents in artificial societies[END_REF] appears to be a promising way to allow each agent to build its own network of trust. Evaluating the TrustNet before making interactions can help one agent to compute the trustworthiness of its partners and thus to decide which reliable partner to interact with. The main aim of this research on robustness is to provide tools to agents and MAS to define an efficient communication strategy.

This paper is organized as follows. Section II introduces preliminary definitions used in the whole paper: cognitive dissonance, disturbance, trust. An example of application is detailed in Section III. Section IV introduces the basic ontology and details the various kinds of information which are stored and exchanged by agents. The study of information exchanges between agents and their impact on stored data is presented in Section V before describing a method to compute the reliability of information in Section VI. Section VII presents some preliminary results on a simplified model using trust tables. Finally Section VIII concludes this paper and presents the future planned works.

II. PRELIMINARY DEFINITIONS A. Cognitive dissonance and disturbance

In his cognitive dissonance theory, Festinger [START_REF] Festinger | A Theory of Cognitive Dissonance[END_REF] considers that two elements of cognition (which includes perceptions, mental attitudes and behaviors) are dependent if there exists a link between their object, or independent, otherwise. Two dependent cognitions are called 'consonant' when one involves or supports the other. Conversely, two cognitions are called 'dissonant' when one involves or supports the opposite of the other [START_REF] Pasquier | Cohérence et conversations entre agents : vers un modèle basé sur la consonance cognitive[END_REF]. The theory assumes that agents in a dissonant state will try to reduce this dissonance, for example by changing or forgetting some mental attitudes.

By applying Festinger's theory to MAS context, we call 'dissonant act', any action performed against the achievement of the community's objective and 'dissonant agents', the ones who perform these dissonant acts, i.e. agents giving false or inexact information in our context. A piece of information is called dissonant with relation to a set of information, when one element of this information set and the new piece of information are inconsistent. In that case, we say that both information are dissonant. Moreover we call 'disturbed multiagents system' a system that includes one or several dissonant agents. By applying the cognitive dissonance assumption described above, we assume that the community purpose is to reduce the dissonance, for example by eliminating false or incoherent information or by avoiding to interact with dissonant agents.

B. Trust

In order to reduce dissonance, we give to agents the ability to reason about other agents and to choose with which they want to interact. For this purpose one of the most efficient tools is to introduce concepts of trust and reputation. We use trust in order to identify and isolate untrustworthy agents, to evaluate an interaction utility and to decide whether and with which to interact [START_REF] Muller | Computational trust and reputation models[END_REF]. In the sequel we refer to the following definition of trust and reputation because it appears to be general and complete enough to take into account several existing models of trust: "Trust is a measurable level of the subjective probability with which an agent A assesses that another agent B will perform a particular action in a favorable way to A, both before A can monitor such action (or independently of its capacity ever to be able to monitor it) and in a context in which it affects its own action" [START_REF] Huynh | Trust and reputation in open multi-agent systems[END_REF] Moreover Marsh [START_REF] Marsh | Formalising trust as a computational concept[END_REF] has proposed one of the first trust models. His model takes into account only direct information to compute trust. Up to now lot of trust and reputation models have been published [START_REF] Sabater | Review on computational trust and reputation models[END_REF]. Our work is based on the model suggested by Schillo et al. [START_REF] Schillo | Using trust for detecting deceitful agents in artificial societies[END_REF]. In this model, agents communicate factual information but also trusts they have in other agents. Agents can thus build a network of trust values transmitted by witnesses, called the 'TrustNet'. The final trust value of an agent towards another one is thus an aggregate of direct experiences and testimonies.

III. ONGOING EXAMPLE: DANGER MAPPING

In this paper we consider the following example, that we will refer to as the 'Mapping example' in the sequel. Let a swarm of perfectly localized robots modeled by agents patrolling in an urban zone affected by a natural disaster. The objective of each agent is to build the most complete, precise and reliable map of the environment by using least resources possible. In particular, its job consists in detecting dangerous places and their danger level represented by a color. To map a territory, robots can detect directly the state of one zone thanks to their sensors. They can also communicate with other robots to exchange knowledge necessary to build the danger map. We assume that each robot has a local perception and that the communication in the system is also local. Among agents, we assume that some can transmit false or inaccurate information.

In the sequel, we assume that agents map the territory as a grid. Coordinates (x, y) of each patch are thus integers. We also use following notations: AGENT denotes the set of agents and COLOR the set of colors that robots can assign to a place depending on its danger level.

IV. INFORMATION

Among information that agents store into their memory and communicate to other agents, we distinguish two kinds of information: information about the environment and information about agents. We also split the former in two sets depending on the information origin.

A. Information about the environment 1) Direct information: The direct information set, noted D X , represents information collected by agent X via its sensors.

In the Mapping example, we denote direct information:

• < (x, y), color > an explored zone (x, y) ∈ Z 2 , with a danger level coded color ∈ COLOR; • < ¬(x, y) > a location (x, y) without danger. For instance, the set D A = {< (x 1 , y 1 ), blue >} means that agent A has detected a dangerous blue level zone on location (x 1 ,y 1 ).

2) Indirect information: The indirect information set, noted IN D X , represents information that agent X receives from other agents. An element of indirect information is composed of two parts: a first one representing information about the environment that is domain-dependent (coordinates and color in the Mapping example) and the list of agents by which information have traveled since they have been collected from the environment. We call this list the 'information path' and the first agent which has collected it, the 'information source'.

In the Mapping example, we note indirect information:

• < (x, y), color , [A 1 ...A n ] > with (x, y) ∈ Z 2 , color ∈ COLOR, A i ∈ AGENT . • < ¬(x, y), [A 1 ...A n ] >a location (x,

B. Information about agents

Agents store trust on other agents in a graph called Trust-Net. TrustNet [START_REF] Schillo | Using trust for detecting deceitful agents in artificial societies[END_REF] is a directed graph where nodes represent agents and edges carry information about agents' trust estimation. We note a TrustNet built by the agent X as T N X =< {N ode X }, {< Arc X , V alue X >} >. An example of TrustNet is proposed in Figure 1

V. INFORMATION EXCHANGE BETWEEN AGENTS

After the previous static description of the various kinds of information manipulated by agents, we address in this section the issue of their dynamics due to interactions between agents, i.e. how these information are stored and how they alter previous stored information. In a disturbed MAS, we consider that agents may distort every kind of information except the information path because this data can be obtained from the system.

When two agents meet, they share information on the environment and information on the agents. When an agent receives information from another one, it compares each received information with its own information to adjust the confidence it has in this agent. We distinguish three kinds of compared information that we can relate to Festinger's distinction on cognitions: independent information, i.e. information that are incomparable (in the Danger Mapping example, this includes information about different coordinates); consonant information, i.e. information that are dependent (i.e. same coordinates) and provide the same information (i.e. same color); dissonant information, i.e. information that are dependent (i.e. same coordinates) and provide different information (i.e. different colors).

In the sequel, we consider two agents A and B exchanging information. We only study A's side of the interaction because this interaction is symmetric. The process described below is the same for the agent B.

A. Update of information about the environment

We consider that A meets B at the instant t. (D X ) t and (IN D X ) t represent sets of direct and indirect information of X at this time point.

¥ ¦ §¨© ¥ ¥ ¦ §¨© Figure 2.

Influence of B's information on A's information sets

Agent A merges information received from B with its own information and produces updated information sets (D A ) t+1 and (IN D A ) t+1 . We argue that B's direct and indirect information sets do not alter D A because D A represents the information collected directly from the environment. To compute (IN D A ) t+1 , agent A merges all information received from B with its own existing indirect information set by a fusion function Ψ Data . Figure 2 illustrates these influence relations. We thus have:

• (D A ) t+1 = (D A ) t • (IN D A ) t+1 = Ψ Data ((IN D A ) t , (D B ) t , (IN D B ) t )
To highlight main aspects of function Ψ Data , here is a simple example of information exchange, with (D A ) t = {< (x, y), blue >}, red,[C,D]>,<(x,y),yellow,[B]>,<(x,y),blue,[B,F]>}.

B. Update of information about agents

Besides exchanging information about the environment, agents also share information on agents, i.e. their TrustNet. The update of A's TrustNet will be computed in two steps: the update of its trust α in B and the update of its own TrustNet using a fusion function Ψ TrustNet taking as parameters both TrustNets (T N A ) t and (T N B ) t and the trust α:

(T N A ) t+1 = Ψ TrustNet ((T N A ) t , (T N B ) t , α).
(T N A ) t+1 =< {Node * }, {< (Arc * , Value * ) >} > from (T N A ) t and (T N B ) t . We can write N ode * = N ode A ∪ N ode B and (Arc * ,V alue * ) = Ψ ArcV alue ((T N A ) t , (T N B ) t , α)
where Ψ ArcV alue merges values on the arcs of both graphs, considering the trust A has on B. Due to space limitation, we do not detail this function here.

VI. COMPUTING THE INFORMATION RELIABILITY

Via communication, agents receive a lot of information. Among these information, some can be false or inaccurate. Agents have thus to make a selection among this flow and determine which are reliable and which are not. In order to calculate the information reliability, each agent uses a probability tree to represent information in its memory. The advantage is that its use is simple and effective. In the Mapping example, each agent uses a tree such as the one in Figure 4 to store the information about one zone. Each node stores an elementary information. T and F represent respectively the existence and the non-existence of danger; black , blue, yellow , red belongs to COLOR and represents the set of danger levels of the zone. We associate to each edge of the tree the pair (γ, {A i ...A n }), where γ is the probability of the information to be true (its reliability) and {A i ...A n } the information sources. Each time a new information comes, the agent updates these values in the tree.

A major issue is to determine the information reliability when two groups of agents give different information about the same patch, i.e. to calculate the reliability of dissonant information. We need to compare the impact of both groups. For this purpose, we assign a trust weight T W to each group depending on individual trusts as follows. We introduce two thresholds: an upper threshold (U pp) and a lower threshold (Low ). Between Upp and 1, agents are regarded as "reliable". Between Upp and Low , agents are under observation. Between 0 and Low , agents are regarded as "unreliable".

To evaluate the trust weight of an agent's community, we first split the community in three sets (reliable, unreliable and others). A weight is assigned to each member of a set (for example an arbitrary +λ for one reliable agent, -1 for one unreliable agent, +1 to one other agent). Then, considering the cardinal of these sets, we calculate the balanced sum:

T W (A 1 , ..., A n ) = λ * Card ({A i |i ∈ [1, n], U pp ≤ Trust(A i )}) +Card ({A i |i ∈ [1, n], Low < Trust(A i )< Upp}) -Card ({A i |i ∈ [1, n], Trust(A i ) ≤ Low })
whereTrust(A i ) is the trust in agent A i calculated according to the TrustNet. This computation method aims at creating an equilibrium between the number of quality of agents in a group and their quantity. The Trust Weight expresses the weight of the group on the reliability of information transmitted by this group in relation with the trust weight of other groups. Formally, we note p i a piece of information brought by a group G i of G 1 , ..., G n , the set of groups having transmitted a dissonant piece of information. We compute the reliability of a piece of information as follows:

reliability (p i ) = T W (G i ) k∈[1,n] T W (G k )

VII. SOME RESULTS

Preliminary simulations under NetLogo [START_REF] Wilensky | Netlogo. center for connected learning and computer based modeling[END_REF] have been proceeded on a simplified model of our Mapping application to test the interest of introducing trust to improve the quality of the communication in a disturbed Multi-Agents System. This simplified version deals with danger zones without color codage. It does neither use TrustNet to represent trust, nor makes the distinction between direct and indirect information. Agents only transmit environmental information and trust tables, i.e. a table in which agents store trust values on other agents. When an agent B's trust value falls below the Low threshold in the trust table of an agent A, A does not consider anymore information coming from B. Moreover, A doesn't send any more information to B.

We made some simulation and this strategy appears to have an impact on the volume of communications and on the number of steps the system goes to reach its objective: get a complete and trustful map of the environment. In Figure 5, we compare the number of meetings which is also the number of communications in the MAS if there is no communication policy and the number of sent messages when each agent interrupts communication with untrustful agents without impairing the performance of the MAS. These results are significant and encourage us to develop a more accurate model using no more trust tables but TrustNets and a distinction between direct and indirect information.

VIII. CONCLUSION AND FUTURE WORKS

This article addresses the problem of dissonance in a disturbed MAS where information collection or transmission can be altered by unreliable agents. We show how each agent associates a reliability to information and a trust to other agents to improve its perception of the world. The advantage of our approach is that each agent can distinguish direct (collected on the environment) and indirect (collected by exchanging information with other agents) information, not only in its stored data, but also in the data transmitted by agents it meets. Considering these information, agents can build and update finely tuned information probability trees and more accurate TrustNets. Actually, variations of one agent's confidence in another agent depend at the same time on the quality of information transmission by the other agent and on the distance between information gathered by both agents. This process finally helps an agent to enforce its communication with trusted agents and to reject communication from untrusted agents, so to reduce dissonance in the disturbed MAS.

Our work is focusing on the best way to merge information in the direct and indirect databases stored by an agent and to compute trust variations on the arcs of the TrustNet based on the confrontation of information.

Future work will focus on the MAS self-organization about communication management, the structuring of agents communities according to their reliability and the limits of dissonance a disturbed MAS can support.

  y) without danger transmitted through A 1 ...A n-1 from the source A n . So, the set IN D A = {< (x 1 , y 1 ), blue, [C, D] >} means that agent A has received an information transmitted by C (C got this information from D before) about a dangerous blue level zone on the location (x 1 , y 1 ).

Figure 1 .

 1 Figure 1. Example of TrustNet

  (IN D A ) t = {< (x, y), red, [C, D] > }, (D B ) t = {< (x, y), yellow >}, (IN D B ) t = {< (x, y), blue, [F ] >}. As explained above, D A remains unchanged : (D A ) t+1 = {< (x, y), blue >}. Now, A will fusion received information from B with its existing indirect information set. In this simple example, all received information become indirect information for A. It should indicate that B transmitted them. It thus transforms B's direct information in A's indirect one by adding to them a new path initialized with agent B. It moreover adds B to the path of received indirect information. It thus have: (IN D A ) t+1 = {<(x,y),

1 )Figure 3 . Comparing agents databases 2 )

 132 Figure 3. Comparing agents databases

Figure 4 .

 4 Figure 4. Information tree

Figure 5 .

 5 Figure 5. Comparison between the number of sent messages (yellow) and the number of meetings (purple)