Jan Soko Lowski †

Antoni Żochowski
email: zochowsk@ibspan.waw.pl

Elie Cartan

Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spatial dimensions

Keywords: Shape optimization, topological derivative, optimal design, contact problem, compliance optimization, topology optimization

Topological derivatives for elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions of elasticity boundary value problems in domains with small holes, and determine the explicit asymptotic expansions of solutions with respect to small parameter which describes the radius of internal hole. The elastic potentials of Muskhelishvili gives us an explicite solution in the ring C(ρ, R) = {ρ < |x| < R} in the form of complex valued series. The series depends on the small parameter, the radius ρ of the ring, and we are interested in the behavior of the series for the passage ρ → 0. Such analysis leads to the expansion of the elastic energy in the form

where E 1 (R) is used to determine the first order topological derivatives of shape functionals, and E 2 (R) can be used to determine the second order topological derivatives of 1 shape functionals. In the paper the first order term E 1 (R) is given, however the method is general and can be used to determine the subsequent terms of the energy expansion and the topological derivatives of higher order.

Introduction

In the engineering literature there are many results concerning the shape optimization of contact problems in elasticity. The boundary variations technique for such problems is described in [START_REF] Soko | Introduction to Shape Optimization[END_REF] in the framework of nonsmooth analysis combined with the speed method. Nonsmooth analysis is necessary since the shape differentiability of solutions to contact problems is obtained only in the framework of Hadamard differentiability of metric projections onto polyhedric sets in the appropriate Sobolev spaces. However, to our best knowledge, there is no numerical methods for simultaneous shape and topology optimization [START_REF] Soko | Optimality conditions for simultaneous topology and shape optimization[END_REF] of contact problems. The main difficulty in analysis of contact problems is associated with the nonlinearity of the nonpenetration condition over the contact zone which makes the boundary value problem nonsmooth. In the paper we propose a method for numerical evaluation of topological derivatives for such problems.

In a series of previous works [START_REF] Soko | On topological derivative in shape optimization[END_REF], [START_REF] Soko | Topological derivative for optimal control problems[END_REF], [START_REF] Soko | Topological derivatives for elliptic problems[END_REF], [START_REF] Soko | Topological derivatives of shape functionals for elasticity systems[END_REF] the authors introduced the notion and developed a method of its efficient computing for the so-called topological derivative. This derivative is applicable to domain functionals defined as integrals of some functions depending on solutions to elliptic boundary value problems e.g. of the Poisson equation or of the elasticity boundary value problems. This derivative approximates the effect of making a small hole in a domain, allowing thus to consider topology changes.

The approach proved to be promising and has been used for shape optimization, the identification of inclusions [START_REF] Jackowska-Strumi L Lo | On numerical solution of shape inverse problems[END_REF], and extended to other similar problems [START_REF] Lewinski | Justification of the bubble method for the compliance minimization problems of plates and spherical shells[END_REF], [START_REF] Lewinski | Energy change due to appearing of cavities in elastic solids[END_REF]. Recently it has been generalized to simultaneous topology and shape optimization [START_REF] Soko | Optimality conditions for simultaneous topology and shape optimization[END_REF], giving rise to new necessary optimality conditions. We refer also e.g. to [START_REF] Lewinski | Energy change due to appearing of cavities in elastic solids[END_REF], [START_REF] Garreau | The topological asymptotic for PDE systems: the elasticity case[END_REF], [START_REF] Novotny | Topological Sensitivity Analysis Applied to Topology Design of Kirchhoff's Plate Bending Problem[END_REF], for some application of the topological sensitivity analysis in the linear case. Numerical methods are presented in [START_REF] Allaire | Structural optimization using topological and shape sensitivity via a level set method[END_REF], [START_REF] Amstutz | A new algorithm for topology optimization using a level-set method[END_REF], [START_REF] Eschenauer | Bubble method for topology and shape optimization of structures[END_REF], [START_REF] Fulmanski | A levelset method in shape and topology optimization for variational inequalities[END_REF], [START_REF] Laurain | Singularly perturbed domains in shape optimization[END_REF]. Asymptotic analysis, in particular in elasticity, is described in [START_REF] Berezin | Remark on the Schrödinger equation with singular potential[END_REF], [START_REF] Ammari | Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of Inhomogeneities of Small Diameter[END_REF], [START_REF] Il'in | Matching of Asymptotic Expansions of Solutions of Boundary Value Problems[END_REF], [START_REF] Mazja | The asymptotic behavior of energy integrals under small perturbations of the boundary near corner points and conical points[END_REF], [START_REF] Nazarov | Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions[END_REF], [START_REF] Nazarov | Asymptotic analysis of shape functionals[END_REF], [START_REF] Nazarov | Self adjoint extensions for the Neumann Laplacian in application to shape optimization[END_REF], [START_REF] Nazarov | The topological derivative of the Dirichlet integral due to formation of a thin ligament[END_REF], [START_REF] Nazarov | Self adjoint extensions of differential operators in application to shape optimization[END_REF], [START_REF] Nazarov | Selfadjoint extensions for elasticity system in application to shape optimization[END_REF], [START_REF] Nazarov | Modeling of topology variations in elasticity. System modeling and optimization[END_REF], [START_REF] Nazarov | Topological derivative of the energy functional due to formation of a thin ligament on the spatial body[END_REF], [START_REF] Pavlov | The theory of extension and explicitly soluble models. Uspehi Mat[END_REF], [START_REF] Soko | Topological derivative for optimal control problems[END_REF], [START_REF] Soko | Topological derivatives for elliptic problems[END_REF], [START_REF] Soko | Topological derivatives of shape functionals for elasticity systems[END_REF], and of unilateral problems, including the sensitivity analysis, is considered in [START_REF] Argatov | Asymptotic solution of the problem of an elastic body lying on several small supports[END_REF], [START_REF] Argatov | On asymptotic behavior of the energy functional for the Signorini problem under small singular perturbation of the domain[END_REF], [START_REF] Jarušek | Conical differentiability for evolution variational inequalities[END_REF], [START_REF] Rao | Tangent sets in Banach spaces and applications to variational inequalities[END_REF], [START_REF] Soko | Topology optimization for unilateral problems[END_REF].

The knowledge of topological derivatives is required for the optimality conditions of simultaneous shape and topology optimization. The topological derivative of a given shape functional can be determined from the variations of the shape functional created by the variations of the topology of geometrical domains. The topology variations are defined by nucleation of small holes or cavities or more generally small defects in geometrical domains. The modern mathematical background for evaluation of such derivatives by the asymptotic analysis techniques of boundary value problems is established in [START_REF] Nazarov | Asymptotic analysis of shape functionals[END_REF]. In [START_REF] Nazarov | Asymptotic analysis of shape functionals[END_REF] the error estimates for asymptotic approximations of solutions to boundary value problems in singularly perturbed geometrical domains are provided in the weighted Hölder spaces. The asymptotic approximations of solutions are used in order to established the explicit formulae for the topological derivatives of shape functionals.

Shape optimization and topological derivatives

The shape and topology optimization is one of the most important mathematical problems in structural mechanics. There are numerous application in solid mechanics, we refer to the monographs [START_REF] Delfour | Shapes and Geometries, Advances in Design and Control[END_REF], [START_REF] Kanaun | The effective field method in the mechanics of composite materials[END_REF], [START_REF] Mazja | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF], [START_REF] Muskhelishvili | Some Basic Problems on the Mathematical Theory of Elasticity[END_REF], [START_REF] Nazarov | Elliptic Problems in Domains with Piecewise Smooth Boundaries[END_REF], [START_REF] Soko | Introduction to Shape Optimization[END_REF], for the specific examples. In the paper we consider the elasticity boundary values problems, therefore we restrict ourselves to the shape functional in the form of the elastic energy. This corresponds to the minimization of the energy or equivalently maximization of the compliance.

Contact problems

Our goal is to establish the conical differentiability with respect to the topology variations of solutions for two dimensional contact problem in the elasticity. Let us consider the bounded domain Ω with the boundary ∂Ω = Γ 0 ∪ Γ c . On Γ 0 the displacement vector of the elastic body is given, on Γ c the frictionless contact conditions are prescribed. To specify the week formulation we need an expression for the symmetric bilinear form and for the convex set K ⊂ H 1 (Ω) 2 .

The method of analysis is the same as in the case of Signorini problem for Laplacian. We start with the formulation of the free boundary problem in unperturbed domain Ω. The form of variational inequality is straightforward.

Contact problem in Ω Find u = u(Ω) = (u 1 , u 2 ) and σ = (σ) ij , i, j = 1, 2, such that -div σ = f in Ω , (1) Cσ -ǫ(u) = 0 in Ω , (2)
u = 0 on Γ 0 , (3)
uν ≥ 0, σ ν ≤ 0, σ ν uν = 0 σ τ = 0 on Γ c . (4)
Here

σ ν = σ ij ν j ν i , σ τ = σν -σ ν = σ i τ 2 i=1 , σν = {σ ij ν j } 2 i=1 , ǫ ij (u) = 1 2 (u i,j + u j,i ), i, j = 1, 2, ǫ(u) = (ǫ ij ) 2 i,j=1 , {Cσ} ij = c ijkℓ σ kℓ , c ijkℓ = c jikℓ = c kℓij , c ijkℓ ∈ L ∞ (Ω).
The Hooke's tensor C satisfies the ellipticity condition

c ijkℓ ξ ji ξ kℓ ≥ c 0 |ξ| 2 , ∀ξ ji = ξ ij , c 0 > 0, (5)
and we have used the summation convention over repeated indices.

When the topology of Ω is changed, we have the following contact problem in the domain Ω ρ with the small hole B(ρ).

Contact problem in Ω ρ . Find u = u(Ω ρ ) = (u 1 , u 2 ) and σ = (σ) ij , i, j = 1, 2, such that -div σ = f in Ω ρ , (6) Cσ -ǫ(u) = 0 in Ω ρ , (7)
u = 0 on Γ 0 , (8) σν
= 0 on Γ ρ , (9)
uν ≥ 0, σ ν ≤ 0, σ ν uν = 0 σ τ = 0 on Γ c . (10)
We assume for simplicity that the case of isotropic elasticity is considered, thus the symmetric bilinear form associated with the boundary value problem (1)-( 4) is given by

a(u, u) = Ω (λ + µ)(ǫ 11 + ǫ 22 ) 2 + µ(ǫ 11 -ǫ 22 ) 2 + µγ 2 12 dx, (11)
where γ 12 = 2ǫ 12 and λ, µ are Lame constants. The problem ( 6)-( 10) is approximated by the problem with modified bilinear form in the following way.

Approximation of contact problem in Ω ρ .

Let us surround the hole B(ρ) with the circle Γ R = ∂B(R) where R is fixed. Then in the set Ω R = Ω \ B(R) the solution to the problem ( 6)-( 10) may be approximated by the solution v of another problem in the whole domain Ω with boundary conditions as in ( 1)-( 4), but a modified bilinear form of the energy functional

a(ρ; v, v) = a(v, v) + ρ 2 b(v, v) + o(ρ 2 ) in H 1 (Ω) 2 . ( 12
)
The derivative b(v, v) of the bilinear form a(ρ; v, v) with respect to ρ 2 at ρ = 0+ is given by the expression

b(v, v) = -2πe v (0) - 2πµ λ + 3µ σ II δ 1 -σ 12 δ 2 , (13)
where all the quantities are evaluated for the displacement field v according to formulae given below in terms of the line integrals over Γ R . Hence, we can determine the bilinear form a(ρ; v, w) for all v, w, from the equality

2a(ρ; v, w) = a(ρ; v + w, v + w) -a(ρ; w, w) .
In the same way the bilinear form b(v, w) is determined from the formula for b(v, v).

The convex set is defined in this case by

K = {v ∈ H 1 (Ω) 2 |vν ≥ 0 on Γ c , v = g on Γ 0 } . ( 14
)
Let us consider the following variational inequality which provides a sufficiently precise for our purposes approximation u ρ of the solution u(Ω ρ ) to contact problem ( 6)-( 10),

u ρ ∈ K : a(ρ; u, v -u) ≥ L(ρ; v -u) ∀v ∈ K . ( 15
)
The result obtained is the following, for simplicity we assume that the linear form L(ρ; •) is independent of ρ.

Theorem 1 For ρ sufficiently small we have the following expansion of the solution u ρ with respect to the parameter ρ at 0+,

u ρ = u(Ω) + ρ 2 q + o(ρ 2 ) in H 1 (Ω) 2 , ( 16
)
where the topological derivative q of the solution u(Ω) to the contact problem is given by the unique solution of the following variational inequality

q ∈ S K (u) = {v ∈ (H 1 Γ 0 (Ω)) 2 |vν ≤ 0 on Ξ(u) , a(0; u, v) = 0} (17) a(0; q, v -q) + b(u, v -q) ≥ 0 ∀v ∈ S K (u) . ( 18
)
The coincidence set Ξ(u) = {x ∈ Γ s |u(x).ν(x) = 0} is well defined [START_REF] Rao | Tangent sets in Banach spaces and applications to variational inequalities[END_REF] for any function u ∈ H 1 (Ω) 2 , and u ∈ K is the solution of variational inequality (14) for ρ = 0.

Remark 1 In the linear case, it can be shown that u(Ω ρ )u ρ = o(ρ 2 ) in the norm of appropriate weighted space. We refer the reader to [START_REF] Nazarov | Asymptotic analysis of shape functionals[END_REF] for the related error estimates in the Hölder weighted spaces. In general, we cannot expect that u ρ is close to u(Ω ρ ) in the vicinity of B ρ , therefore the weighted spaces should be used for error estimates.

For the convenience of the reader we provide the explicit formulae for the terms in b(v, v) appearing in [START_REF] Il'in | Matching of Asymptotic Expansions of Solutions of Boundary Value Problems[END_REF]. They are

2πe v (0) = λ + µ π 2 R 6 Γ R (v 1 x 1 + v 2 x 2 ) ds 2 + + µ π 2 R 6 Γ R (1 -9k)(v 1 x 1 -v 2 x 2 ) + 12k R 2 (v 1 x 3 1 -v 2 x 3 2 ) ds 2 + (19) + µ π 2 R 6 Γ R (1 + 9k)(v 1 x 2 + v 2 x 1 ) - 12k R 2 (v 1 x 3 2 + v 2 x 3 1 ) ds 2 , with σ II = µ πR 3 Γ R (1 -9k)(v 1 x 1 -v 2 x 2 ) + 12k R 2 (v 1 x 3 1 -v 2 x 3 2 ) ds, σ 12 = µ πR 3 Γ R (1 + 9k)(v 1 x 2 + v 2 x 1 ) - 12k R 2 (v 1 x 3 2 + v 2 x 3 1 ) ds,
and

δ 1 = 9k πR 3 Γ R (v 1 x 1 -v 2 x 2 ) - 4 3R 2 (v 1 x 3 1 -v 2 x 3 2 ) ds, δ 2 = 9k πR 3 Γ R (v 1 x 2 + v 2 x 1 ) - 4 3R 2 (v 1 x 3 2 + v 2 x 3 1 ) ds.
Here k = λ + µ λ + 3µ .

Correction to the energy functional

In this subsection we shall sketch the derivation of formulae for b(v, v) as given in [START_REF] Soko | Modelling of topological derivatives for contact problems[END_REF].

Let us consider the contribution, in the absence of volume forces, of the energy integral over the circle surrounding the origin (i.e. the potential location of the small hole)

e R (u) = 1 2 B(R) (σ : ǫ) dx = 1 2 Γ R u T (σ.n) ds (20)
to the global elastic energy.We shall leave the displacement on Γ R unchanged and consider the distortion to the stress field caused by introducing the small hole, denoted here by σ:

δe R = 1 2 Γ R u T ( σ.n) ds. (21)
This distortion may be explicitely computed using the well known solution of the plane elasticity system in R 2 with small circular void. To this end we need expressions for the values of strains in the center of the circle in terms of the values of displacement on Γ R . These are calculated as follows (derivation is given in [START_REF] Soko | Modelling of topological derivatives for contact problems[END_REF]).

Let us define I 1 (k, l) and I 2 (k, l) as

I 1 (k, l) = 1 α(k, l) Γ R u 1 x k 1 x l 2 ds , I 2 (k, l) = 1 β(k, l) Γ R u 2 x k 1 x l 2 ds , (22)
where

α(k, l) = R k+l+2 2π 0 cos k+1 φ sin l φ dφ , β(k, l) = R k+l+2 2π 0 cos k φ sin l+1 φ dφ ,
whenever these expressions make sense, i.e. if k is odd and l even or vice versa. Observe that α(k, 0) = β(0, k) and

α(1, 0) = πR 3 , α(3, 0) = 3 4 πR 5 , α(1, 2) = 1 4 πR 5 , α(5, 0) = 5 8 πR 7 , α(3, 2) = 1 8 πR 7
and so on. Furthermore, let

δ 1 = 9k ([I 1 (1, 0) -I 2 (0, 1)] -[I 1 (3, 0) -I 2 (0, 3)]) , δ 2 = 9k ([I 1 (0, 1) + I 2 (1, 0)] -[I 1 (0, 3) + I 2 (3, 0)]) . (23)
In terms of these symbols the formulae for the exact values of strain components at the point x 0 = 0 read as follows:

ǫ 11 + ǫ 22 = I 1 (1, 0) + I 2 (0, 1) , ǫ 11 -ǫ 22 = I 1 (1, 0) -I 2 (0, 1) -δ 1 , γ 12 = I 1 (0, 1) + I 2 (1, 0) + δ 2 . ( 24
)
These values are used in [START_REF] Mazja | The asymptotic behavior of energy integrals under small perturbations of the boundary near corner points and conical points[END_REF], which may be rewritten as

δe R = - 1 2 πρ 2 (λ + µ)(ǫ 11 + ǫ 22 ) 2 + µ(ǫ 11 -ǫ 22 ) 2 + µγ 2 12 + + 1 - 1 k + ρ 2 R 2 1 k (σ II δ 1 -σ 12 δ 2 ) . (25)
Then due to Hooke's law

δe R = -πρ 2 e u (0) - 1 2 πρ 2 1 - 1 k + ρ 2 R 2 1 k (σ II δ 1 -σ 12 δ 2 ) . ( 26
)
which leads immediately to [START_REF] Il'in | Matching of Asymptotic Expansions of Solutions of Boundary Value Problems[END_REF].

In order to make clear the origin of ( 19) we collect below the dependences given by ( 23), [START_REF] Nazarov | Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions[END_REF] and write down the explicit expression for the terms appearing in [START_REF] Nazarov | Elliptic Problems in Domains with Piecewise Smooth Boundaries[END_REF]:

ǫ 11 + ǫ 22 = 1 πR 3 Γ R (u 1 x 1 + u 2 x 2 ) ds, ǫ 11 -ǫ 22 = 1 πR 3 Γ R (1 -9k)(u 1 x 1 -u 2 x 2 ) + 12k R 2 (u 1 x 3 1 -u 2 x 3 2 ) ds, γ 12 = 1 πR 3 Γ R (1 + 9k)(u 1 x 2 + u 2 x 1 ) - 12k R 2 (u 1 x 3 2 + u 2 x 3 1 ) ds, δ 1 = 9k πR 3 Γ R (u 1 x 1 -u 2 x 2 ) - 4 3R 2 (u 1 x 3 1 -u 2 x 3 2 ) ds, δ 2 = 9k πR 3 Γ R (u 1 x 2 + u 2 x 1 ) - 4 3R 2 (u 1 x 3 2 + u 2 x 3 1 ) ds.
The above derivation has one drawback, namely due to the use of the particular solution in the infinite medium with a small hole it gives only the first term in the asymptotic expansion of the energy. In the next section we propose the general procedure, so we are able to produce as well the higher order terms in the asymptotic expansion.

Explicit elasticity solution in the ring

Let us consider the plane elasticity problem in the ring centered around zero with external radius R and internal radius ρ. We use polar coordinates (r, θ) with e r pointing outwards and e θ perpendicularly in the counter-clockwise direction. Assume that the displacement on the outer boundary is given, while the inner boundary is free. We want to compare solution to such a problem to the one defined in the full circle, with the same displacement data. To this end we shall construct the exact representation of both solutions, using the complex variable method of [START_REF] Muskhelishvili | Some Basic Problems on the Mathematical Theory of Elasticity[END_REF]. It has been shown there that

σ rr -iσ rθ = 2Re φ ′ -e 2iθ (zφ ′′ + ψ ′ ) σ rr + iσ θθ = 4Re φ ′ 2µ(u r + iu θ ) = e -iθ (κφ -z φ′ -ψ) (27)
where φ, ψ are given by complex series

φ = A log(z) + k=+∞ k=-∞ a k z k ψ = -κ Ā log(z) + k=+∞ k=-∞ b k z k (28)
Here µ -Lame constant, ν -Poisson ratio, κ = 3 -4ν in the plain strain case, and κ = (3ν)/(1 + ν) for plane stress.

The displacement data are given in the form of Fourier series,

2µ(u r + iu θ ) = k=+∞ k=-∞ A k e ikθ (29)
The traction-free condition on some circle means σ rr = σ rθ = 0. After taking into account simple relations

z = r 2 1 z , z = re iθ , e -iθ = r 1 z we get for displacements the formula 2µ(u r + iu θ ) = 2κAr log(r) 1 z - Ā1 r z+ + p=+∞ p=-∞ [κra p+1 -(1 -p)ā 1-p r -2p+1 -b-(p+1) r -2p-1 ]z p (30)
Hence, comparing ( 29) and [START_REF] Nazarov | Selfadjoint extensions for elasticity system in application to shape optimization[END_REF] gives, assuming substitution r := R, p = -1: 2κAr log(r) + (κa 0 -b0 ) -2ā 2 r 2 = A -1

p =0: κra 1 -rā 1 - 1 r b-1 = A 0 p =1: -Ā + κr 2 a 2 -b-2 1 r 2 = A 1 p ∈{-1, 0, +1}: κr p+1 a p+1 -(1 -p)ā 1-p r -p+1 -b-(p+1) r -(p+1) = A p (31)
Similarly we obtain representation of tractions on some circle

σ rr -iσ rθ = 2A 1 z + (κ + 1) 1 r 2 Āz+ + p=+∞ p=-∞ [(1 + p)(1 -p)a p+1 + (1 -p)ā 1-p r -2p -(p -1) 1 r 2 b p-1 ]z p (32)
and the resulting traction-free condition, assuming substitution r := ρ,

p = -1: 2A + 2ā 2 r 2 + 2 1 r 2 b -2 = 0 p =0: a 1 + ā1 + 1 r 2 b -1 = 0 p =1: (κ + 1) 1 r 2 Ā = 0 p ∈{-1, 0, +1}: (1 + p)(1 -p)a p+1 + (1 -p)ā 1-p r -2p -(p -1) 1 r 2 b p-1 = 0 ( 33
)
Denote d 0 = κa 0 -b0 . For the full circle we must eliminate singularities, i.e. b -k = a -k = A = 0 for k = 1, 2, . . . and then using (31) obtain

d 0 0 = A -1 + 2 κ Ā1 Re a 0 1 = 1 (κ -1)R Re A 0 , Im a 0 1 = 1 (κ + 1)R Im A 0 a 0 k = 1 κR k A k-1 , k = 2, 3, . . . b 0 k = - 1 R k [(k + 2) 1 κ A k+1 + Ā-(k+1) ], k = 1, 2, . . . ( 34
)
Let us check the formulas for uniformly stretched (compressed) circle. Then A 0 = 2µu r (R) and other A k vanish. Hence only

a 0 1 = 2µ (κ -1)R u r (R)
is different from 0 and ψ = 0, φ = a 0 1 z. According to e.g. [START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF] the solution should be u r (r) = u r (R)r/R. Indeed, using [START_REF] Nazarov | Self adjoint extensions for the Neumann Laplacian in application to shape optimization[END_REF] we get

2µ(u r + iu θ ) = r 1 z (κa 0 1 z -ā0 1 z] = 2µ r R u r (R).
The other test of ( 31) is applying it to the ring with different displacement conditions at r = ρ and r = R. Then these formulas allow, after longer calculations, to obtain the same solution as in [START_REF] Gross | The second fundamental problem of elasticity=20 applied to a plane circular ring[END_REF].

In further analysis we consider the ring and assume ρ < 0.5R. Then from (31) for r = R and (33) for r = ρ, by considering first p ∈ {-1, 0, 1}, we get A = 0 and

d 0 = A -1 + 2R 4 κR 4 + ρ 4 Ā1 a 2 = R 2 κR 4 + ρ 4 A 1 Re a 1 = R (κ -1)R 2 + 2ρ 2 Re A 0 , Im a 1 = 1 κ + 1 Im A 0 b -1 = -2ρ 2 Re a 1 = - 2ρ 2 R (κ -1)R 2 + 2ρ 2 Re A 0 b -2 = -ρ 4 ā2 = - ρ 4 R 2 κR 4 + ρ 4 Ā1
Here we may again test the correctness of formulas. From [START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF] we get the the solution for the uniformly stretched ring with unloaded inner boundary: It reads, in the plain strain case,

u r (r) = ρ 2 ρ 2 + (1 -2ν) 1 r + 1 -2ν ρ 2 + (1 -2ν) r u R (1)
Again, A 0 = 2µu r (R = 1), and from ( 27)

a 1 = 1 (κ -1) + 2ρ 2 2µu R (1), b -1 = - 2ρ 2 (κ -1) + 2ρ 2 2µu R (1). Hence φ = a 1 z, ψ = b -1 1 z and 2µu r (r) = r 1 z κa 1 z -za 1 -b -1 1 r 2 z = = (κ -1) 1 (κ -1) + 2ρ 2 r + 2ρ 2 (κ -1) + 2ρ 2 1 r 2µu R (1).
Taking into account that κ = 3 -4ν we get the same result.

Observe, that

d 0 -d 0 0 = -ρ 4 2 κ(κR 4 + ρ 4 ) Ā1 a 1 -a 0 1 = -ρ 2 2 (κ -1)R((κ -1)R 2 + 2ρ 2 ) Re A 0 a 2 -a 0 2 = -ρ 4 1 κR 2 (κR 4 + ρ 4 ) A 1 (35)
There remains to compute the rest of terms. Taking p = +k, k = 2, 3, . . . in (31) for r = R and (33) for r = ρ gives

κa k+1 R k+1 + (k -1)ā -(k-1) R -(k-1) -b-(k+1) R -(k+1) = A k (k + 1)a k+1 ρ 2(k+1) + ā-(k-1) ρ 2 + b k-1 ρ 2k = 0 (36)
and for p = -k, k = 2, 3, . . .

κa -(k-1) R -(k-1) -(k + 1)ā k+1 R k+1 -bk-1 R k-1 = A -k -(k -1)a -(k-1) ρ 2 + āk+1 ρ 2(k+1) + b -(k+1) = 0 (37)
We eliminate a -(k-1) and b -(k+1) using

a -(k-1) b -(k+1) = -(k + 1)ρ 2k , -ρ 2(k-1) -k 2 ρ 2(k+1) , -(k -1)ρ 2k • āk+1 bk-1 = T k (ρ) • āk+1 bk-1 (38)
and get the system which may be rewritten as

S k (ρ) • a k+1 b k-1 = A k Ā-k (39)
with entries

S k (ρ) 11 = κR k+1 -(k 2 -1)R -(k-1) ρ 2k + k 2 R -(k+1) ρ 2(k+1) S k (ρ) 12 = -(k -1)(R -(k-1) ρ 2(k-1) -R -(k+1) ρ 2k ) S k (ρ) 21 = -(k + 1)(R k+1 + κR -(k-1) ρ 2k ) S k (ρ) 22 = -R k-1 -κR -(k-1) ρ 2(k-1) (40)
In fact the formulas [START_REF] Soko | Topological derivative for optimal control problems[END_REF], [START_REF] Soko | Topological derivatives of shape functionals for elasticity systems[END_REF] are correct also for k = 0, 1 and ρ > 0.

The matrix S k (ρ) is a perturbation of S k (0), which would produce the solution for the full circle, namely

a 0 k+1 , b 0 k-1 . Observe that T k (0) = 0. Direct computations lead to estimates |a 3 -a 0 3 | ≤ Λ |A 2 |ρ 4 + |A -2 |ρ 2 (41)
and for k = 4, 5, . . .

|a k -a 0 k | ≤ Λ |A k-1 |ρ 3(k-1)/2 + |A -(k-1) |ρ 3(k-2)/2 (42)
where the exponent k/2 has been used to counteract the growth of k 2 in terms like

k 2 ρ k/2 . Similarly |b 1 -b 0 1 | ≤ Λ |A 2 |ρ 4 + |A -2 |ρ 2 (43)
and for k = 2, 3, . . .

|b k -b 0 k | ≤ Λ |A k+1 |ρ 3(k+1)/2 + |A -(k+1) |ρ 3k/2 (44)
From relation [START_REF] Soko | Topological derivative for optimal control problems[END_REF] we get another estimate

|a -k | ≤ Λρ 2k |A k+1 | + |A -(k+1) | , k = 1, 2, . . . ( 45
) end |b -k | ≤ Λρ 2(k-1) |A k-1 | + |A -(k-1) | , k = 3, 4, . . . ( 46
)
Here Λ is a constant independent from ρ and A i . Observe that the corrections proportional to ρ 2 are present only in a 1 , a 3 , b -1 , a -1 . The rest is of the order at least O(ρ 3 ) (in fact O(ρ 4 )).

The condition for A i , namely

k=+∞ k=-∞ √ 1 + k 2 |A k | 2 ≤ Λ 0 (47)
together with bounds ( 35),( 41),( 42),( 43), ( 44),( 45),(46) ensure, that the expression for elastic energy concentrated in the ring splits into the one corresponding to the full circle, correction proportional to ρ 2 and the rest, which is uniformly of the order O(ρ 3 ) for all u satisfying u 2

H 1/2 (Γ R ) ≤ Λ 0 4 Numerical illustration Rugby ball. Let us take u r = s 0 cos 2 θ = 1 2 s 0 + 1 2 s 0 cos 2θ. Hence A k = [. . . , 1 2 µs 0 , 0, A 0 = µs 0 , 0, 1 2 µs 0 , . . .]
The resulting distortion for ρ = 0.2 and r = 0.3 are shown in Fig. 1 (solid line -undeformed, dashed -deformed ring, dotted -deformed ball):

Bubble. Now we take u r = s 0 sin 4θ. Hence A k = [. . . , µs 0 i, 0, 0, 0, A 0 = 0 , 0, 0, 0, -µs 0 i, . . .]

The resulting distortions for ρ = 0.2 and r = 0.3 are shown in Fig. 2 using the same types of lines.

Asymptotics. In the second numerical experiment -bubble -only A -4 and A 4 were nonzero, what means, that the difference between positions of the contour r = 0.3 for full circle and the ring should behave like ρ 6 . In the first experiment it should be ρ 2 . In Fig. 3 we show the slopes of this difference for ρ = [0.05, 0.10, 0.15, 0.20, 0.25] in log-log graph (solid -rugby, dashed -bubble): Comparison of distortions. The deformations for ρ = 0.2 and several intermidiate radii (dashed -undeformed, solid -deformed contours) are shown in Fig. 4. It may again be seen that the influence of the outer boundary distortion vanishes quicker for the bubble case, where A k with smaller indices are nonzero.

where

φ 1 = a 1 -1 1 z + a 1 1 z + a 1 3 z 3 , ψ 1 = b 1 -1 1 z + b 1 1 z. (55)
As a result

σ 1 rr -iσ 1 rθ = 2ℜ[-a 1 -1 1 z 2 + a 1 1 + 3a 1 3 z 2 ] -e 2iθ (6a 1 3 |z| 2 + 2a 1 -1 z z 3 + b 1 1 -b 1 -1 1 z 2 ). ( 56
)
Computing corrections to coefficients. The matrix S k (ρ) may be for k = 2 expressed as

S 2 (ρ) = S 2 (0) + ρ 2 S 1 2 + O(ρ 4 ). Using (40) S 2 (0) a 0 3 b 0 1 = A 2 Ā-2
we may get

a 1 3 b 1 1 = -S -1 2 (0)S 1 2 a 0 3 b 0 1 ( 57
)
where

S 2 (0) = κR 3 , 0 -3R 3 , -R S 1 2 = 0 , -1/R 0 , -κ/R . Similarly T 2 (ρ) = 0 + ρ 2 T 1 2 + O(ρ 4 ), T 1 2 = 0 , -1 0 , 0 .
Hence, using [START_REF] Soko | Topological derivative for optimal control problems[END_REF],(57) we get finally

a 1 -1 = -b0 1 , a 1 3 = 1 κR 4 b 0 1 b 1 1 = 3 + κ 2 κR 2 b 0 1 , a 1 1 = - 2 (κ -1)R 2 ℜa 0 1 (58) b 1 -1 = -2ℜa 0 1
Since, as is obvious, we need only A 0 , A 2 , A -2 and

A k = µ π 2π 0 (u r + iu θ )e -ikθ dθ
as well as

u r + iu θ = (u 1 + iu 2 )e -iθ
Here values of displacements are taken as u i (R cos θ, R sin θ). After discretization these integrals constitute weighted sums of values of u i at certain points on Γ R . If we assume piecewise linear approximation over triangles, then it is well known that

u h i (x) = x T   x 1 1 x 1 2 1 x 2 1 x 2 2 1 x 3 1 x 3 2 1   -1 U i = x T M -1 U i = (M -T x) T U i = c T U i
where u h i (x) is a value of the approximation of u i at a point x inside the triangle defined by vertices x 1 , x 2 , x 3 and U i = [U 1 i , U 2 i , U 3 i ] T is a vector of the values of u i at these vertices. Observe that c is a vector of weights with which nodal values enter into the expression for u h i (x). Let now U = [u 1 1 , u 1 2 , . . . , u K 1 , u K 2 ] T be a vector of nodal values of u for the global triangulation. Then we may write down the following formulae µ π Taking into account (62) we may conclude that the first term in the correction of energy is a well defined quadratic form. Similar, only more complicated expressions may be obtained for further terms corresponding to ρ 4 and higher. The derivation described above is new and while it gives the same result as in Section 2 for the first term, it is much more general.

Figure 1 :

 1 Figure 1: Rugby-like distortion and the detail near the central hole (exaggerated).

Figure 2 :

 2 Figure 2: Bubble-like distortion and the detail near the central hole (exaggerated).

Figure 3 :

 3 Figure 3: The slope of diminishing influence of the hole for rugby and bubble (below) type of distortions.

Figure 4 :

 4 Figure 4: The pattern of distortions for both experiments.

2π 0 u 1 µ π 2π 0 u 2 µ π 2π 0 u 1 µ π 2π 0 u 2 µ π 2π 0 u 1 U µ π 2π 0 u 2 U µ π 2π 0 u 1 U µ π 2π 0 u 2 23 U 2 |A 2 |

 01020102010201022322 cos θ dθ = c T 11 U sin θ dθ = s T 21 U cos 3θ dθ = c T 13 U sin 3θ dθ = s T 23 U sin θ dθ = s T 11 cos θ dθ = c T 21 sin 3θ dθ = s T 13 cos 3θ dθ = c T Here s ij , c ij are sparse vectors of weights with which nodal values of u enter into appropriate integrals. In this notation (ℜA 0 ) 2 = (c 11 + s 21 ) T U 2 = (c 13 + s 23 ) T U 2 + (c 23s 13 ) T U 2 |A -2 | 2 = (c 11s 21 ) T U 2 + (c 21 + s 11 ) T U 2 ℜ(A 2 A -2 ) = U T (c 13 + s 23 )(c 11s 21 )U -U T (c 23s 13 )(c 21 + s 11 )U (64)

Explicit expansion of elastic energy

The elastic energy contained in the ring has the form

Since

Thus the problem reduces to finding σ 1 (u). From ( 27), [START_REF] Nazarov | The topological derivative of the Dirichlet integral due to formation of a thin ligament[END_REF] we know that σ ρ (u) is a linear function of infinite vectors a, b, while σ 0 (u) is the same function of a 0 , b 0 . Here a 0 , b 0 are computed for B(R), while a, b correspond to C(ρ, R). In order obtain σ 1 (u) it is enough to express a, b as

Let us observe as well that

The analysis of formulae (34) for a 0 , b 0 and their counterparts a, b leads to the conclusion, that the only nonzero terms in a 1 , b 1 will be a

Taking into account that A = 0 in [START_REF] Nazarov | The topological derivative of the Dirichlet integral due to formation of a thin ligament[END_REF] for our problem,

we obtain

Now we shall return to the integral (53). Using (56) we get

where

From ( 53) is obvious that only a part of displacement will contribute to the integral, namely

Now we are able to collect all formulae and obtain the final expression, using [START_REF] Pavlov | The theory of extension and explicitly soluble models. Uspehi Mat[END_REF],(60),(61


Céline Lacaux

Jean-Michel Loubes

Hurst exponent estimation of Fractional Lévy Motion

come

Introduction

In this paper we present results on an asymptotic analysis of a wavelet type estimator of the self-similarity (Hurst) parameter of a Real Harmonizable Fractional Lévy Motion (RHFLM). In particular, we show consistency of this estimate in a noisy regression framework. This enables to detect RHFLM in noisy data and to use it in practical settings.

It is well known that standard Brownian Motion fails in explaining certain statistical time series arising from finance and turbulence theory. Hence Mandelbrot and Van Ness, in [START_REF] Mandelbrot | Fractional Brownian motion, fractionnal noises and applications[END_REF], introduced a stochastic motion of a quite different nature, the Fractional Brownian Motion (in short FBM). The FBM of index H is the only centered Gaussian field, vanishing at zero, with stationary increments and self-similar with index H. Its Hurst exponent governs its properties. The main difference between both processes is that, now the increments are not independent and the process can model short or long range dependent data.

Nevertheless, in image modeling, in finance or in biology, the processes are rarely Gaussian, which prevents the use of Gaussian fields. Hence, Benassi et al., in [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF], introduce Real Harmonizable Fractional Lévy Motions (in short RHFLM) to model such processes. Let us recall that a RHFLM X H of index H (0 < H < 1) is defined as the stochastic integral

X H (x) = R d e -ix•ξ -1 ξ H+d/2 L(dξ), x ∈ R d ( 1
)
where • is the Euclidean norm and L(dξ) is a Lévy random measure in the sense of [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF].

Such processes are non Gaussian, locally asymptotically self-similar with Hurst exponent H and are well fitted to mimic most of the irregular phenomena that can be observed in turbulence experiments, provided the parameter H is well chosen. So, estimating the Hurst exponent of a RHFLM is the key issue in order to analyze data and to model real observations by such a process. More precisely, we want to be able to estimate the Hurst exponent of a RHFLM observed only at discrete times in a white noise framework. In their work, [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF] propose an estimator of Hurst exponent based on some generalized quadratic variation of the process. This estimator has first been introduced by [START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF] in a Gaussian framework. Such estimators have also been studied in [START_REF] Benassi | Identification of Filtered White Noises[END_REF] in the case of some self-similar Gaussian random fields or by [6], [5], [START_REF] Ayache | On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion[END_REF], [START_REF] Ayache | Regularity and identification of generalized multifractional Gaussian processes[END_REF] and [START_REF] Lacaux | Real harmonizable multifractional Lévy motions[END_REF] in the case of multifractional random fields.

However, such estimation techniques are unable to handle noisy observations and to estimate this exponent when data are blurred by a Gaussian white noise, as we will see later in this paper. Our purpose is hence to construct a robust estimator and to detect RHFLM in noisy data.

For this, we consider wavelet analysis of this stochastic process. Work on properties of wavelet coefficients of fractional Brownian motion was pioneered in the papers by [START_REF] Flandrin | Wavelets for scaling processes[END_REF] or [START_REF] Flandrin | The many roads to time-frequency[END_REF]. Statistical properties of wavelet type estimators for Hurst exponent were studied in [START_REF] Delbeke | A wavelet based estimator for the parameter of selfsimilarity of fractional Brownian motion[END_REF] or [START_REF] Bardet | Statistical study of the wavelet analysis of fractional Brownian motion[END_REF] in the case of FBM. Such estimators have also been introduced in the case of linear fractional stable motion in [START_REF] Delbeke | Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion[END_REF], [START_REF] Stoev | Estimation of the self-similarity parameter in linear fractional stable motion[END_REF] and [START_REF] Pipiras | Asymptotic normality for wavelet-based estimators of fractional stable motion[END_REF]. In [START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens[END_REF] are highlighted the properties of wavelet coefficients for self-similar processes or long-range dependent processes. For multifractal processes, such estimators are studied in [START_REF] Jaffard | On lacunary wavelet series[END_REF], [START_REF] Gamboa | Bayesian estimation of a multifractal function[END_REF] or [START_REF] Gamboa | Parametric estimation of a multifractal function[END_REF] for example. They are based on a regression of the log-variance of the wavelet coefficients versus scale. Other authors study also this issue, for instance in [START_REF] Kurchenko | Confidence intervals and rate of convergence for the estimates of Hurst parameter of the FBM[END_REF] or [START_REF] Hall | Semiparametric bootstrap approach to hypothesis tests and confidence intervals for the Hurst coefficient[END_REF].

The aim of this paper is to introduce an estimator of the Hurst exponent based on wavelet type coefficients and prove its asymptotic behaviour in the case of RHFLMs. Contrary to other work, the coefficients of a RHFLM are not Gaussian neither independent. Hence we are facing a difficult issue since work in this direction either relies on independence of the coefficients or its Gaussian properties to prove consistency of the estimator, see for instance the work by [START_REF] Morales | Analysis of wavelet-based estimators of the Hurst parameter[END_REF]. Indeed, in such papers, the authors often generalize Gaussian type limit theorems to the case of weak dependent random variables using results in [START_REF] Doukhan | Models, inequalities, and limit theorems for stationary sequences[END_REF] for instance. In [START_REF] Pipiras | Asymptotic normality for wavelet-based estimators of fractional stable motion[END_REF], the asymptotic normality of the estimator relies on properties of stable moving average sequence.

Unfortunately, in the case of coefficients of a RHFLM, we are not in such cases and do not have powerfull probabilistic tools at hand. Hence we rely directly on the properties of RHFLM to get asymptotic results. Nevertheless, we prove almost sure consistency of the moment type estimator in the presence of Gaussian noise. This enables us to compute the estimator and analyze its performance for simulated data.

The paper falls into the following parts. Section 2 is devoted to a wavelet type representation of RHFLM. In Section 3, we construct estimators of the Hurst exponent of a RHFLM. Then, in Section 4, we provide an estimator of the Hurst exponent in a regression framework. Section 5 deals with a numerical study of our estimators for simulated data.

RHFLM and wavelet bases

RHFLM is one of the canonical example of fractional non Gaussian process, with a large presence in both the theoretical literature and applications. Wavelets have become a standard tool for the modeling and analysis of such signals.

Let us first recall the definition and the properties of a RHFLM.

RHFLM

Benassi et al. [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF] have defined RHFLMs by substituting in the harmonizable representation of the FBM to the Wiener measure W (dξ) a Lévy random measure L(dξ). Heuristically, a Lévy random measure is linked with the increments of a Lévy process. Also, the non-Brownian part of a Lévy random measure is defined thanks to a Poisson random measure. We first recall the definition of a Lévy random measure L(dξ) in the sense of [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF].

The non-Brownian part M (dξ) of the Lévy random measure L(dξ) is represented by a Poisson random measure N (dξ, dz) on R × C whose mean measure n(dξ, dz) = dξ ν(dz) is such that ν({0}) = 0 and

∀ p 2, C |z| p ν(dz) < +∞.
Here, ν(dz) is a non vanishing rotationally invariant measure, i.e.

P (ν(dz)) = dθ ν ρ (dρ), ( 2
)
where dθ is the uniform measure on [0, 2π) and

P ρe iθ = (θ, ρ) ∈ [0, 2π) × R + * . The random measure M (dξ) is defined by R f (ξ) M (dξ) = R×C [f (ξ)z + f (-ξ)z] (N -n)(dξ, dz), where f ∈ L 2 (R d ). If f (ξ) = f (-ξ), then R f (ξ) M (dξ) is a real-valued symmetric infinitely divisible random variable and for every u ∈ R E e iu f (ξ) M (dξ) = exp R d ×C e 2iuℜ(f (ξ)z) -1 -2iuℜ (f (ξ) z) dξ ν(dz) . (3)
Here, M (dξ) is only the non-Brownian part of the Lévy random measure L(dξ). Finally, L(dξ) = aM (dξ) + bW (dξ), where (a, b) ∈ R 2 and W (dξ) is a Wiener measure independent of M (dξ). Taking L(dξ) = W (dξ), the field X H defined by ( 1) is a FBM. The Wiener measure W (dξ) is a complex random measure which ensures that X H is a real-valued field, see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF][START_REF] Cohen | From self-similarity to local self-similarity : the estimation problem[END_REF] for reference on complex Gaussian random measure.

If f ∈ L 2 (R) and f (ξ) = f (-ξ), f (ξ) L(dξ) is a symmetric real-valued random variable, E e iu f (ξ) L(dξ) = exp - b 2 u 2 f 2 2 2 E e iau f (ξ) M (dξ) , u ∈ R (4)
and

E R f (ξ) L(dξ) 2 = A f 2 L 2 (R) (5)
with A = a 2 C |z| 2 ν(dz) + b 2 . The isometry property (5) allows us to evaluate the second order moment of the wavelet type coefficients of a RHFLM. In the following, we assume that L(dξ) is a non vanishing measure, i.e. (a, b) = (0, 0).

A RHFLM with index H, defined by

X H (x) = R e -ixξ -1 |ξ| H+1/2 L(dξ), x ∈ R
has stationary increments and the same structure of covariance as a FBM. Note that the Brownian part of X H is a FBM with index H independent from its non-Brownian part.

In addition, as in the case of FBM, RHFLMs have locally Hölder sample paths and their pointwise Hölder at point x is equal almost surely to H. Whereas a FBM is selfsimilar with index H, in general, a RHFLM is only locally self-similar and looks like locally a FBM with index H. An important difference between RHFLM and FBM is that as soon the RHFLM is non-Gaussian, it does not have moment of every order. However, as noticed in the following, the wavelet type coefficients of a RHFLM have moments of every order. This key property allows us to study, in Section 3, moment type estimators based on the wavelet type coefficients of a RHFLM.

Weakness of quadratic variation estimator

Consider the following standard regression model with N + 1 observations. We observe at discrete times l/N, l = 0, . . . , N a noisy RHFLM

Y H l N = X H l N + σ N ε l , l = 0, . . . , N, (6)
where ε l are an i.i.d sample of Gaussian random variables N (0, 1) independent from X H and σ N is the noise level.

In their work, [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF] consider for K > 0 and a sequence a k , k = 0, . . . , K the quadratic variation

V N = 1 N -K + 1 N -K p=0 K k=0 a k X H k + p N 2 = 1 N -K + 1 N -K p=0 [∆X p,N ] 2 .
If observed without observation errors, this technique enables to build a weakly consistent estimate of Hurst exponent. When observed in practice with Gaussian errors, let us apply such method and define the noisy quadratic variation as

W N = 1 N -K + 1 N -K p=0 [∆Y p,N ] 2 (7) = 1 N -K + 1 N -K p=0 ∆X p,N + σ N K k=0 a k ε k+p 2 . Write ξ p = K k=0 a k ε k+p ∼ N (0, K k=0 a 2 k )
. Now, if we consider the expectancy of ( 7), we get that

E(W N ) = E(V N ) + σ 2 N E(ξ 2 1 ) = N -2H c(H) + σ 2 N K k=0 a 2 k ,
where c(H) > 0 only depends on H.

As a result, in order to get consistency of the noisy quadratic variation, we need to ensure that

σ 2 N N = O(N 1-2H ), which implies that σ N = O(1/N H ), 0 < H < 1.
In a standard regression framework, when the RHFLM is observed at equispaced times, the noise level is of order σ N = O(1/ √ N ). Hence the method described in [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF] does not provide a consistent estimator of the Hurst exponent when H > 1/2 in the regression framework. Moreover, when estimating Hurst exponent, we can not consider a method depending on the values of the parameter of interest. As a result we pay attention in the next section to wavelet type estimators.

Wavelet bases

Let ψ(•) be a function with compact support, continuous derivative and r vanishing moments, i.e. ∀m = 0, . . . , r, t m ψ(t) dt = 0.

We assume that ψ = 0, i.e. that its support suppψ = ∅. Define the rescaled function at scale j and location k as ψ jk (t) = 2 j/2 ψ(2 j tk). Then given a process (Y (t)) t∈R observed at continuous time, the corresponding wavelet type coefficients of the process are defined by

w jk = Y (t)ψ jk (t)dt = 2 -j/2 Y u + k 2 j ψ(u)du. (8)
When we observe Y at discrete times, the coefficients w jk are not known. In this case, we will approximate them thanks to a discretization of the integral [START_REF] Benassi | Identification of Filtered White Noises[END_REF]. For ease of writing, we assume that the compact support of ψ is included in [0, 1] and that we observe Y at discrete times l/N , l = 0 . . . N . However, as in [START_REF] Bardet | Statistical study of the wavelet analysis of fractional Brownian motion[END_REF], up to change the discretization, all the following results can be stated for ψ a wavelet function with compact support. We refer to [START_REF] Mallat | A wavelet tour of signal processing[END_REF] and [START_REF] Daubechies | Ten lectures on wavelets[END_REF] for general references about wavelet bases and their properties. In this framework, we will replace the coefficients w jk by the corresponding coefficients of the discretized process defined by

w n jk = 1 2 j/2 n n p=1 Y p + nk 2 j n ψ( p n ). (9)
Remark that w n jk is a discretization of constant step of the integral [START_REF] Benassi | Identification of Filtered White Noises[END_REF]. The parameter n will be chosen such that w n jk can be computed knowing Y at times l/N , l = 0, . . . , N . Let us first assume that we observe a RHFLM in continous time. Some key properties of its wavelet type coefficients are given in the following lemma. As in the case of LFSM ( [START_REF] Delbeke | A wavelet based estimator for the parameter of selfsimilarity of fractional Brownian motion[END_REF]), these coefficients can be rewritten as stochastic integral with respect to the Lévy random measure L(dξ). The coefficients of a LFSM are stable random variables and then do not have a finite second order moment. In our frawemork, w jk is an infinitely divisible random variable with moments of every order. Since a RHFLM X H has stationary increments, its coefficients w jk are stationary in k for each fixed j according to [START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens[END_REF]. We provide here a direct proof of this result. Note that our framework is not studied in [START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens[END_REF] since X H is not (in general) a self-similar process. Then, contrary to [START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens[END_REF], w jk is not a self-similar process in j. However, we can establish the asymptotics of w jk as j → +∞.

Lemma 2.1 (wavelet type coefficient of RHFLM). Let (X H (t)) t∈R be a RHFLM with Hurst exponent H. Define

w jk = 2 j/2 X H (t)ψ(2 j t -k)dt, (10)
where the function ψ has r ≥ 1 vanishing moments. We have the following properties:

1. w jk = R ψ jk (ξ) |ξ| H+1/2 L(dξ) a.s. where f (ξ) = R exp(itξ)f (t)dt.
2. (Stationarity) w jk ∼ w j0 , for k = 0, . . . , 2 j -1, but the coefficients are not independent.

3. (Moments) Set N 0 ∼ N (0, E(w 2 00 )). For all integer p 0, E(w 2p+1 jk ) = 0 and as j → +∞, for each fixed k

2 pj+2pH E(w 2p jk ) → E(N 2p 0 ). 4. (Asymptotic Normality) As j → +∞, √ 2 j(1+2H) w jk (D)
-→ N 0, E(w 2 00 ) .

Proof.

• In view of the definition of the process X H ,

w jk = R ψ jk (t) R e -itξ -1 |ξ| H+1/2 L (dξ) dt.
Since ψ(•) has r ≥ 1 vanishing moments, R ψ jk (t)dt = 0 [START_REF] Delbeke | Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion[END_REF] and then the first statement is immediate provided we can exchange the integral in ξ with the integral in t. However, this exchange is not trivial since the stochastic integral in ξ is defined in a L 2 framework. We first point out that since ψ is a function with compact support, ψ jk is a C ∞ -function which vanishes at infinity. Define

g jk (ξ) = ψ jk (ξ) |ξ| H+1/2 , ξ ∈ R\{0}.
Since ψ satisfies [START_REF] Delbeke | Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion[END_REF], g is a square integrable function and

α jk = R ψ jk (ξ) |ξ| H+1/2 L(dξ)
is well defined. Then, using the isometry property (5) and the independence of the random measures M (dξ) and W (dξ), one easily proves owing to the Fubini Theorem that E |w jkα jk | 2 = 0, which leads to the first statement.

• By definition of ψ jk ,

w jk = 2 -j/2 R ψ(2 -j ξ) |ξ| H+1/2 exp 2 -j ikξ L(dξ)
Recall that the control measure ν(dz) is invariant by rotation. Then, (3) and ( 4) gives

w jk (D) = 2 -j/2 R ψ(2 -j ξ) |ξ| H+1/2 L (dξ) = w j0 .
• Since ψ has r ≥ 1 vanishing moments, R tψ(t)dt = 0, and then, as ξ → 0, ψ jk (ξ) = O(ξ 2 ) and g jk ∈ L p (R) for every p ≥ 2. The wavelet type coefficient w jk has moments of every order in view of Proposition 2.2 in [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF]. We first point out that by definition of L (dξ), w jk is a real-valued symmetric random variable which implies ∀p ∈ N, E w 2p+1 jk = 0.

In order to obtain the asymptotic behaviour of E w 2p jk as j → +∞, we apply Proposition 2.2 in [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF] which links this moment to the deterministic L 2q -norms

g jk 2q 2q = R |g jk (ξ)| 2q dξ = 2 j(1-2q-2qH) g 00 2q 2q .
Let us first assume that L(dξ) = M (dξ). Then Proposition 2.2 in [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF] leads to 2 ,

E(w 2p jk ) = 2 -2pj-2pH p m=1 2 jm (2π) m Lm m q=1 (2l q )! g 00 2lq 2lq +∞ 0 ρ 2lq ν ρ (dρ) (l q !)
where Lm stands for the sum over the set of partitions L m of {1, . . . , 2p} in m subsets K q such that the cardinality of K q is 2l q with l q 1. Therefore, as j → +∞,

2 pj+2pH E(w 2p jk ) → (2π) p Lp p q=1 (2l q )! g 00 2lq 2lq +∞ 0 ρ 2lq ν ρ (dρ) (l q !) 2 . ( 12
)
Note that by definition of Lp , in [START_REF] Delbeke | A wavelet based estimator for the parameter of selfsimilarity of fractional Brownian motion[END_REF], for each q, l q = 1, which leads to

2 pj+2pH E(w 2p jk ) → (2π) p 2 p g 00 2p 2 +∞ 0 ρ 2 ν ρ (dρ) 2p card(L p ).
Hence, as j → +∞,

2 pj+2pH E(w 2p jk ) → card(L p ) E w 2 00 p .
Since card(L p ) = (2p)!/(2 p p!), as j → +∞,

2 pj+2pH E(w 2p jk ) → E N 2p 0 .
Therefore we have proved the third statement in the case where L(dξ) = M (dξ). Note that this statement is evident when L(dξ) = W (dξ). Hence, the independence between W (dξ) and M (dξ) gives the conclusion for any Lévy random measure L(dξ) = aM (dξ) + bW (dξ).

• By stationarity (see second statement), we can assume k = 0. Equations ( 3) and ( 4) gives the characteristic function of w j0 . A simple change of variable (λ = 2 -j ξ) and a dominated convergence argument leads to the conclusion.

Hence a RHFLM is indexed by the single parameter H. It controls both the correlation structure of the process and the smoothness of its sample paths. In the two following sections, we define a moment based estimator and present main results concerning its asymptotic behaviour in both cases where the random process is observed directly or in a regression framework.

Estimation procedure and properties without observation noise

The moment based estimator derives from the observation that the statistical second order moment of the wavelet type coefficients w jk of a RHFLM obey a certain scaling property. Namely, we get

E(w 2 jk ) = 2 -j(1+2H) C 2 H , (13)
where C H is given by

C 2 H = a 2 + b 2 C |z| 2 ν (dz) R | ψ(λ)| 2 |λ| 1+2H dλ = A R | ψ(λ)| 2 |λ| 1+2H dλ = 0
since ψ is a continous function with suppψ = ∅.

If we observe a continous path of the RHFLM, hence it is possible to compute directly the true wavelet type coefficients of the random process. However, as in [START_REF] Bardet | Statistical study of the wavelet analysis of fractional Brownian motion[END_REF], we also tackle the case where the RHFLM is observed at discrete times, which induces in the estimation, a discretization error. In both cases, we construct a consistent estimator.

Continuous time observations without noise

Suppose we observe the wavelet type coefficients of a RHFLM with unknown Hurst exponent. The coefficients are defined as a growing array of random variables w jk , k = 0, . . . , 2 j -1, j = 0, . . . , J, where J is the maximum number of levels in the wavelet type expansion. Theorem 3.1 (Consistency of moments wavelet type estimator). Consider the Hurst exponent estimator of a RHFLM defined as

ĤJ = -1 2J log 2   2 J -1 k=0 w 2 Jk   (14)
Provided ψ has r ≥ 1 vanishing moments, the following asymptotics holds ĤJ J→+∞ -→ H, a.s.

Proof. Define for all J > 0

V J = 2 J -1 k=0 w 2 Jk , ṼJ = 2 2JH V J .
Hence,

E(V J ) = 2 -2JH C 2 H , E ṼJ = C 2 H = E w 2 00 . Applying Formula 20 in [7], Var(V J ) = 2B2 -J-4JH | ψ(λ)| 4 |λ| 2+4H dλ +2A 2 2 -J-4H 2 J -1 l=-2 J +1 e ilλ | ψ(λ)| 2 |λ| 1+2H dλ 2 +2 -2J-4JH B 2 J -1 l=-2 J +1 e 2ilλ | ψ(λ)| 4 |λ| 2+4H dλ = (I) + (II) + (III). with A = 4a 2 C |z| 2 ν(dz) + b 2 and B = 4a 2 C |z| 2 ν(dz).
For the second term, we first point out that ψ is continuously differentiable. Then using that lim |λ|→+∞ ψ(λ) = 0 and ψ(0) = 0, partial integration yields that

e ilλ | ψ(λ)| 2 |λ| 1+2H dλ = -1 il e ilλ d dλ | ψ(λ)| 2 |λ| 1+2H dλ.
Hence we can conclude that

e ilλ | ψ(λ)| 2 |λ| 1+2H dλ 2 c 1 + l 2 ,
for c a given finite positive constant and every λ ∈ R.

Let us recall that tψ(t)dt = 0, so that ψ(ξ) = O |ξ| 2 as ξ → 0. The third term is handled the same way (owing two integrations by parts), which enables us to conclude that

e 2ilλ | ψ(λ)| 4 |λ| 2+4H dλ c 1 + l 2 , ∀l ∈ R.
Finally, we obtain that the order of the variance is given by the terms (I) and (II) so

VarV J ∼ D H 2 -J-4JH , with 0 < D H = 2B R | ψ(λ)| 4 |λ| 2+4H dλ + 2A 2 ∞ l=-∞ R e ilλ | ψ(λ)| 2 |λ| 1+2H dλ 2 = 2B R | ψ(λ)| 4 |λ| 2+4H dλ + 4A 2 ∞ l=1 R cos (lλ)| ψ(λ)| 2 |λ| 1+2H dλ 2 + 2C 4 H < +∞ This implies after renormalization that Var ṼJ +∞ ∼ D H 2 -J => +∞ J=1
Var ṼJ < +∞.

The Borel Cantelli lemma yields that

2 2JH V J J→+∞ -→ E(w 2 00 ) = 0, a.s .
Finally we obtain the result

-1 2J log 2 (V J ) J→+∞ -→ H a.s .

Discretized version

Suppose we observe X H (l/N ), l = 0 . . . N . The size of data is then N + 1 and the wavelet type coefficients w jk can not be evaluated. However, the discretized coefficients

w n jk = 1 2 j/2 n n p=1 X H p + nk 2 j n ψ( p n ), k = 0 . . . 2 j -1, j = 0 . . . J
can be computed taking n = n(j) such that 2 -j N/n(j) ∈ N. Also, we replace w Jk by w n Jk in ( 14) and then obtain a new estimator which can be computed. Next theorem gives its consistency. Note that the parameter of discretization n depends on j and that the minimal size of data we need is 2 J n(J) + 1. In the following, we then take N = 2 J n(J).

Theorem 3.2. Consider the Hurst exponent estimator of a RHFLM defined as

ĤJ = -1 2J log 2   2 J -1 k=0 (w n Jk ) 2   (15)
with n = n(J) = 2 J and with size of data N + 1 = 2 2J + 1. Provided ψ has r ≥ 1 vanishing moment, the following asymptotics holds

ĤJ J→+∞ -→ H, a.s.
Before proving Theorem 3.2, we state a result about the difference w jk -w n jk in norm L 2 . In particular, a rate of convergence of w n jk in norm L 2 is given for each fixed j and k. Moreover, from the result established, we will deduced a comparison between the estimators defined by ( 14) and ( 15) and how to link n and J in [START_REF] Flandrin | Wavelets for scaling processes[END_REF], and then the size of data N + 1 and J, in order to obtain the asymptotic behaviour of (15). Lemma 3.3. There exists a constant C > 0 such that for every (j, k, n), j ∈ N\{0}, k = 0 . . . 2 j -1, n ∈ N\{0},

2 2jH E w jk -w n jk 2 ≤ C 1 2 j n 2H + 2 2jH 2 j n 2 . Proof of Lemma 3.3. Let f H (x, ξ) = e -ixξ -1 |ξ| H+1/2 and h n (k, ξ) = 1 0 f H (u + k, ξ)ψ(u) du - 1 n n p=1 f H p n + k, ξ ψ p n . Since w jk -w n jk = 2 -j/2 R h n k, 2 -j ξ L(dξ),
in view of ( 5),

E w jk -w n jk 2 = 2 -j A R h n k, 2 -j ξ 2 dξ = 2 -j-2jH A R |h n (k, λ)| 2 dξ.
Then let us write

h n (k, λ) = n p=1 h n,p (k, λ), with h n,p (k, λ) = p/n (p-1)/n f H (u + k, λ)ψ(u) -f H p n + k, λ ψ p n du.
By the Minkowski inequality,

2 2jH E w jk -w n jk 2 ≤ 2 -j A n p=1 h n,p (k, •) L 2 (R) 2 . (16)
Moreover, the Cauchy-Schwarz inequality implies that

|h n,p (k, λ)| 2 ≤ 1 n p/n (p-1)/n f H (u + k, λ)ψ(u) -f H p n + k, λ ψ p n 2 du.
Therefore, applying the Minkowski inequality, we obtain:

h n,p (k, •) L 2 (R) ≤ A n,p (k) + B n,p (k)
where

A n,p (k) = 1 √ n p/n (p-1)/n f (u + k, •) -f p n + k, • 2 L 2 (R) ψ 2 p n du 1/2 and B n,p (k) = 1 √ n p/n (p-1)/n f (u + k, •) 2 L 2 (R) ψ(u) -ψ p n 2 du 1/2 .
Let us first point out that

f (u + k, •) -f p n + k, • 2 L 2 (R) = f u - p n , • 2 L 2 (R) = u - p n 2H f (1, •) 2 L 2 (R) .
As a result, there exists a finite constant M 1 such that

A n,p (k) = M 1 n H+1 ψ p n ,
for every n, p, j and k.

In addition,

B n,p (k) = 1 √ n f (1, •) L 2 (R) p/n (p-1)/n |u + k| 2H ψ(u) -ψ p n 2 du 1/2
Then owing to a Taylor expansion, one proves that there exists a constant M 2 such that for every n, p, j and k,

B n,p (k) ≤ M 2 (k + 1) H n 2 sup (p-1)/n≤x≤p/n |ψ ′ (x)| ≤ 2 jH M 2 n 2 sup (p-1)/n≤x≤p/n |ψ ′ (x)|.
Then, in view of ( 16),

2 jH E w jk -w n jk 2 ≤ 2 -j A M 1 n 1+H n p=1 ψ p n + 2 jH M 2 n 2 n p=1 sup (p-1)/n≤x≤p/n |ψ ′ (x)| 2 .
We then point out that Let us now prove Theorem 3.2.

Proof of Theorem 3.2. Let ṼJ,n = 2 2JH 2 J -1 k=0 (w n Jk ) 2 .
We compare ṼJ,n to ṼJ defined in proof of Theorem 3.1. Define DJ,n = ṼJ,n -ṼJ .

Since DJ,n 2 ≤ 2 2JH 2 J -1 k=0 (w Jk -w n Jk ) 2 , by lemma 3.3, E DJ,n 2 ≤ C 1 n 2H + 2 2JH n 2 ,
where C is a finite constant. Taking n = n(J) = 2 J , the Borel Cantelli lemma yields that as J → +∞, DJ,2 J → 0, a.s.

Recall that , as J → +∞, ṼJ → E w 2 00 a.s.

Then, ṼJ,2 J → E w 2 00 a.s, which leads to the convergence of ĤJ defined by [START_REF] Flandrin | Wavelets for scaling processes[END_REF].

Regression framework

Suppose we observe noisy data from a RHFLM X H

Y H l N = X H l N + σ N ε l , l = 0, . . . , N, (17)
where ε l are an i.i.d sample of Gaussian random variables N (0, 1) and σ N is the noise level. We point out that 2 ). This framework is the usual regression setting well studied in statistics, which models natural observations of a phenomenom, observed at discrete times.

σ N = O(N - 1
The discretized coefficients w n jk of X H can not be observed. However, the discretized coefficients of Y H

d n jk = 1 2 j/2 n n p=1 Y H p + nk 2 j n ψ( p n
), k = 0 . . . 2 j -1, j = 0 . . . J, can be computed, taking N = 2 2J and n = n(j) such that 2 -j N/n(j) ∈ N as in Section 3.

Theorem 4.1. Consider the estimator of the Hurst coefficient of a RHFLM defined as

ĤJ = -1 2J log 2   2 J -1 k=0 (d n Jk ) 2   (18)
with n = n(J) = 2 J and N = 2 2J . Provided ψ has at least r ≥ 1 vanishing moments, the following asymptotics holds ĤJ J→+∞ -→ H, a.s.

Proof. We proceed as in the proof of Theorem 3.2. Let us recall that

ṼJ,n = 2 2JH 2 J -1 k=0 (w n Jk ) 2
and define

WJ,n = 2 2JH 2 J -1 k=0 (d n Jk ) 2 .
Then,

E WJ,n -ṼJ,n 2 ≤ 2 2JH 2 J -1 k=0 E (d n Jk -w n Jk ) 2 .
By definition of Y H ,

d n Jk -w n Jk = 1 2 J/2 n σ N n p=1 ψ p n ε p+nk ∼ N 0, σ 2 N 2 J n 2 n p=1 ψ 2 p n .
Hence,

E WJ,n -ṼJ,n 2 ≤ 2 2JH σ 2 N n 2 n p=1 ψ 2 p n .
Since lim n→+∞ As a result, we obtain consistency of the estimators in the practical case where the RHFLM is observed at discrete times. This result can be used to build an estimator of the Hurst exponent when studying the outcome of an experiment though to behave like a RHFLM. Once this parameter is estimated it becomes possible to try to model the data by a realization of a RHFLM. Hence, it should be of interest to construct a test based on the estimator of the Hurst exponent. So the asymptotic distribution of the estimator is needed. However, such a result is very difficult to obtain. On the one hand, the nature of the observations prevents the use of Central Limit type theorems. On the other hand, evaluating the distance between the distribution of the coefficients of a RHFLM and a Gaussian distribution is also far beyond the scope of this paper due since the calculations can only be handled using (3), preventing the use of weak dependency theorems. Nevertheless a consistency result is a first step in a modeling attempt with RHFLM.

Numerical study

In this section, consider the issue of estimating the Hurst exponent of a RHFLM observed with a Gaussian white noise. In the simulations we present in Figure 1 and 2 simulated data in straight lines, obtained using the procedure described in [START_REF] Lacaux | Series representation and simulation of multifractional Lévy motions[END_REF], together with the noisy paths in dotted lines.

First we aim at studying the rate of convergence of the wavelet type estimator. The estimator of the Hurst exponent is constructed using rescaled Daubechies wavelet Db(4) with r = 4 vanishing moments. We construct the estimator of the Hurst exponent for the two previous RHFLMs with the two different values of the Hurst exponent H = .4 and H = .7. The data are dyadic N = 2 2J and we increase the data from J = 2 to J = 7. For the noisy data, we perform 30 replications of the estimation procedure and plot the boxplots for the absolute error for different values of J in Figure 3 and Figure 4. In both cases, we can see that the convergence is achieved with N > 2 8 observations. Such study highlights consistency of the estimate. But, due to the relative slowness of the rate of convergence together with the lack of an asymptotic distribution law, it is difficult to estimate the efficiency of such an estimation procedure for real data. However we can compare the efficiency of our estimator with the estimator given in [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF]. We compute this estimator and the estimator constructed in Section 3, when H = .4 for two cases, whether or not the RHFLM is observed with a Gaussian white noise. We take N = 2 10 observations and consider 30 replications. We plot the distribution of both estimators, first the distribution of the wavelet type estimator and then the distribution of the quadratic variation based estimator. We observe in Figure 5 that without noise, the two estimators give the same kind of results. But, as expected, in a noisy setup, in Figure 6, the estimator based on quadratic variation does not concentrate around the true value of the Hurst parameter. This result is also highlighted by the boxplot of the absolute error of the two estimators, in Figure 7. The absolute error of estimation is important in the noisy setup for the quadratic type estimator.

Finally, we study the relative influence in the choice of the mother wavelet by computing the distribution of the absolute estimation error of the estimation of H = .4 with noisy data for 30 simulations and two different sets of coefficients, obtained for a rescaled [6] A. Benassi, S. Cohen, and J. Istas. Identifying the multifractional function of a Gaussian process. Statistic and Probability Letters, 39:337-345, 1998.

1 0

 1 |ψ ′ |(u) du < +∞, which concludes the proof.

= R ψ 2 2 ≤ 2 2 J

 2222 (u)du = 1, there exists a constant C ∈ (0, +∞) such that for every J and n,E WJ,n -ṼJ,n 2JH Cσ 2 N n .Also, taking n = n(J) = 2 J , the Borel Cantelli Lemma yields that WJ,2 J -ṼJ,2 J J→+∞ -→ 0 a.s., as soon as σ N = O(N -1/2 ) with N = 2 2J . Then, since (see proof of Theorem 3.log 2 2 -2JH WJ,2 JJ→+∞-→ H a.s., as soon as σ N = O(N -1/2 ) with N = 2 J n(J) = 2 2J .

Figure 1 :

 1 Figure 1: Raw and Noisy Data with H = .4

Figure 2 :Figure 3 :

 23 Figure 2: Raw and Noisy Data with H = .7

Figure 4 :Figure 5 : 4 )Figure 6 :

 4546 Figure 4: Absolute Estimation Error for noisy data with H = .7

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Boxplot for quadratic variation estimator and wavelet type estimator (H = .4)


Master RSN Aurore Bellens
email: aurore.bellens@orange.fr

Natalia Grabar
email: natalia.grabar@univ-lille3.fr

Nathalie Valmond Le Banc

Eric Kergosien
email: eric.kergosien@univ-lille3.fr

Fabien Eloire
email: fabien.eloire@univ-lille1.fr

Marta Severo Dicen

Social Media and European Cultural Routes: Instagram Networks on the Via Francigena

Keywords: J.4 [Computer Applications]: Social And Behavioral Sciences -Sociology Algorithms, Management, Experimentation, Human Factors, Theory Cultural Routes, Instagram, Social Networks, Cultural Heritage, Spatial Feature, Text Mining

In the last few years, European Cultural Routes have become important tourist destinations. What makes this kind of tourism interesting is both the complexity of values and significances that motivate it and the variety of social actors (walkers, inhabitants, institutions, commercial actors, etc.) that are involved. Empirical studies on this phenomenon are still limited because they are really time-consuming and resource-demanding. Yet, social media data seem to provide new perspectives on this field by generating digital traces of the tourists' behavior. This paper aims at exploring how social media data can be employed in order to study tourism on European Cultural Routes. We will consider the case study of the platform Instagram on the route "Via Francigena".

INTRODUCTION

European Cultural Routes are those itineraries based on a historic route, a cultural concept, figure or phenomenon, conveying a transnational importance and significance for the understanding and respect of common European values. Since 1987, they have been protected by an official Programme established by the Council of Europe (http://culture-routes.net/). Until now, 33 itineraries have been awarded with this European label and several others are currently applying for a similar certification. Among them, we can mention the Way of St. James to Santiago de Compostela, the Via Francigena between Rome and Canterbury, the Cluniac Sites in Europe, the Viking Routes, etc.

One of the main goals of Programme is favouring touristic attraction and facilitating management of these trails [START_REF] Denu | The evolution of the Council of Europe Cultural Routes programme[END_REF]. As a matter of fact, in the last few years, the number of visits of the cultural routes has strongly increased. What makes this kind of tourism interesting is both the complexity of values and significances that motivate the visits and the variety of social actors (walkers, inhabitants, institutions, commercial actors, etc.) that are involved [START_REF] Mariotti | Tourism and Cultural Routes: clusters, cultural districts and tourism systems[END_REF]. Yet, empirical studies on this phenomenon are still limited because they are really time-consuming and resource-demanding. For the most famous itineraries, such as the Way of St. James and the Via Francigena, the institutions that manage the routes provide few statistical data mainly about frequentation: how many pilgrims, their age, gender, etc. [3]. Occasionally, more interesting information about the walkers and their motivation or social phenomena along the trail have been collected thanks to fieldworks on small portions of the trail [START_REF] Conti | The European Cultural Routes as engine for sustainable development[END_REF]. Yet, results of these studies can hardly be generalized on all the trail. Today, social media data seem to provide new perspectives on this field of study. By catching digital traces of the tourists' actions, they are expected to provide new information on social interactions and opinions related to places [START_REF] Edwards | Digital social research, social media and the sociological imagination: Surrogacy, augmentation and reorientation[END_REF]. This paper aims at exploring how social media data can be used in order to study tourism related to European Cultural Routes. This paper presents a preliminary case study, based on the use of the platform Instagram on the route "Via Francigena". Then, methods developed for this case will be reproduced on other itineraries.

Via Francigena is a 1800 km itinerary stretching from Rome to Canterbury across Italy, Switzerland, France and UK. It has been taken by more than 50,000 pilgrims in 2014. While in the past centuries it was mainly a pilgrimage route, today people walking along it have most of the time different motivations: culture, tourism, wellness, environment or spiritual ones.

In Section 2, we introduce the context and the rationale of our work. Section 3 presents our methodology. Section 4 introduces the preliminary results of the project: it first describes social interactions along the Via Francigena and then presents an approach to automatically mine territories along the Via Francigena from Instagram data. Finally, section 5 gives some concluding remarks and perspectives for future research works.

STATE-OF-THE-ART

Several scholars have already validated the use of social media data for studying phenomena related to tourism [START_REF] Chareyron | Big data: A new challenge for tourism[END_REF]. Indeed, the interest of social media is that any interaction based on them leaves traces that can easily be saved, stored and massively retrieved. This intrinsic traceability of digital platforms promises, if it is controlled by an appropriate methodology, to provide a new source of data for the study of social facts [START_REF] Lazer | Life in the network: the coming age of computational social science[END_REF]. In the field of tourism, geotagged photos published on social network sites are surely the more exploited objet for studying people' activities and behaviour related to touristic places [START_REF] Kisilevich | Eventbased analysis of people's activities and behavior using Flickr and Panoramio geotagged photo collections[END_REF]. More particularly, considering the case of cultural routes, Chayeron et al. [START_REF] Chareyron | Mining tourist routes using flickr traces[END_REF] showed that the analysis of digital traces retrieved on the photo-sharing platform Flickr can help to rebuild the spatial path of a given tourist and to identify the most popular tourist points.

Although there are numerous empirical studies of this type, we agree with [START_REF] Chareyron | Big data: A new challenge for tourism[END_REF] that the use of social media data raises several methodological and theoretical issues and, consequently, more research is still necessary. In particular, they risk to produce quite naïve findings that can be already obtained by mainstream tourist studies. This article will contribute to this field of research by investigating how social media data can be used for studying social phenomena related to complex geographical and social touristic objects, such as the cultural routes. Yet, this study doesn't require big data techniques, because the corpus is not so vast and various. As said, we will consider the case of the Via Francigena through the analysis of the traces left by users of Instagram.

Among the existing social network platforms, visual social media such as Flickr, Panoramio and Instagram have emerged as the most suitable sites for studying social phenomena related to touristic places. Lately, Instagram has become the social platform most used by tourists for sharing pictures of the top moments of their trips. From the social scientist's viewpoint, the interest of Instagram, as all the other visual social media, is that each publication is linked to space and time. Indeed, it is connected to a specific couple of geographical coordinates and fixed in a precise time span. Thanks to this feature, Hochman and Manovich [START_REF] Hochman | Zooming into an Instagram City: Reading the local through social media[END_REF], several others after them [START_REF] García-Palomares | Identification of tourist hot spots based on social networks: A comparative analysis of European metropolises using photosharing services and GIS[END_REF], have elaborated advanced techniques to identify social and cultural patterns based on large visual datasets especially for studying urban tourism.

Even if Instagram is a visual social media based on photo sharing, it is important to highlight that photos are generally accompanied by a textual caption (Figure 1). Caption may include two types of items very useful for tourism research's sake: hashtags and mentions. Hashtags have been already used for studying exchanges content [START_REF] Highfield | A methodology for mapping Instagram hashtags[END_REF] while mentions allow performing social network analysis on Instagram [START_REF] Manikonda | Analyzing user activities, demographics, social network structure and usergenerated content on instagram[END_REF]. In particular, several studies related to social media textual data have described the particular way of expressing named entities in natural language from social network data. Among these works, a first family has focused on the identification and the categorization of named entities (person, place et organization) using Wikipedia [START_REF] Pospecu | Mining Tourist Information from User-Supplied Collections[END_REF] [START_REF] Oussalah | Identifying and Extract Named Entities from Wikipedia Database Using Entity Infoboxes[END_REF]. In these works, the authors seek to take into account only toponym names as places (for examples Rome, Paris, London, etc.). Despite significant works in spatial entities (SE) recognition, a second smaller family has focused on the identification of more complex SE. The TEXT2GEO method [START_REF] Tahrat | Text2Geo : from textual data to geospatial information[END_REF] presents an hybrid approach that combines natural language patterns and a machine learning approach. The combination of both methods significantly improves precision and recall results 1 . However, The main disadvantage of machine learning approaches is that a part of data has to be indexed manually, and that is a complex task and timeconsuming.

In this paper, we will focus on these two items: (i) on hashtags, especially to investigate more deeply the spatial entities related to the cultural route cited in the captions of photos related to the Via Francigena; (ii) on mentions, in order to investigate the dynamics of social connections related to the cultural route. In the next paragraphs we briefly present the methodology employed for studying these two aspects and the obtained results.

METHODOLOGY

In this paper, we study the content related to the Via Francigena published on Instagram between September 2011 and April 2016. Thank to the tool Hashtag Instagram Explorer [START_REF] Borra | Digital Methods Initiative[END_REF] developed by the DMI of the University of Amsterdam, we could extract and analysed 7 034 photos (and related captions) including the following hashtags:

#viafrancigena, #francigena, #camminafrancigena, #francigenatrip and #visitfrancigena. We carried out two techniques of analysis: (i) social network analysis and (ii) automated textual analysis.

As regards the first analysis, we looked at whether it was possible to reconstitute social networks between Instagram users based on mentions included in captions. Indeed, Instagram allows users to identify (and "mention") other users related to the posted photo, and, consequently, create a link with them. Thus it is possible to retrace these links by building a "co-mention" network. In our case, all Instagram accounts can be either receiver or sender.

In order to extract mentions included in our corpus, we, first, developed a Python program capable to identify and extract the name(s) of other user(s) included in each 'post'. Second, we build a directed graph where nodes are the users and the arcs correspond to mentions of the users. Finally, we used the software Gephi and Pajek to visualize the graph and analyse it, thanks to a social network analysis approach. We considered the following measures: density, centrality, reciprocity and cluster measures.

As regards the second analysis, we seek to identify complex spatial entities (SE) from Instagram textual data in order to apprehend territories that Via Francigena crosses. To achieve this, we propose a semi-automatic method by combining a morphosyntactic approach with a lexicon approach based on the list the Via Francigena's stops and on Wikipedia data.

To create the list of SE from Wikipedia, we have selected Wikipedia articles related to places located in regions that Via Francigena crosses. Each SE is enriched with several metadata extracted from the article (city, region, description, geocoordinates, hyperlink to the article). Then, the indexation of Instagram data using the lexicon allowed us to generate a list of SE useful to describe the Via Francigena. For each SE, one (or two) label is attached to describe its type (stop or other place from Wikipedia).

Finally, a disambiguation step is applied on spatial entities, identified using Wikipedia data. From the list of SE, a SE is validated if its region or city name is identified in one or more Instagram message. Geo-coordinates extracted from Wikipedia allow calculating the distance between each SE and the stops. We have carried out experiments on Instagram data in order to identify the SEs that are related to the Via Francigena stops (5, 10, 15, 20, 30, 50, 100km). A correct distance is validated if the SE candidate is related to the cultural route into Instagram textual data. Then, a SE is validated if it is located less than 15 kilometers from the nearest stop. For example, the SE named « passo della cisa » is identified into 3 messages and characterized by Wikipedia as a place from a region that Via Francigena crosses. This SE is validated because it is located at 6 kilometers from the stop called "Berceto". The results were evaluated in terms of precision and recall.

DISCUSSION 4.1 Studying social interactions along the Via Francigena

In this section, we are interested in the networks that users of Instagram, that have used different hashtags linked to Via Francigena (between August 2014 and April 2016, a total of 4 619 photos), have created when they 'mention' each other within their 'caption'. As said, our data were composed of two kinds of data: first, a list of users, which will be represented by nodes in our graph; and second, a list of directed arcs (based on co-mentions within 'posts'), which will be represented by links in our graph. The network aggregates 304 nodes and is composed of 442 arcs (see Figure 2).

Figure 2. The co-mention network of users using hashtags related to the Via Francigena

The network average degree for a node, that is the mean number of links received and sent by one node, is equal to 18, but this number masks large disparities, because, for instance, the more central node (that is franigenatoscana) has a degree centrality of 122 links, whereas 30% of nodes just have one link. The density of this network2 is very weak: 0.5%. What is interesting to note is that we clearly have a 'core-periphery' network structure: 84 nodes (28%) belongs to what we call the 'periphery', because they own only one tie within the network; whereas the 'core' is composed of 220 nodes (72%). The fact that users practice 'auto-mention' in the network, lead to the existence of some isolated nodes, which lower the density measure.

In order to analyze the core of the network, we decided to delete the periphery, that is to say 84 nodes (see Figure 3). The density remains weak, 0.7%, but quite higher than that of the global network. Regarding the reciprocity phenomenon 3 , it appears that less than 5% of the links are reciprocal, which is very low compared to the networks of the real life where reciprocity is a quite common feature of social relations. It indicates that these kinds of digital networks allow people to express their emotions about their experience on the cultural route, but that, objectively, they don't foster contacts and exchanges with others. Similarly, more than 20% of links just arise from people who auto-mention themselves outside any other kind of contact. So mentions are not really meant to create connections with other walkers but mostly to build their own visibility and digital reputation. This structural description of the network can be fruitfully completed by considering nodes' features: who are the people or entities that have participated to generate this kind of network related to "Via Francigena" cultural route? We have built a typology of users dividing them in four classes according to their nature: class 1 (yellow nodes) is composed of individuals (physical persons, including some politicians); class 2 (green nodes), of tourism associations or institutions; class 3 (pink nodes), of firms involved in the tourism sector (for instance: hotels and restaurants); and class 4 (blue nodes), of Italian tourism websites.

Core

Periphery

Core + Periphery Class 1 (yellow) Individuals

Table 1. The 4 classes of nodes within Instagram's co-mention network

Individuals (class 1) represent more than 75% of the network users. In most cases, they are tourists or pilgrims who have walked on Via Francigena: they are not Instagram's specialists but they just share pictures from their different trips. One also finds globe-trotters, such as for instance italian_backpacker. Finally, in this class 1, we also find two local politicians: the first one is Nicola Zingaretti (nicolazingaretti on Instagram), president of the Latium region in Italy ; and the second one is Enrico Rossi (rossipresidente on Instagram), president of the Tuscany region. It is interesting to note that our methodology based on co-mention networks allows to easily identify 'false' individuals, i.e. computer robots programmed to be "social capitalists" [19] searching for e-reputation 4 , which appear on Instagram5 but disappear from our network because they use only popular hashtags to create links and increase their visibility but never use the co-mention's strategy.

Among associations and institutions (class 2), one finds, for instance, the official tourist offices' pages of Lazio, Tuscia and Pavia regions; but also the visittuscany page, which is subsidized by the Tuscany region and European Union, whose aim is to promote events and good addresses (i.e. hotels, restaurants) and whose strategy is to re-post, through its profile, pictures from blogs or individuals. Among firms (class 3) one finds lodges or restaurants, which are mentioned by walkers, staying or eating in these places; one also finds firms proposing visits of the region, as chiantiprivatetour, which mentions the most popular associations or institutions as to benefit from their visibility. Finally, among tourism websites (class 4), one finds groups of amateurs, whose aim is to share their passion for the region or Via Francigena. For instance, francogenatoscana which re-posts the pictures of walkers thanks to the hashtag #francigenatoscana.

In terms of degree centrality, seven accounts are really central: francigenatoscana (122 received or sent links), an account which invites people to post photos of Tuscany with the ashtag #francigenatoscana to allow them to repost these photos, igersviterbo (111), an account which aims to discover the city of Viterbo, tusciafotografia (105), corresponding to a tourism website to discover Tuscany, slowtravelfest (85), a music festival, visitttuscany (84), the official account of the Tuscany Region, ristorantedabenedetta (80) a restaurant in the city of Vetralla, and volgotoscana (57), an association which aims at discovering the Toscana Region. Interestingly, none of these accounts belongs to individuals' class 1: t the three most mentioned users are tourism websites, others two are firms and the last two are in class 2 'associations or institutions'.

All these seven main accounts belong the core-network and play an intermediary role, in the sense of the betweenness centrality 6 . But it is interesting to note that two nodes, that are not central in terms of degree, are, however, central in terms of betweenness. The first one is Viefrancigene_aevf, which is the official account of EAVF 7 . The posts in which it is mentioned are those concerning events on the Via Francigena, or those of walkers posting photos of landscapes or monuments of the Via Francigena. The second account is Ilariadibagio, which is an Italian photographer, whose account contains essentially pictures of landscapes. She mentions only three other users: a magazine, a tourism website and a festival.

Coming back to the four classes we have qualitatively identified, they are different, not only because of their intrinsic characteristics, but also because of their structural position within the network. The structural analysis of the network (see Figure 4 8 ) offers a better visibility concerning the structure of the flows of mentions within and between each class. Some classes are not central in the network, such as class 1 and 3, whereas others are prominent because they send and/or, especially, receive many links, such as class 4 and 2. This structural result is not accidental, but can be explained in terms of the roles played by the different kinds of actors within the network. Individuals (from class 1) are not central because their posts are seldom, but as they are numerous, they actively participate to the centrality of associations-institutions (from class 2) and tourism websites (from class 4) by diffusing information of these two kinds of actors. Thus, the structural role of individuals is to improve the visibility of others. Yet, it is not their main goal, but an effect of their willingness to share their experience of the cultural route, by posting links related to it. Conversely, the centrality of classes 2 and 4 is generated by the fact that these actors receive links three times more than they send. Finally, the firms (from class 3) send links more than they receive, which indicates that they try to generate activity and obtain visibility around their profile, in a commercial perspective. This strategy seems not entirely efficient at that time because they remain, in average, little central in the network.

Although network density is weak, class 1 is the denser one (0.6% within the core network): indeed, individuals often practice the 'co-mention' (which relates to reciprocity phenomenon) between them, compared to the actors of other classes (52 comentions on 90 within the whole network). It is due to the fact that some people mention others with whom they walk and have an Instagram account, such as soniapallai who belongs to a group of hikers and who mentions her companions in several of her posts. Her pictures can be group photos, but also landscape photos. This confirms that Via Francigena's walkers are not necessarily pilgrims or foreigners, but also local inhabitants which walk, for pleasure, with their friends. Moreover, individuals are those who have the most numerous links with the actors of other classes. If they mention a lot tourism websites (from class 4) and associations-institutions (from class 2), this is due to the fact that some accounts of these classes re-post some individuals' pictures. Thus, walkers employ hashtags from tourism websites and associations-institutions: it is 7 European Association of Via Francigena, i.e. the association in charge of the Via Francigena. 8 In this graph, we have grouped all users of the same class in a single node (procedure named "shrink" in Pajek software). The size of the nodes is proportional to the number of users of each class (see Table 1), and the weight of links corresponds to the total number of links between each class of users.

a very simple way for them to share their experience of Via Francigena. Conversely, tourism websites and associations-institutions don't make links between each other (see Figure 4): this may be explained by the fact that these accounts are competitors in the war for the e-reputation. This can be the case, for instance, of accounts that have a similar goal such as two cyclists' clubs. Or, this can be the case of websites financed by advertising, such as eventi_della_tuscia 9 which revenues is subject to its audience (in terms number of visitors, numbers of clicks on advertising).

To conclude, social network analysis offers an original point of view to understand how different kinds of actors mobilize social media to achieve their goals. In this case, Instagram's data allow us to build a network of co-mentions, which reveals some density, centrality, reciprocity and clustering phenomena. In further researches, we plan to build the same kind of networks from Twitter and Facebook, to see if there are common users, roles and practices on these social media.

Mining spatial entities along the Via Francigena

The corpus employed in the second analysis is constituted by 2598 Instagram posts (in Italian, English and French) published between September 2011 and January 2016. As previously stated, our methodology is organized in two phases: (i) extraction of stops and SE related to the Via Francigena from Wikipedia and (ii) disambiguation of SE.

Based on these first exploratory treatments, we can provide some statistical figures. Firstly, 2099 of 2598 posts contain one or more relevant SE. In them, 12917 occurrences of SE have been identified: 4359 related to Via Francigena stages and 8558 related to other ES identified on Wikipedia. Figures 5 and6 highlight the most popular ES respectively stages and SE from Wikipedia. The Wikipedia data allowed finding other types of SE than cities. For instance, in Figure 6 Bolsena Lake appears 86 times in the corpus. We also can identify churches and cathedrals that are situated near a stop. Moreover, as regards the mentions of countries, not surprisingly we noticed that the country appearing most often is Italy which represent 95% of the countries mentions. The other countries are very poorly represented with 3% for Switzerland and 1% for France and England. For the regions of Italy, the region of Tuscany is the most popular with 85% of the SE, 6% for Lazio, 5% for the Aosta Valley and 1% for the Piemonte, Lombardia, Emilia-Romagna and Liguria.

To evaluate our processes, we calculated the precision and recall on a sample of 50 posts. A group of three annotators manually annotated all SE from the 50 posts. This allowed us to evaluate the quality of our variant SE extraction method. On the sample of 50 posts, the recall is 1 and the precision is 0,97. Indeed, irrelevant SE are identified. For example, "Fondo", which is a part of the city "Traversella" is also a common noun, and the meaning is "the lake bottom".

As a final step, a demonstrator has been developed as a web application based on Java/Javascript and Google Maps API technologies. An extract is shown in figure 7. This demonstrator allows a user to visualize the Via Francigena cultural route (green color); and to show on the map relevant information related to SE identified using Wikipedia (red color). Links to complete description of SE available on Wikipedia are being integrated.

CONCLUSIONS

In the last few years, social media data have emerged as an original and rich source of information for investigating social phenomena related to tourism. Even if numerous empirical studies have been recently developed in this field, several methodological and theoretical aspects need further exploration. In this paper, we present the preliminary results of the PEPS Study on "Digital representations of European Cultural Routes" that will contribute to this field of research. Particularly, it focuses on the use of social media data, notably Instagram data, for studying European Cultural Routes. The case of the Via Francigena has been considered.

Based on the preliminary results presented here, we may conclude that the interest of this case is double. Firstly, while previous studies have mainly focused on specific tourist points of interest (notably cities), it shows the potential of social media data for investigating a complex touristic objet such as a cultural route. Text related to photos on Instagram has allowed us to identify most popular stops and localities related to the cultural route. In particular, we could develop based on Wikipedia in order to identify geographical places mentioned on Instagram posts. Secondly, this case study shows the interest to analyze textual data in visual social media. In particular, the analysis of mentions included in Instagram posts may allow investigating social interaction related to the cultural route. Yet, as a conclusion, it is also important to remind the limits and precautions necessary for using social media database [START_REF] Zeynep | Big questions for social media big data: Representativeness, validity and other methodological pitfalls[END_REF]. Consequently, even if we already obtained interesting results, they need to be validated through further research. Firstly, findings of this case study will be tested through in-deep interviews to some of the identified Instagram users. Secondly, we will incorporate the usage of wikidata and commons to improve the SE extraction process 10 . Thirdly, the methods developed here will be tested on other cultural routes.

Figure 1 .

 1 Figure 1. Example of Instagram photo with caption including mentions.

Figure 3 .

 3 Figure 3. Core of the co-mention network related to the Via Francigena.

Figure 4 .

 4 Figure 4. Intra-and inter-class links: a structural approach of the co-mention network.

Figure 5 .Figure 6 .

 56 Figure 5. Distribution of the Via Francigena stops by counting the number of occurrences in Instagram data

Figure 7 .

 7 Figure 7. Mapping of SE related to Via Francigena from Instagram data

I.e. The difference between the number of observed links and the number of links if all the nodes were connected.

I.e. When A sends a link to B and B sends also a link to A.

These robots are usually hidden behind 'false' young American profiles, who post at least one time per hour.

We tried to contact them several times but they never answered us, except the fact that they sent us a subscription proposal to our profile, after we had subscribed ourselves to their profile.

Betweenness centrality is the number of times a user is a path between others actors within the network.

http://www.eventidellatuscia.it

ACKNOWLEDGMENTS

This research has been founded by the University of Lille and CNRS through a grant PEPS/University of Lille. More information about the global project "Digital representations of European Cultural Routes" at the url www.itinerairesculturels.fr.