ADAPTIVE DENSITY ESTIMATION UNDER WEAK DEPENDENCE GANNAZ IR ÈNE (1) AND WINTENBERGER OLIVIER (2)

Keywords: 1991 Mathematics Subject Classification. Primary 62G07; Secondary 60G10, 60G99, 62G20 adaptive estimation, cross validation, hard thresholding, near minimax results, nonparametric density estimation, soft thresholding, wavelets, weak dependence

 propose near-minimax estimators fn based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators fn depend on weak dependence properties of the sequence (Xt) t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.

Introduction

Let (X t ) t∈Z be a real valued time series admitting a common marginal density f that is compactly supported. The general purpose of this paper is to estimate f by wavelet estimators f n constructed from n observations (X 1 , . . . , X n ). In their seminal paper [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] showed that projection-like linear estimators are not optimal: introduction of nonlinearity via thresholds of wavelet coefficients is investigated. Wavelets thresholding provides estimators which adapt themselves to the unknown smoothness of f , we refer to [START_REF] Vannucci | Nonparametric density estimation using wavelets[END_REF] [START_REF] Vannucci | Nonparametric density estimation using wavelets[END_REF] for a survey of the use of wavelet bases in density estimation. The present work extends near minimax results of soft and hard-threshold estimators from the independent and identically distributed (iid for short) framework, see Theorem 5 in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], to cases where weak dependence between variables occurs.

Our main assumptions give bounds for covariance terms as decreasing sequences which tend to zero when the gap between the past and the future of the time series goes to infinity. In order to give examples satisfying these conditions, we introduce coefficients that give bounds of covariance terms and that are computable for a large class of models. Weak dependent coefficients, introduced by Doukhan and Louhichi (1999) [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF], as mixing ones, are well-adapted to that purpose. Using β-mixing coefficients Tribouley and Viennet (1998) [START_REF] Tribouley | L p -adaptive density estimation in a β-mixing framework[END_REF] proposed minimax estimators with respect to the Mean Integrated Square Error (MISE for short). [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] (3) obtained near-minimax results using α-mixing coefficients. The loss of a logarithmic factor in the convergence rate in this last paper is balanced by the generality of the context, as the class of α-mixing models is larger than the one of β-mixing models.

However, α and β-mixing coefficients are not easy to compute for some models and are useless for others; [START_REF] Andrews | Non strong mixing autoregressive processes[END_REF] (1) proved that the mixing coefficients of the stationary solution of the AR(1) model

X t = 1 2 (X t-1 + ξ t )
, where (ξ t ) t∈Z iid with a Bernouilli law of parameter 1/2, (

do not tend to zero as the gap from the past to the future of the time series goes to infinity. Mixing coefficients do not behave nicely in this case as, through a reversion of time, the Markov chain solution of (1.1) is a dynamical system, i.e. X t-1 = T (X t ) for some transformation T , namely T (x) = 2x1 1 0≤x<1/2 + (2x -1)1 1 1/2≤x≤1 . So called weak dependence coefficients have been recently developed to deal with such processes, see Dedecker et al. [START_REF] Dedecker | Weak Dependence: Models, Theory and Applications[END_REF], Maume-Deschamps [START_REF] Maume-Deschamps | Exponential inequalities and functional estimations for weak dependent data ; applications to dynamical systems[END_REF] and references therein. Introduced by [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF] in ( 6), φ-weak dependence coefficients give sharp bounds on the covariance terms of dynamical systems, such as the stationary solution to (1.1). Using these coefficients, we prove near-minimax results of thresholded wavelet estimators for dynamical systems called expanding maps. To our knowledge, only non adaptive density estimation has been studied in this non-mixing context, see for instance Bosq et Guegan (2), Prieur [START_REF] Prieur | Applications statistiques de suites faiblement dépendantes et de systèmes dynamiques[END_REF] and Maume-Deschamps [START_REF] Maume-Deschamps | Exponential inequalities and functional estimations for weak dependent data ; applications to dynamical systems[END_REF].

The advantage of our approach is also to treat in one draw many other contexts of dependence. We prove that near-minimaxity still holds for a very large class of models using λ-weak dependence coefficients, defined by Doukhan and Wintenberger, 2007, (14). We pay for generality by adding up conditions on the joint densities of the couples (X 0 , X r ) for all r > 0. These conditions are not restrictive as it is satisfied for many econometric models such as ARMA, GARCH, ARCH, LARCH, MA models.

The estimation scheme is based on Donoho et al.'s procedures developed in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] for the iid case, and it is adaptive with respect to the regularity of f . Soft and hard-threshold levels (λ j ) j 0 ≤j≤j 1 are chosen equal to K j/n for some K > 0. Note that the constant K and the highest resolution level j 1 depend on the weak dependence properties of the observations. If weak dependence properties are known, estimators are near minimax: same rates as in the iid setting are achieved for mean-L p errors with 1 ≤ p < ∞. If weak dependence properties are unknown, we develop cross-validation procedures to approximate threshold levels λj and the highest resolution level j 1 . We check on simulations that corresponding estimators are adaptive with respect to the regularity of f and to weak dependence properties of (X t ) t∈Z . The order of errors of approximations in the dependence cases are very close to the one in the iid cases.

We believe that we obtain such good results as we work on simulations of processes satisfying our main Assumption (D). This assumption consists of the exponential decay of the covariance terms. We give in this paper some simulations study of dynamical systems that do not satisfy this Assumption (D). Comparing the behavior of our estimators with the kernel ones show that ours are less efficient when covariance terms do not decrease exponentially fast. We also prove that the error terms of our estimators are unbounded in some cases, due to terms of covariances that decrease too slowly. This is a restriction of our procedure, based on thresholding wavelet coefficients.

The paper is structured as follows. In Section 2, we give notation, we introduce estimation procedures and we formulate weak dependence assumptions. Main results are given in Section 3 and examples of models in Section 4. Cross-validation procedures and their accuracy on simulations are developed in Section 5. Proofs are relegated in the last Section.

Preliminaries

2.1. Notation. We restrict ourselves to the estimation of a density f which is compactly supported. We suppose that f is supported by [-B, B] for some B > 0. For all p ≥ 1, L p denotes the space of all functions f such that

f p p = |f (x)| p dx < ∞.
2.2. Density estimators. Throughout the paper, we work within an r-regular orthonormal multiresolution analysis of L 2 (endowed with the usual inner product), associated with a compactly supported scaling function φ and a compactly supported mother wavelet ψ. Without loss of generality, we suppose that the support of functions φ and ψ is included in an interval [-A, A] for some A > 0. Let us recall that φ and ψ generate orthonormal basis by dilatations and translations: for a given primary resolution level j 0 , the functions {φ j,k : x → 2 j/2 φ(2 j x -k)} k∈Z and {ψ j,k : x → 2 j/2 ψ(2 j x -k)} k∈Z are such that the family

{φ j 0 ,k , k ∈ Z, ψ j,k j ≥ j 0 , k ∈ Z} is an orthonormal base of L 2 . Any function f ∈ L 2 can thus be decomposed as f = k∈Z α j 0 ,k φ j 0 ,k + ∞ j=j 0 k∈Z β j,k ψ j,k , where α j,k = f (x)φ j,k (x)dx, β j,k = f (x)ψ j,k (x)dx.
The nonlinear estimator developed in ( 8) is defined by the equation

f n = k∈Z α j 0 ,k φ j 0 ,k + j 1 j=j 0 k∈Z γ λ j ( β j,k )ψ j,k ,
where αj,k = n -1 n i=1 φ j,k (X i ) and βj,k = n -1 n i=1 ψ j,k (X i ), and where γ λ is a threshold function of level λ. The authors consider both hard and soft thresholding functions, corresponding respectively to γ λ (β) = β1 1 |β|>λ and γ λ (β) = (|β| -λ) + sign(β). If no distinction is done in the sequel, both hard and soft thresholding estimators are concerned.

Let s, π, r be three positive real numbers satisfying s + 1/2 -1/π > 0. We assume that f belongs to the Besov ball B s π,r (M 1 ) on the real line, i.e. f s,π,r ≤ M 1 where

f s,π,r = |α 0,0 | +   j∈N 2 j(sπ+π/2-1) k∈Z |β j,k | π r/π   1/r
.

Approximation errors of an estimator f n are expressed as E f n -f p p for p ≥ 1. Associated minimax rates are the best convergence decrease α of the worst approximation error we may achieve over all estimators f n :

inf fn sup f ∈B s π,r (C) E f n -f p p = O n -pα .
The minimax rate α is determined in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] as:

α = α + = (s/(1 + 2s) if ǫ ≥ 0, α -= (s -1/π + 1/p)/(1 + 2s -2/π) if ǫ ≤ 0, where ǫ = sπ -(p -π)/2. (2.1)
2.3. Weakly dependent assumption. Throughout the paper, the symbol δ denotes with no distinction φ or ψ and δ j,k (x) = δ j,k (x) -E δ j,k (X 0 ) for all integers j ≥ 0 and k. Define for all positive integers u, v the quantities

C j,k u,v (r) = sup max s i+1 -s i =s u+1 -su=r {|Cov( δ j,k (X s 1 ) • • • δ j,k (X su ), δ j,k (X s u+1 ) • • • δ j,k (X s u+v ))|}. (2.2)
Functions φ and ψ play a symmetric role through δ in this setting. As stressed in [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF], bounds on covariance terms C j,k u,v (r) are useful to extend asymptotic results from the iid case. Now we can state the main assumption of this paper:

(D): There exists a sequence ρ(r) such that for all r ≥ 0, all indexes j, k, u, v, we have

C j,k u,v (r) ≤ (u+v+uv)/2 (2 j/2 M 2 ) u+v-2 ρ(r) where M 2 is a constant satisfying δ ∞ ≤ M 2 . (D1)
Moreover, there exist real numbers a, b, C 0 > 0 depending only on δ, f and on the dependence properties of (X t ) t∈Z such that

ρ(r) ≤ C 0 exp(-ar b ) for all r > 0. ( D2
)
In Section 4, we give explicit conditions on the stationary process (X t ) t∈Z in order that it satisfies Assumption (D). When it is possible, the values of the constants a, b, C 0 and M 2 are given.

Main results

Let (X t ) t∈Z be a stationary real valued time series.

3.1.

A useful lemma. Under similar conditions than Assumption (D), moments inequalities of even orders and Bernstein's type inequalities for the sums n i=1 δ j,k (X i ) are respectively given in ( 9) and [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF]. The following Lemma recall these inequalities applied on the quantities β j,k . They remain valid for α j,k as φ and ψ play the same role in (D). Lemma 3.1. If (D) holds then for all even integer q ≥ 2, for all λ ≥ 0 and for j, k such that 0 ≤ j ≤ log n and 0 ≤ k ≤ 2 j -1:

E| β j,k -β j,k | q ≤ C 1 n -q/2 , (3.1)
P | β j,k -β j,k | ≥ λ ≤ 2 exp - nλ 2 C 2 + C 3 (2 j /n) b/(2(1+2b)) ( √ nλ) (2+3b)/(1+2b) , (3.2)
where C 1 , C 2 and C 3 are constants depending on q and on the constants of Assumption (D): a, b, C 0 and M 2 .

The proof of this Lemma is given in Section 6.1. Notice that C 3 depends deeply on the dependence context through C 0 , see Propositions 4.1 and 4.2 for more details.

3.2.

Near-minimax results of thresholded wavelet estimators. Following results are extensions to weak dependence settings of Theorem 5 of (8). Theorem 3.1. Suppose that f ∈ B s π,r (M 1 ) with 1/π < s < N/2 where N is the regularity of the function ψ. If (D) holds, then for each 1 ≤ p < ∞ there exists a constant C(N, p, a, b, C 0 , M 1 , M 2 , A, B) such that

E[ f n -f p p ] ≤ C        log n n pα if ǫ = 0 log n n pα (log n) (p/2-(1∧π)/r) + if ǫ = 0
where the minimax rate α and the parameter ǫ are given in (2.1). Here j 0 is chosen as the smallest integer larger than log(n)(1 + N ) -1 , j 1 is the largest integer smaller than log(n log -2/b-3 (n)) and λ j = K j/n for a sufficiently large constant K > 0.

A sketch of the proof of this Theorem is given in Section 6.2.

We refer to [START_REF] Meyer | Wavelets and Operators[END_REF] for the definition of the parameter N , the regularity of the wavelet function ψ. The condition s < N/2 ensures the sparsity of the wavelet coefficients of the density f . This condition is not restrictive as N can be chosen sufficiently large in practice.

The estimators f n are the same than in the iid case given in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], except for the highest resolution level j 1 which is smaller here. This restriction is needed in the weak dependence context due to the Bernstein's type inequality (3.2) which is not as sharp as the one in the iid case. But this restriction does not perturb the rate of convergence which is the same as the one obtained in the iid case by [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF].

The constant K > 0 plays a key role in the asymptotic behavior of f n . From (3.2), we infer that the constant K depends on the parameters a, b, C 0 , M 2 through C 1 , C 2 , C 3 in an intricate way. Then how sufficiently large the constant K must be deeply depends on the dependence structure of observations (X 1 , . . . , X n ). Then contrarily to the iid case, we are not able to develop direct procedures based on the observations (X 1 , . . . , X n ) that chose a convenient parameter K like in [START_REF] Juditsky | On minimax density estimation on R[END_REF]. In Section 5, we propose cross validation procedures to determine the threshold levels λ j when the weak dependence properties of the process (X t ) t∈Z are unknown.

Examples

In this Section, we give examples of models that satisfy Assumption (D) using φ and λ-weak dependence coefficients introduced respectively in ( 6) and [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF]. For each of them, we proceed in two steps: Firstly we give sufficient conditions on these coefficients for ensuring Assumption (D) in Propositions 4.1 and 4.2 and secondly we give in Subsections 4.2 and 4.4 examples of models satisfying such conditions.

A Lipschitz function h : R u → R for some u ∈ N * is a function such that Lip (h) < ∞ with Lip (h) = sup (a 1 ,...,au) =(b 1 ,...,bu) |h(a 1 , . . . , a u ) -h(b 1 , . . . , b u )| |a 1 -b 1 | + • • • + |a u -b u | .
As Lipschitz functions play an important role in weak dependence contexts we restrict ourselves to the cases where N > 4. This assumption on the regularity of the wavelet functions implies that φ and ψ can be chosen as Lipschitz functions, as established in [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. Note also that f ∞ < ∞ as f ∈ B s π,r (M 1 ), see equation [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF] in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF]. For convenience, we denote with no distinction ψ j,k (x) -β j,k and φ j,k (x) -α j,k as δ j,k (x) for any integers j ≥ 0 and k.

Weak dependence coefficients is to generalize mixing ones. Let us recall that α-mixing coefficients can be defined in a similar way by two equations:

α(r) = sup 1 ≤ ℓ i + r ≤ j1 ≤ • • • ≤ j ℓ g ∞ ≤ 1 E|E (g(X i 1 , . . . , X iu )|σ({X j , j ≤ i})) -E (g(X i 1 , . . . , X iu )) |, α(r) = sup (u, v) ∈ N * × N * i1 ≤ • • • ≤ iu ≤ iu + r ≤ iu+1 ≤ • • • ≤ iu+v f ∞, g ∞ ≤ 1 |Cov(f (X i 1 , . . . , X iu ), g(X i u+1 , . . . , X i u+v ))|.
These coefficients measure the dependence between the past and the future values of the process (X t ) t∈Z as the gap r between past and future goes to infinity. As these coefficients are often too restrictive, the authors of (7) release them by considering a supremum taken on functions with bounded variations or on Lipschitz bounded functions rather than uniformly bounded functions. These different choices of functions sets lead to different coefficients of weak dependence, namely φ and λ respectively. We give hereafter the precise definition of φ and λ-weak dependence coefficients.

4.1. φ-weak dependence. Bounded variations functions are defined as follows: let BV and BV 1 denote the sets of functions g supported on [-A, A] satisfying respectively g BV < +∞ and g BV ≤ 1 where

g BV = |g(-A)| + sup n∈N sup a 0 =-A<a 1 <•••<an=A n i=1 |g(a i ) -g(a i-1 )|.
Let (Ω, A, P) be a probability space, M a σ-algebra of A and let X = (X 1 , . . . , X v ), for v ≥ 1, be a collection of real valued random variables X i defined on A. We define the coefficient φ as it was introduced in ( 7) by the equation:

φ(M, X 1 , . . . , X v ) = sup g 1 ,...,gv∈BV 1 v i=1 g i (x i )P X|M (dx) - v i=1 g i (x i )P X (dx) ∞ ,
where dx = (dx 1 , . . . , dx v ). The coefficients φ(r) are now defined by the equation

φ(r) = max 1≤ℓ 1 ℓ sup i+r≤j 1 ≤•••≤j ℓ φ(σ({X j ; j ≤ i}), X j 1 , . . . , X j ℓ ) .
These coefficients are multivariate extensions of φ 1 (r) defined in [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF], see also [START_REF] Maume-Deschamps | Exponential inequalities and functional estimations for weak dependent data ; applications to dynamical systems[END_REF]. Instead of mixing coefficients, they efficiently treat the dependence structure of dynamical systems and associated Markov chains. A process (X t ) t∈Z is said to be φ-weakly dependent if the series φ(r) goes to 0 when r goes to infinity, i.e. when the gap between observations from the future and observations from the past goes to infinity. Introducing φ-weakly dependent processes is useful in the present framework due to the following links with the Assumption (D):

Proposition 4.1. Assume that (X t ) t∈Z is a process such that there exist a, b, c > 0 satisfying

φ(r) ≤ c v exp(-ar b ), (4.1)
then Assumption (D) holds with

M 2 = 2( δ ∞ + ALip δ) and C 0 = 4c( δ ∞ + ALip δ) f ∞ δ 1 .
The proof of this Proposition is given in Subsection 6.3.

4.2.

Examples of φ-weakly dependent processes. Following the work of ( 7), we give a general class of models where (4.1) is satisfied.

Lemma 4.1. Assume that (X t ) t∈Z is a process satisfying the Markov property, taking values in [0, 1] and such that there exist constants a, b, c > 0 satisfying, for any functions g, k with g ∈ BV 1 and E|k(X 0 )| < ∞, and for all r ≥ 0,

|Cov(k(X 0 ), g(X r ))| ≤ E|k(X 0 )| exp(ar -b ), (4.2) E(g(X r |X 0 = •) BV ≤ c, (4.3)
then φ v (r) ≤ c v exp(-ar b ) and the conclusions of Proposition 4.1 follow.

The proof of this Lemma is given in Subsection 6.3. Various examples of processes satisfying conditions of Lemma 4.1 are given in [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF]. We recall here the case of Markov chains obtained by time reversing expanding maps, as they are extensions of Andrew's example (1.1).

Let us define stationary Markov chains (X t ) t∈Z associated with dynamical systems through a reversion of the time as non degenerate stationary solutions of the recurrent equation

X t = T i (X t-i ), ∀t ∈ Z, i ∈ N (4.4)
where

T : [0, 1] → [0, 1] is a deterministic function.
Remark 1. For such Markov chains, mixing coefficients are useless. Future values write simply as functions of past values via (4.4) and then it is easy to check that α(r) are constant from their definitions. Thus α(r) does not tend to 0 and α-mixing coefficients do not evaluate the dependence of such processes.

A Markov chain (X t ) t∈Z is associated with an expanding map through a reversion of the time if T satisfies • (Regularity) The function T is differentiable, with a continuous derivate T ′ and there exists a grid 0 =

a 0 ≤ a 1 • • • ≤ a k = 1 such that |T ′ (x)| > 0 on ]a i-1 , a i [ for each i = 1, . . . , k. • (Expansivity)
For any integer i, let I i be the set on which the first derivate of T i , (T i ) ′ , is defined. There exists a > 0 and s > 1 such that inf

x∈I i {|(T i ) ′ (x)|} > as i . • (Topological mixing) For any nonempty open sets U , V , there exists i 0 ≥ 1 such that T -i (U ) ∩ V = ∅ for all i ≥ i 0 .
Under these three conditions a non degenerate stationary solution (X t ) t∈N to (4.4) exists and has remarkable properties, see [START_REF] Viana | Stochastic dynamics of deterministic systems[END_REF] for a nice survey. For instance, the process (X t ) t∈Z satisfies the conditions of Proposition 4.1 for some a, c > 0 and b = 1, see [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF]. Moreover, the marginal distribution is absolutely regular and its distribution belongs to BV . Noticing that B 1 1,1 ⊂ BV ⊂ B 1 1,∞ , see e.g. ( 8), Theorem 3.1 provides adaptive estimators of the marginal density of (X t ) t∈Z in that context and extends Theorem 2.2 of (25) where rates of the MISE for non adaptive estimators are given in such context. 4.3. λ-weak dependence. The stationary process (X t ) t∈Z is λ-weakly dependent, as defined in [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF], if there exists a sequence of non-negative real numbers λ(r) satisfying λ(r) → 0 as r → ∞ and such that:

Cov h (X i 1 , . . . X iu ) , k X i u+1 , . . . , X i u+v ≤ (u k ∞ Lip(h) + v h ∞ Lip(k) + uvLip (h)Lip (k)) λ(r) for all p-tuples, (i 1 , . . . , i p ) with i 1 ≤ • • • ≤ i u ≤ i u + r ≤ i u+1 ≤ • • • ≤ i p , and for all h ∈ Λ u and h ∈ Λ v where Λ u = {h : R u → R, Lip (h) < ∞, h ∞ < ∞} , for any u ≤ 1.
The λ-weak dependence provides simple bounds of the covariance terms:

Cov δ j,k (X i 1 ) • • • δ j,k (X iu ), δ j,k (X i u+1 ) • • • δ j,k (X i u+v ) ≤ (u + v + uv) δ j,k u+v-2 ∞ (Lip δ j,k ) 2 λ(r).
The right hand side term is bounded by (2 j/2 2 δ ∞ ) u+v-2 Lip δ 2 2 3j λ(r). At this point, we do not achieve Assumption (D1) as this bound depends on j via an extra term 2 3j . An additional assumption is needed to ensure Assumption (D):

(J): The joint densities f r of (X 0 , X r ) exist and are bounded for r > 0.

Proposition 4.2. Assume that (X t ) t∈Z is a λ-weakly dependent process satisfying (J) and such that there exist a ′ , b ′ , c ′ > 0 and a ′′ , b ′′ , c ′′ > 0 with:

λ(r) ≤ c ′ exp(-a ′ r b ′ ) and f r ∞ ≤ c ′′ exp(a ′′ r b ′′ ) for all r > 0. If b ′′ < b ′ then (D) holds for some C 0 > 0 with M 2 = 2 δ ∞ , a = a ′ /4 and b = b ′ .
The proof of this Proposition is given in Subsection 6.3.

4.4.

Examples of λ-weakly dependent processes. Firstly we give in this Section two generic λ-weakly dependent models, Bernoulli shifts and random processes with infinite connections. Secondly we detail the conditions of Proposition 4.2 in three more specific models.

Let H : R Z → [0, 1] be a measurable function. A Bernoulli shift with innovations ξ t is defined as [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF], such Bernoulli shifts are λ-weakly dependent with λ(r) ≤ 2v([r/2]). If (ξ t ) t∈Z is iid, (v(r)) r>0 is a non-increasing sequence satisfying E H (ξ j , j ∈ Z) -H ξ ′ j , j ∈ Z ≤ v(r) for all r > 0, where the iid sequence (ξ ′ j ) j∈Z is such that ξ ′ j = ξ j for |j| ≤ r and ξ ′ j independent of ξ j otherwise. If the weak dependence coefficients λ ξ (r) refer to (ξ t ) t∈Z , we can compute these of (X t ) t∈Z . More precisely, the result in [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF] states that if there exists ℓ ≥ 0 such that

X t = H ((ξ t-i ) i∈Z ) , t ∈ Z. According to
|H(x) -H(y)| ≤ b s ( z ℓ ∨ 1)|x s -y s |,
for some sequence b j ≥ 0 satisfying j |j|b j < ∞ and where x, y ∈ R Z coincide except for the index s ∈ Z and x = sup

i∈Z |x i |, if E|ξ 0 | m ′ < ∞ for some m ′ ≥ ℓ + 1 then (X t ) t∈Z is λ-weakly dependent with λ(k) ≤ c inf r≤[k/2]   |j|≥r |j|b j + (2r + 1) 2 λ ξ (k -2r) m ′ -1-ℓ m ′ -1+ℓ   , for some c > 0.
Different values of b in Assumption (D2) may arise naturally when realizing the minimum of the equation above, see below for some classical examples.

Another approach is the one of random processes with infinite connections considered in [START_REF] Doukhan | A fixed point approach to model random fields[END_REF]. Let F : [0, 1] Z/{0} × R → [0, 1] be measurable. Under suitable conditions on F , the stationary solution of the equation X t = F ((X j , j = t), ξ t ), a.s., exists and is λ-weakly dependent. We refer the reader to [START_REF] Doukhan | A fixed point approach to model random fields[END_REF] for more details and we end the Section with some specific λ-weakly dependent models.

4.4.1. Infinite moving average. A Bernoulli shift is an infinite moving average process if 4.4.2. LARCH(∞) model. Let (ξ t ) t∈Z be an iid centered real valued sequence and a, a j , j ∈ N * be real numbers. LARCH(∞) models are solutions of the recurrence equation

X t = i∈Z α i ξ t-i . ( 4
X t = ξ t   a + j =0 a j X t-j   . (4.6)
The stationary solution of (4.6) satisfies the condition on λ in Proposition 4.2 with b ′ = 1/2 if there exists K, α > 0 and α < 1 such that a j ≤ Kα |j| for all j = 0. Then Assumption (D2) is ensured with b = 1/2. See [START_REF] Doukhan | A LARCH(∞) vector valued process[END_REF] for applications of this model in econometrics.

4.4.3. Affine model. Let us consider the stationary solution (X t ) t∈Z of the equation

X t = M (X t-1 , X t-2 , . . .)ξ t + f (X t-1 , X t-2 , . . .),
where M and f are both Lipschitz functions. This model contains various time series processes such as ARCH, GARCH, ARMA, ARMA-GARCH, etc. If the ξ t are iid random variables with a bounded marginal density, then (J) holds and the joint densities are uniformly bounded, as stated in the Appendix of [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF]. Moreover if the functions M and f have exponentially decreasing Lipschitz coefficients, then conditions of Proposition 4.2 hold with b ′ = 1/2, b ′′ = 0 and Assumption (D2) follows with b = 1/2.

Cross-validation procedures and simulations

The aim of this Section is to evaluate the applicability and the quality of the procedure on simulated data. Even if the estimators are adaptive with respect to the regularity of the density function, a constant appears in the threshold levels that we cannot calibrate if the dependence properties of the observations are unknown. Then we develop a cross-validation scheme in order to apply concretely the estimator on simulated data. We investigate several examples of dependence for the simulated observations that satisfy the convergence result of this paper. We also give counter-examples where the estimators fail to converge near-minimaxly. 5.1. Cross-validation procedures. According to Theorem 3.1, let us fix j 0 as the smallest integer larger than log(n)(1 + N ) -1 and define f HT CV n and f ST CV n respectively as the hard and soft-thresholding estimators associated with threshold levels (( λ j ) j 0 ≤j≤j * ). Here j * = log 2 n and (( λ j ) j 0 ≤j≤j * ) are determined by cross-validation procedures: λ j = Argmin λ CV j (λ) where cross validation criterion are respectively defined by the equations HTCV:

CV j (λ) = k∈Z 1 1 {| β j,k |≥λ}   β 2 j,k - 2 n(n -1) 1≤i =h≤n ψ j,k (X i )ψ j,k (X h )   , STCV: CV j (λ) = k∈Z 1 1 {| β j,k |≥λ}   β 2 j,k - 2 n(n -1) 1≤i =h≤n ψ j,k (X i )ψ j,k (X h ) + λ 2   .
These criterion are obtained by approximating the coefficients β j,k by βj,k and the products

β j,k β j,k ′ by h =i ψ j,k (X i )ψ j,k (X h )/(n(n -1)) in the Integrated Square Error.
The estimator j 1 is defined as the smallest integer such that CV j ( λ j ) = 0 for all j 1 ≤ j ≤ j * . Notice that cross-validation procedures may consider larger resolution levels than the estimators f n as j * is larger than j 1 given in Theorem 3.1.

Different dependent samplings satisfying (D).

To illustrate the behavior of this crossvalidation scheme, we simulate three different weak-dependence cases with the same marginal absolutely continuous distribution F . The simulations were carried out as follows:

Case 1: Independent observations are given by X i = F -1 (U i ) where the U i are simulations of independent variables, uniform on [0,1].

Case 2: A φ-weakly dependent process is obtained by the equation

X i = F -1 (G(Y i )) for i = 1, . . . , n with G(x) = 2 x(1 -x)/π and (Y i ) i=1,.
..,n given by

Y 1 = G -1 (U 1 ) and, recursively, Y i = T i-1 (Y 1 ) for 2 ≤ i ≤ n with T (x) = 4x(1 -x).
Note that for all 1 ≤ i ≤ n the Y i admits the repartition function G the invariant distribution of T . Moreover the sequence (Y 1 , . . . , Y n ) satisfies the assumptions of Proposition 4.1, see [START_REF] Prieur | Applications statistiques de suites faiblement dépendantes et de systèmes dynamiques[END_REF] for details.

Case 3: A λ-weakly dependent process resulting from the transform

X i = F -1 (G(Y i )) of variables (Y t ) t∈Z which are solution of Y t = 2(Y t-1 + Y t+1 )/5 + 5ξ t /21
where (ξ t ) t∈Z is an iid sequence of Bernoulli variables with parameter 1/2. The stationary solution of this equation admits the representation

Y t = j∈Z a j ξ t-j ,
where a j = 1/3(1/2) |j| . This solution belongs to [0, 1] and its marginal distribution G is the one of (U +U ′ +ξ 0 )/3 where U and U ′ are independent variables following U([0, 1]). For

1 ≤ i ≤ n, the solution Y i is approximated by Y (N ) i for 1 ≤ j ≤ N and j-N ≤ i ≤ n+N -j where Y (j) i
is generated according to the convergent algorithm given in [START_REF] Doukhan | A fixed point approach to model random fields[END_REF]: the initial values Y (0) i are fixed equal to 0 and, given simulated variables (ξ i ) -N ≤i≤n+N , we define recursively Y

(j) i = 2(Y (j-1) i-1 + Y (j-1) i+1 )/5 + 5ξ i /21 for 1 ≤ i ≤ N and i -N ≤ t ≤ n + N -i.
Error of approximation are negligible as decreasing exponentially fast with the parameter N that we fix N = n, see Lemma 6 of ( 12) for more details. Moreover, Proposition 2.1 of (5) ensures that there exists a, C > 0 such that λ(r) ≤ C exp(-ar) for the process (Y t ) t∈Z and consequently for (X t ) t∈Z .

Two different density functions are considered. The first one is a mixture of a sinus function and a uniform distribution, presenting a discontinuitie, and the second one is a mixture of two gaussian distributions. The calculations were carried out on MATLAB on a Unix environment. We considered n = 2 10 observations repeated M = 500 times for each of the three weak dependence cases. Once the data simulated we applied cross-validation procedures. The usual DWT algorithm proposed by [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF] and implemented in the Wavelab toolbox by Donoho and his collaborators (available on [START_REF] Wavelab | [END_REF]) only gives values of wavelet functions on an equidistant grid. As one needs to compute these values at given data points, we consider an equidistant grid of I points with the number of points I huge with respect to the number of observations n. Then we approximate the values ψ j,k (X i ) by ψ j,k ([X i I]/I) where [x] denotes the closest integer from any real number x. The wavelets used for the decomposition are Daubechies Symmlets with N = 8 zero-moments. Notice that another possible scheme is the algorithm of (28) that gives directly the values ψ j,k (X i ). But as it needs much more calculus time it has not been used here for convenience.

In Figures 1 and2 are represented the estimators f HT CV n and f ST CV n and the true density function f in different weakly dependent cases. The quality of the estimators is visually good. According to Figures 1 and2 the weak dependence properties of the simulated data do not seem to affect both procedures of estimation. Density estimators presented in Figures 1 and2 do not detect the discontinuity in the density. Actually, for any finite data set, not enough simulated values are concentrated around the discontinuity to allow estimators to detect it. In Figure 3 threshold levels are represented with respect to resolution levels. Their behaviors are similar in all cases: the threshold levels increase with respect to resolution levels. For small resolutions, both HTCV and STCV procedures are close as λ 2 j is negligible in CV j (λ). For high resolution it is also the case as λ 2 j is big enough to kill almost all β j,k . Moreover these figures tend to confirm that threshold levels do not depend on weak dependence. Finally remark that the curves do not behave in square root of j as theoretical threshold levels given in Theorem 3.1.

After looking at the threshold levels values, we give in Figure 4 the frequencies of the β j,k that are less than λ j with respect to j. As these frequencies are not discretized in two values 0 and 1, we can infer that both f HT CV n and f ST CV n are not equivalent to linear estimators. It is encouraging for both methods as the results in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] state that linear estimators are not nearminimax. One can also see in these figures that frequencies of effective thresholds are the same among the weak dependence cases. Finally, means of higher resolution levels are given in last Table 2. According to Theorem 3.1, the values of this parameter do depend on the cases of weak dependence. But no significant differences appear on simulations. Table 2. Means of j 1 on 500 simulations of n = 2 10 observations.

Mean of j 1

Comparison with kernel estimators.

As results computed in the last Subsection are systematically better for the STCV estimators than for the HTCV ones, we only present in the sequel results for the STCV estimators. Let us now consider the case of a density function that is a mixture of normal distributions. Like in [START_REF] Hardle | Wavelets Approximation and Statistical Applications[END_REF], we compare the quality of the wavelet estimator f ST CV with linear kernel estimators. The kernel used is Epanechnikov's one and we computed two choices for its width parameter: firstly we consider the width given by the rule of thumb of Matlab, (more precisely, the width is equal to (q 3 -q 1 )/(2 * 0.6745) * (4/(3 * n)) 1/5 , where q 1 and q 3 denote respectively the first and the third quartile of the empirical distribution) and secondly we consider the width obtained by cross validation on the mean integrated squared error risk.

In Figure 5 are represented the means of wavelet and kernel estimators in the different cases. Like in the last Subsection, there is no visual difference between the different cases of dependence. In this Figure, it appears that the mean of the kernel estimators fails to detect the two modes of the density when the width is choosing according to the rule of thumb. The bandwidth is overestimated in this case. The quality of the wavelet estimators f ST CV n and the kernel estimators with the width from a cross validation procedure are visually equivalent. To analyse more precisely approximations realized by f ST CV n and by kernel estimators, we represent in Figure 6 the evolution of the mean L p risk with respect to p for the three estimators in each case of dependence, i.e. E( g -f p p ) 1/p with g equals to one of the three estimators. Even if these risks are close to each others, kernel estimator with cross validation bandwidth has the smallest risk for small values p ≤ 4. Yet, approximations of this kernel estimator clearly get worse with higher values of p, while risks of f ST CV n seem relatively stable for different value of p. Concerning kernel estimator with the width parameter taken according to the rule of thumb, the L p risk is worse for small values of p but comparable with the one of wavelet estimator for higher values, even if the modes of the density are not detected.

These graphs show that an advantage of f ST CV n is that its mean L p risk seems stable for high values of p. Nevertheless, the mean of the L 2 risk is larger than the one of kernel estimators with cross-validation width and the computation time higher. One possible way to improve the quality of approximation for the cross validation procedure may be to consider different levels of thresholding at each resolution level. We do not investigate this axis of research as then the time of computation exploded.

Different dependent samplings that do not satisfy (D).

In this Subsection, we discuss the necessity of Assumption (D). For this, we study the convergence of the density estimators on some dynamical systems that do not satisfy this assumption. More precisely, we focus on Liverani-Saussol-Vaienti maps, see [START_REF] Liverani | A probabilistic approach to intermittency[END_REF], defined as the solution of X t = T i (X t-i ) with

T (x) = x(1 + 2 α ′ x α ′ ), 0 ≤ x ≤ 1/2 for some 0 < α ′ < 1 2x -1, 1/2 < x ≤ 1.
The process (X t ) t∈Z is stationary and such that the covariance terms Cov(f (X 0 ), g(X r )) are of order r 1-1/α ′ , see [START_REF] Young | Recurrence times and rates of mixing[END_REF] and refinement in [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF]. Thus the Assumption (D) is not satisfied in this case and we have the non-minimaxity of any thresholded wavelets estimators:

Proposition 5.1. Suppose that the father wavelet φ is such that φ > 0 and that the assumptions of Theorem 3.1 are satisfied. If 1 > α ′ ≥ 1/(2α + 1) with α defined by (2.1), then for the thresholded estimators of the marginal density of the Liverani-Saussol-Vaienti map of index α there exists some C > 0 such that:

n 2α E[ f n -f 2 2 ] ≥ C,
for n sufficiently large. The same result also holds for the cross validation thresholded estimator f ST CV n .

The proof of this Proposition is given in 6.

To simulate these dynamical systems, we simulate Z 0 according to the Lebesgue measure on [0, 1], then we apply recursively T to determine Z i and finally we set (X 1 , . . . , X n ) = (Z n+1 , . . . , Z 2n ). This approximation of the stationary solution does not affect the study of the convergence rates as the dynamical system Z is ergodic in mean with rate O(n 1-1/α ′ ), see Theorem 5 of [START_REF] Young | Recurrence times and rates of mixing[END_REF].

The analytic expression of the density f is unknown but it is proved to be continuous, locally Lipschitz and to behave like x -α ′ as x → 0, see [START_REF] Liverani | A probabilistic approach to intermittency[END_REF] and [START_REF] Young | Recurrence times and rates of mixing[END_REF]. In particular, f is unbounded on [0, 1]. Then we restrict our study on [0.01, 1] where f is bounded. As the true density f is unknown, we compare here the estimators f ST CV n with other estimators. In Figure 7 Visually, means of both estimators are closed to each other. To detect some difference between the estimators behavior, we decide to compute the moments of order k = 1, . . . , 20 of the estimators integrated on [0, 1]: For small values α ′ ≤ 0.02 and k ≤ 4, the moments of both estimators have similar values. But as α ′ growths, all the moments of f ST CV n explode more rapidly than the ones of kernel estimators as k increases. Previous simulations studies show no behavior difference for f ST CV n between independent cases and dependent cases satisfying (D). In dependent cases that do not satisfy (D), the behavior of f ST CV n depends on the decrease rates of covariance terms. On the contrary, the behavior of the kernel estimators with rule of thumb width is more stable when (D) is not satisfied.

1 0.01 E g k (t)

Proofs

In this Section are collected all the proofs of this paper. This proof is essentially based on Lemmas 6.1 and 6.2. The first one, Lemma 6.1, is simply the Theorem 2 of (9) that we recall here for completeness: Lemma 6.1. (Doukhan & Louhichi (1999)) Let (Z i ) 1≤i≤n be centeres variables. Let q be an even integer and n ≥ 2. Suppose that for all p = 2, . . . , q and for all 1 ≤ s 1 ≤ s p ≤ n satisfying max s i+1 -s i = s u+1 -s u = r, there exists V p,n such that:

n n-1 r=0 (r + 1) p-2 Cov(Z s 1 • • • Z su , Z s u+1 • • • Z sp ) ≤ V p,n .
Then, we have

E| n i=1 Z i | q ≤ (2q -2)! (q -1)! V q/2 2,n ∨ V q,n , (6.1)
We refer the reader to [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] for the proof of this result. We apply this result on Z i = ψ j,k (X i )β j,k , when (D) holds. We first determine the bounds V p,n as under Assumption (D) we have:

n-1 r=0 (r + 1) p-2 Cov(Z s 1 • • • Z su , Z s u+1 • • • Z sp ) ≤ n-1 r=0 (r + 1) p-2 ρ(r)p 2 (2 j/2 M 2 ) p-2 .
Here we need the following analytic Lemma to bound the quantity ∞ r=0 (r + 1) p ρ(r): Lemma 6.2. If (D2) is satisfied, i.e. if ρ(r) ≤ C 0 e -ar b , then for all integer p we have

∞ r=0 (r + 1) p ρ(r) ≤ C 1 C p 2 (p!) 1/b , (6.2)
with some constants C 1 and C 2 that are depending on a and b.

Applying this result, whose proof is given at the end of this subsection, we directly obtain the new bound:

n-1 r=0 (r + 1) p-2 Cov(Z s 1 • • • ≤ C 1 p 2 (2 j/2 M 2 C 2 ) p-2 ((p -2)!) 1/b We set V p,n = C 1 p 2 (2 j/2 M 2 C 2 ) p-2 ((p -2)!) 1/b n
, and, applying Lemma 6.1, we obtain that:

E| n i=1 (Z i -EZ 0 )| q ≤ (2q -2)! (q -1)! (C 1 n) q/2 ∨ C 1 (2 j/2 M 2 C 2 ) q-2 ((q -2)!) 1/b .
Dividing by n -q and noticing that 2 j/2 ≤ n for 0 ≤ j ≤ log n, we derive that:

E| β j,k -β j,k )| q ≤ n q/2 (2q -2)! (q -1)! (C 1 ) q/2 ∨ C 1 (M 2 C 2 ) q-2 ((q -2)!) 1/b ,
which corresponds to the inequality (3.1). In particular, for p = 2 we have

E| n i=1 (Z i -EZ 0 )| 2 ≤ 4C 1 n.
Now the inequality (3.2) of Lemma 3.1 is a direct application of Theorem 1 in [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF] with

S n = n i=1 (Z i -EZ 0 ), t = nλ, ν = 0, µ = 1/b, A n = 4C 1 n and B n = 2M 2 C 2 2 (2+b)/b 2 j/2
. We refer the reader to [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF] for the definition of the parameters t, ν, µ, A n and B n .

Proof of Lemma 6.2. Define g(x) = (1 + x)e -ax b for all x ≤ 0. Studying its derivative, we can easily see that it exists x a,b such that the function g decreases on [x a,b , +∞). If we denote k ≥ 1 the smallest integer greater than x a,b , we can infer from (D2) that

∞ r=0 (r + 1) p ρ(r) ≤ C 0 k-1 r=0 (r + 1) p ρ(r) + ∞ k-1 (x + 1) p exp(-ax b )dx ≤ C 0 C a,b + ∞ 0 (x + 1) p exp(-ax b )dx .
With a convex inequality on x → x p , we achieve the bound

∞ r=0 (r + 1) p ρ(r) ≤ C 0 C a,b + 2 p-1 ∞ 0 x p exp(-ax b )dx + ∞ 0 exp(-ax b )dx . Then, writing u = ax b , ∞ r=0 (r + 1) p ρ(r) ≤ C 0 C a,b + 2 p-1 b -1 a -(p+1)/b Γ p + 1 b + a -1/b Γ 1 b ,
with Γ defined by Γ(x) = ∞ 0 u x-1 exp(-u)du for all x > 0. Let ⌈x⌉ denotes the largest integer smaller than x > 0 and note Γ = sup x∈]0,1] Γ(x). Using the inequalities Γ((p + 1)/b) ≤ Γ⌈(p + 1)/b⌉! ≤ Γe p/b (p !) 1/b and the fact that this last bound is also available for Γ(1/b), we have:

∞ r=0 (r + 1) p ρ(r) ≤ C 0 C a,b + 2 p-1 b -1 a -(p+1)/b + a -1/b Γe p/b (p !) 1/b ≤ C 0 C a,b + 2 -1 b -1 a -1/b Γ 2e 1/b (a -1/b ∨ 1) p (p !) 1/b .
The Lemma comes immediately by choosing the appropriated constants. 6.2. Proof of Theorem 3.1. The proof of Theorem 3.1 is very similar to the one in the iid case given in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF]. In the sequel, C denotes a positive real number that does not depend on n, j nor k. Its value may vary from an equation to another. Let us fix 1 ≤ p < ∞ and consider only the cases where 1 ≤ π ≤ p. The cases where π > p follow from the case π = p applying Jensen's inequality on the error term E f n -f p p . Theorem 3.1 provides the convergence rate for f belonging to the Besov Ball B s π,r (M 1 ). In particular f ∈ L 2 and it can be written as:

f = 2 j 0 -1 k=0 α j 0 ,k φ j 0 ,k E j 0 f + ∞ j=j 0 2 j -1 k=0 β j,k ψ j,k D j 0 f
. We decompose the estimators f n of f in the same way:

f n = 2 j 0 -1 k=0 α j 0 ,k φ j 0 ,k E j 0 f + j 1 j=j 0 2 j -1 k=0 γ λ j ( β j,k )ψ j,k D j 0 f
, where the γ λ j denotes without distinction the soft and hard-threshold function.

Thanks to Minkowski's inequality, the risk of f n is divided in two terms:

E[ f n -f p p ] ≤ 2 p-1    E[ E j0 f -E j0 f p p ] T 1 + E[ D j 0 ,j 1 f -D j 0 f p p ] T 2    .
To study the convergence rates of these terms the main tools are the following Lemmas given respectively in [START_REF] Meyer | Wavelets and Operators[END_REF] and [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF]. Here for any p ≥ 1 we denote • ℓp the ℓ p -norm defined by a p ℓp = i |a i | p for any sequence of real number (a i ) i≥0 .

Lemma 6.3. Let δ denote with no distinction φ and ψ. For any 1 ≤ p ≤ ∞, there exists c 1 , c 2 > 0 such that for all j ≥ 0 and all sequence (a k ) 0≤k≤2 j -1 we have

c 1 2 j(p/2-1) a p ℓp ≤ 2 j -1 k=0 a k δ j,k p p ≤ c 2 2 j(p/2-1) a p ℓp .
Lemma 6.4. For any 1 ≤ p < ∞ there exists a constant C > 0 such that for all 0 ≤ j -≤ j + and any triangular arrays (a j,k ) j -≤j≤j + , 0≤k≤2 j -1 we have

j + j=j - 2 j -1 k=0 a j,k δ j,k p p ≤ C                j + j=j - 2 j(p/2-1) a j p ℓp , if 1 ≤ p ≤ 2,   j + j=j - 2 jβp/(p-2)   (p-2)/2 j + j=j - 2 j(p/2-1-βp/2) a j p ℓp , if p > 2.
Here β is an arbitrary real number.

6.2.1. Bias of scale estimation T 1 . Lemma 6.3 for δ = φ and a k = α j 0 ,k -α j 0 ,k yields the existence of C > 0 such that

T 1 = E 2 j 0 -1 k=0 ( α j 0 ,k -α j 0 ,k )φ j 0 ,k p p ≤ C 2 j 0 (p/2-1) 2 j 0 -1 k=0 E| α j 0 ,k -α j 0 ,k | p .
Thanks to Lemma 3.1, the term T 1 is bounded by

T 1 ≤ C 2 j 0 p/2 n -p/2 .
Note that the choice of j 0 in Theorem 3.1 implies that the order of the bound is (2 j 0 /n) p/2 ≤ Cn -pN/(2+2N ) negligible compare with n -ps/(1+2s) thanks to the hypothesis N > 2s. Note that α

- α + = 1 + ǫ 2spπ(1 + 2(s -1/π)) , (6.3)
in order that if ǫ > 0 then α + < α -, if ǫ < 0 then α + > α -and if ǫ = 0 then α + = α -. We conclude that for all possible choices of ǫ the term T 1 is negligible.

6.2.2. Details term T 2 . The proof is based on multiple applications of Lemma 6.4 with a j,k = γ λ j ( β j,k )-β j,k and different j -and j + . Studying the expectation of the loss, according to Lemma 6.4 and from the linearity of the expectation, the key terms of the bounds are

E γ λ j ( β j,k ) -β j,k p ℓp = 2 j -1 k=0 E|γ λ j ( β j,k ) -β j,k | p .
The following Lemma gives upper bounds for the terms E|γ λ j ( β j,k ) -β j,k | p for any j, k: Lemma 6.5. Under the assumptions of Theorem 3.1, there exists a constant C > 0 such that for all j 0 ≤ j ≤ j 1 and all 0 ≤ k ≤ 2 j -1 we have

E|γ λ j ( β j,k ) -β j,k | p ≤ C(B1 ∧ B2 ∧ B3),
where B1=|β j,k | p , B2=|β j,k | π λ p-π j and B3=λ p j .

Proof. From the definition of the hard and soft-threshold functions γ λ j we have :

|γ λ j ( β j,k ) -β j,k | p ≤ 2 p-1 | β j,k -β j,k | p + |λ j | p 1 1 {| β j,k |>λ j } + |β j,k | p 1 1 {| β j,k |≤λ j } . (6.4)
The idea is to introduce the difference | β j,k -β j,k | and bound it using Lemma 3.1. More precisely, we have

1 1 {| β j,k |>λ j } ≤ 1 1 {| β j,k -β j,k |≥λ j /2} + 1 1 {|β j,k |≥λ j /2} ,
and the expectation of the first term of the bound in (6.4) is bounded by:

E| β j,k -β j,k | 2p 1/2 P(| β j,k -β j,k | ≥ λ j /2) 1/2 + |λ j | p P(| β j,k -β j,k | ≥ λ j /2) + (E| β j,k -β j,k | p + |λ j | p )1 1 {|β j,k |≥λ j /2} . Inequality (3.1) in Lemma 3.1 provides that E| β j,k -β j,k | p ≤ Cn -p/2
and consequently the expectation terms of the sum are smaller than |λ j | p . Applying (3.2) of Lemma 3.1 with λ = λ j satisfying λ j √ n = Kj, we infer the existence of a constant c > 0 such that

P(| β j,k -β j,k | ≥ λ j /2) 1/2 ≤ C2 -cKj from the inequality (2 j/2 / √ n) b ≤ (log n) -2b-3 ≤ Cj -2b-3 . Using 1 1 {| β j,k |≤λ j } ≤ 1 1 {| β j,k -β j,k |≥λ j } + 1 1 {|β j,k |≤2λ j }
, the expectation of the second term of the bound in (6.4) is lower than

|β j,k | p P(| β j,k -β j,k | ≥ λ j ) + 1 1 {|β j,k |≤2λ j } .
As above the probability term is bounded by 2 -cKj . Using all the previous bounds in inequality (6.4) Thanks to a suitable choice of K it is obvious that (λ p j + |β j,k | p )2 -cKj is smaller than (B1 ∧ B2 ∧ B3). The result of Lemma 6.5 follows.

The end of the proof of Theorem 3.1 is based on successive uses of B1, B2 and B3 depending on the resolution levels j.

Let us first consider the highest multi resolution levels j > j + where B1 is the most efficient bound given in Lemma 3.1. The integer j + will be fixed later. From Minkowski's inequality we obtain the following bound for T 2 , up to a constant:

E j + j=j 0 2 j -1 k=0 (γ λ j ( β j,k ) -β j,k )ψ j,k p p T 21 + E j + <j≤j 1 2 j -1 k=0 γ λ j ( β j,k )ψ j,k - j + <j 2 j -1 k=0 β j,k ψ j,k p p T 22 .
The term T 22 is bounded applying Lemma 6.4 with j -= j + +1, j + = ∞ and a j,k = γ λ j ( β j,k )-β j,k for j + < j ≤ j 1 and 0 ≤ k ≤ 2 j -1, a j,k = -β j,k for j > j 1 and 0 ≤ k ≤ 2 j -1. Using Lemma 6.5 with B1, we achieve that E a j p ℓp ≤ C β j p ℓp for all j > j + and we have:

T 22 ≤ C            j>j + 2 j(p/2-1) β j p ℓp , if 1 ≤ p ≤ 2,   j>j + 2 jβp/(p-2)   (p-2)/2 j>j + 2 j(p/2-1-βp/2) β j p ℓp , if p > 2.
The Sobolev inclusion B s π,r ⊂ B s ′ p,r ⊂ B s ′ p,∞ with s ′ = s -1/π + 1/p leads us to choose β = -2s ′ . Noting that j>j + 2 -j2s ′ p/(p-2) ≤ C2 -j + s ′ p2/(p-2) we obtain the inequalities:

T 22 ≤ C2 -j + s ′ p ∞ j=j + 2 j(s ′ p+p/2-1) β j p ℓp ≤ C f p s ′ ,p,∞ 2 -j + s ′ p .
We can choose j + as the largest integer such as

2 j + ≤ n log n α s ′ .
We have to check that j + ≤ j 1 for n sufficiently large, i.e. s ′ > α. When ǫ ≤ 0 we have α = s ′ /(1 + 2(s -1/π)) and α < s ′ because s > 1/π by hypothesis. When ǫ > 0, equality (6.3) implies that α + ≤ α -< s ′ and then obviously s ′ > α for all the possible values of ǫ. With this choice of j + , the rate of convergence of T 22 is then the one stated in the Theorem.

We study the convergence of T 21 using the bounds B2 and B3 given in Lemma 6.5. Let us investigate further these two cases in order to compare their efficiency. On the one hand, using B2 and noting that f s,π,∞ ≤ f s,π,r < ∞, we obtain the inequality: E γ λ j ( β j. ) -β j. On the other hand, using B3 we have directly E γ λ j ( β j. ) -β j. p ℓp ≤ C2 j λ p j . The rate of these two upper bounds are equivalent when λ 2 j = 2 -j(2s+1) . Replacing λ j by its value, it follows that B2 is more efficient for j > j -with j -the largest integer such that

2 j -≤ n log n 1 1+2s
.

We check that j -≤ j + for all n and all possible values of ǫ as α = s/(1 + 2s) > s ′ /(1 + 2s) if ǫ ≥ 0 and s ′ /α = 1 + 2s -2/π < 1 + 2s if ǫ ≤ 0.

We decompose again 21 using Minkowski's inequality. It gives the following upper bound, up to a constant: Let us choose β = 1/2 in order to obtain the inequalities

T 211 ≤ C(log n/n) -p/2 2 j -p/2 ≤ C(log n/n) p(1-1/(2s+1))/2 ≤ C(log n/n) pα if ǫ ≥ 0.
This term is negligible if ǫ < 0 using (6.3).

To conclude, it remains to bound T 212 using Lemma 6.4 and B2. We use that λ From now we have to distinguish the cases where ǫ = 0 to these where ǫ = 0.

• If ǫ = 0: Let us take β = -ǫ/p. Then if ǫ < 0, we obtain the inequalities:

T 211 ≤ C(log n/n) (p-π)/2 2 -j + ǫ ≤ C(log n/n) αǫ/s ′ +(p-π)/2 , and we conclude using the equality αǫ/s ′ + (p -π)/2 = pα. If ǫ > 0, then T 211 ≤ C(log n/n) (p-π)/2 2 -j -ǫ ≤ C(log n/n) p(1-1/(1+2s))/2 ≤ C(log n/n) pα .

• If ǫ = 0: Then p/2 = πs + π/2 and as sπ > 1 we deduce that p > 2. Moreover we notice that (p -π)/2 = αp and thus 2 j(p/2-1) E γ λ j ( β j. ) -β j p ℓp ≤ Cλ Let us denote t j = 2 j(πs+π/2-1) 2 j -1 k=0 β j π ℓπ . Notice that from the definition of the Besov norms we have j t r/π j ≤ C f r s,π,r . From Lemma 6.4 with β = 0 we achieve T 211 ≤ C(log n/n) αp j p/2-1 + j -<j≤j + t j , as p > 2.

If r ≤ π then j -<j≤j + t j ≤ C and the result of Theorem 3.1 follows. If π < r we use Holder's inequality at the powers r/π and r/(r -π): Denoting with no distinction ψ j,k (x) -β j,k and φ j,k (x) -α j,k as δ j,k (x) for any j, k, we collect here some inequalities useful in this Section: E| δ j,k (X 0 )| ≤ 2 f ∞ δ 1 2 -j/2 , E| δ j,k (X 0 )| 2 ≤ f ∞ , δ j,k ∞ ≤ 2 δ ∞ 2 j/2 , Lip δ j,k ≤ Lip δ2 3j/2 and δ j,k BV ≤ ( δ ∞ + ALip δ)2 j/2+1 for all j ≥ 1. The last assertion comes from the fact that δ j,k is a bounded Lipschitz function supported by [(-A + k)2 -j , (A + k)2 -j ].

Proof of Proposition 4.1. As for any j, k the function δ j,k has bounded variations we have

Cov δ j,k (X s 1 ) • • • δ j,k (X su ), δ j,k (X s u+1 ) • • • δ j,k (X s u+v ) ≤ vE δ j,k (X s 1 ) • • • δ j,k (X su ) δ j,k v BV φ v (r).
Noticing that δ j,k ∞ ≤ δ j,k BV , it follows

C j,k u,v (r) ≤ v δ j,k u+v-2 BV δ j,k BV E| δ j,k (X 0 )| φ v (r)
≤ (u + v + uv)(2 j/2+1 c( δ ∞ + ALip δ)) u+v-1 2 f ∞ δ 1 2 -j/2 exp(-ar b ).

Then Proposition 4.1 is proved.

Proof of Lemma 4.1. The proof is very close to the one given in [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF]. First notice that (4.3) for r = 0 implies that c ≥ 1 = sup g∈BV 1 g BV . From (4.2) we infer that φ(σ(X 0 ), X r ) ≤ exp(ar -b ) applying Lemma 4 of ( 6) on (4.2). From the Markov property we get φ(σ({X j , j ≤ 0}), X r ) = φ(σ(X 0 ), X r ). Now for any ℓ ≤ 1, for all r ≤ i 1 ≤ • • • ≤ i ℓ consider any g i j ∈ BV 1 for 1 ≤ j ≤ ℓ.

Collecting these facts and using that |S j 0 | is up to a constant equal to 2 j 0 , we obtain that n 2α E( f n -f 2 2 ) is larger than some positive constant and the result of Proposition 5.1 is proved.

. 5 )

 5 If ξ t are iid random variables satisfying E|ξ 0 | ≤ 1 then (X t ) t∈Z is λ-weakly dependent with λ(r) ≤ 4 |j|>[r/2] |a j |. If a j ≤ Kα |j| for j = 0, K > 0 and 0 < α < 1 then the condition on λ in Proposition 4.2 holds with b ′ = 1. Then Assumption (D2) is ensured with b = 1.

5. 3 ..

 3 Comparison of f HT CV n and f ST CV n The first density considered is a mixture between a sinus function and a uniform distribution.

Figure 1 .

 1 Figure 1. Examples of estimators f HT CV n obtained on 2 10 observations. The true distribution is represented in dashed lines. Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

  0

Figure 2 .

 2 Figure 2. Examples of estimators f ST CV n obtained on 2 10 observations. The true distribution is represented in dashed lines. Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

Figure 3 .

 3 Figure 3. Means of the proportions of the threshold levels obtained by cross-validation with respect to the resolution levels for hard-thresholding (left) and soft-thresholding (right). Case 1 corresponds to the solid line, Case 2 to the dashed line and Case 3 to the dotted line.

Figure 4 .

 4 Figure 4. Means of the proportions of thresholded coefficients with respect to the resolution levels for hard-thresholding (left) and soft-thresholding (right). Case 1 corresponds to the solid line, Case 2 to the dashed line and Case 3 to the dotted line.

Figure 5 .

 5 Figure 5. Means of estimators f ST CV n obtained from 2 10 observations on 500 simulations. The mean of the wavelet estimators is represented in dashed lines while the means of the kernel estimators are represented respectively in line with dots for the rule of thumb's width (kernel estimator 1) and in dots for the cross-validation width (kernel estimator 2). Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

Figure 6 .

 6 Figure 6. Evolution of the L p risk of estimators f ST CV n and kernel estimators obtained on 2 10 observations. The wavelet estimator is represented in dashed lines while the kernel estimators are represented respectively in dots line for the rule of thumb width (kernel estimator 1) and in dots for the cross-validation width (kernel estimator 2). Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

  , we plot the mean of M = 100 estimators f ST CV n and Epanechnikov kernel estimators given by Matlab's rule of thumb for 9 different values of α ′ . for α'=0.9

Figure 7 .

 7 Figure 7. Means of the estimators f ST CV n obtained on 2 10 observations and 500 simulations. Means of the kernel estimators are represented in dashed lines.

  function g is alternatively f ST CV n or kernel estimators.

9 Figure 8 . 6 . 1 .

 9861 Figure 8. Moments of the estimators f ST CV n obtained on 2 10 observations and 500 simulations. In dashed lines are represented the moments of the kernel estimators.

  leads to an upper bound for E|γ λ j ( β j,k )-β j,k | p with the expression B(p, j, k)+ (λ p j + |β j,k | p )2 -cKj where B(p, j, k) = λ p j 1 1 {|β j,k |≥λ j } + |β j,k | p 1 1 {|β j,k |≤λ j } .We investigate three ways for bounding B(p, j, k), using the fact that 1 1 {a≤b} ≤ (b/a) α for all a, b > 0 and all α ≥ 0: B1: B(p, j, k) ≤ C|β j,k | p using the indicator in the first term of the sum, B2: B(p, j, k) ≤ C|β j,k | π λ p-π j with a = λ j , b = |β j,k | and α = π in the first term and a = |β j,k |, b = λ j and α = p -π in the second term of the sum, B3: B(p, j, k) ≤ Cλ p j using the indicator in the second term of the sum.

212 . 2 2

 2122 j ( β j,k ) -β j,k )ψ j,j ( β j,k ) -β j,k )ψ j,k p p TWe control the term T 211 using B3 according to the discussion above and applying Lemma 6.4 with λ 2 j ≤ C log n/n:T 211 ≤ C(log n/n) p/jp(1-β)/2 , if p > 2.

2 j 2 T 2  2 j

 2222 ≤ C log n/n and we let appear the symbol ǫ= sπ -(p -π)/211 ≤ C(log n/n) (p-π)/-<j≤j + 2 -j(ǫ+pβ/2) , if p > 2.

≤ 6 . 3 .

 63 C(log n) 1-π/r . Proofs of results given in Sections 5 and 4. In this Section are collected the proof of Proposition 4.1, Lemma 4.1, Proposition 4.2 and Propostion 5.1.

Table 1

 1 gives approximations by Monte-Carlo method of the MISE. MISE values of the estimators have the same order whereas the weak dependent cases and fST CV

	n	is preferable in all the

Table 1 .

 1 MISE approximated by MC on 500 simulations of samples of size n = 2 10 .

Acknowledgements. We thank Karine Tribouley and Vincent Rivoirard for their helpful advices to one of the author. We also thank referees for having pointed out lapses in the first version of this paper.

Let us prove that we can restrict ourselves to the ℓ-uplets satisfying i 1 < • • • < i ℓ . On the one hand, if i j = i j ′ = r then we have E(g r (X r )g r (X r )|X i 1 = •) BV ≤ c g 2 r BV from (4.3). As from assumption g r ∞ ≤ g r BV ≤ 1 then g 2 r BV ≤ g r BV + g r BV ≤ 2 and we achieve the bound E(g r (X r )g r (X r )|X i 1 = •) BV ≤ 2c. On the other hand, if i j < i j ′ then, from the Markov property, we get the equation

We proceed in two steps. Firstly from (4.3), we infer thet

has variations bounded by c. Notice also that g i j ,i j ′ ∞ ≤ g i j ′ ∞ ≤ 1 and therefore we deduce the bound g i j g i j ,i j ′ BV ≤ g i j BV + g i j ,i j ′ BV ≤ 1+c. Secondly, using (4.3) on g i j ,i j ′ / g i j ,i j ′ BV , we infer that E(g i j (X i j )g

As c ≥ 1, this bound is larger than the one in the case i j = i j ′ . Now, by straightforward recurrences in the worst cases i 1 < • • • < i ℓ , we get that

for all x, we have almost surely the equation

From the definition of coefficients φ we get

For all 1 ≤ ℓ ≤ v this bound holds uniformly for all g i j ∈ BV 1 , 1 ≤ j ≤ ℓ, using the definition of φ v (r) we conclude that φ v (r) ≤ c v φ(σ({X j , j ≤ 0}), X r ) ≤ c v exp(-ar b ).

Proof of Proposition 4.2. We use the direct bound given in the proof of Lemma 1 of ( 23) under (J):

Noticing that γ ∧β ≤ γ 1/4 β 3/4 for any positive numbers γ and β, we combine the two bounds on the covariance terms and we infer that (D1) is satisfied with

Assumptions of Proposition 4.2 on respective decrease and increase rates of λ(r) and f r ∞ yield the existence of C 0 > 0 such that ρ(r) ≤ C 0 exp(-a ′ r b ′ /4).

Proof of Propostion 5.1. We give the proof for f n but it also holds for f ST CV n . We begin with recalling the result of Corollary 7.1 in (15): Lemma 6.6 [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF]. For any Lipschitz function δ 1 , bounded measurable function δ 2 such that δ 1 , δ 2 = 0 in a neighborhood of 0, then for any 0 < α ′ < 1, there exists some constant

We use the decomposition of f ∈ L 2 in the orthogonal basis of wavelets functions and we obtain

If we develop E( α j 0 -k -α j 0 -k ) 2 using the covariance terms and denoting δ j,k (x) = φ j,k (x) -α j,k for x ∈ [0.01, 1] and null elsewhere, it comes:

We want to apply Lemma 6.6 with δ 1 = δ 2 = δ j 0 ,k . We check easily the assumptions of this Lemma because of the definition of δ j 0 ,k , resulting from the fact that we estimate the density on [0.01, 1]. Moreover φ j 0 ,k = 2 -j 0 /2 φ with φ > 0 from assumption and then the covariance terms Cov( δ j 0 ,k (X 0 ), δ j 0 ,k (X r )) are equivalent to C2 -j 0 r 1-1/α ′ for some C > 0 as r goes to infinity.

Let n 0 be such that for all n ≥ n 0 , u r,n = n-r n 2 Cov( δ j 0 ,k (X 0 ), δ j 0 ,k (X r )) is nonnegative. For some m > m ′ > 2, we decompose the sum of covariance terms, for n sufficiently large, in four sums:

where [a] denotes the integer part of a. The first term goes to 0 with rate n. Then, by definition of n 0 , the second and the last terms are nonnegative. Concerning the sum

The minimax rate α is such that α < 1/2 and then by hypothesis α ′ ≥ 1/(2α+1) > 1/2. Consequently, when n goes to infinity, the sum

r=[n/m] u r,n is larger than a partial sum equivalent to C2 -j 0 n 1-1/α ′ for some C > 0 as n → ∞. As we assume 2α ≥ 1/α ′ -1 we obtain the existence of some C > 0 such that n 2α 2 j 0 n-1 r=1 n -r n 2 Cov( δ j 0 ,k (X 0 ), δ j 0 ,k (X r )) ≥ C, for n sufficiently large.


Philippe Weber

Christophe Simon

Dynamic evidential networks in system reliability analysis: A Dempster Shafer Approach

Nowadays, complex manufacturing processes have to be dynamically modeled to estimate their reliability. Moreover the results computed with classical methods need to be reinforced by managing the uncertainty. To address these difficulties, this paper presents a new method for modeling and analyzing the system reliability based on Dynamic Evidential Networks (DEN). This method allows modeling the influence of time and uncertainty on the failure and degradation of the system. The DEN graphical structure provides an easy way to specify the dependencies and, hence, to provide a compact representation of the system based on the Dempster Shafer theory. In addition, the DEN formalism is associated to simulation tools that enable an efficient processing for the models. A small system is used to compare the reliability estimations obtained by the proposed DEN model and those obtained by the classical Markov Chain.

I. INTRODUCTION

One of the main challenges of the Extended Entreprise is to maintain and optimize dynamically the quality of the services delivered by industrial objects along their life cycle. The goal is to design decision-making aid systems to maintain the process in operation. Nevertheless, a decision is taken without a perfect perception of all the system states. This partial perception argues in favor of using a probabilistic estimation of the system state. Nevertheless, several alternatives mathematical representations of both stochastic and epistemic uncertainties have been proposed, including evidence theory, possibility theory and fuzzy set theory [START_REF] Helton | An exploration of alternative approches to the representation of uncertainty in model predictions[END_REF], [START_REF] Klir | General information theory: aims, results end open problems[END_REF]. Evidence theory is a promising alternative allowing a fuller representation of the implications of the knowledge uncertainty than in the probabilistic representation [START_REF] Hoyland | System reliability theory: models and statistical methods[END_REF]. Works on system safety and Evidential Networks (EN) have been developed recently [START_REF] Simon | Bayesian networks inference algorithm to implement dempster shafer theory in reliability analysis[END_REF]. The authors used the exact inference algorithm proposed by Jensen [START_REF] Jensen | An Introduction to Bayesian Networks[END_REF] to compute the marginal mass distributions in the network.

The purpose of this paper is to introduce Dynamic Evidential Networks (DEN) in a way similar to the Markov Chains (MC) [START_REF] Padhraic | Belief networks, hidden markov model, and markov random fields: A unifying view[END_REF] to deals with epistemic uncertainty. The considered problems are those involving systems whose dynamics can be modeled as stochastic processes. The current system state determine the probability distribution over the next states. In the work reported here a study dedicated to the comparison between MC and DEN for system reliability estimation is proposed.

Section 2 presents the limits of MC model when epistemic uncertainty is present in the knowledge. Section 3 presents the Dempster Shafer theory and the new Dynamic Evidential Network (DEN) formalism. The proposed methodology is an original formalization of a system reliability model by DEN proposed in section 4. A simulation of a classical system is developed in section 4 to compare MC and DEN models. Finally, section 5 presents conclusion and perspectives.

II. PROBLEM STATEMENT

To take the uncertainty into account, it is possible to consider the process state as a random variable that takes its values from a finite state space corresponding to the possible process states. A MC allows modeling the dynamics of sequences taken by these states [START_REF] Boutillier | Decision theoretic planning: structural assumptions and computational leverage[END_REF]. The uncertainty taken into account by MC is the stochastic one. But, as databases of reliability are used to define the failure rates of the components [START_REF] Ccps | Offshore reliability data handbook[END_REF]- [START_REF] Hauge | Reliability Data for Safety Instrumented Systems[END_REF], and the values of failure rates are associated to several classes of components, some databases [START_REF] Ccps | Offshore reliability data handbook[END_REF], [12], [START_REF] Ccps | Guidelines for process equipement reliability data with data tables[END_REF] supply the lower and upper values, mean or median values or error factors. Besides, Kletz [START_REF] Kletz | HAZOP and HAZAN: Identifying and assessing process industry hazards[END_REF] pointed out that the parameter in reliability may change with a factor 3 to 10. Some authors [START_REF]Reliability Prediction of Electronic Equipment[END_REF], [START_REF] Villemeur | Reliability, availability, maintainability and safety assessment: methods and techniques[END_REF] recommend to introduce a correction coefficient to modify the failure rate of the databases according to the operational conditions of the component. This imprecision in the knowledge of failure rates induces an epistemic uncertainty which is not suitably managed by stochastic models like MC.

When upper and lower bounds of the failure rates (λ X , λ X ) of a component X are given, it is easy to define the bounds of the component reliability at a mission time T i :

P X , P X = 1 -exp([λ X , λ X ].T i ) (1)
Where P X is the lower probability of the hypothesis (X = {U p}) and P X the corresponding upper probability.

In the following, a method based on Dempster Shafer theory to manage the epistemic uncertainty is presented to model system reliability. It is based on Dynamic Evidential Networks.

III. EVIDENTIAL NETWORK THEORY

The Evidential Network model is based on graph theory and Dempster Shafer theory. Each node represents a variable and arcs indicate direct conditional relations between the connected nodes. The Dynamic Evidential Network (DEN) takes into account the time by defining different nodes to model variables with respect to different time slices.

A. Evidence theory

In the Dempster-Shafer theory, let us consider the set of r mutual exclusive hypothesis called the frame of discernment:

Ω = {H 1 , H 2 , . . . H r } (2)
The basic belief assignment follows the function:

m : 2 Ω → [0, 1] (3)
with

∑ A i ∈2 Ω m(A i ) = 1 and 2 Ω = {{H 1 }, . . . , {H r }, {H 1 , H 2 }, . . . , {H 1 . . . H r }, { / 0}} Each A i ∈ 2 Ω with m(A i ) > 0 is call a focal element.
The constraint m( / 0) = 0 is not necessary but this condition supposes that all hypotheses are known. Therefore Ω is assumed to be exhaustive [START_REF] Smets | The transferable belief model[END_REF]. The epistemic uncertainty is modeled by focal elements describing ambiguity between several hypotheses. For instance, A i = {H 1 , H 2 } denotes that the hypothesis H 1 or H 2 is respected but it is not known which one exactly.

B. Plausibility and belief functions

The belief mass distribution is used to compute the upper and lower bounds of probability. This interval contains the probability of a set of hypotheses (focal elements) and, is limited by two non-additive measures: the belief (Bel) and plausibility (Pls) of the focal element [START_REF] Klir | Fuzzy Sets and Fuzzy Logic[END_REF].

The measure of the belief Bel(A i ) in the focal element A i can be interpreted as the total amount of justified support given to A i . It defines the lower bound of probability that the focal element exists. It is computed by the sum of all belief masses of a subset of hypotheses B that contribute to the existence of A i with B included in A i .

Pls(A i ) = ∑ B|A i ∩B = / 0 m (B) (4)
The measure of plausibility Pls(A i ) quantifies the maximum amount of specific support that could be given to A i , if justified by additional information. This upper limit is computed by the sum of all the belief masses of the subsets of hypotheses B such that the intersection between B and A i is not empty.

Bel(A i ) = ∑ B|B⊆A i m (B) (5)
As Dempster Shafer Theory defines capacities of order ∞ and it verifies the assumptions of regular monotone measures (see [START_REF] Klir | General information theory: aims, results end open problems[END_REF]), the following property is given:

Bel(A i ) ≤ P(A i ) ≤ Pls(A i ) (6)
The property eq.6 is well known, Shafer has already defined it in his works in 1976 [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF].

C. The Evidential Network

Evidential Networks are directed acyclic graphs used to represent uncertain knowledge in Artificial Intelligence [START_REF] Simon | Bayesian networks inference algorithm to implement dempster shafer theory in reliability analysis[END_REF]. A EN is defined as a couple: G = ((N, A), M), where (N, A) represents the graph; N is a set of nodes; A is a set of arcs; M represents the set of conditional mass distributions that quantify dependencies between the variables.

A discrete random variable is represented by a node X ∈ N with a finite number of focal elements. The focal set of X is defined by

S X = {A X 1 , A X 2 . . . A X i . . .} with A X i ∈ 2 Ω * and Ω * = {H X 1 , ..., H X r }.
The vector m(X) denotes a belief mass distribution over the focal set S X , where m(A X i ) is the belief mass assignment to the focal element A X i . In the graph depicted in figure 1, the nodes X and Y are linked by an arc. If (X,Y ) ∈ A and (Y, X) / ∈ A then X is considered as a parent of Y . The parent set of a node Y is defined as pa(Y ) = X.

M (Y |pa(Y ) ) =     m A Y 1 A X 1 • • • m A Y P A X 1 . . . . . . m A Y 1 A X Q • • • m A Y P A X Q     (7)
Concerning the root nodes, i.e. the node without parents, the CMT contains only one row describing the a priori mass distribution of each focal element.

Various inference algorithms can be used to compute marginal mass distributions in the Evidential Network. The most classical one relies on the use of a junction tree. This algorithm is used in Bayesian Networks and more explications can be found in [5, pp.76]. Inference in EN allows taking into account any focal element of a variable observation (an event or knowledge) for the updating of the mass distribution of each variable.

Inference computes node mass distributions knowing the focal set of one or several variables. Without any event observation, the computation is based on the a priori mass distribution of the root nodes. When observations are made, the knowledge is integrated in the network and the mass distributions over the variables focal elements are updated.

Knowledge is formalised as evidence. A hard evidence of the random variable X indicates that the mass assigned to its focal elements is concentred only to one focal element of

S X : {A X 1 , ..., A X Q }.
For instance m(A X 1 ) = 1 and m(A X q =1 ) = 0. Nevertheless, when this knowledge is uncertain, soft evidence can be used (see [START_REF] Valtorta | Soft evidential update for probabilistic multiagent systems[END_REF]). A soft evidence for a node X is defined as any evidence that enables to update the prior mass distribution values for the focal elements of X. For example, the belief mass is assigned to the focal element A X 1 and A X Q with the same mass and not in the other focal elements: m(A X 1 ) = 0.5, m(A X Q ) = 0.5 and m(A X q ={1,Q} ) = 0. To compute belief and plausibility measures, we use equations 5 and 4. As these measures are non additives, they cannot be computed in one node if the tool used verifies the additivity axioms. Simon and Weber [START_REF] Simon | Bayesian networks inference algorithm to implement dempster shafer theory in reliability analysis[END_REF] have proposed to compute these measures on a variable in the network with two separated nodes as shown in figure 2. The conditional mass table of a belief node follows the table I where we introduce the doubt modality as the contrary of believe.

I CMT OF Bel(A X j ) A X j A X i ∈ 2 Ω X Believe Doubt . . . . . . . . . A X i 1 if A X j ⊆ A X i ; 0 else 0 if A X j ⊆ A X i ; 1 else . . . . . . . . .
On the same way the node that computes the plausibility measure contains the conditional mass table given in table II where disbelief is introduced as the contrary of plausibility.

TABLE II CMT OF Pls(H

X i ) A X j A X i ∈ 2 Ω X Plausibility Disbelie f . . . . . . . . . A X i 1 if A X j ∩ A X i = / 0 0 else 0 if A X j ∩ A X i = / 0 1 else . . . . . . . . .
Besides, we can note that these tables can be used to compute plausibility and belief measures on each state of a component or node of the network in order to give information on probability boxes that bound the real value of the probability for components or subsystems (cf. eq.6).

D. Dynamic Evidential Network

Dynamic Evidential Network is an EN including a temporal dimension. This new dimension is managed by time-indexed variables. The variable is represented at time k by a node X k ∈ N with a finite number of focal elements S X k : {A

X k 1 , ..., A X k Q }. m(A X k 1 )
denotes the belief mass assigned to the focal element A X 1 at time step k. Several time stages are represented by several sets of nodes N 0 , . . . , N k . N k includes all the random variables relative to the time slice k.

An arc linking two variables belonging to different time slices represents a temporal variation of the belief mass and models the dependence between these variables. Then, DEN allows to model random variables and their impacts on the future distribution of other variables. Defining these impacts as transition-belief masses between the focal elements of the variable at time step k and those at time step k + 1 leads to the definition of CMT relative to inter-time slices, as it is defined in eq.7.

M (X k+1 |X k ) =      m A X k+1 1 A X k 1 • • • m A X k+1 Q A X k 1 . . . . . . m A X k+1 1 A X k Q • • • m A X k+1 Q A X k Q      (8
) With this model, the future slice (k + 1) is conditionally independent of the past given the present (k), which means that the CMT M(X k+1 |pa (X k+1 )) respects properties similar to the Markov properties. from an observed situation at time step k = 0, the belief mass distribution m(X k ) over n i focal sets is computed by the DEN inference. To compute m(X k+T ), the proposed solution keeps a compact network form. This solution is based on iterative inferences. The notion of time is introduced through inference. Indeed, it is possible to compute the belief mass distribution of any variable X i at time step k + 1 based on the belief mass distribution corresponding to time step k. The belief mass distributions at time step k + 2 . . . are computed using successive inferences. Then, a network with only two time slices is defined. The first slice contains the nodes corresponding to the current time step (k), the second one those of the following time step (k + 1). Observations, introduced as hard evidence or belief mass distributions, are only realized in the current time slice (k). The time increment is carried out by setting the computed marginal belief masses of the node at time step k + 1 as observations for its corresponding node in the previous time slice.

Moreover, the CMT is equivalent to the Markovian model of the variable X if pa(X k+1 ) = X k and S X k = S X k+1 and without mass assignment to epistemic focal elements.

IV. DEN TO MODEL RELIABILITY

A. Evidential Network to model system reliability

As mentioned previously, the probist reliability [START_REF] Cai | System failure engineering and fuzzy methodology: An introductory overview[END_REF] of a system with low complexity may be modeled by a variable X with two hypotheses. The corresponding frame of discernment is then described by:

Ω = {U p, Down} (9)
The set of focal elements is defined by 2 Ω = {{U p}, {Down}, {U p, Down}, { / 0}}. In probist reliability, the constraint m( / 0) = 0 is assumed, thus the possibility to allocate a quantity to the focal set {U p, Down} softens the probability framework. The role of this focal element is to characterize the ignorance on the real probability distribution over the system states without commitment. It means that the system can be in the state {U p} or {Down}. Of course, it should not be considered that the system can be in the two states simultaneously. This ignorance is an epistemic uncertainty and is described by the epistemic state {U p, Down}.

When P X is the lower probability of the hypothesis {U p} and P X the corresponding upper probability, the upper and lower bounds of the reliability (P X , P X ) are easily translated in the EN to define the a priori belief mass assignment of X: m({U p}) = P X m({Down}) = 1 -P X m({U p, Down}) = P X -P X [START_REF] Goble | Safety Instrumented Systems Verification-Practical Probabilistic Calculations[END_REF] Then applied the eq.4 and 5 Pls({U p}) = P X and Bel({U p}) = P X .

B. Dynamic Evidential Network to model entities Markovian behavior

The reliability is modeled as a DEN made of two nodes as presented in figure 3. A model of component X reliability with uncertainty in failure rate is easily translated into a DEN model. Two nodes are defined to model the random variable at time slices (k) and (k + 1) : X k and X k+1 with the focal sets {{U p}, {Down}, {U p, Down}}.

The upper and lower bounds of the failure rates (λ X , λ X ) of the component X are used to define the Conditional belief Mass Table : M(X k+1 |X k ). The parameters of M(X k+1 |X k ) are defined from the failure rates (λ X , λ X ) (eq.11).

Explanations: 1) If the component is up then it may fall down but there is uncertainty in the failure probability. To determine the belief mass that the component is down we propose to model the uncertainty using the epistemic hypothesis {U p, Down}. Then the transition belief mass m({Down} k+1 {U p} k ) = λ X corresponds to the mini- mum of belief that the component fall down (it is sure that the component is down after the transition time). The transition mass m({U p, Down} k+1 {U p} k ) = λ X -λ X corresponds to the uncertainty of the com- ponent hypothesis after the transition time. There is an ambiguity between the hypotheses {U p} and {Down}. Therefore m({U p} k+1 {U p} k ) = 1 -λ X corresponds to the belief mass transition that it is sure that there is no failure. This mass is computed by:

m({U p} k+1 {U p} k ) = 1 -(λ X + λ X -λ X ) = 1 -λ X (12)
2) When the component is down, it is impossible that the component becomes up because no reparation is considered in reliability analysis. Therefore, m{U p} = 0, m({U p} k+1 {Down} k ) = 0 and m({Down} k+1 {Down} k ) = 1.

C. EN to model independent failure of components

A Fault Tree (FT) allows describing the logic of the propagation of failures through the system. This method allows to model the reliability of the system assuming the hypothesis of independence between the events (failures) affecting the entities. The paper [START_REF] Simon | Bayesian networks inference algorithm to implement dempster shafer theory in reliability analysis[END_REF] shows the equivalence between FT and EN in the case of uncertainty. The CMTs are used to define the OR/AND gates.

V. APPLICATION

The proposed method is applied to a classical example of reliability analysis. This example allows comparing easily the proposed method based on DEN with those using MC.

Figure 4 describes the system. Three valves V i are used to distribute a fluid. Each valve has one failure modes. The upper and lower bounds of the failure rates are given below:

λ 1 = 2.10 -3 λ 2 = 3.10 -3 λ 3 = 4.10 -3 λ 1 = 1.10 -3 λ 2 = 2.10 -3 λ 3 = 3.10 -3
An equivalent model of this MC is realized by means of the DEN depicted in figure 6 including the uncertainty on failure rates.

Following equation 11 for temporal nodes, the temporal conditional mass table of each valves is computed. Figure 7 M shows the CMT for the temporal node between V k 1 and V k+1 1 .

(X k+1 |X k ) =      m {U p} k+1 {U p} k m {Down} k+1 {U p} k m {U p, Down} k+1 {U p} k m {U p} k+1 {Down} k m {Down} k+1 {Down} k m {U p, Down} k+1 {Down} k m {U p} k+1 {U p, Down} k m {Down} k+1 {U p, Down} k m {U p, Down} k+1 {U p, Down} k      =   1 -λ X λ X λ X -λ X 0 1 0 0 λ X 1 -λ X   (11)
The propagation through the Dynamic Evidential Network model allows taking into account the dependency between the failure modes for the computation of the system reliability R(k). Thus, the node R(k) computes the reliability of the system from the mass distributions over the states of valves according to the conditional mass table given in figure 8.

Inferences are realized thanks to BayesiaLab c software that uses an iterative procedure (http://www.bayesia.com). BayesiaLab c is used to simulate the DEN modeling R(k) of the system over 1200 time steps depicted in figure 9. The results of MC are also added to this figure for comparison.

The results obtained by means of DEN are close to those obtained with two different simulations of the MC model, the first simulation with lower bound of the failure rates and the second with the upper bound of the failure rates. They are in fact more precise in DEN. Indeed, the differences are due to the approximation made assuming in the Markov model that the simultaneous failures cannot occurred, this hypothesis is not assumed in the DEN model.

VI. CONCLUSION AND FURTHER WORK

In this paper, we have presented Evidential Networks based on the Dempster Shafer theory to model the reliability of systems. A dynamic approach is derived to model the temporal evolution of complex systems with uncertainty. The correspondence between Markov Chain and DEN is presented and applied to the system reliability estimation.

The proposed method seems to be a good solution to model the reliability of complex systems. Indeed, the number of states needed to model a complex system with MC increases exponentially (a state for each combination of elementary states). As the DEN representation is based on the modeling of process entities, the obtained model is more compact and readable than MC. This paper shows that DEN represents a very powerful tool for decision-making aid in maintenance. This modeling method is very interesting when human are taking into account in the model as human actions introduce uncertainty.

Fig. 1 .

 1 Fig. 1. A basic Evidential Network. The set M represents the Conditional Mass distribution Tables (CMT). Then, each node has an associated CMT. For instance, in figure 1, the nodes X and Y are defined over the focal sets S X : {A X 1 , ..., A X Q } and S Y : {A Y 1 , ..., A Y P }. Then, the CMT of Y is defined by the conditional belief mass table m(Y |X ) over each Y focal elements knowing the focal elements of its parent X. This CMT is defined as a matrix:

Fig. 2 .

 2 Fig. 2. Computing nodes of Bel and Pls measures

Fig. 3 .

 3 Fig. 3. A DEN for the random variable X.

3 )

 3 In the epistemic uncertainty, the component is up or down. Let us consider that the component is up. There exists a transition mass modeling the failure of the component after the transition time and leading to the hypothesis {Down}. This belief transition mass allocated to {Down} corresponds to the minimum of belief that the component fail down i.e. λ X . If the component is down then the component stay down. This is the reason why m({Down} k+1 {U p, Down} k ) = λ X and m({U p} k+1 {U p, Down} k ) = 0.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Valve system.

Fig. 6 .Fig. 7 .

 67 Fig. 6. DEN model.

Fig. 8 .

 8 Fig. 8. CMT of R(k).

Fig. 9 .

 9 Fig. 9. DEN vs MC simulations.

TABLE