Véronique Maume-Deschamps
email: veronique.maume@univ-lyon1.fr

Didier Rullière
email: didier.rulliere@univ-lyon1.fr

Antoine Usseglio-Carleve

D Rulli Ère

Spatial Expectile Predictions for Elliptical Random Fields

Keywords: Elliptical distribution, Expectile regression, Extremal expectile, Spatial prediction, Kriging. AMS Classification : 60G15, 60G60, 62H11, 62M30

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Introduction

Kriging, introduced by [START_REF] Krige | A statistical approach to some basic mine valuation problems on the witwatersrand[END_REF], and formalized by [START_REF] Matheron | Traité de géostatistique appliquée[END_REF], aims at predicting the conditional mean of a random field (Z t ) t∈T given the values Z t1 , ..., Z t N of the field at some points t 1 , ..., t N ∈ T , where typically T ⊂ R d . When using Kriging techniques, for any x ∈ T , the conditional mean of Z x given Z t1 , ..., Z t N is approximated by a linear combination of Z t1 , ..., Z t N where the weight vector is the solution of a least square minimization problem (see Ligas and Kulczycki, 2010, for example). It seems natural to predict, in the same spirit as Kriging, other functionals by linear combinations. In a previous work (see [START_REF] Maume-Deschamps | Spatial Quantile Predictions for Elliptical Random Fields[END_REF], we focused on quantiles. In this paper, we apply the same methodology to conditional expectiles in order to get spatial expectile predictions. However, as we will see, more technical details are involved, and some numerical algorithms will be required.

In 1978, Koenker and Bassett proposed a conditional quantile estimation as an affine combination of Z t1 , ..., Z t N , called Quantile Regression (cf. [START_REF] Koenker | Regression Quantiles[END_REF]. More recently, some papers propose an Expectile Regression, using the same approach (see [START_REF] Yang | Flexible Expectile Regression in Reproducing Kernel Hilbert Space[END_REF] or [START_REF] Sobotka | Geoadditive expectile regression[END_REF], for example). The weight vector is the solution of a minimization problem, with an asymmetric loss function. In the case where α = 1 2 , it corresponds exactly to the conditional mean regression, or Kriging. Otherwise, it is more difficult to get explicit formulas. The Expectile Regression approach usually requires time consuming simulations to compute expectations. Moreover, in a non-gaussian setting, the conditional expectile may not be expressed as a linear combination of the covariates, thus the consistency of the estimation by expectile regression is not guaranteed. In this paper, we focus on elliptical random fields. Elliptical distributions, formalized by [START_REF] Cambanis | On the theory of elliptically contoured distributions[END_REF], have the advantage of being stable under affine transformations. Therefore, explicit iterative algorithms for the expectile regression may be obtained for consistent elliptical distributions (cf. [START_REF] Kano | Consistency Property of Elliptical Probability Density Functions[END_REF]).

Nevertheless, the expectile regression is generally not equal to the conditional expectile and the difference may be large, especially for extreme levels of expectile. This is why we propose a new dedicated expectile prediction that is adapted to extremal levels.

The paper is organized as follows. In Section 2, we give some definitions, properties and examples of elliptical distributions satisfying the consistency property. For these models, we give formulas for conditional expectiles in Section 3. Section 4 is devoted to expectile regression for consistent elliptical random fields: iterative algorithms are obtained. In Section 5, we propose some extremal predictions and prove asymptotic equivalences when the expectile level is close to 0 or 1. Section 6 provides a numerical study. In particular, we emphasize the fact that expectile regression is generally not appropriate, especially for high level expectiles. We illustrate this point on several examples.

Elliptical distributions

In this section, we recall some useful properties and classical examples of elliptical distributions. Most results may be found, for instance, in [START_REF] Frahm | Generalized Elliptical Distributions: Theory and Applications[END_REF]. As these results are classical, we use here almost the same formulation as in our previous article Maume- [START_REF] Maume-Deschamps | Spatial Quantile Predictions for Elliptical Random Fields[END_REF].

Definition 2.1. Let X be a d-dimensional random vector. X is elliptical if and only if there exists a unique µ ∈ R d , a semi-positive definite matrix Σ ∈ R d×d , and a function Φ : R + → R such that the characteristic function of (X -µ) is E [exp(it(X -µ))] = Φ(t Σt) . For such an elliptical random vector, we write X ∼ E d (µ, Σ, Φ)

It seems important to note that Σ is not necessarily the covariance matrix of X. More precisely, Σ is proportional to the covariance matrix K of the random vector X, when it is defined, i.e there exists a positive coefficient τ such that:

(2.1) Σ = τ K For example, for Gaussian distributions, τ = 1, i.e Σ = K. But this is not always the case for all elliptical distributions: for Student distributions with ν > 2 degrees of freedom, τ = ν-2 ν . Furthermore, K may not exist (e.g. for Cauchy distributions). In the present paper, we do only consider the case of non-degenerated distributions, i.e. we assume that the matrix Σ is invertible.

The following representation theorem is central in the theory of elliptical distributions. This result may be found in [START_REF] Cambanis | On the theory of elliptically contoured distributions[END_REF].

Theorem 2.1 [START_REF] Cambanis | On the theory of elliptically contoured distributions[END_REF]). The random vector X is elliptical, X ∼ E d (µ, Σ, Φ), if and only if d) ,

(2.2) X = µ + RΛU (
where ΛΛ T = Σ, U (d) is a d-dimensional random vector uniformly distributed on S d-1 (the unit sphere of dimension d), and R is a non-negative random variable independent of U (d) .

The representation of Theorem 2.1 is not unique (see [START_REF] Cambanis | On the theory of elliptically contoured distributions[END_REF] for details). Given µ and Σ, the elliptic random vector X is characterized by the non-negative random variable R, called the radius of X. We now recall the consistency property of an elliptical distribution. The related definitions and properties may be found in [START_REF] Kano | Consistency Property of Elliptical Probability Density Functions[END_REF].

Definition 2.2. Let X ∼ E d (µ, Σ, Φ). X is said to be consistent if Φ is dimension-free, i.e. if Φ does not depend on d. [START_REF] Kano | Consistency Property of Elliptical Probability Density Functions[END_REF] established the following relation between Definition 2.2 and the radius R.

Proposition 2.2. Let X ∼ E d (µ, Σ, Φ), and let R denote its radius. X is consistent if and only if:

(2.3) R d = χ d ,
where χ d is the square root of a χ 2 distributed random variable with d degrees of freedom, is a non-negative random variable whose law does not depend on d, and χ d , and U (d) are mutually independent.

Theorem 2.3 (Elliptical density). Let X ∈ R d be an elliptical random vector, X ∼ E d (µ, Σ, Φ), and let R be the corresponding radius of X.

(2.4)

f X (x) = c d | det(Λ)| g d (x -µ)Σ -1 (x -µ)
where

c d g d (t) = Γ( d 2 ) 2π d 2 √ t -(d-1) f R ( √ t), and f R (t) is the p.d.f of R.
The coefficient c d is called the normalization constant and the function g d is called the generator of X. Table 1 provides some examples of elliptical distributions, associated with their coefficients and generators. Most of them may easily be found in the literature: Kotz distribution is introduced in [START_REF] Nadarajah | The Kotz Type Distribution with applications[END_REF], Student in [START_REF] Nadarajah | Multivariate T-Distributions and Their Applications[END_REF], Laplace in [START_REF] Eltoft | On the multivariate Laplace distribution[END_REF] and [START_REF] Kozubowski | Multivariate generalized Laplace distribution and related random fields[END_REF], Unimodal Gaussian Mixture in [START_REF] Fraley | Model-based clustering, discriminant analysis, and density estimation[END_REF] and Slash in [START_REF] Gómez | A new family of slash-distributions with elliptical contours[END_REF].

In Table 1,

K m (x) = π 2 I-m(x)-Im(x) sin(mπ)
denotes the modified Bessel function of the second kind with order m, where

I m (x) = +∞ k=0 1 k!Γ(k + m + 1)
x 2 2k+m (see [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]), and χ 2 m (x) denotes the c.d.f of the χ 2 distribution with m degrees of freedom, evaluated at x. Remark that the Cauchy distribution corresponds to a Student distribution with ν = 1 degree of freedom.

In this paper, we focus on consistent elliptical distributions. Unfortunately, Logistic and Kotz distributions

Distribution Coefficient c d Generator g d (t) Gaussian 1 (2π) d 2 exp(-t 2 ) Student, ν > 0 Γ( d+ν 2 ) Γ( ν 2 ) 1 (νπ) d 2 1 + t ν -d+ν 2 Logistic Γ( d 2 ) (2π) d 2 +∞ 0 x d 2 -1 . e -x (1+e -x ) 2 dx -1 exp(-1 2 t) (1+exp(-1 2 t)) 2 Kotz, q, r, s > 0 sΓ( d 2 ) π d 2 Γ( 2q+d-2 2s ) r 2q+d-2 2s t q-1 exp(-rt s ) Unimodal GM 1 (2π) d 2 n k=1 π k θ d k exp - θ 2 k 2 t Laplace, λ > 0 2 λ(2π) d 2 K d 2 -1 √ 2 λ t √ λ 2 t d 2 -1 Slash 2 a 2 -1 aΓ( d+a 2 ) π d 2 χ 2 d+a (t) t d+a 2
Table 1. Some classical d-dimensional elliptical distributions with corresponding normalisation constants and generators do not have this property (except the Kotz distribution with s = q = 1, and r = 1 2 , i.e the Gaussian distribution). For consistent models, the non negative random variable is given in Table 2.

Distribution Gaussian 1 Student, ν > 0 χν √ ν Unimodal Gaussian Mixture n k=1 π k δ θ k Laplace, λ > 0 1 √ E( 1 λ ) Slash Beta (a, 1)
Table 2. Some classical consistent d-dimensional elliptical distributions with corresponding random variable

We have seen that an elliptical distribution is characterized by parameters µ, Σ, and by either the characteristic function Φ, the radius R or the generator g d . For this reason, we define the distribution of an elliptical random vector by any of these three possible characterizations, using indifferently the notations

X ∼ E d (µ, Σ, Φ), X ∼ E d (µ, Σ, R) or X ∼ E d (µ, Σ, g d ).
At last, in order to emphasis the role played by the radius and the dimension, we also use the denomination (R, d)-elliptical, as defined hereafter.

Definition 2.3. An elliptical random vector of R d with radius R is called (R, d)-elliptical.

The following proposition, from [START_REF] Hult | Multivariate extremes, aggregation and dependence in elliptical distributions[END_REF], is the basis of our study.

Proposition 2.4 (Affine transformation). Let X a consistent (R, d)-elliptical random vector with parameters µ and Σ. Then for any c ∈ R d , c T X is (R, 1)-elliptical with parameters c T µ and c T Σc. Proposition 2.4 implies that an affine transformation of a (R, d)-elliptical random vector is a (R, 1)elliptical random variable. The proposition below is a direct consequence of this result (see [START_REF] Hult | Multivariate extremes, aggregation and dependence in elliptical distributions[END_REF] for a proof).

Proposition 2.5 (Subvectors distributions). Let X = (X 1 , X 2 ) T be a consistent (R, d)-elliptical random vector with X 1 ∈ R d1 , X 2 ∈ R d2 , d 1 + d 2 = d and parameters µ and Σ. Let us write:

Σ = Σ 11 Σ 12 Σ 21 Σ 22 , µ = µ 1 µ 2 .
Then X 1 and X 2 are respectively (R, d 1 )-and (R, d 2 )-elliptical with parameters µ 1 , Σ 11 and µ 2 , Σ 22 , respectively.

Remark that a p-dimensional subvector of a (R, d)-elliptical random vector with the consistency property is (R, p)-elliptical. As a consequence, all marginals are (R, 1)-elliptical. The following proposition gives some indications concerning the conditional distributions of elliptical vectors. The proof is already given in [START_REF] Maume-Deschamps | Spatial Quantile Predictions for Elliptical Random Fields[END_REF].

Proposition 2.6 (Conditional distribution). Let X = (X 1 , X 2 ) T be a consistent (R, d)-elliptical random vector with with X 1 ∈ R d1 , X 2 ∈ R d2 , d 1 + d 2 = d and parameters µ and Σ. Let us write:

(2.5) Σ = Σ 11 Σ 12 Σ 21 Σ 22 , µ = µ 1 µ 2 .
The conditional distribution X 2 |(X 1 = x 1 ) has parameters:

(2.6)

µ 2|1 = µ 2 + Σ 21 Σ -1 11 (x 1 -µ 1 ) Σ 2|1 = Σ 22 -Σ 21 Σ -1 11 Σ 12 Furthermore, X 2 |(X 1 = x 1
) is elliptical, with radius R * given by:

(2.7) R * d = R √ 1 -B R √ BU (d) = C -1 11 (x 1 -µ 1 )
where C 11 is the Cholesky root of Σ 11 , and B ∼ Beta( d1 2 , d2 2 ).

At last, the conditional density of X 2 |(X 1 = x 1 ) is given by: q1) , and

(2.8) f X2|X1 (x 2 |x 1 ) = c 2|1 |Σ 2|1 | 1 2 g d q 1 + (x 2 -µ 2|1 ) T Σ -1 2|1 (x 2 -µ 2|1 ) with c 2|1 = c d c d 1 g d 1 (
q 1 = (x 1 -µ 1 ) T Σ -1 11 (x 1 -µ 1 ).
We have introduced the main definitions and properties of elliptical distributions. With these tools, we can define the notion of elliptical random fields. Indeed, a random field (X(t)) t∈T is R-elliptical if ∀n ∈ N, ∀t 1 , ..., t n ∈ T , the vector (X(t 1 ), ..., X(t n )) is (R, n)-elliptical. Obviously, it implies that all the k-dimensional subvectors of (X(t 1 ), ..., X(t n )) are (R, k)-elliptical. This assumption corresponds to consistent elliptical distributions properties given in Propositions 2.4, 2.5 and in [START_REF] Kano | Consistency Property of Elliptical Probability Density Functions[END_REF]. We thus focus our study on elliptical distributions with the consistency property. In the following section, we focus on conditional expectiles of elliptical distributions, applied to our problem of spatial prediction. We consider the following context: (X(t)) t∈T is an R-elliptical random field defined on some metric space T . We consider N observations at locations t 1 , ..., t N ∈ T , called (X(t 1 ), ..., X(t N )). Given X(t 1 ), ..., X(t N ), our aim is to predict, at a site t ∈ T , a functional of the distribution of X(t): the further defined expectile. Notice that the vector (X(t), X(t 1 ), ..., X(t N )) is (R, N + 1)-elliptical. Thus, we can denote X 2 = X(t) ∈ R and X 1 = (X(t 1 ), ..., X(t N )) ∈ R N and restrict ourselves to the study of the conditional distribution of the random variable X 2 given the random vector X 1 .

Theoretical Expectiles

3.1. General expression. Expectiles, introduced by [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF], may be seen as a generalization of quantiles. In this part, we introduced these quantities with the notion of elicitability (see e.g. [START_REF] Ziegel | Coherence and ellicitability[END_REF]). Indeed, let X be a random variable. The α-quantile q α of X is given by the minimization problem:

(3.1) q α (X) = arg min q∈R E (α -1)(X -q)1 {X-q<0} + α(X -q)1 {X-q>0}
Equation (3.1) easily leads to the relationship P (X ≤ q α (X)) = α for continuous distributions. The α-expectile e α (X) is defined as the solution of a similar minimization problem:

(3.2) e α (X) = arg min e∈R E (1 -α)(X -e) 2 1 {X-e<0} + α(X -e) 2 1 {X-e≥0}
The former definition assumes the existence of the first two moments of X, but other definitions can be proposed, involving only the first moment (see e.g. [START_REF] Bellini | Generalized quantiles as risk measures[END_REF]). In this part, we give a general expression of conditional expectiles in our elliptical context. Consider the respective cumulative distribution functions Φ R and Φ * R ,

(3.3) Φ R (x) = P RU (1) ≤ x , Φ R * (x) = P R * U (1) ≤ x ,
where U (1) is 1 or -1 with probability 1 2 . Φ R and Φ R * are respective cumulative distribution functions of the reduced centered (R, 1)-elliptical random variable and (R * , 1)-elliptical random variable. With this notation, we can introduce the following definition.

Definition 3.1. Let Ψ R : R * →] -∞, 0[∪]1, +∞[ be such that:

(3.4) Ψ R (x) = Φ R (x) + 1 x +∞ x yc 1 g 1 (y 2 )dy .
This function is introduced mainly to simplify future equations and relationships. From Equation (3.4), we can deduce several properties. Lemma 3.1. Ψ R satisfies the following properties:

• Ψ R (-x) = 1 -Ψ R (x), ∀x ∈ R * . • Ψ R : R * →] -∞, 0[∪]1, +∞[ is bijective and decreasing. • Ψ -1 R (1 -α) = -Ψ -1 R (α), ∀α ∈ R * .
The proof requires very classical and simple calculations, then we do not develop it in details.

We will see that the general expression of expectile is related to the function Ψ R . We introduce a last lemma before giving this expression.

Lemma 3.2. Let X ∼ E 1 (µ, σ 2 , g 1 ) be an elliptical random variable. We have the following relationship:

(3.5) E[X1 {X≥0} ] = µΦ R µ σ + σ +∞ -µ σ yc 1 g 1 (y 2 )dy
Proof. We recall that density of X is given by:

f X (x) = c1 σ g 1 (x-µ) 2 σ 2
Then: We are now able to give a general expression for the α-expectile of a univariate elliptical distribution.

E[X1 {X≥0} ] = +∞ 0 x c 1 σ g 1 (x -µ) 2 σ 2 dx = µ +∞ -µ σ c 1 g 1 y 2 dy + σ +∞ -µ σ yc 1 g 1 y 2 dy Since Φ R (x) = x -∞ c 1 g 1 y 2 dy and 1 -Φ R (x) = Φ R (-x) (
Proposition 3.3. Let X a (R, 1)-elliptical random variable, with parameters µ and σ 2 . The α-expectile e α of X is given by:

(3.6) e α (X) = µ + σΨ -1 R α 2α-1 , α = 1 2 µ , α = 1 2
Proof. We have to solve the minimization problem:

e α (X) = arg min x∈R E[S α (X -x)]
By deriving the loss function, we get:

(

1 -α)E[X -e α ] + (2α -1)E (X -e α )1 {X-eα≥0} = 0
Using Lemma 3.2, we have the equality:

E[(X -e α )1 {X-µ≥0} ] = (µ -e α )Φ R µ -e α σ + σ +∞ -µ-eα σ yc 1 g 1 (y 2 )dy
We add this term in the previous equation:

(

1 -α)(µ -e α ) + (2α -1)   (µ -e α )Φ R µ -e α σ + σ +∞ -µ-eα σ yc 1 g 1 (y 2 )dy    = 0
A level of α = 1 2 leads to the obvious solution e α = µ. Then, we consider now α = 1 2 (then e α = µ), the equation may be written as follows:

Φ R e α -µ σ + σ e α -µ +∞ eα -µ σ yc 1 g 1 (y 2 )dy = α 2α -1 Hence Ψ R eα-µ σ = α 2α-1
, and

e α = µ + σΨ -1 R α 2α-1
Let us focus now on the conditional expectiles of an elliptical vector. Be X ∈ R N +1 a consistent (R, N + 1)-elliptical vector, X 1 ∈ R N and X 2 ∈ R subvectors of X. Since X 2 |X 1 is (R * , 1)-elliptical, where R * is given in Proposition 2.6, it is not difficult to get a general expression for e α (X 2 |X 1 ).

Proposition 3.4. Let X = (X 1 , X 2 ) be a (R, N + 1)-elliptical random vector, with parameters µ and Σ. The α-expectile e α of X 2 |(X 1 = x 1 ) is given by:

(3.7) e α (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R * α 2α-1 , α = 1 2 µ 2|1 , α = 1 2
where µ 2|1 and σ 2|1 = Σ 2|1 are given in Equation (2.6), and R * in Equation (2.7).

Proof. Obvious with Proposition 3.3 and Proposition 2.6.

Our formulas involve the terms

Ψ -1 R α 2α-1 and Ψ -1 R * α 2α-1 , which are respectively the solutions of Φ R (x) + 1 x +∞ x yc 1 g 1 (y 2 )dy = α 2α -1 and Φ R * (x) + 1 x +∞ x yc * 1 g * 1 (y 2 )dy = α 2α -1
. Then, in the next subsection, we propose some algorithms to compute these terms.

3.2. Algorithms. We propose some algorithms to compute the standardized conditional expectile Ψ -1 R α 2α-1 , and thus the conditional expectile e α (X 2 |X 1 = x 1 ) of Equation (3.7). Obviously, the same algorithms will

apply for Ψ -1 R * α 2α-1 , using R * instead of R.
We consider two kinds of algorithms: a MM algorithm (see [START_REF] Hunter | A Tutorial on MM Algorithms[END_REF]) and a fixed-point algorithm. We also study the speed of convergence for these two algorithms.

Let us introduce the following definitions and properties that may be found in [START_REF] Frontini | Some variant of Newton s method with third-order convergence[END_REF]. Consider a sequence e (k) k∈N and a target value e * ∈ R. The sequence e (k) k∈N is said to converge to e * ∈ R with order p ∈ [1, ∞) if for all k, e (k) -e * ≤ k where k is a positive sequence satisfying

(3.8) ∃c > 0 : lim k→+∞ k+1 p k = c .
The coefficient c is called the asymptotic factor. In particular, one says that the convergence is linear if p = 1 and c < 1, or quadratic if p = 2. Now, consider a fixed-point algorithm e (k+1) = f e (k) , where e (k+1) , e (k) ∈ R, and f : R → R is a continuous and differentiable function, k ∈ N. The convergence of the algorithm is insured by [START_REF] Frontini | Some variant of Newton s method with third-order convergence[END_REF].

|f (e)| < 1, ∀e ∈ R. If furthermore f is p-times differentiable with ∀m < p, f (m) (e * ) = 0 and f (p) (e * ) = 0, then the convergence is of order p with an asymptotic factor c = f (p) (e * ) p! (see
3.2.1. MM algorithm. We now present an approach based on a MM algorithm, in order to compute

Ψ -1 R α 2α-1
and the associated expectile of Equation (3.7). A similar approach has been proposed in [START_REF] Yang | Flexible Expectile Regression in Reproducing Kernel Hilbert Space[END_REF], and some results thus explicitly refer to this paper.

In the following, we denote by S α the function

S α (x) = (1 -α)x 2 1 {x<0} + αx 2 1 {x≥0} .
Lemma 3.5 [START_REF] Yang | Flexible Expectile Regression in Reproducing Kernel Hilbert Space[END_REF]). The function S α is Lipschitz:

(3.9) |S α (a) -S α (b)| ≤ 2 max{1 -α, α}|a -b|, ∀a, b ∈ R
Hence the following quadratic upper bound:

(3.10) S α (a) ≤ S α (b) + S α (b)(a -b) + max{1 -α, α}(a -b) 2 , ∀a, b ∈ R
Using this result from Yang et al. ( 2015), we propose a new MM algorithm, detailed hereafter.

Proposition 3.6 (MM algorithm). The following sequence e (k) k∈N converges to

Ψ -1 R α 2α-1 : (3.11)              e (0) = 0 e (k+1) = e (k) - e (k) max{1-α,α} α -(2α -1)Φ R e (k) + + 2α -1 max{1 -α, α} +∞ e (k) yc 1 g 1 (y 2 )dy
Furthermore, the convergence is linear with an asymptotic factor

(3.12) c = 1 - 1 max{1 -α, α} α -(2α -1)Φ R Ψ -1 R α 2α -1
Proof. Let X be the (R, 1)-elliptical random variable with parameters 0 and 1. At the k th iteration, we call R (k) = X -e (k) . It follows for any e ∈ R:

X -e = R (k) -(e -e (k) )
Thanks to Lemma 3.5, we have the following upper bound:

E S α R (k) -(e -e (k) ) ≤ E S α (R (k) ) -(e -e (k) )E S α (R (k) ) + max{1 -α, α} e -e (k) 2 Let us denote Q e, e (k) = E S α (R (k) ) -(e -e (k) )E S α (R (k) ) + max{1 -α, α} e -e (k) 2 . When trying to minimize E S α R (k) -(e -e (k)
) , the principle of MM Algorithm is to minimize its upper bound, and to choose e (k+1) = arg min e∈R Q e, e (k) . We easily get:

e (k+1) = e (k) + 1 2 max{1-α,α} E S α X -e (k) It remains to calculate E S α X -e (k) . E S α X -e (k) = 2(1 -α)E X -e (k) + 2(2α -1)E X -e (k) 1 {X-e (k) ≥0}
Lemma 3.2 leads to Equation (3.11). Now, let us prove the convergence of the algorithm. It is sufficient to have |ϕ (x)| < 1, ∀x.

ϕ (x) = 1 - α -(2α -1)Φ R (x) max{1 -α, α} .
Clearly, since 0 < Φ R (x) < 1, ∀x ∈ R, 0 < ϕ (x) < 1. Hence the convergence. Furthermore, since ϕ (x) = 0, ∀x ∈ R, the convergence is linear in the sense of Equation (3.8), with asymptotic factor ϕ

Ψ -1 R α 2α-1 , hence Equation (3.12).
As expected, lim α→1 c = 1. Indeed, when α is close to 1, e α (X) is huge, and its calculation requires more iterations. On the other hand, a level of α = 1 2 gives c = 0, since the sequence in the algorithm is constant, e (k) = 0 for all k.

3.2.2. Fixed-point algorithm. It can be shown that e α (X) satisfies a fixed-point relationship. In this paragraph, we detail this relationship and the corresponding fixed-point algorithm.

Proposition 3.7 (Fixed-point algorithm). The following sequence e (k) k∈N converges to

Ψ -1 R α 2α-1 : (3.13)          e (0) = 0 e (k+1) = (1 -2α) +∞ e (k) yc 1 g 1 (y 2 )dy (2α -1)Φ R e (k) -α
Furthermore, the convergence is quadratic (in the sense of Equation (3.8) with p = 2), with an asymptotic factor

(3.14) c = (2α -1)c 1 g 1 Ψ -1 R α 2α-1 2 2 (2α -1)Φ R Ψ -1 R α 2α-1 -α
Proof. Let X the (R, 1)-elliptical random variable with parameters 0 and 1. Since µ = 0 and σ = 1, Ψ -1 R α 2α-1 = e α (X) and thus:

Ψ -1 R α 2α -1 = arg min e∈R E [S α (X -e)] Then, Ψ -1 R α 2α-1 is such that E S α X -Ψ -1 R α 2α-1 = 0, i.e: 2(1 -α)E X -Ψ -1 R α 2α -1 + 2(2α -1)E X -Ψ -1 R α 2α -1 1 {X-Ψ -1 R ( α 2α-1
)≥0} = 0 With the previous equation, and the formula of Lemma 3.2, we get the following fixed-point relation

Ψ -1 R α 2α -1 = (1 -2α) +∞ Ψ -1 R ( α 2α-1 ) yc 1 g 1 (y 2 )dy (2α -1)Φ R Ψ -1 R α 2α-1 -α .
As in the MM algorithm given above, we have a recursive algorithm e (k+1) = ϕ(e (k) ). Then, the convergence

is insured by ϕ Ψ -1 R α 2α-1 < 1. The quadratic convergence is insured if ϕ Ψ -1 R α 2α-1 = 0 and ϕ Ψ -1 R α 2α-1 = 0. We have ϕ (x) = (2α -1)c 1 g 1 (x 2 ) x(2α -1)Φ R (x) -αx + (2α -1) +∞ x yc 1 g 1 (y 2 )dy [(2α -1)Φ R (x) -α] 2 . Using Equation (3.1), we deduce ϕ Ψ -1 R α 2α-1 = 0.
With this relationship, we not only show that the algorithm converges, but this convergence is at least quadratic. We straightforward calculations, and get:

ϕ Ψ -1 R α 2α -1 = (2α -1)c 1 g 1 Ψ -1 R α 2α-1 2 (2α -1)Φ R Ψ -1 R α 2α-1 -α . If α = 1 2 , ϕ Ψ -1 R α 2α-1
= 0 and the convergence is quadratic. Furthermore, the asymptotic factor c is easily deduced, with the relationship c = ϕ (Ψ -1 R ( α 2α-1 ))

2

, hence Equation (3.14).

Considering the convergence rate, this fixed-point algorithm is more efficient than the MM algorithm. In the following, unless specified, we thus only consider this fixed-point algorithm.

Examples.

In this subsection, we give some examples of theoretical conditional expectiles calculated by our algorithms for Gaussian, Student and Gaussian Mixture distributions. The difficulty is to calculate the terms Φ R * e (k) and

+∞ e (k)
yc * 1 g * 1 (y 2 )dy. In the general case, we have no guarantee of obtaining closedform formulas for these quantities. Nevertheless, the calculation is possible in the three examples mentioned above. As the fixed-point algorithm (3.13) is asymptotically faster than the MM algorithm (3.11), we just give results for the fixed-point algorithm, but the MM algorithm works as well.

Recall that in all cases, we have the relationship

e α (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R * α 2α -1 .
Then, for all the examples, we give an algorithm to calculate Ψ -1

R * α 2α-1 .
3.3.1. Gaussian example. The gaussian case, as usual, is the most simple case. Since R * d = R, calculations are very simple, and lead to the following sequence which converges to Ψ -1

R * α 2α-1 : (3.15)    e (0) = 0 e (k+1) = (1-2α)ϕ(e (k) ) (2α-1)Φ(e (k) )-α 3.3.2. Student example.
In the Student case, the conditional radius R * is not equal to R, but calculations are still possible. Indeed, using Equations (2.8), Algorithm (3.13) becomes

e (k+1) = (1 -2α) +∞ e (k) yc 2|1 g N +1 q 1 + (x 2 -µ 2|1 ) T Σ -1 (x 2 -µ 2|1 ) dy (2α -1)Φ ν e (k) -α ,
where c 2|1 and g n are given in Proposition 2.6 and Table 1 . Hence Ψ -1

R * α 2α-1 is obtained by: (3.16)        e (0) = 0 e (k+1) = (2α-1) Γ ( ν+N +1 2 ) Γ ( ν+N 2 ) √ π √ ν+q 1 1-ν-N 1+ 1 ν ( e (k) ) 2 ( 1+ 1 ν q 1) 1-ν-N 2 (2α-1)Φ ν+N e (k) ν+N ν+q 1 -α
3.3.3. Gaussian Mixture example. We do the same kind of calculations as in the Gaussian and Student cases, and get the following algorithm.

(3.17)

       e (0) = 0 e (k+1) = (1-2α) n i=1 πiθ N -1 i exp - θ 2 i 2 q1 ϕ(θie (k) ) (2α-1) n i=1 πiθ N i exp - θ 2 i 2 q1 Φ(θie (k) )-α n i=1 πiθ N i exp - θ 2 i 2 q1
Some numerical applications and illustrations are given in Section 6.

In some other cases, it is difficult to express Φ R * e (k) or

+∞ e (k)
yc * 1 g * 1 (y 2 )dy. Obviously, one can approximate numerically these values, but this approximation may lead to a poor prediction of the expectile, especially for extreme levels of α. This is why we propose two prediction methods. The first one, introduced in the next section, is called expectile regression.

Expectile Regression

Expectile Regression, introduced by [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF], by analogy to Quantile Regression (see [START_REF] Koenker | Regression Quantiles[END_REF]), is an usual way to estimate conditional expectiles. If

X 1 ∈ R N and X 2 ∈ R, the α-expectile of X 2 |(X 1 = x 1 ) is approximated by (4.1) êα (X 2 |X 1 = x 1 ) = β * T x 1 + β * 0
, where β * and β * 0 are solutions of the following minimization problem (4.2)

(β * , β * 0 ) = arg min β∈R N ,β0∈R E S α (X 2 -β T X 1 -β 0 ) .
and where the scoring function S α (see [START_REF] Ziegel | Coherence and ellicitability[END_REF]) is

(4.3) S α (x) = (1 -α)x 2 1 {x<0} + αx 2 1 {x>0} .
Obviously, if α = 1 2 , the scoring function S 1 2 (x) is the least square loss function, and the minimization 4.2 gives the kriging vector β

* = E[X 1 X T 1 ] -E[X 1 ]E[X 1 ] T -1 (E[X 1 X 2 ] -E[X 1 ]E[X 2 ]), and β * 0 = E[X 2 ] -β * T E[X 1 ]. If α = 1
2 , the problem is more difficult to solve, and we usually need simulations and stochastic algorithms. Fortunately, in our elliptical context, we do not need it to express β * and β * 0 . The following lemma will be useful to calculate β * . Lemma 4.1. Let (X 1 , X 2 ) a consistent (R, 2)-elliptical random vector, with parameters µ = (µ 1 , µ 2 ) and

Σ = σ 2 1 ρσ 1 σ 2 ρσ 1 σ 2 σ 2 2
. We have the following relationship:

(4.4) E[X 1 X 2 1 {X2≥0} ] = µ 1 µ 2 -ρ σ 1 σ 2 µ 2 2 Φ R µ 2 σ 2 + (µ 1 σ 2 + ρσ 1 µ 2 ) +∞ - µ 2 σ 2 yc 1 g 1 (y 2 )dy + ρσ 1 σ 2 +∞ - µ 2 σ 2 y 2 c 1 g 1 (y 2 )dy
Proof. We write:

E[X 1 X 2 1 {X2≥0} ] = +∞ 0 x 2 f X2 (x 2 )   +∞ -∞ x 1 f X1,X2 (x 1 , x 2 ) f X2 (x 2 ) dx 1   dx 2
We have

+∞ -∞ x 1 f X1,X2 (x 1 , x 2 ) f X2 (x 2 ) dx 1 = E[X 1 |X 2 ].
Using Equation (2.6), we get:

E[X 1 X 2 1 {X2≥0} ] = +∞ 0 x 2 f X2 (x 2 ) µ 1 + ρ σ1 σ2 (x 2 -µ 2 ) dx 2
We recall that density of

X 2 is f X2 (x 2 ) = c1 σ2 g 1 (x2-µ2) 2 σ 2 2
. Equation (4.4) is obtained by using Lemma 3.2.

We are now able to give the expression of β * , and the result is exactly the same we have calculated in [START_REF] Maume-Deschamps | Spatial Quantile Predictions for Elliptical Random Fields[END_REF], i.e β * corresponds to the vector of simple Kriging weights. For the sake of simplicity, let ρ j be the correlation coefficient between X 1j and the random variable X 2 -β * T X 1β * 0 . Indeed, X 1j is (R, 1)-elliptical with parameters µ 1j and σ 2 1j . Furthermore, X 2 -β * T X 1 -β * 0 is (R, 1)-elliptical too, with parameters µ 2 -β * T µ 1 -β * 0 and (-β * , 1) T Σ(-β * T , 1). Hence the (R, 2)-elliptical vector X 1j , X 2 -β * T X 1 -β * 0 admits as second parameter the matrix:

σ 2 1j ρ j σ 1j (-β * , 1) T Σ(-β * T , 1) ρ j σ 1j (-β * , 1) T Σ(-β * T , 1) (-β * , 1) T Σ(-β * T , 1)
Proposition 4.2 (Explicit form of β * ). ∀α ∈ [0, 1], the optimal β * is given by:

(4.5) β * = Σ -1 11 Σ 12
Proof. We recall the minimization problem which verifies (β * , β * 0 ): arg min

β∈R N ,β0∈R (1 -α)E (X 2 -β T X 1 -β 0 ) 2 -(1 -2α)E (X 2 -β T X 1 -β 0 ) 2 1 {X2-β T X1-β0≥0}
A quick gradient approach gives the following equation system

(1 -α)E X 1 (X 2 -β * T X 1 -β * 0 ) + (1 -2α)E X 1 (X 2 -β * T X 1 -β * 0 )1 {X2-β * T X1-β * 0 ≥0} = 0 (1 -α)E X 2 -β * T X 1 -β * 0 + (1 -2α)E (X 2 -β * T X 1 -β * 0 )1 {X2-β * T X1-β * 0 ≥0} = 0 We know that X 2 -β * T X 1 -β * 0 is (R, 1)-elliptical with parameters µ = µ 2 -β T µ 1 -β 0 and Σ = Σ 22 - 2β T Σ 12 + β T Σ 11 β. If we denote σ = √
Σ, and use Lemmas 3.2 and 4.1, we get the following system, ∀j ∈ {1, ..., N }

                   (1 -α)µ + (1 -2α)µΦ R µ σ + (1 -2α)σ +∞ µ σ yc 1 g 1 (y 2 )dy = 0 (1 -2α) µ 1j µ -ρ j σ1j σ µ 2 Φ R µ σ + (1 -2α) (µ 1j σ -ρ j σ 1j σ) +∞ µ σ yc 1 g 1 (y 2 )dy +(1 -2α)ρ j σ1j σ +∞ 0 x 2 2 f X2 (x 2 )dx 2 = 0
Adding the first equation and the second one, it remains

-(1-2α)ρ j σ 1j σ µ 2 Φ R µ σ -(1-2α)ρ j σ 1j σ +∞ µ σ yc 1 g 1 (y 2 )dy+(1-2α)ρ j σ 1j σ +∞ 0 x 2 2 f X2 (x 2 )dx 2 +(1-α)ρ j σ 1j σ = 0
Obviously, ρ j = 0. We have seen in Maume-Deschamps et al. ( 2016)(Proposition 4.2), that ρ j = 0, ∀j was equivalent to β * = Σ -1 11 Σ 12 . In the following remark, we emphasis the role played by the affine constant β * 0 in the affine model of Equation (4.1). Indeed, considering a linear model without this constant would lead to highly undesirable properties.

Remark 1 (Linear models pitfalls). Consider a linear expectile regression given by:

êlinear α (X 2 |X 1 = x 1 ) = β * T x 1 ,
where β * is the solution of the following minimization problem β * = arg min β∈R N E S α (X 2 -β T X 1 ) , and where the scoring function S α is defined as previously by Equation (4.3). This corresponds to the case where β * 0 = 0. Then (i) êlinear α (X 2 |X 1 = x 1 ) does not depend on α: the linear expectile regression is obviously not suited for other levels than α = 1/2. (ii) If Σ 12 = 0 R N , which means that X 2 and X 1 are not correlated, then êlinear α (X 2 |X 1 = x 1 ) = 0, while in the uncorrelated case, we would expect to predict the conditional expectile by the unconditional one e α (X 2 ). (iii) Consider an elliptical random field (X(t)) t∈T and let X 1 = (X(t 1 ), ..., X(t n )) ∈ R N , X 2 = X(t) ∈ R, so that Σ 12 depends on t. Assume that there exists t 0 ∈ T such that lim The result of Remark 1 means that a linear model êα (X 2 |X 1 = x 1 ) = β * T x 1 will always return the same value, ∀α ∈ [0, 1]. It is a reason why we need the term β * 0 , if we hope approximate e α (X 2 |X 1 = x 1 ). To calculate it, we give the following lemma.

Lemma 4.3. X 2 -β * T X 1 is (R, 1)-elliptical with parameters µ 2 -Σ 21 Σ -1 11 µ 1 and Σ 2|1 = σ 2 2 -Σ 21 Σ -1 11 Σ 12 . Proof. X 2 -β * T X 1 ,
as affine transformation of a consistent (R, d)-elliptical random vector, is obviously (R, 1)-elliptical. Furthermore, with the expression of β * given in Proposition 4.

2, E[X 2 -β * T X 1 ] = µ 2 - Σ 21 Σ -1
11 µ 1 . The second parameter equals (-β * , 1) T Σ(-β * , 1), thus:

(-β * , 1) T Σ(-β * , 1) = σ 2 2 -2 n i=1 β * i ρ ix σ 1i σ 2 + n j=1 β * j n i=1 β * i ρ ij σ 1i σ 1j ,
or, in matrix form:

(-β * , 1) T Σ(-β * , 1) = σ 2 2 -2β * T Σ 12 + β * T Σ 11 β * Thanks to Equation (4.5), we have β * = Σ -1
11 Σ 12 . Then the following equation holds:

(-β * , 1) T Σ(-β * , 1) = σ 2 2 -2Σ 21 Σ -1 11 Σ 12 + Σ 21 Σ -1 11 Σ 11 Σ -1 11 Σ 12 = σ 2 2 -Σ 21 Σ -1 11 Σ 12
With the help of Lemma 4.3, we can give the Expectile Regression Predictor, and its distribution in the following theorem.

β * = Σ -1 11 Σ 12 β * 0 = µ 2 -Σ 21 Σ -1 11 µ 1 + σ 2|1 Ψ -1 R α 2α-1
The Expectile Regression Predictor with level α ∈ [0, 1] is given by:

(4.7) êα (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R α 2α -1 Furthermore, (4.8) êα (X 2 |X 1 ) ∼ E 1 µ 2 + σ 2|1 Ψ -1 R α 2α -1 , Σ 21 Σ -1 11 Σ 12 , g 1 Proof. β * is given in Equation (4.5). β * 0 satisfies: β * 0 = arg min β0∈R E S α (X 2 -β * T X 1 -β 0 ) Thus, β * 0 is the α-expectile of the random variable X 2 -β * T X 1 . We have seen in Lemma 4.3 that X 2 - β * T X 1 is (R, 1)-elliptical with parameters µ 2 -Σ 21 Σ -1
11 µ 1 and σ 2|1 . Then, using the expectile formula of Equation (3.6), we get:

β * 0 = µ 2 -Σ 21 Σ -1 11 µ 1 + σ 2|1 Ψ -1 R α 2α -1 We can now express our Expectile Regression Predictor of X 2 given X 1 = x 1 : êα (X 2 |X 1 = x 1 ) = β * x 1 + β * 0 = µ 2 + Σ 21 Σ -1 11 (x 1 -µ 1 ) + σ 2|1 Ψ -1 R α 2α -1
We recognize, on the left, the expression of µ 2|1 given in Equation (2.6).

Since êα (X

2 |X 1 ) = µ 2 + Σ 21 Σ -1 11 (X 1 -µ 1 ) + σ 2|1 Ψ -1 R α 2α-1 , with X 1 ∼ E N (µ 1 , Σ 11 , g N ). We get, us- ing Proposition 2.4, êα (X 2 |X 1 ) ∼ E 1 µ 2 + σ 2|1 Ψ -1 R α 2α-1 , (Σ 21 Σ -1 11 )Σ 11 (Σ -1 11 Σ 12 ), g 1
Hence the result.

We have given in Theorem 4.4 our first expectile predictor. As an illustration, we propose to calculate this predictor in several cases. 4.1. Examples. In this subsection, we apply the Expectile Regression Predictor êα (X 2 |X 1 ) given in Theorem 4.4 on several examples, in order to compare it with the theoretical conditional expectiles e α (X 2 |X 1 ). We recall the formulas:

   e α (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R * α 2α-1 êα (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R α 2α-1
Obviously, if R = R * , we do not have the same algorithms to compute e α (X 2 |X 1 ) and êα (X 2 |X 1 ). We have seen in Section 3 that we were able to compute e α (X 2 |X 1 ) in the Gaussian, Student and Gaussian Mixture cases. In the Laplace and Slash cases, we cannot compute e α (X 2 |X 1 ), but we are able to compute êα (X 2 |X 1 ). In the following, we thus apply the Expectile Regression Predictor for these last two cases.

4.1.1. Laplace example. The Laplace case is interesting, because we have no explicit algorithm to calculate Ψ -1 R * α 2α-1 , since Φ R * is unknown in this case. However, it is possible to calculate Ψ -1 R α 2α-1 . Indeed, the following algorithm converges to this value.

(4.9)

                 e (0) = 0 e (k+1) = (1-2α) exp -2 λ e (k) λ 2 +e (k) (2α-2)-(1-2α) exp -2 λ e (k)
, if e (k) > 0

e (k+1) = (1-2α) exp 2 λ e (k) λ 2 -e (k) (2α-1) exp 2 λ e (k) -2α
, if e (k) ≤ 0 4.1.2. Slash example. Consider the Slash distribution as in Table 2. As in the Laplace case, we have no explicit algorithm for Ψ -1

R * α 2α-1 , but we can approximate Ψ -1 R α 2α-1 : (4.10)                    e (0) = 0 e (k+1) = (1-2α)2 a 2 -1 a a-1 Γ ( 1+a 2 ) √ π   χ 2 1+a ( e (k) ) 2 ( e (k) ) a-1 + 2 1-a 2 Γ ( 1+a 2 ) exp -( e (k) ) 2 2   (2α-1)   Φ(e (k) )-sgn(e (k) ) 2 a 2 -1 Γ ( 1+a 2 ) √ π χ 2 1+a ( e (k) ) 2 | e (k) | a   -α , if e (k) = 0 e (k+1) = (2α -1) a a-1 2 π , if e (k) = 0
Numerical applications of this example are proposed in Section 6.

We have seen that, in general, our Expectile Regression Predictor is not equal to the theoretical conditional expectiles. The difference between these quantities is an error term, which may be huge, especially for high levels of α (see Section 6). This is why, in the following section, we propose another predictor for extreme levels of expectiles.

Extremal Expectiles

In the previous part, we have proposed some algorithms to calculate the Expectile Regression Predictor. Let us recall:

   e α (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R * α 2α-1 êα (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R α 2α-1
Notice that the difference lies in the use of the radius R * or R. Whereas the distribution of R * is in general hard to obtain, the one of R may be known or estimated. Our aim is thus to etablish a relation between

Ψ -1 R * α 2α-1 and Ψ -1 R α 2α-1
for extremal values of α, i.e for α → 0 or α → 1. We have done a similar study in Maume-Deschamps et al. ( 2016), with the same kind of assumption(but for Ψ R ):

Assumption 1. Their exist 0 < < +∞ and γ ∈ R such that: q1) (see Equation 2.8). Then, the coefficients γ and satisfy:

(5.1) lim x→+∞ Ψ R * (x) Ψ R (x γ ) = , where Ψ = 1 -Ψ. We recall Ψ R (x) = Φ R (x) + 1 x +∞ x yc 1 g 1 (y 2 )dy, and Ψ R * (x) = Φ R * (x) + 1 x +∞ x yc * 1 g N +1 (q 1 + y 2 )dy, with c * 1 = c N +1 c N g N (
(5.2) lim

x→+∞ x γ-1 +∞ x yc * 1 g N +1 (q 1 + y 2 )dy γ +∞ x γ yc 1 g 1 (y 2 )dy = Proposition 5.1.
Gaussian, Student, Unimodal Gaussian Mixture and Slash distributions satisfy Assumptions 1 and 2 with coefficients γ and given in Table 3.

The proof is detailed in Appendix section 8.

Distribution γ Gaussian 1 1 Student, ν > 0 N +ν ν Γ( ν+N +1 2 )Γ( ν 2 ) Γ( ν+N 2 )Γ( ν+1 2 ) 1 + q1 ν N +ν 2 ν N 2 +1 ν+N ν-1 ν+N -1 Unimodal Gaussian Mixture 1 min(θ1,...,θn) N exp - min(θ 1 ,...,θn) 2 2 q1 n k=1 π k θ N k exp - θ 2 k 2 q1 Slash N a + 1 2 1-a 2 (a-1)Γ( N +1+a 2 )q N +a 2 1 a( N a +1)(N +a-1)Γ( N +a 2 )Γ( 1+a 2 )χ 2 N +a (q1)
Table 3. Coefficients γ and for classical consistent elliptical distributions, where q 1 = (x 1 -µ 1 ) T Σ -1 11 (x 1 -µ 1 ).

Thanks to Equation 5.2, we have, under Assumption 1, the following equivalence:

(5.3) Ψ R * (x) ∼ x→+∞ Ψ R (x γ )
Our aim is now to get an equivalence relationship between functions Ψ -1 R * and Ψ -1 R * . For that purpose, we refer to the paper of [START_REF] Djurcić | Strong Asymptotic Equivalence and Inversion of Functions in the Class Kc[END_REF], which gives some conditions to obtain the equivalence of inverse functions if these functions are equivalent.

Definition 5.1. A function f is a ϕ-function if f : [0, +∞[→ [0, +∞[, f (0) = 0, f is continuous, non decreasing on [0, +∞[, and f → +∞ when x → +∞.
Clearly, our two equivalent functions Ψ R * (x) and Ψ R (x γ ) are not ϕ-functions for several reasons:

lim x→0 Ψ R * (x) = lim x→0 Ψ R (x) = -∞, lim x→+∞ Ψ R * (x) = lim x→+∞ lΨ R (x γ ) = 0 -.
We have to transform these functions in order to find an equivalence. Let us define first some more definitions and properties. The following is the definition of a general class of functions K c , which contains in particular Regularly Varying functions. The results will thus be more general than those derived from Karamata's theorem. Definition 5.2. K c is the set of all ϕ-functions f with the property:

(5.4) lim

x→+∞ λ→1 f (λx) f (x) = 1
In order to inverse the equivalence 5.3, we do the following assumption.

Assumption 2. Let Ω and Ω * be

(5.5) Ω(x) = -1 Ψ R (x γ ) Ω * (x) = -1 Ψ R * (x)
Then Ω -1 or Ω -1 * belongs to the class K c .

Using Definitions 5.1, 5.4, and Assumptions 1, 2, we are able to inverse equivalence 5.3, hence the following proposition.

Proposition 5.2. Under Assumptions 1 and 2, we have

(5.6) Ψ -1 R * α 2α -1 ∼ α→1 Ψ -1 R 1 - α -1 (2α -1) 1 γ Remark that Ψ -1 R * 1 -α-1 (2α-1)
corresponds to an expectile of level (2α-1) +1-α (2α-1) +2(1-α) . Obviously, if = 1, this level is α. We now define two predictors êα↑ and êα↓ . Using our several analytical results, we will prove later that they are equivalent to the theoretical expectile for extreme values of α.

Definition 5.3 (Extremal Expectiles Predictors). Define

(5.7)

     êα↑ (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R 1 -α-1 (2α-1) 1 γ êα↓ (X 2 |X 1 = x 1 ) = µ 2|1 -σ 2|1 Ψ -1 R 1 - α (2α-1) 1 γ
Using Equation (5.6), we are able to prove the asymptotic equivalences between our predictors êα↑ , êα↓ and the theoretical expectiles, respectively for α → 1 and α → 0.

Theorem 5.3 (Equivalence with theoretical expectiles). Under Assumptions 1 and 2, Equation (5.6), the first equivalence is immediate. Concerning the second one, i.e when α → 0, we use the symmetry properties of elliptical distributions: we know that e α (X

(5.8)    êα↑ (X 2 |X 1 = x 1 ) ∼ α→1 e α (X 2 |X 1 = x 1 ) êα↓ (X 2 |X 1 = x 1 ) ∼ α→0 e α (X 2 |X 1 = x 1 ) Proof. We recall the relationship e α (X 2 |X 1 = x 1 ) = µ 2|1 + σ 2|1 Ψ -1 R * α 2α-1 . With
2 |X 1 = x 1 ) = µ 2|1 -σ 2|1 Ψ -1 R * 1-α 1-2α . Or if α = 1 -α, we have e α (X 2 |X 1 = x 1 ) = µ 2|1 -σ 2|1 Ψ -1 R * α 2α -1 , with α → 1. Using Equation (5.6), we get e α (X 2 |X 1 = x 1 ) ∼ α →1 µ 2|1 -σ 2|1 Ψ -1 R 1 - α -1 (2α -1) 1 γ .
Replace α by 1 -α to the second equivalence.

From Theorem 5.3, we can deduce that êα↑ , êα↓ are approximations of e α , respectively when α is close to 1 or 0. We propose, in Appendix section 8, to check if Assumptions 1 and 2 are fulfilled, and the calculation of and γ for each example. In Section 6, we propose some numerical applications and graphical illustrations. Let us now consider some numerical illustrations on the examples we have given, in order to compare our extremal predictors with the Expectile Regression Predictor and the theoretical expectiles.

Numerical study

In order to give a visual overview of the predictors, we have plotted in Figure 1, the conditional expectiles of an elliptical process observed at N = 5 points. We call X 1 ∈ R 5 the covariates vector. For x ∈ R, X 2 denotes the process at x and the aim is to predict the expectile of X 2 |X 1 = x 1 . For simplicity, we assume that the process is centered, and stationary (matrices Σ and Σ 11 are obtained through an exponential kernel). But our results would be applicable without these assumptions. Parameters for the Gaussian Mixture example are θ 1 = 1, θ 2 = 2 and p = 0.9.

Of course, for the Gaussian process, the curves coincide. For the other examples, Expectile Regression Predictors seem very far from the theoretical curves, especially in the Slash case. On the other hand, Extremal predictors perform significantly better; they look closer to the target conditional expectiles here. We propose to use the following RM SE in order to quantify the difference between expectile regression and theoretical expectile. We also consider the RM SE for the Extremal Predictor:

(6.1) RM SE(ê α ) = 1 n n i=1 e α (X (i) 2 |X 1 = x 1 ) -êα (X (i) 2 |X 1 = x 1 ) 2 .
(6.2) RM SE( êα ) = 1 n n i=1 e α (X (i) 2 |X 1 = x 1 ) -êα (X (i) 2 |X 1 = x 1 ) 2 .
The RMSE measures the average error in the prediction of the conditional expectiles. As expected, the error of êα is increasing with α, and may be huge, like in the Slash case. On the other hand, the error of êα tends to 0 when α tends to 1. This trend is highlighted by the ee-plots (theoretical expectile vs expectile prediction plots) in Figure 2.

Conclusion

In this paper, we focused on conditional expectiles prediction, for elliptical random fields with the consistency property. We have shown that theoretical expectiles were relying on a radius R * whose distribution was, in the general case, difficult to obtain. We thus have proposed two different methods to predict conditional expectiles. The first one is to use expectile regression, i.e to express the conditional expectile as an affine transformation of the observed values. This approach is widely used in the literature but it often requires a large number of simulations, especially for extreme levels of expectile (when α → 0 or α → 1). We have seen, in a first time, that we can obtain some iterative algorithms in our case of consistent elliptical random fields. Furthermore, we have given the distribution of the expectile regression (Theorem 4.4). We have seen that expectile regression is not suited to non Gaussian distributions. A second predictor is given in order to cope with expectile regression problems for extremal expectile levels. We have shown that the proposed extremal expectile predictor is equivalent to the true conditional expectile for extreme expectile levels. We have also illustrated on several numerical examples the better performance of this predictor for extreme levels. As a perspective, these prediction methods require the knowledge of the distribution of the covariates vector X 1 . We have not explored the prediction procedure when the X 1 's distribution is estimated (parametrically e.g.). Finally, we would like to emphasize that we have given examples in dimension d = 1, but all the results may be used in higher dimensions.

Appendix

Proof of Proposition 5.1.

Gaussian example. Firstly, we have to calculate the limit in Equation (5.2), with g n

(t) = exp -t 2 , ∀n ∈ N, c 1 = 1 √ 2π and c * 1 = 1 √ 2π exp q1 2 . Then, we consider lim x→+∞ x γ-1 +∞ x y exp - y 2 2 dy γ +∞ x γ y exp - y 2 2 dy .
If we take γ = 1, we directly get the limit equal to 1. Now, we have to prove that Ω -1 belongs to the class K c . For that purpose, we consider the limit lim

x→+∞ λ→1 Ω -1 (λx) Ω -1 (x) .
We have previously seen that Ω

-1 (x) = Ψ -1 R 1 + 1 x 1 γ .
Then we have to consider the limit of

Ψ -1 R (1+ 1 λx ) Ψ -1 R (1+ 1 x )
.

With δ = λ -1 and y = 1 x , this limit becomes lim (y,δ)→(0,0)

Ψ -1 R 1 + y (δ+1) Ψ -1 R 1 + y .
In order to calculate this kind of limit, we move to polar coordinates, i.e we take δ = r cos(θ), y = r sin(θ), and verify whether the limit when r → 0 exists and is not related to θ. Hence

f (θ) = lim r→0 Ψ -1 R 1 + r sin(θ) (r cos(θ)+1) Ψ -1 R 1 + r sin(θ)
The two terms both tend to 0, then we can use the l'Hôpital's rule, and

f (θ) = lim r→0 sin(θ)(r cos(θ)+1)-r sin(θ) cos(θ) (r cos(θ)+1) 2 sin(θ) Ψ R Ψ -1 R 1 + r sin(θ) Ψ R Ψ -1 R 1 + r sin(θ) (r cos(θ)+1)
The ratio on the left clearly tends to 1. Furthermore, we have seen in the proof of Lemma 3.1 that

Ψ R (x) = - 1 x 2 +∞
x yc 1 g 1 (y 2 )dy. In the Gaussian case, this value is equal to -1 x 2 ϕ(x), and = 1. Then, we get

f (θ) = lim r→0 Ψ -1 R 1 + r sin(θ) r cos(θ)+1 2 Ψ -1 R (1 + r sin(θ)) 2 ϕ Ψ -1 R (1 + r sin(θ)) ϕ Ψ -1 R 1 + r sin(θ) r cos(θ)+1
The ratio on the left is clearly equal to f (θ) 2 . Concerning the term on the right, we apply once again the l'Hôpital's rule. Since ϕ (x) = -xϕ(x), it leds to

f (θ) = f (θ) 2 lim r→0 Ψ -1 R (1 + r sin(θ)) 3 Ψ -1 R 1 + r sin(θ) r cos(θ)+1 3 = 1 f (θ) = 1
Student example. At a first stage, we calculate the limit in Equation (5.2) with g n

(t) = 1 + t ν -n+ν 2 , ∀n ∈ N, c 1 = Γ( 1+ν 2 ) Γ( ν 2 ) 1 √ νπ and c * 1 = Γ( N +1+ν 2 ) Γ( N +ν 2 ) 1 √ νπ 1 + q1 ν N +ν 2 . It remains lim x→+∞ x γ-1 +∞ x yc * 1 1 + q1+y 2 ν -N +1+ν 2 dy γ +∞ x γ yc 1 1 + y 2 ν -1+ν 2 dy = .
The two functions under the integrals are polynomials, then easy to integrate, hence

lim x→+∞ x γ-1 c * 1 ν ν+N -1 1 + q1+x 2 ν 1-N -ν 2 γc 1 ν ν-1 1 + x 2γ ν 1-ν 2 = .
Then, if we take γ = N +ν ν , and replace c 1 and c * 1 by their values, we get

Γ ν+N +1 2 Γ ν 2 Γ ν+N 2 Γ ν+1 2 1 + q 1 ν N +ν 2 ν N 2 +1 ν + N ν -1 ν + N -1 = .
Now, we have to check if Assumption 2 is fulfilled, i.e if the function Ω -1 belongs to the class K c . Let us calculate the limit lim

x→+∞ λ→1 Ω -1 (λx) Ω -1 (x) .
Using the same changes of variables as in the Gaussian case, we consider

f (θ) = lim r→0 Ψ R Ψ -1 R 1 + r sin(θ) Ψ R Ψ -1 R 1 + r sin(θ) (r cos(θ)+1) In the Student case, Ψ R (x) = -1 x 2 Γ( 1+ν 2 ) Γ( ν 2 ) 1 √ νπ ν ν-1 1 + x 2 ν 1-ν 2 . Then, it only remains f (θ) = lim r→0 Ψ -1 R 1 + r sin(θ) (r cos(θ)+1) 2 Ψ -1 R 1 + r sin(θ) 2   1 + Ψ -1 R (1+ r sin(θ) ) 2 ν 1 + Ψ -1 R (1+ r sin(θ) (r cos(θ)+1) ) 2 ν   1-ν 2
On the left, the ratio tends to f (θ) 2 . The term on the right equals lim r→0

Ψ -1 R (1+ r sin(θ) ) 2 Ψ -1 R (1+ r sin(θ) (r cos(θ)+1) ) 2 1-ν 2 = f (θ) ν-1 .
Finally, we get the relationship

f (θ) = f (θ) ν+1 , ν > 0 Hence f (θ) = f (θ) 1 γ = 1.
Unimodal Gaussian Mixture example. Let us calculate the limit in Equation (5.2). In the Unimodal GM case,

g d (t) = n k=1 π k θ d k exp - θ 2 k 2 t , c 1 = 1 √ 2π and c * 1 = 1 √ 2π n k=1 π k θ N k exp - θ 2 k 2 q1
. The calculation of limit (5.2)

gives lim x→+∞ x γ-1 n k=1 θ N -1 k c * 1 exp - θ 2 k 2 q 1 exp - θ 2 k 2 x 2 γ n k=1 π k 1 θ k c 1 exp - θ 2 k 2 x 2γ
.

From now, let us consider that γ = 1. Asymptotically, we only consider the terms exp -

k * exp -θ k * 2 q 1 exp -θ k * 2 x 2 c 1 π k * θ -1 k * exp -θ k * 2 x 2 .
After simplifications, and replacing c 1 and c * 1 by their values, we get

= θ N k * exp -θ k * 2 q 1 n k=1 π k θ N k exp - θ 2 k 2 q 1 .
We have just seen that Assumption 1 is satisfied. We have to check now if Assumption 2 is fulfilled. As usual, we consider the limit

f (θ) = lim r→0 Ψ R Ψ -1 R 1 + r sin(θ) Ψ R Ψ -1 R 1 + r sin(θ) (r cos(θ)+1)
In the UGM case, we have the relationship Ψ R (x) = -1

x 2 n k=1 π k θ k ϕ (θ k x). Rewriting Ψ R (x) in the previous limit, we get

f (θ) = lim r→0 Ψ -1 R 1 + r sin(θ) (r cos(θ)+1) 2 Ψ -1 R 1 + r sin(θ) 2 n k=1 π k θ k ϕ θ k Ψ -1 R 1 + r sin(θ) n k=1 π k θ k ϕ θ k Ψ -1 R 1 + r sin(θ) (r cos(θ)+1)
.

The term on the left is equal to f (θ) 2 . In order to calculate the limit of the second ratio, we consider only the leading terms in the numerator and denominator, i.e the terms with the biggest - .

We have calculated, in the Gaussian case, this kind of limit, and proved that it was equal to 1. Slash example. We consider the limit in Equation (5.2), with g n (t) =

χ 2 n+a (t) t n+a 2 , ∀n ∈ N * , c * 1 = 1 √ π Γ( N +1+a 2 ) Γ( N +a 2 ) q N +a 2 1 χ 2 N +a (q1) and c 1 = 2 a 2 -1 aΓ( 1+a 2 ) √ π
. Using integrations by parts, we get = lim

x→+∞ x γ-1 c * 1 N +a-1   q 1 + x 2 1-N -a 2 χ 2 N +1+a q 1 + x 2 + 2 1-N -a 2 exp - q 1 +x 2 2 Γ( N +1+a 2 )   γ c1 a-1 x γ(1-a) χ 2 1+a (x 2γ ) + 2 1-a 2 exp -x 2 2 Γ( 1+a 2 )
.

Asymptotically, it only remains = lim x→+∞ (a -1)x γ-1 c * 1 x 1-N -a γc 1 (N + a -1)x γ(1-a) . Using γ = N a + 1, and replacing c * 1 and c 1 by their values, we directly obtain the value of given in Table 3. Concerning Assumption 2, we consider

f (θ) = lim r→0 Ψ R Ψ -1 R 1 + r sin(θ) Ψ R Ψ -1 R 1 + r sin(θ) (r cos(θ)+1) .
In the Slash case, Ψ R (x) = -1

x 2 2 a 2 -1 a a-1

Γ( 1+a 2 ) √ π χ 2 1+a (x 2 ) |x| a-1 + 2 1-a 2 Γ( 1+a
2 ) √ 2πϕ(x) . Hence the limit

f (θ) = lim r→0 Ψ -1 R 1 + r sin(θ) (r cos(θ)+1) 2 Ψ -1 R 1 + r sin(θ) 2 χ 2 1+a Ψ -1 R (1+ r sin(θ) ) 2 |Ψ -1 R (1+ r sin(θ) )| a-1 + 2 1-a 2 Γ( 1+a 2 ) √ 2πϕ Ψ -1 R 1 + r sin(θ) χ 2 1+a Ψ -1 R (1+ r sin(θ) (r cos(θ)+1) ) 2 |Ψ -1 R (1+ r sin(θ) (r cos(θ)+1) )| a-1 + 2 1-a 2 Γ( 1+a 2 ) √ 2πϕ Ψ -1 R 1 + r sin(θ) (r cos(θ)+1)
The ratio on the left is obvously equal to f (θ) 2 . Concerning the term on the right, the ratios χ 2 1+a (x) x clearly tends to 0 when x → ∞. Then, the limit may be written more easily

f (θ) = f (θ) 2 lim r→0 ϕ Ψ -1 R 1 + r sin(θ) ϕ Ψ -1 R 1 + r sin(θ) (r cos(θ)+1)
We have already calculated this limit in the Gaussian case, and seen the relationship f (θ) = 1 f (θ) = 1, hence Assumption 2.

Proof of Proposition 5.2. We firstly introduce two lemmas.

Lemma 8.1 [START_REF] Djurcić | Strong Asymptotic Equivalence and Inversion of Functions in the Class Kc[END_REF]). Suppose that f and g are two strictly increasing ϕ-functions, and that at least one of the functions f -1 , g -1 belongs to the class K c , and f (x) ∼ x→∞ g(x). Then f -1 (x) ∼ The proof of Lemma 8.2 is straightforward, using properties given in Lemma 3.1. Using these two lemmas, we can now give the proof of Proposition 5.2. or Ω -1 * belongs to the class K c . Then, we can apply Lemma 8.1. Hence the equivalence Ω -1 (x) ∼

x→∞ Ω -1 * (x). In other words,

Ψ -1 R * 1 + 1 x ∼ x→∞ Ψ -1 R 1 + 1 x 1 γ
If we do the change of variable 1 + 1 x = α 2α-1 , we get the result (5.6).

  by the symmetry properties of elliptical distributions), we get Equation (3.5).

t→t0||Σ 12

 12 || = 0, for a given norm ||.||, then lim t→t0 êlinear α (X 2 |X 1 = x 1 ) = 0, while, as before, we would expect lim

  Theorem 4.4 (Expectile Regression Predictor).

Figure 1 .

 1 Figure 1. Expectile Regression Predictor dotted, theoretical expectiles in solid lines, and Extremal Expectile Predictors dashed, for levels α = 0.9995 and 0.0005

Figure 2 .

 2 Figure 2. Theoretical expectiles vs expectile predictions for Gaussian, Student, UGM and Slash examples. Expectile regression appears on the left of each pair of panels, extremal predictor on the right.

  .e the smallest θ k . Let k * such that θ k * = min{θ 1 , ..., θ n }. We have = lim x→+∞ c * 1 π k * θ N -1

  the smallest θ k . Let k * be such that θ k * = min{θ 1 , ..., θ n }. It remains f (θ) = f (θ) 2 lim r→0 ϕ θ k * Ψ -1 R 1 + r sin(θ) ϕ θ k * Ψ -1 R 1 + r sin(θ) (r cos(θ)+1)

  x→∞ g -1 (x) Lemma 8.2. If γ > 0, then Ω and Ω * are ϕ-functions. Furthermore, under Assumption 1, we have

Proof.R

  By quick calculations, we get Ω -1 (x) = Ψ -1 we have Ω(x) ∼ x→∞ Ω * (x) (Lemma 8.2), with Ω and Ω * ϕ-functions. With Assumption 2, Ω -1

Table 4 .

 4 Table4is a summary of the RMSE for all treated examples, and different levels of α. Obviously, we only consider the cases α ≥ 1 r( êα ) = RM SE( êα ) and r(ê α ) = RM SE(ê α ), for different levels of α, and different consistent elliptical distributions.

	2

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR)..


Zeynep Baharoglu
email: baharogl@pasteur.fr

Didier Mazel

Influence of very short patch mismatch repair on SOS inducing lesions after aminoglycoside treatment in Escherichia coli

Keywords: SOS, antibiotics, sub-MIC, VSPR, DNA repair

Low concentrations of aminoglycosides induce the SOS response in Vibrio cholerae

but not in Escherichia coli. In order to determine whether a specific factor present in E. coli prevents this induction, we developed a genetic screen where only SOS inducing mutants are viable. We identified the vsr gene coding for the Vsr protein of the very short patch mismatch repair (VSPR) pathway. The effect of mismatch repair (MMR) mutants was also studied. We propose that lesions formed upon aminoglycoside treatment are preferentially repaired by VSPR without SOS induction in E. coli and by MMR when VSPR is impaired.

Introduction

Low concentrations of antibiotics are now known to lead to bacterial resistance development [START_REF] Baharoglu | Multiple Pathways of Genome Plasticity Leading to Development of Antibiotic Resistance[END_REF]. Sub-minimal inhibitory concentrations (sub-MICs) of several antibiotics induce the SOS response [START_REF] Baharoglu | Vibrio cholerae triggers SOS and mutagenesis in response to a wide range of antibiotics: a route towards multiresistance[END_REF]. This temporary induction of SOS allows bacteria to mutate, evolve and adapt to their environment. Thus understanding the mechanisms leading to SOS induction is central to understand the connections between antibiotic treatments, bacterial stress responses and resistance development. Interestingly, aminoglycosides induce SOS in Vibrio cholerae and not in Escherichia coli [START_REF] Baharoglu | Vibrio cholerae triggers SOS and mutagenesis in response to a wide range of antibiotics: a route towards multiresistance[END_REF]. The SOS regulon and DNA repair mechanisms are very similar in both bacteria. One hypothesis that could explain the lack of SOS induction in E. coli is that specific factor(s), absent from V. cholerae, prevents SOS induction by sub-MIC aminoglycosides. In order to test this hypothesis, we submitted E. coli to mariner transposition in mass and selected for mutants which became able to induce SOS in response to tobramycin sub-MIC (TOB). We chose to use low doses of tobramycin (100 fold below the MIC) because this antibiotic was used in our previous studies, the same effect was also observed with other aminoglycosides [START_REF] Baharoglu | Identification of genes involved in low aminoglycoside-induced SOS response in Vibrio cholerae: a role for transcription stalling and Mfd helicase[END_REF][START_REF] Baharoglu | RpoS Plays a Central Role in the SOS Induction by Sub-Lethal Aminoglycoside Concentrations in Vibrio cholerae[END_REF][START_REF] Baharoglu | Vibrio cholerae Triggers SOS and Mutagenesis in Response to a Wide Range of Antibiotics: a Route towards Multiresistance[END_REF]. Our screen led to the identification of the very short patch mismatch repair pathway as an E. coli specific response to TOB treatment, without impacting the induction of SOS by DNA damaging agents. S1 shows oligonucleotide sequences.

Materials and methods

Plasmid

Strain constructions. MG1655 vsr::km (B599) and MG1655 ∆dam::Tn9 (B371) were constructed by P1 transduction of vsr::km and ∆dam::Tn9 from strains Keio JW1943 and M0099 Fermentas . MG1655 mutS::Tn10 vsr::km was constructed by P1 transduction of vsr::km into MG1655 mutS strain (2580). Antibiotics were used at the following concentrations: 25µg/ml kanamycin (km) and chloramphenicol (Tn9); 15 µg/ml tetracycline (Tn10).

Construction of the E. coli::mariner insertion library was performed as described [START_REF] Baharoglu | Identification of genes involved in low aminoglycoside-induced SOS response in Vibrio cholerae: a role for transcription stalling and Mfd helicase[END_REF] into MG1655 ∆dapA E. coli strain. The library was spread on several Petri dishes containing 100 µg/ml spectinomycin + 60 µg/ml DAP in order to achieve a library size of 10.000 clones.

Identification of mutants: the mariner transposon insertion locus was determined by sequencing as described in a previous study [START_REF] Baharoglu | Identification of genes involved in low aminoglycoside-induced SOS response in Vibrio cholerae: a role for transcription stalling and Mfd helicase[END_REF]. Antibiotics concentrations used for the screen: TOB: tobramycin 0.01 µg/ml; MMC: mitomycin C 0.02 µg/ml, LEV: levofloxacin 0.01 µg/ml.

Flow cytometry experiments were performed as described [START_REF] Baharoglu | Vibrio cholerae triggers SOS and mutagenesis in response to a wide range of antibiotics: a route towards multiresistance[END_REF]. Statistical significance. Each flow cytometry measurement was independently performed on at least four independent samples. When necessary, in order to test if the means of two groups were different, we first tested the equality of their variance with a Fisher test. When the variances of two groups were found to be significantly different, an unequal variance t test was performed. Otherwise a simple t test was performed. The chosen significance threshold was 0.05 for all tests. The corresponding P values are given in Table S2.

Results

In order to search for genes that prevent SOS induction by AGs at low antibiotic concentrations in E. coli, we used a random gene inactivation strategy using mariner transposon insertions as described previously [START_REF] Chiang | Construction of a mariner-based transposon for epitopetagging and genomic targeting[END_REF][START_REF] Baharoglu | Identification of genes involved in low aminoglycoside-induced SOS response in Vibrio cholerae: a role for transcription stalling and Mfd helicase[END_REF]. The E. coli MG1655 ∆dapA (8120) was used as the host for our genetic screen (Fig. 1A). dapA encodes the dihydrodipicolinate synthase, an enzyme essential for the synthesis of diaminopimelic acid (DAP), essential for the peptidoglycan reticulation. Thus, MG1655 ∆dapA only grows in medium supplemented with DAP [START_REF] Richaud | Directed evolution of biosynthetic pathways. Recruitment of cysteine thioethers for constructing the cell wall of Escherichia coli[END_REF], and dies without addition of DAP (Fig. 1B). In parallel, we constructed a plasmid where the dapA gene is under control of an SOS regulated promoter, the promoter of the V. cholerae integrase gene intIA. In our setup, induction of SOS leads to transcription from intIA promoter [START_REF] Baharoglu | Conjugative DNA transfer induces the bacterial SOS response and promotes antibiotic resistance development through integron activation[END_REF][START_REF] Guerin | The SOS response controls integron recombination[END_REF] and biosynthesis of DAP. In order to verify that SOS induction results in viability of the MG1655 ∆dapA carrying the plasmid Pint_dapA (Fig. 1B), we grew the strain in LB supplemented with DAP in order to allow growth in the absence of SOS induction. We plated a 10 -6 dilution of the overnight culture on several antibiotics at sub-MIC (and without DAP). In E. coli, mitomycin C and levofloxacin are known SOS inducers whereas tobramycin is not. We confirmed that MG1655 ∆dapA with plasmids carrying Pint_dapA form colonies when SOS is induced (MMC and LEV) and that they are not viable in the absence of SOS induction (TOB) (Fig. 1B). The recN promoter was also tested but we chose to pursue with the intIA promoter as it gave better results (Fig. 1B), probably because the basal expression of recN is higher than intIA.

We then transformed our MG1655 ∆dapA mariner insertion library with the plasmid Pint-dapA and selected viable colonies on sub-MIC TOB in absence of DAP in the medium (Fig. 1A). Sub-MIC TOB does not induce SOS in wild type E. coli [START_REF] Baharoglu | RpoS Plays a Central Role in the SOS Induction by Sub-Lethal Aminoglycoside Concentrations in Vibrio cholerae[END_REF][START_REF] Baharoglu | Vibrio cholerae triggers SOS and mutagenesis in response to a wide range of antibiotics: a route towards multiresistance[END_REF], thus if clones are able to grow on this medium, this means that DapA is expressed and that SOS is induced in these conditions. With such a screen after transposition, we expected to get colonies where inactivation of a gene would lead to SOS-dependent expression of dapA from the intIA promoter (or that transposition created a new promoter upstream of the dapA gene). We obtained two viable colonies on medium supplemented with sub-MIC TOB and devoid of DAP. The sequencing of flanking regions in both colonies led to the identification of the vsr gene (b1960) in these two clones. Vsr is involved in very short patch repair (VSPR) in E. coli [START_REF] Macintyre | The Vsr endonuclease of Escherichia coli: an efficient DNA repair enzyme and a potent mutagen[END_REF]. VSPR pathway comprises the Vsr endonuclease, DNA polI (polA), DNA ligase, MutL and MutS [START_REF] Robertson | Reconstitution of the very short patch repair pathway from Escherichia coli[END_REF]. The identification of vsr in our inactivation screen suggests that in the absence of VSPR, SOS is induced by sub-MIC TOB in E. coli. This induction was confirmed by flow cytometry using the gfp reporter under the control of the SOS dependent recN promoter (Fig. 2) that we developed and used in our previous studies [START_REF] Baharoglu | Conjugative DNA transfer induces the bacterial SOS response and promotes antibiotic resistance development through integron activation[END_REF][START_REF] Baharoglu | Transformation-induced SOS regulation and carbon catabolite control of the V. cholerae integron integrase: connecting environment and genome plasticity[END_REF][START_REF] Baharoglu | RpoS Plays a Central Role in the SOS Induction by Sub-Lethal Aminoglycoside Concentrations in Vibrio cholerae[END_REF][START_REF] Baharoglu | Vibrio cholerae triggers SOS and mutagenesis in response to a wide range of antibiotics: a route towards multiresistance[END_REF]. In order to confirm the result obtained in our genetic screen, we constructed a MG1655 ∆vsr strain and verified that inactivation of vsr leads to SOS induction by TOB, and that complementation with a plasmid expressing vsr restores the wild type phenotype (Fig. 2A, ∆vsr p_Vsr), showing that in the absence of VSPR, SOS is induced by sub-MIC TOB in E. coli.

VSPR is involved in the repair of T-G mismatches methylated by Dcm and competes

with mismatch repair (MMR) [START_REF] Bhagwat | Cooperation and competition in mismatch repair: very shortpatch repair and methyl-directed mismatch repair in Escherichia coli[END_REF]. Furthermore, VSPR in E. coli requires the presence of the Dam protein, because E. coli ∆dam strain has decreased levels of Vsr protein [START_REF] Bell | Very-short-patch repair in Escherichia coli requires the dam adenine methylase[END_REF]. We tested and observed that SOS is induced by TOB in the MG1655 ∆dam strain (Fig. 2A), which is consistent with the fact that decreased VSPR leads to SOS induction. Note that the basal level of SOS (in LB) is high in the ∆dam mutant compared to wild type. This is because E. coli ∆dam undergoes massive DNA double strand break formation, which is a powerful SOS inducer [START_REF] Robbins-Manke | Analysis of global gene expression and double-strand-break formation in DNA adenine methyltransferase-and mismatch repair-deficient Escherichia coli[END_REF].

On the other hand, VSPR is also influenced by MutS levels. MutS is part of the Damdirected mismatch repair pathway in E. coli. VSPR is decreased in a mutS+ strain carrying MutS+ in multicopy plasmids [START_REF] Lieb | Very short patch repair of T:G mismatches in vivo: importance of context and accessory proteins[END_REF][START_REF] Macintyre | The Vsr endonuclease of Escherichia coli: an efficient DNA repair enzyme and a potent mutagen[END_REF], meaning that a high level of MutS inhibits VSPR, possibly because mismatch repair competes with VSPR when MutS levels are high. However, the presence of MutS is important for optimal VSPR [START_REF] Robertson | Reconstitution of the very short patch repair pathway from Escherichia coli[END_REF]. MutS and MutL are proposed to recruit Vsr on the mismatch [START_REF] Polosina | The Escherichia coli mismatch repair protein MutL recruits the Vsr and MutH endonucleases in response to DNA damage[END_REF] [START_REF] Drotschmann | The Escherichia coli MutL protein stimulates binding of Vsr and MutS to heteroduplex DNA[END_REF]. As a result, VSPR is not only inhibited by high MutS levels, but also decreased in a mutS deficient background.

To address whether MutS can influence SOS induction by TOB through VSPR modulation, we first measured the effect of TOB on SOS in MG1655 deficient for mutS and we found that SOS is induced by TOB in the absence of MutS (Fig. 2A). Moreover, upon over-expression of Vsr in the mutS background, SOS goes back to wild type levels, meaning that in the absence of mutS, SOS is induced by TOB because of decreased Vsr levels. We then over-expressed MutS+ from a plasmid in MG1655 and observed that over-expression of MutS+ which decreases VSPR, also induces SOS in the presence of TOB (Fig. 2B,MutS+).

These observations are consistent with a possible role of MutS on VSPR in the presence of TOB.

To further test this hypothesis, we decided to use previously characterized dominant negative mutants of mutS which decrease MMR efficiency [START_REF] Wu | Dominant negative mutator mutations in the mutS gene of Escherichia coli[END_REF]. These mutants, expressed from plasmids, were classified in two categories : class I mutants which decrease MMR without affecting VSPR in a wild type (MutS proficient) (or ∆mutS) background and class II mutants which decrease both MMR and VSPR [START_REF] Lieb | Interaction of MutS and Vsr: some dominantnegative mutS mutations that disable methyladenine-directed mismatch repair are active in very-short-patch repair[END_REF]. Some class I mutants even increased VSPR. It was thus concluded that MutS protein lacking functions required for MMR can still participate in VSPR. We decided to test S621N and T669I class I mutants and the G619D class II mutant for their effect on SOS. Whether in a mutS+ or a ∆mutS background, class I mutants are expected to enhance VSPR and thus prevent SOS induction by TOB whereas the class II mutant should decrease VSPR thus allow SOS induction by TOB. Fig. 2B shows the results obtained with these mutants in wild type and ∆mutS backgrounds. As expected, in strains over-expressing VSPR proficient class I mutants S621N and T669I, SOS levels are similar to the ones with the empty plasmid in the wild type E. coli strain, whereas overexpression of MutS+ which decreases VSPR shows a high SOS induction. Conversely, in the strain over-expressing the VSPR deficient class II mutant G619D, SOS is induced by TOB.

We obtained similar results in the ∆mutS background: VSPR proficient class I mutants S621N and T669I have lower SOS levels than VSPR deficient class II mutant G619D and the MutS+ over-expressing strains. Altogether, our data suggest a key role for VSPR to repair the DNA damage that would otherwise induce the SOS response in the presence of low concentrations of TOB. Note that the double ∆vsr ∆mutS mutant has a high basal level of SOS. Even though we do not have an exact explanation for this observation, we can hypothesize that since the double mutant has a slight growth defect compared to the wild type or single mutants in LB, this could influence SOS induction, independently of tobramycin treatment, as observed.

Discussion

SOS induction can be seen as a bacterial adaptation mechanism, which has to be tightly controlled by bacteria in order to prevent decrease of fitness in conditions where it is not needed. The differences in the conditions inducing SOS between bacteria may account for their different lifestyles and studying these conditions is important to understand different defense strategies developed by bacteria.

In E. coli, inactivation of vsr leads to SOS induction by TOB. Interestingly, the VSPR repair pathway is absent from V. cholerae [START_REF] Banerjee | An orphan DNA (cytosine-5-)-methyltransferase in Vibrio cholerae[END_REF] where TOB induces SOS (although a Dcm homologue exists). On the other hand, MMR is present and functional in both bacteria. We propose that in E. coli, unlike in V. cholerae, DNA lesions during TOB treatment can be repaired by VSPR in a way which does not induce SOS. One can even think that when active, VSPR may prevent the binding of RecA on DNA and subsequent SOS induction. However, transduction and conjugation experiments involving recombination of incoming homologous DNA in the chromosome did not yield any difference in recombination efficiency between wild type and ∆vsr strains (data not shown).

Interestingly, we have shown that inactivation of mutS in V. cholerae abolishes SOS induction by TOB [START_REF] Baharoglu | Identification of genes involved in low aminoglycoside-induced SOS response in Vibrio cholerae: a role for transcription stalling and Mfd helicase[END_REF], which means that MutS is involved in TOB-mediated SOS induction in V. cholerae. In E. coli, mutS phenoype is more complex to interpret as both its inactivation as well as its over-expression induce SOS. This difference can likely be explained by the presence of VSPR in E. coli (and its absence from V. cholerae). Importantly, our observation that MutS presence or absence influences SOS induction in both bacteria shows that MMR is involved in the repair of TOB-mediated lesions. We propose that in E. coli, TOB-mediated DNA lesions are preferentially repaired by VSPR without inducing SOS. In the absence of VSPR, the repair of these lesions engage SOS induction. This can possibly be due to single MG1655 ∆dapA carrying the specified plasmid was grown in LB supplemented with DAP in order to allow growth in the absence of SOS induction. A 10 -6 dilution of the overnight culture was plated on several antibiotics at sub-MIC (and without DAP). The growth of colonies was monitored. "+" means growth of colonies. "-" means no colony development. TOB: tobramycin 0.01µg/ml; MMC: mitomycin C 0.02 µg/ml, LEV: levofloxacin 0.01 µg/ml

Legend to figures

  construction. [pACYC184-Pint-dapA]: dapA gene fused to the SOS regulated V. cholerae intIA gene promoter was amplified using oligonucleotides 1514/1515 on E. coli MG1655 genomic DNA; cloned into pTOPO Stratagene plasmid (ampicilin 100 µg/ml); extracted by EcoRI and cloned into pACYC184 (tetracycline 15 µg/ml). [pACYC184-PrecN-dapA]: same as Pint-dapA with oligonucleotides 1513/1515. pB453. vsr gene of E. coli fused to the constitutive bla promoter was amplified using oligonucleotides 2435/2436 on MG1655 and cloned into pTOPO. pB421, pB449 and pB451 were constructed by site-directed mutagenesis by whole plasmid amplification using respectively oligonucleotides 2429/2430, 2439/2440 and 2441/2442 on plasmid pB420 carrying wild type mutS gene under the control of its own promoter (chloramphenicol 25 µg/ml). The PCR product was digested with DpnI and circularized in competent TOP10 cells. Table

Figure 1 :

 1 Figure 1: The genetic screen. A. Experimental setup. B. Validation of the host strain.

Figure 2 :

 2 Figure 2: SOS induction levels after sub-MIC tobramycin treatment. A. Effect VSR. B.

Table 1 : Strains and plasmids

 1

	Strains	Genotype	Resistance	Construction
	8120	MG1655 ∆dapA		Laboratory collection
	9404	MG1655 ∆dapA pACYC184-Pint-dapA	Tetracyclin	This study
	B599	MG1655 vsr::km	Kanamycin	This study
	B371	MG1655 ∆dam::Tn9	C h l o r a m p h e n i c o l	T h i s s t u d y
	2580	MG1655 mutS::Tn10	Tetracyclin	Laboratory collection
	B600	MG1655 mutS::Tn10 vsr::km	T e t r a c y c l i n , K a n a m y c i n	T h i s s t u d y
	Plasmids
	pB453	pTOPO-Pbla-vsr	Ampicillin	This study
	pB420	pACYC184-mutS+	Chloramphenicol	[16]
	pB421	pACYC184-mutSG619D	C h l o r a m p h e n i c o l	T h i s s t u d y
	pB449	pACYC184-mutSS621N	Chloramphenicol	This study
	pB451	pACYC184-mutST669I	C h l o r a m p h e n i c o l	T h i s s t u d y
	p8356	pSW23T_sfiA::PrecN-gfp	Chloramphenicol	[7, 8]

stranded DNA gaps formed by MMR action. Conversely, in V. cholerae where VSPR is absent, the repair always involves SOS induction.