S A Padoan

M Ribatet

S A Sisson
email: scott.sisson@unsw.edu.au

Likelihood-based inference for max-stable processes

Keywords: Composite likelihood, Extreme value theory, Max-stable processes, Pseudo-likelihood, Rainfall, Spatial Extremes

The last decade has seen max-stable processes emerge as a common tool for the statistical modelling of spatial extremes. However, their application is complicated due to the unavailability of the multivariate density function, and so likelihood-based methods remain far from providing a complete and flexible framework for inference. In this article we develop inferentially practical, likelihood-based methods for fitting max-stable processes derived from a composite-likelihood approach. The procedure is sufficiently reliable and versatile to permit the simultaneous modelling of joint and marginal parameters in the spatial context at a moderate computational cost. The utility of this methodology is examined via simulation, and illustrated by the analysis of U.S. precipitation extremes.

Introduction

A common objective of spatial analysis is to quantify and characterise the behavior of environmental phenomena such as precipitation levels, windspeed or daily temperatures. A number of generic approaches to spatial modelling have been developed (e.g. [START_REF] Barndorff-Nielsen | Stochastic Methods in Hyrdology[END_REF]; [START_REF] Cressie | Statistics for Spatial Data[END_REF]; [START_REF] Ripley | Spatial statistics[END_REF]), but these are not necessarily ideal for handling extremal aspects given their focus on mean process levels. Analyses of spatial extremes are useful devices for understanding and predicting extreme events such as hurricanes, storms and floods. In light of recent concerns 1 over climate change, the use of robust mathematical and statistical methods for such analyses has grown in importance While the theory and statistical practice of univariate extremes is well developed, there is much less guidance for the modelling of spatial extremes. This is problematic as many environmental processes have a natural spatial domain. We consider a temporal series of componentwise maxima of process measurements recorded at k = 1, . . . , K locations, within a contiguous region. Observations {y n,k } each denote the maximum of m samples over n = 1, . . . , N temporal blocks. For example, for daily observations, m = 366 implies the {y n,k } describe process annual maxima.

The spatial analogue of multivariate extreme value models is the class of maxstable processes [START_REF] De Haan | A spectral representation for max-stable processes[END_REF][START_REF] De Haan | Stationary min-stable stochastic processes[END_REF][START_REF] Resnick | Extreme values, point processes and regular variation[END_REF]. Maxstable processes have a similar asymptotic motivation to the univariate Generalised Extreme Value (GEV) [START_REF] Von Mises | La distribution de la plus grande de n valeurs[END_REF][START_REF] Jenkinson | The frequency distribution of the annual maximum (or minimum) values of meteorological elements[END_REF]), providing a general approach to modelling process extremes incorporating temporal or spatial dependence. Statistical methods for max-stable processes and data analysis of practical problems are discussed by [START_REF] Smith | Max-stable processes and spatial extremes[END_REF], [START_REF] Coles | Regional modelling of extreme storms via max-stable processes[END_REF], [START_REF] Coles | Directional modelling of extreme wind speeds[END_REF] and [START_REF] Coles | Modelling extremes of the areal rainfall process[END_REF]. Standard likelihood methods for such models are complicated by the intractability of the multivariate density function in all but the most trivial cases. This presents an obstacle in the use of max-stable processes for spatial extremes.

There is a lack of a proper inferential framework for the analysis of spatial extremes (although De [START_REF] De Haan | Spatial extremes: Models for the stationary case[END_REF] describe some non-parametric estimators). In this article we develop flexible and inferentially practical methods for the fitting of max-stable processes to spatial data based on non-standard, composite likelihood-based methods [START_REF] Lindsay | Composite likelihood methods[END_REF]. An appealing feature of this approach is that the estimation of GEV marginal parameters can be performed jointly with the dependence parameters in a unified framework. Accordingly, there is no need for separate estimation procedures. With highly-structured problems such as maxstable processes, this approach produces flexible and reliable results with a moderate computational cost.

The article is organised as follows: Section 2 reviews the theory of max-stable processes and its relationship to spatial extremes. Our composite likelihood approach is developed in Section 3 and Section 4 evaluates the method's performance through a number of simulation studes. We conclude with an illustration of a real extremal data analysis of U.S. precipitation levels.

2 Max-stable processes and spatial extremes

Max-stable processes

Max-stable processes provide a natural generalisation of extremal dependence structures in continuous spaces. From this, closed-form bivariate distributions can be derived.

Definition.

Let T be an index set and { Ỹi (t)} t∈T , i = 1, . . . , n be n independent replications of a continuous stochastic process. Assume that there are sequences of continuous functions a n (t) > 0 and b n (t) ∈ R such that

Y (t) = lim n→∞ max n i=1 Ỹi (t) -b n (t) a n (t)
, t ∈ T.

If this limit exists, the limit process Y (t) is a max-stable process de Haan (1984).

Two properties follow from the above definition De Haan and [START_REF] De Haan | Limit theory for multivariate sample extremes[END_REF]. Firstly, the one-dimensional marginal distributions belong to the class of generalised extreme value distributions (GEV), Y ∼ GEV(µ, λ, ξ) with distribution function

F (y; µ, λ, ξ) = exp -1 + ξ(y -µ) λ -1/ξ + , -∞ < µ, ξ < ∞, λ > 0,
where a + = max(0; a) and µ, λ and ξ are respectively location, scale and shape parameters [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF]. Secondly, for any K = 2, 3, . . ., the K-dimensional marginal distribution belongs to the class of multivariate extreme value distributions. W.l.o.g. if a n (t) = n, b n (t) = 0 ∀t, then the corresponding process, {Z(t)} t∈T , has unit Fréchet margins, with distribution function F (z) = exp(-1/z), z > 0. This process is obtainable as standardisation of {Y (t)} t∈T through

{Z(t)} t∈T ≡ 1 + ξ(t)(Y (t) -µ(t)) λ(t) 1/ξ(t) + t∈T
, where µ(t), ξ(t) and λ(t) > 0 are now continuous functions. The process Z is still a max-stable process. If Z is also stationary, the process may be expressed through it's spectral representation [START_REF] De Haan | Stationary min-stable stochastic processes[END_REF].

In detail, let {X j , U j } j≥1 be a Poisson process, Π, on R n × R + , with counting measure Π(•) := Σ j I (X j ,U j ) (•) and intensity measure ν(dx) × u -2 du, where I (X j ,U j ) (A) is the indicator function of the random number of points falling in a bounded set A ⊂ R n × R + and ν is a positive measure. For a nonnegative measurable (for fixed

t ∈ T ) function f (x-t) such that R n f (x-t)ν(dx) = 1, ∀t ∈ T the stochastic process Z(t) := max j=1,2,... {U j f (X j -t)}, t ∈ T, (1)
is a stationary max-stable process de [START_REF] De Haan | A spectral representation for max-stable processes[END_REF]. [START_REF] Smith | Max-stable processes and spatial extremes[END_REF] reinterprets this process in terms of environmental episodes such as storm phenomena, in which U, X and f represent respectively storm magnitude, the center and the shape. [START_REF] Schlather | A dependence measure for multivariate and spatial extreme values: Properties and inference[END_REF] term this the storm profile model.

For a finite set of indexes t 1 , . . . , t K ∈ T and positive thresholds z 1 , . . . , z K for K ∈ N, the distribution of the random vector Z(t 1 ), . . . , Z(t K ) is de Haan ( 1984)

Pr{Z(t k ) ≤ z k , k = 1, . . . , K} = exp - R n max 1≤k≤K f (x -t k ) z k ν(dx) . (2)
It then follows that the marginal distributions are unit Fréchet:

Pr{Z(t) ≤ z} = exp -z -1 R n f (x -t)ν(dx) = exp(-1/z).
Alternative spectral representations of max-stable processes exist [START_REF] Schlather | Models for stationary max-stable random fields[END_REF].

Extremal coefficients

Given n independent realisations of a random vector Y ∈ R d , the joint distribution of componentwise maxima satisfies [START_REF] De Haan | Limit theory for multivariate sample extremes[END_REF][START_REF] Resnick | Extreme values, point processes and regular variation[END_REF] Pr max

k max j=1,...,n Y (j) k /n ≤ z = Pr max j=1,...,n Y (j) 1 /n ≤ z θ = exp(-θ/z), z > 0,
for k = 1, 2, . . . , K and common threshold z, where the rightmost term is a Fréchet(θ) distribution. The parameter 1 ≤ θ ≤ K is the extremal coefficient and it measures the extremal dependence between the margins, an important practical quantity in applications [START_REF] Smith | Max-stable processes and spatial extremes[END_REF]. The information in the extremal coefficient reflects the practical number of independent variables. If K is finite then θ = 1 indicates complete dependence, whereas θ = K demonstrates full independence.

In the max-stable process framework, from (2), for all z > 0 we have Pr{Z(t k ) ≤ z, k = 1, . . . , K} = exp(-θ/z), and so

θ = R n max 1≤k≤K {f (x -t k )} ν(dx).
where θ again represents the effective number of independent variables. [START_REF] Schlather | A dependence measure for multivariate and spatial extreme values: Properties and inference[END_REF] discuss the extremal coefficient within a max-stable context.

Spatial models

Suppose now that T ⊆ R 2 and that {X j } j≥1 are random points in R 2 . While for K > 2, the general K-dimensional distribution function under the max-stable process representation (1) permits no analytically tractable form, a class of bivariate spatial models is available when the storm profile model, f , is a bivariate Gaussian density and µ is a Lebesgue measure [START_REF] Smith | Max-stable processes and spatial extremes[END_REF][START_REF] De Haan | Spatial extremes: Models for the stationary case[END_REF]. In this case, for locations t i and t j the bivariate distribution function of

{Z(0), Z(h)} is Pr{Z(0) ≤ z i , Z(h) ≤ z j } = exp - 1 z i Φ a(h) 2 + 1 a(h) log z j z i - 1 z j Φ a(h) 2 + 1 a(h) log z i z j , (3)
where h = (t j -t i ) ⊤ , 0 is the origin, Φ is the standard Gaussian distribution function, a(h) = (h T Σ Σ Σ -1 h) 1/2 and Σ Σ Σ is the covariance matrix of f , with covariance σ 12 and standard deviations σ 1 , σ 2 > 0. A derivation of (3) is in Appendix A.2. A general max-stable process with a Gaussian storm profile model, f , is termed a Gaussian extreme value process [START_REF] Smith | Max-stable processes and spatial extremes[END_REF], whereas the specific model ( 3) is the Gaussian extreme value model [START_REF] Coles | Regional modelling of extreme storms via max-stable processes[END_REF].

Second-order partial derivatives of (3) yield the 2-dimensional density function

f (z i , z j ) = exp - Φ(w(h)) z i - Φ(v(h)) z j Φ(w(h)) z 2 i + ϕ(w(h)) a(h)z 2 i - ϕ(v(h)) a(h)z i z j Φ(v(h)) z 2 j + ϕ(v(h)) a(h)z 2 j - ϕ(w(h)) a(h)z i z j + v(h) ϕ(w(h)) a(h) 2 z 2 i z j + w(h) ϕ(v(h)) a(h) 2 z i z 2 j , ( 4
)
where ϕ is the standard Gaussian density function, w(h) = a(h)/2 + log(z j /z i )/a(h) and v(h) = a(h) -w(h). The derivation of (4) is in Appendix A.3. Observe that a(h) measures the strength of extremal dependence: a(h) → 0 represents complete dependence, and (in the limit) a(h) → ∞ indicates complete independence. In accordance with spatial models, the extreme dependence between Z(0) and Z(h) decreases monotonically and continuously with h = t j -t i De [START_REF] De Haan | Spatial extremes: Models for the stationary case[END_REF], and for fixed h the dependence decreases monotonically as a(h) increases. Accordingly, characterisation of extremal dependence is determined by the covariance, Σ Σ Σ, which is therefore of interest for inference.

Due to high-dimensional distributional complexity the study of extremal dependence is commonly limited to pairwise components through the extremal coefficients

θ(h) = R 2 max{f (x), f (x -h)}dx, 1 ≤ θ(h) ≤ 2.
The dependence on h is explicit. Specifically for the Gaussian extreme value model, θ(h) = 2Φ(a(h)/2), following an argument along the lines of Appendix A.2. Alternative models result by considering e.g. exponential or t storm profile models De Haan and Pereira ( 2006), or stationary Gaussian process profile models [START_REF] Schlather | Models for stationary max-stable random fields[END_REF].

Likelihood-based inference

The analysis of spatial extremes is concerned with the joint modelling of a spatial process at large numbers of data-recording stations in a fixed region. As discussed in Section 2, the lack of closed-form distribution for max-stable processes in greater than K = 2 dimensions precludes straightforward use of standard maximum likelihood methods for this class of models. We now develop inferentially practical, likelihood-based classes of max-stable processes derived from a composite-likelihood approximation [START_REF] Lindsay | Composite likelihood methods[END_REF][START_REF] Varin | On composite marginal likelihoods[END_REF]. The procedure is sufficiently reliable and versatile to permit the simultaneous and consistent modelling of joint and marginal parameters in the spatial context at a moderate computational cost.

Composite likelihoods

For a parametric statistical model F with density function family

F = {f (y; ψ ψ ψ), y ∈ Y ⊆ R K , ψ ψ ψ ∈ Ψ ⊆ R d },
and a set of marginal or conditional events {I k : k ∈ K} (for some K ⊆ N) subset of some sigma algebra on Y, the composite log-likelihood is defined by

ℓ C (ψ ψ ψ; y) = k∈K log f (y ∈ I k ; ψ ψ ψ),
where log f (y ∈ I k ; ψ ψ ψ) is the log-likelihood associated with event I k . First-order partial derivatives of ℓ C (ψ ψ ψ; y) with respect to ψ ψ ψ yield the composite score function

D ψ ψ ψ ℓ C (ψ ψ ψ; y), from which maximum composite likelihood estimator of ψ ψ ψ, if unique, is obtained by solving D ψ ψ ψ ℓ C ( ψ ψ ψ MCLE ; y) = 0. Similarly, second-order partial derivatives of D ψ ψ ψ ℓ C (ψ ψ ψ; y) yield the Hessian matrix H ψ ψ ψ ℓ C (ψ ψ ψ; y) (Appendix A.1).
The key utility of the composite log-likelihood is it's ability, under the usual regularity conditions, to provide consistent and unbiased parameter estimates when standard likelihood estimators are not available. Under appropriate conditions [START_REF] Lindsay | Composite likelihood methods[END_REF][START_REF] Cox | A note on pseudolikelihood constructed from marginal densities[END_REF] the maximum composite likelihood estimator is consistent and asymptotically distributed as

ψ ψ ψ MCLE ∼ N(ψ ψ ψ, Ĩ(ψ ψ ψ) -1 ) with Ĩ(ψ ψ ψ) = H(ψ ψ ψ) J(ψ ψ ψ) -1 H(ψ ψ ψ), where H(ψ ψ ψ) = E{-H ψ ψ ψ ℓ C (ψ ψ ψ; Y)} and J(ψ ψ ψ) = V{D ψ ψ ψ ℓ C (ψ ψ ψ; Y)}
are analogues of the expected information matrix and the variance of the score vector. Although the maximum composite likelihood estimator can be unbiased, it may not be asymptotically efficient in that Ĩ(ψ ψ ψ) -1 , the inverse of the Godambe information matrix [START_REF] Godambe | An optimum property of regular maximum likelihood estimation[END_REF], may not attain the Cramér-Rao bound [START_REF] Cox | A note on pseudolikelihood constructed from marginal densities[END_REF].

3.2

The pairwise setting for spatial extremes form (3) implies a pairwise composite log-likelihood may be constructed as

ℓ P (ψ ψ ψ; y) = N n=1 K i=1 K-1 j=i+1 log f (y n,i , y n,j ; ψ ψ ψ), (5)
where each f (y n,i , y n,j ; ψ ψ ψ) is a bivariate marginal density based on data at locations i and j, taken over all distinct location pairs. In order to characterized limiting behavior by a max-stable process (1) we require unit Fréchet marginal distributions. Accordingly, we consider the bijection (Y i , Y j ) = g(Z i , Z j ) with inverse function given by

Z i = 1 + ξ i (Y i -µ i ) λ i 1/ξ i + Z j = 1 + ξ j (Y j -µ j ) λ j 1/ξ j + (6)
where Z i ≡ Z(t i ) and Y j ≡ Y (t j ), and for each marginal Y , the constants µ, ξ and λ > 0 ensure that Z is unit Fréchet distributed. The resulting bivariate density is

f Y i ,Y j (y i , y j ) = f Z i ,Z j g -1 (y i , y j ) |J(y i , y j )|,
where f Z i ,Z j (z i , z j ) denotes the density of the Gaussian extreme value model ( 4), and

|J(y i , y j )| = 1 λ i λ j 1 + ξ i (y i -µ i ) λ i 1/ξ i -1 + 1 + ξ j (y j -µ j ) λ j 1/ξ j -1 + .
This change of variable permits the use of GEV marginals (over unit Fréchet) without reforming the problem definition. Hence, the pairwise log-likelihood (5) allows simultaneous assessment of the tail dependence parameters (3) between pairs of sites and also the location, scale and shape parameters of the marginal distribution at each location. The parameters can not be estimated as an analytical solution of the composite score equation. Nonetheless quasi-Newton numerical maximization routines (e.g. Broyden, 1967) can be applied in order to obtain maximum likelihood estimates.

Variances of parameter estimates are provided through inverse of the Godambe information matrix, with estimates of the matrices H(ψ ψ ψ) and J(ψ ψ ψ) given by

H( ψ ψ ψ MCLE ) = - N n=1 K i=1 K-1 j=i+1 H ψ ψ ψ log f (y n,i , y n,j ; ψ ψ ψ MCLE ) and J( ψ ψ ψ MCLE ) = N n=1 K i=1 K-1 j=i+1 D ψ ψ ψ log f (y n,i , y n,j ; ψ ψ ψ MCLE ) D ψ ψ ψ log f (y n,i , y n,j ; ψ ψ ψ MCLE ) ⊤ ,
each evaluated at the composite maximum likelihood value. In practice the matrix H is obtained straightforwardly with the numerical maximization routine employed for likelihood maximization. An explicit expression for J is derived in Appendix A.4.

In principle, estimating unique marginal parameters for each location ensures correct model application by respecting marginal constraints, though computational issues arise for large numbers of parameters. Alternatively, as is common in the modelling of univariate extremes, we may describe the GEV parameters through parsimonious regression models, which may be functions of space, environmental and other covariates and random effects. Specifically, we may express each parameter as

η(x) ≡ h{f (x)} = Xβ β β, ( 7
)
where h is a link function, x = (x 1 , . . . , x d ) is a vector of predictors, X is a N × (d + 1) design matrix, and β β β is a (d + 1) × 1 vector of unknown parameters.

For further flexibility, a non-parametric approach may provide a useful alternative. Non-parametric modelling of univariate extreme value responses has been recently proposed by Chavez-Demoulin and [START_REF] Chavez-Demoulin | Generalised additive modelling of sample extremes[END_REF], [START_REF] Yee | Vector generalised linear and additive extreme value models[END_REF] and [START_REF] Padoan | Mixed model-based additive models for sample extremes[END_REF]. The work of [START_REF] Kammann | Geoadditive models[END_REF], who developed a non-parametric spatial regression with Gaussian response, may be extended to the current spatial extremes setting.

Model selection

There are two model selection approaches under the composite likelihood framework. For nested models [START_REF] Varin | On composite marginal likelihoods[END_REF] the p-dimensional parameter, ψ ψ ψ is partitioned as ψ ψ ψ = (ψ ψ ψ ′ , ψ ψ ψ ′′ ), where ψ ψ ψ ′ is q-dimensional, and testing ψ ψ ψ ′ = ψ ψ ψ 0 versus a two-sided hypothesis proceeds via the composite likelihood ratio test statistic

W (ψ ψ ψ 0 ) = 2{ℓ C ( ψ ψ ψ) -ℓ C (ψ ψ ψ 0 , ψ ψ ψ ′′ (ψ ψ ψ 0 ))}.
Under the null Kent (1982)

W ∼ q j=1 ν i χ 2 i , (8)
where χ 2 i are independent χ 2 1 random variables, and ν 1 ≥ . . . ≥ ν q are the eigenvalues of (H

ψ ψ ψ ′ ψ ψ ψ ′ ) -1 Ĩψ ψ ψ ′ ψ ψ ψ ′ .
Here, H ψ ψ ψ ′ ψ ψ ψ ′ and and Ĩψ ψ ψ ′ ψ ψ ψ ′ respectively denote the information matrix and the Godambe information matrix, each restricted to those elements associated with parameter ψ ψ ψ ′ . (Dependence on ψ ψ ψ under the null is omitted for brevity.) Hypothesis testing based on (8) either approximates null distribution using estimates of the eigenvalues ν i [START_REF] Rotnitzky | Hypothesis-testing of regression parameters in semiparametric generalised linear models for cluster correlated data[END_REF], or adjusts the composite likelihood such that the usual asymptotic χ 2 q null is preserved [START_REF] Chandler | Inference for clustered data using the independence loglikelihood[END_REF]. The composite likelihood information criterion (CLIC) [START_REF] Varin | A note on composite likelihood inference and model selection[END_REF], useful in the case of non-nested models, performs model selection on the basis of expected Kullback-Leibler divergence between the true unknown model and the adopted model (Davison, 2003, p. 123). In the composite likelihood context, this is the AIC under model mis-specification [START_REF] Takeuchi | Distribution of information statistics and criteria for adequacy of models (in japenese)[END_REF]; (Davison, 2003, p. 150-152). Model selection is based on the model minimising

-2{ℓ C ( ψ ψ ψ MCLE ; Y) -tr[ J( ψ ψ ψ MCLE ) H( ψ ψ ψ MCLE ) -1 ]},
where the second term is the usual composite log-likelihood penalty term.

Simulation Study

We now evaluate the utility of the composite likelihood in the spatial extremes context. We examine various forms of extremal dependence with the Gaussian storm profile (4) characterised through the covariance, Σ Σ Σ, including directional and strength of dependence variations (Table 1 and Figure 1). The covariance has direct meteorological interpretation and defines the extremal dependence directly. The K site locations are uniformly generated over a 40 × 40 region. Given the moderate computational demand for large site numbers, likelihood maximisation (and other) routines have been implemented in C and collected in the forthcoming R package SpatialExtremes. Table 2 summarizes estimator performance based on moderately sized datasets: K = 50 sites and N = 100 observations. Estimate means and standard errors over 500 data replications are reported, indicating good correspondence to the true values. There is no evidence of bias, even in cases where poorer performance may be expected such as very strong or weak dependence. Overall, the Godambe standard errors and sample standard deviations are consistent, though there is some discrepancy in the case of strong dependence. Here, the eigenvalues of Σ Σ Σ -1 4 are 5 ± √ 10/7500 and so we are near the parameter space boundary. Consequently, for some of the 500 replications, the asymptotic normality of ψ ψ ψ MCLE fails, and so the Godambe standard errors are not relevant.

Table 3: Normalised mean squared error of extremal coefficient estimates based on 500 data simulations with K = 50 sites and N = 100 observations. Estimators are the composite MLE and those proposed by [START_REF] Smith | Max-stable processes and spatial extremes[END_REF] and [START_REF] Schlather | A dependence measure for multivariate and spatial extreme values: Properties and inference[END_REF] Table 3 depicts normalised mean squared errors for three different estimators of the extremal coefficient functions: the composite MLE derived from the Gaussian extreme value model, and those proposed by [START_REF] Smith | Max-stable processes and spatial extremes[END_REF] and [START_REF] Schlather | A dependence measure for multivariate and spatial extreme values: Properties and inference[END_REF]. Normalized mean square errors are used to prevent the largest extremal coefficients from dominating. From Table 3, in general it is clear that the composite likelihood estimator is the most accurate. The standard deviation in the strong dependence case again suffers from the failure of the asymptotic normality of ψ ψ ψ MCLE .

Estimator performance for a range of dataset sizes (N = 10, 50, 100, 500) and site numbers (K = 10, 50, 100) is listed in Table 4, under 500 data replications of spatial dependence model Σ Σ Σ 3 . As expected, the simulations indicate that there is some bias and larger variance for small N, and negligible bias and small variance for large N. Observe that, for fixed sample size, the number of sites does not impact the estimation results. This behavior is illustrated in Figure 2, highlighting, in particular, good estimator performance with increasing N.

Table 4: Composite MLEs for a varying number of sites (K) and observations (N ), based

on 500 simulations of spatial extreme data using the Gaussian extreme value model with covariance Σ Σ Σ 3 (Table 1). Standard deviations are reported in parantheses.

K N = 10 N = 50 N = 100 N = 500 True σ 2 1 10 245 (120.2) 207 (43.8) 205 (31.7) 199 (13.3) 200 50 244 ( 90.4) 208 (37.5) 200 (28.3) 199 (11.4) 100 239 ( 94.3) 205 (37.8) 202 (30.4) 199 (11.5) σ 2 2 10 353 (159.1) 305 (63.9) 301 (44.8) 298 (19.5) 300 50 353 (131.6) 309 (56.6) 303 (44.3) 298 (16.9 Finally, we consider model selection under misspecification. Figure 3 illustrates power curves (likelihood ratio tests) and rejection rates (CLIC) for two hypotheses, each versus their complement. Namely, H 0 : σ 2 1 = 200 fixing σ 2 2 = 300 and σ 12 = 150 (top plots), and H 0 : σ 2 1 = σ 2 2 = 200 fixing σ 12 = 0 (bottom plots). All resulting curves are near-quadratic and, for the pairwise likelihood ratio based tests, rejection rates are close to the confidence level α = 0.05 when the null hypothesis is true. In this study, contrary to the results derived by [START_REF] Chandler | Inference for clustered data using the independence loglikelihood[END_REF], the adjustment of the W statistic (8) [START_REF] Rotnitzky | Hypothesis-testing of regression parameters in semiparametric generalised linear models for cluster correlated data[END_REF] appears to have slightly more power -even when testing mutiple parameters. In contrast the CLIC statistic demonstrates poor performance, with the rejection rate under the true model reaching only 20%.

Application to U.S. precipitation data

We illustrate the developed methodology in an analysis of U.S. precipitation data. These data consist of 46 gauging stations with daily rainfall records over a period of 91 years (Figure 4). We express the model GEV parameters as simple linear functions of longitude, lattitude and altitude. Further model variations, including additional environmental covariates, and regressions of the covariance Σ Σ Σ on these covariates to examine spatial variations in extremal dependence were not considered.

Exploratory analyses of the functional complexity of the GEV parameters was performed by fitting independent GEV models to the data from each station and evaluating appropriate surface responses using standard techniques (e.g. ANOVA, Fisher tests). After identifying an upper limit to model complexity, we perform model selection on the pairwise composite likelihood of the Gaussian extreme value process using the methodologies outlined in Section 3.3.

Table 5 summarizes some of the different models investigated. According to the CLIC criterion, model M 5 (in bold) is the preferred choice, although models M 2 and Table 5: Some Gaussian extreme value processes and their corresponding maximised negative composite log-likelihood, degrees of freedom and the CLIC score.

Model

-ℓ M 6 also appear competitive. Despite the relatively poor performance of the CLIC criterion for model selection (see Section 4), the same conclusions are obtained from the deviance based tests, with p-values around 0.9. For model M 5 , the covariance Σ Σ Σ is estimated as σ 2 1 = 0.06323 (0.06463), σ 12 = 0.01334 (0.03661) and σ 2 2 = 0.02581 (0.02538), where the parantheses indicate standard errors.

P ( ψ ψ ψ MCLE ; y) d.f. CLIC M 0 : µ(x) = α 0 + α 1 (lat) + α 2 (alt) + α 3 (lon) 412,110.2 12 12,848,229 σ(x) = β 0 + β 1 (lat) + β 2 (alt) + β 2 (lon) ξ(x) = γ 0 M 1 : µ(x) = α 0 + α 1 (lat) + α 2 (alt) 412,110.9 11 16,096,068 σ(x) = β 0 + β 1 (lat) + β 2 (alt) + β 3 (lon) ξ(x) = γ 0 M 2 : µ(x) = α 0 + α 1 (lat) + α 2 (alt) + α 3 (lon) 412,113.3 11 1,008,997 σ(x) = β 0 + β 1 (lat) + β 2 (alt) ξ(x) = γ 0 M 3 : µ(x) = α 0 + α 1 (lat) + α 3 (lon) 412,234.1 11 1,926,389,242 σ(x) = β 0 + β 1 (lat) + β 2 (alt) + β 3 (lon) ξ(x) = γ 0 M 4 : µ(x) = α 0 + α 1 (lat) + α 2 (alt) + α 3 (lon) 412,380.5 11 33,209,042 σ(x) = β 0 + β 1 (lat) + β 3 (lon) ξ(x) = γ 0 M 5 : µ(x) = α 0 + α 1 (lat) + α 2 (alt) 412,113.3 10 1,008,261 σ(x) = β 0 + β 1 (lat) + β 2 (alt) ξ(x) = γ 0 M 6 : µ(x) = α 0 + α 1 (lat) 412,237.3 9 1,086,347 σ(x) = β 0 + β 1 (lat) + β 2 (alt) ξ(x) = γ 0
Figure 5 (right plot) illustrates the spatial variation of pointwise 50-year return level estimates. Comparison to the regional elevation map (left plot) indicates that the most extreme precipitation events occur in mountainous regions. Figure 6 (left plot) also depicts the strength of spatial dependence through the extremal coefficient function. There is clear evidence of anisotropy, with stronger dependence in the north-east/south-west direction. Interestingly, this axis corresponds to the shape of the Appalachian Mountains as well as the coastline. Accordingly, this directional extremal dependence may be the consequence of storms following either the coastline or the massif. Finally, conditioning of a fixed site (and for a given threshold) by using the pairwise conditional distribution, the conditional q-year return level estimates can [START_REF] Rotnitzky | Hypothesis-testing of regression parameters in semiparametric generalised linear models for cluster correlated data[END_REF], and CB chol and CB svd [START_REF] Chandler | Inference for clustered data using the independence loglikelihood[END_REF] using Cholesky and singular value decompositions. Right panels: CLIC rejection rates. Test levels are α = 0.05.

Point estimates are based on 1000 data replications.

easily be provided. Figure 6 (right plot) illustrates the spatial variation of conditional pointwise 50-year return level estimates, given the fixed site indicated by the star.

Conclusion

As a natural generalisation of extremal dependence structures, max-stable processes are a powerful tool for the modelling of multivariate extremes. Unfortunately, the intractability of the multivariate density function precludes inference except in trivial cases (e.g. bivariate), or requires additional approximations and immense computational overheads [START_REF] Jiang | The indirect method: Inference based on intermediate statistics -A synthesis and examples[END_REF][START_REF] Sisson | Sequential Monte Carlo without likelihoods[END_REF][START_REF] Peters | On sequential Monte Carlo, partial rejection control and approximate Bayesian computation[END_REF]. This article has developed composite likelihood-based inferential methods for general max-stable processes. Our results demonstrate good applicability in the spatial context. The benefits of this likelihood-based approach are the flexible joint modelling of marginal and dependence parameters, coupled with good estimator behaviour with finite samples, all at moderate computational cost.

Modifications of the model formulation would draw alternative representations of extremal modelling into the composite-likelihood based framework, given the known links between these and block maxima (GEV) approaches (e.g. [START_REF] Coles | An Introduction to Statistical Modelling of Extreme Values[END_REF]). These include threshold excess models for marginals (Davison and Smith, 1990), and the limiting Poisson characterisation of extremes. The obvious practical benefit from these extensions would be the incorporation of more data into the modelling process.

F (z i , z j ) = exp - ∞ -∞ ∞ -∞ max f (x 1 , x 2 ) z i , f (x 1 -t 1 , x 2 -t 2 ) z j dx 1 dx 2 = exp - ∞ -∞ ∞ -∞ f (x 1 , x 2 ) z i I f (x 1 , x 2 ) z i ≥ f (x 1 -t 1 , x 2 -t 2 ) z j dx 1 dx 2 - ∞ -∞ ∞ -∞ f (x 1 -t 1 , x 2 -t 2 ) z j I f (x 1 -t 1 , x 2 -t 2 ) z j ≥ f (x 1 , x 2 ) z i dx 1 dx 2 ,
where f (x 1 , x 2 ) is the bivariate normal density of (X 1 , X 2 ) ∼ N((0, 0) ⊤ , Σ Σ Σ), and for brevity we set h = (t jt i ) ⊤ ≡ (t 1 , t 2 ) ⊤ . Recall from (3) that

a(h) = (h T Σ Σ Σ -1 h) 1/2 = 1 (1 -ρ 2 ) t 2 1 σ 2 1 - 2ρt 1 t 2 σ 1 σ 2 + t 2 2 σ 2 2 1/2
where ρ = σ 12 /σ 1 σ 2 . Consider first the case (t

1 σ 2 -ρt 2 σ 1 ) > 0. Note that f (x 1 , x 2 )/z i ≥ f (x 1 -t 1 , x 2 -t 2 )/z j implies that exp - 1 2(1 -ρ 2 ) x 2 1 σ 2 1 - 2ρx 1 x 2 σ 1 σ 2 + x 2 2 σ 2 2 /z i ≥ exp - 1 2(1 -ρ 2 ) (x 1 -t 1 ) 2 σ 2 1 - 2ρ(x 1 -t 1 )(x 2 -t 2 ) σ 1 σ 2 + (x 2 -t 2 ) 2 σ 2 2 /z i ⇔ x 1 ≤ σ 2 1 σ 2 2(t 1 σ 2 -ρt 2 σ 1 ) t 2 1 σ 2 1 - 2ρt 1 t 2 σ 1 σ 2 + t 2 2 σ 2 2 + σ 2 1 σ 2 (1 -ρ 2 ) t 1 σ 2 -ρt 2 σ 1 log z j z i - x 2 (t 2 σ 2 1 -ρt 1 σ 1 σ 2 ) t 1 σ 2 -ρt 2 σ 1 ⇔ x 1 ≤ c. From this, it follows that ∞ -∞ ∞ -∞ f (x 1 , x 2 ) z i I f (x 1 , x 2 ) z i ≥ f (x 1 -t 1 , x 2 -t 2 ) z j dx 1 dx 2 = 1 z i ∞ -∞ c -∞ 1 2π 1 -ρ 2 exp - 1 2(1 -ρ 2 ) x 2 1 σ 2 1 - 2ρx 1 x 2 σ 1 σ 2 + x 2 2 σ 2 2 dx 1 dx 2 = 1 z i ∞ -∞ ϕ(x 2 ) c -∞ ϕ x 1 -ρσ 1 x 2 /σ 2 σ 1 1 -ρ 2 dx 1 dx 2 = 1 z i ∞ -∞ ϕ(x 2 )Φ c -ρσ 1 x 2 /σ 2 σ 1 1 -ρ 2 dx 2 = 1 z i Φ 1 2 (1 -ρ 2 ) t 2 1 σ 2 1 - 2ρt 1 t 2 σ 1 σ 2 + t 2 2 σ 2 2 1/2 + t 2 1 σ 2 1 - 2ρt 1 t 2 σ 1 σ 2 + t 2 2 σ 2 2 -1/2 log z j /z i (1 -ρ 2 ) -1/2 = 1 z i Φ a(h) 2 + log z j /z i a(h) . Similarly, f (x 1 -t 1 , x 2 -t 2 ) z j ≥ f (x 1 , x 2 ) z i ⇔ x 1 ≥ c. It then follows that ∞ -∞ ∞ -∞ f (x 1 -t 1 , x 2 -t 2 ) z j I f (x 1 , x 2 ) z i ≥ f (x 1 -t 1 , x 2 -t 2 ) z j dx 1 dx 2 = 1 z j ∞ -∞ ∞ c 1 2π 1 -ρ 2 exp - 1 2(1 -ρ 2 ) (x 1 -t 1 ) 2 σ 2 1 - 2ρ(x 1 -t 1 )(x 2 -t 2 ) σ 1 σ 2 + (x 2 -t 2 ) 2 σ 2 2 dx 1 dx = 1 z j ∞ -∞ ϕ(x 2 -t 2 ) ∞ c ϕ (x 1 -t 1 ) -ρσ 1 (x 2 -t 2 )/σ 2 σ 1 1 -ρ 2 dx 1 dx 2 = 1 z j ∞ -∞ ϕ(x 2 -t 2 ) 1 -Φ c -ρσ 1 (x 2 -t 2 )/σ 2 σ 1 1 -ρ 2 dx 2 = 1 z j Φ 1 2 (1 -ρ 2 ) t 2 1 σ 2 1 - 2ρt 1 t 2 σ 1 σ 2 + t 2 2 σ 2 2 1/2 + t 2 1 σ 2 1 - 2ρt 1 t 2 σ 1 σ 2 + t 2 2 σ 2 2 -1/2 log z i /z j (1 -ρ 2 ) -1/2 = 1 z j Φ a(h) 2 + log z i /z j a(h) ,
and the form of the distribution (3) is confirmed. Observe, that the same result is obtained for the case (t 1 σ 2 -ρt 2 σ 1 ) < 0. See also [START_REF] Smith | Max-stable processes and spatial extremes[END_REF] and De [START_REF] De Haan | Spatial extremes: Models for the stationary case[END_REF].

where ψ ψ ψ ⊤ = (σ σ σ, β β β µ , β β β λ , β β β ξ ), σ σ σ ⊤ = (σ σ σ 2 11 , σ σ σ 12 , σ σ σ 2 22 ) and where each parameter β β β is p-dimensional vector of coefficients. The bivariate log-density has the form log f (y i , y

j ; ψ ψ ψ) = A + log(B ⊙ C + D) + E, where A = - Φ(w) z i - Φ(v) z j , B = Φ(w) z 2 i + ϕ(w) a ⊙ z 2 i - ϕ(v) a ⊙ z i ⊙ z j , C = Φ(v) z 2 j + ϕ(v) a ⊙ z 2 j - ϕ(w) a ⊙ z i ⊙ z j , D = v ⊙ ϕ(w) a 2 ⊙ z 2 i ⊙ z j + w ⊙ ϕ(v) a 2 ⊙ z i ⊙ z 2 j and E = log 1 λ i λ j 1 + ξ i y i -µ i 1 λ i 1 ξ i -1 + 1 + ξ j y j -µ j 1 λ j 1 ξ j -1 +
and where

1 ⊤ = (1, . . . , 1), µ i = (X β β β µ β β β µ ) i , ξ i = (X β β β ξ β β β ξ ) i and log(ψ i ) = (X β β β ψ β β β ψ ) i .
The GEV parameters are related to the predictors by the form (7). We assume identity link functions for the location and shape parameters and exponential for the scale. The term E corresponds to the log of the determinat of the Jacobian matrix associated with the transformation (6), see Section 3.2. The first-order derivative term of the square score statistic is defined by

D ψ ψ ψ log f (y i , y j ; ψ ψ ψ) = (D σ σ σ log f (ψ ψ ψ), D β β β µ log f (ψ ψ ψ), D β β β λ log f (ψ ψ ψ), D β β β ξ log f (ψ ψ ψ))
where for brevity we write log f (ψ ψ ψ) ≡ log f (y i , y j ; ψ ψ ψ). Vector differential calculus (e.g. [START_REF] Wand | Vector differential calculus in statistics[END_REF]) leads to

D σ log f (ψ ψ ψ) ≡ - v ⊙ ϕ(w) a ⊙ z i - w ⊙ ϕ(v) a ⊙ z j + C ⊙ (w 2 -1) ⊙ ϕ(v) a 2 ⊙ z 2 j + (1 + w ⊙ v) ⊙ ϕ(w) a 2 ⊙ z i ⊙ z j + B ⊙ (v 2 -1) ⊙ ϕ(w) a 2 ⊙ z 2 i + (1 + w ⊙ v) ⊙ ϕ(v) a 2 ⊙ z i ⊙ z j + (w -2v -w ⊙ v 2 ) ⊙ ϕ(w) a 3 ⊙ z 2 i ⊙ z j + (v -2w -v ⊙ w 2 ) ⊙ ϕ(v) a 3 ⊙ z 2 j ⊙ z i /(B ⊙ C + D) s ⊤ ,
where s ⊤ ≡ (t 2 1 , 2 t 1 t 2 , t 2 2 ), using the results

D σ σ σ w = v a s ⊤ , D σ σ σ Φ(w) = v ⊙ ϕ(w) a s ⊤ , D σ σ σ ϕ(w) = - w ⊙ vϕ(w) a s ⊤ and D σ σ σ v ⊙ ϕ(w) = w ⊙ (1 -v 2 )ϕ(w) a s ⊤ .
The first-order derivativies of v, Φ(v), ϕ(v) and w ⊙ ϕ(v) are the same as the above, substituting v for w. Similarly for the second term we have (1-a⊙w-w 2 )⊙ϕ(v) a 2 ⊙z i ⊙z 3 j -(1-a⊙v-v 2 )⊙ϕ(w)

D β β β µ log f (ψ ψ ψ) ≡ ϕ(w) + a ⊙ Φ(w) a ⊙ z 2 i - ϕ(v) a ⊙ z i ⊙ z j ⊙ z 1-ξ i i λ i (X β β β µ ) i + ϕ(v) + a ⊙ Φ(v) a ⊙ z 2 j - ϕ(w) a ⊙ z i ⊙ z j ⊙ z 1-ξ j j λ j (X β β β µ ) j +     C ⊙ w⊙Φ(v)
a 2 ⊙z j ⊙z 3 i ⊙ z 1-ξ j j λ j (B ⊙ C + D)     (X β β β µ ) j + (ξ i -1) λ i z ξ i (X β β β µ ) i + (ξ j -1) λ j z ξ j (X β β β µ ) j .
Observe that the above expression is obtained by deriving in order the components: D We have

D β β β λ z i ≡ - z i (y i -µ i 1) λ i (X β β β λ ) i and D β β β ξ z i ≡ 1 ξ i z 1-ξ i i (y i -µ i 1) λ i -z i log(z i ) (X β β β ξ ) i .
Finally, D β β β λ E ≡ (ξ i -1)(y i -µ i 1) λ i z i -1 (X β β β λ ) i + (ξ j -1)(y j -µ j 1) λ j z j -1 (X β β β λ ) j and

D β β β ξ E ≡ 1 ξ i (1 -ξ i )(y i -µ i 1) z ξ i i λ i -log(z i ) (X β β β ξ ) i + 1 ξ j
(1 -ξ j )(y j -µ j 1) z ξ j j λ j log(z j ) (X β β β ξ ) j .

In combination we obtain an expression for D ψ ψ ψ log f (y i , y j ; ψ ψ ψ), and from this the squared score statistic.

Figure 1 :

 1 Figure1: A Gaussian extreme value process realisation for Σ Σ Σ 1 , . . . , Σ Σ Σ 5 . Max-stable process simulation routines are available in the RandomFields package in R[START_REF] Schlather | Models for stationary max-stable random fields[END_REF].

Figure 2 :

 2 Figure 2: Composite MLE distribution estimates for N = 50 (dotted line), N = 100 (broken line) and N = 1000 (solid line) observations over K = 10 (top row), K = 50 (centre row) and K = 100 (bottom row) sites. Contours correspond to 0.25, 0.5 and 0.75 percentiles.

Figure 3 :

 3 Figure 3: Left panels: Power curves for the composite likelihood ratio tests; RJ (Rotnitzki

Figure 5 :Figure 6 :

 56 Figure 5: Left: Elevation map (metres) of the region; Right: Pointwise 50-year return level map (cm) estimated from the fitted Gaussian extreme value process.

a 2 ⊙z i ⊙z 2 j

 2

  β β β µ A, D β β β µ log(B ⊙ C + D) and D β β β µ E. These three components have the formD β β β µ f (x) = D z i f (x)D β β β µ z i + D z j f (x)D β β β µ z j , where D β β β µ z i = z 1-ξ i i /λ i X β β β µ .For this reason the derived expressions of D β β β λ log f (ψ ψ ψ) and D β β β ξ log f (ψ ψ ψ) are essentially the same but substituting D β β β µ z i with D β β β λ z i and D β β β ξ z i , and D β β β µ E with D β β β λ E and D β β β ξ E.

Table 1 :

 1 Extremal dependence configurations.

Table 2 :

 2 Composite MLE's based on 500 spatial extreme data simulations (K = 50 sites and N = 100) with the Gaussian extreme value model. True values are in [brackets]. Standard errors obtained through Godambe estimates and sample standard deviation (in parantheses).

		σ 2 1 / s.e.	σ 2 2 / s.e.	σ 2 12 / s.e.
	Σ Σ Σ 1 :	306 [300] / 40.6 (44.7)	306 [300] / 39.8 (41.5)	1 [0] / 27.9 (27.7)
	Σ Σ Σ 2 :	204 [200] / 26.7 (28.5)	305 [300] / 39.6 (39.7)	1 [0] / 21.9 (21.2)
	Σ Σ Σ 3 :	202 [200] / 25.1 (26.1)	300 [300] / 37.3 (37.9)	150 [150] / 25.5 (26.1)
	Σ Σ Σ 4 : 2053 [2000] / 495.2 (300.1) 3065 [3000] / 664.8 (483.1) 1550 [1500] / 412.0 (322.4)
	Σ Σ Σ 5 :	20 [20] / 1.5 (1.6)	30 [30] / 2.3 (2.3)	15 [15] / 1.6 (1.6)

  . Standard deviations are reported in parantheses. The values' order of magnitude is 10 -4 .

		Composite MLE	Smith	Schlather & Tawn
	Σ Σ Σ 1 :	3.1 ( 5.4)	18.4 (26.7)	17.3 (29.9)
	Σ Σ Σ 2 :	2.9 ( 4.8)	19.1 (30.3)	18.9 (28.6)
	Σ Σ Σ 3 :	2.5 ( 4.7)	21.9 (35.7)	21.1 (32.4)
	Σ Σ Σ 4 :	3.0 (35.6)	10.9 (18.9)	6.7 (13.3)
	Σ Σ Σ 5 :	0.3 ( 0.8)	30.3 (44.6)	25.5 (40.7)

  Locations of the 46 gauging stations.

								+ + + + + +	+ + + + + + + + + + + + + + + + + + + + + + + + + +	+ + + + + + + + + + + + + +	+ +
	Latitude (degree)	34 35 36 37 38 39 40	-84	-83	500 1000 1500 Figure 4: 0 -80 -79 -78 Longitude (degree) -82 -81	34 35 36 37 38 39 40	-84	-83	-82 Longitude (degree) -81 -80	-79	-78	12 14 16 18 20 22

Recall, for the spatial setting we have observations {y n,k }, each denoting the maximum of m samples over n = 1, . . . , N blocks and k = 1, . . . K locations in a continuous region. E.g. for daily observations, m = 366 implies the y n,k describe annual maxima. Accordingly, the K univariate marginals are approximately GEV distributed. Despite the intractability of the multivariate max-stable process, availability of the bivariate

Acknowledgements

This work was commenced while SAP was visiting the School of Mathematics and Statistics, University of New South Wales, Sydney, Australia. Their hospitality is gratefully acknowledged. The authors are grateful to Anthony Davison and Stuart Coles for their helpful suggestions, and to Richard Smith for providing the U.S. precipitation data. The work was supported by the CCES Extremes project, http://www.cces.ethz.ch/ projects/hazri/EXTREMES. SAS is supported by the Australian Research Council through the Discovery Project scheme (DP0877432).

Appendix

We present explicit expressions for the distribution function (3), density function (4) and the derivatives required for the estimated covariance matrix in Section 3.2

A.1: Vector notation

Let f be a real-valued function in the d × 1 vector x = (x 1 , . . . , x d ). Then the 1 × d derivative vector, D x f (x), has i-th element ∂f (x)/∂x i . The corresponding Hessian matrix is given by H Scalar functions applied to vectors are also evaluated element-wise. For example, a -1/ξ = (a

A.2: Derivation of the bivariate distribution function

In order to derive the cumulative distribution function (3), considering formula (2) and the assumptions of Section 2.3, we need to solve:

A.3: Derivation of the bivariate density function

In order to derive the bivariate density function (4) we require the second-order derivative of

with respect to z i and z j , where for brevity we set a ≡ a(h), w ≡ w(h) and v ≡ v(h) and write w = a/2 + log(z j /z i )/a and v = a -w. The differentiation gives

First-order differentation gives

using the results

∂Φ(w) ∂z

Second-order differentation yields

Substituting, we obtain the probability density function

A.4: An expression for the squared score statistic

From Section 3.2 the term J(ψ ψ ψ) of the Godambe information matrix can be estimated from

D ψ ψ ψ log f (y i , y j ; ψ ψ ψ) ⊤ D ψ ψ ψ log f (y i , y j ; ψ ψ ψ),


Between two worlds: Italian immigrants and football in America (Argentina, Brazil, USA) Stéphane Mourlane stephane.mourlane@univ-amu.fr Aix Marseille Univ. CNRS, Telemme, Aix-en-Provence I migranti italiani hanno dato un contributo sostanziale alla crescita del calcio in Sud America (Argentina e Brasile) e negli USA. Ma il calcio è stato allo stesso tempo anche una via di integrazione e mantenimento dell'italianità. Le società sportive create da migranti italiani, quando i colori nazionali o il nome si riferiscono al loro paese di origine, dimostrano l'attaccamento dei propri membri alla patria. I club etnici non potevano evitare completamente il processo di integrazione che si verifica quando le generazioni si succedono, né superare i vincoli imposti all'introduzione del calcio nella cultura nazionale del paese di destinazione. Non tutti i giocatori italiani immigrati nelle Americhe sono stati simboli di integrazione. Le scelte di carriera hanno portato alcuni a riattraversare l'oceano e tornare nella patria dei genitori o dei nonni e persino, in alcuni casi, a far parte della nazionale italiana.

Parole chiave: Sport; Calcio; Migrazioni; Italia; America In the 19th century, the British invented football (soccer, in the United States) and, thanks to their imperialist politics, introduced the sport to the rest of the world (Darbon, 2008). The agents of British Imperialism (sailors, merchants, industrialists, diplomatic and colonial civil servants, migrants) were among the first to bring the game to South and North America. However, at the beginning of the 20th century, British influence in North and South America was overshadowed by a growing number of immigrants from Eastern and Southern Europe's working-class neighbourhoods where football was popular (Murray, 1996: 32).

In this trans-Atlantic flow, there was a large number of Italians. Between 1876 and1915, 4.15 million Italians left for the United States, 1.78 million for Argentine and 1.2 million for Brazil. The history of Italian emigration has been the subject of many publications, including research on conditions for the integration and assimilation of migrants (Sanfilippo, 2015). The practice of sports is «commonly seen as a functional means for assimilation and integration, rather than the integration of ethnic boundaries» (Van Rheenen, 2009: 782). These two aspects, assimilation and ethnic identity, are not necessarily poles apart and research in social sciences has shown that they are in fact complementary (Kivisto, 1990;Shenton, 1990). A loyal sentiment of belonging to society and the adoption of national values in the country of destination are not contradictory to maintaining Italianness, since this is seen as a cultural identity. It is possible to take a transnational approach [START_REF] Baily | Immigrants in the Lands of Promise. Italians in Buenos Aires and New York City 1870-1914[END_REF] that takes into account the modalities of "adjustment" (Nelli, 1867). If, as Roger [START_REF] Waldinger | Transnationalism in question[END_REF] has suggested, this approach is less innovative than it might appear to historians, it has the merit of offering a rich and recent historiography for Italian migration (Tirabassi, 2005;Corti, 2013). Here the interest lies in applying this approach to football, which, as a game but also as a social culture, offers a substantial «capacity for mobilization and for demonstration of a sense of belonging» (Bromberger, 1998: 45). Commonly known as "the world game", football also contributes to the development of identity at the local, national, transnational and global levels (Foer, 2004). For Italian migrants in the Americas, football is certainly an integral part of this plurality.

The way multiple identities intertwine varies in terms not only of the period when migration occurred, but also of the role played by football in the social and cultural context of the country of destination. Argentina, Brazil and USA offer an interesting opportunity for a comparative study (Green, 1990) that will contribute to the cultural history of Italian migration.

Ethnic clubs and Italianness

Sports clubs created by Italian migrants, when the national colours or the name refer to their country of origin, demonstrate their members' attachment to the homeland. The first of these clubs, for which football was not the only sport, appeared in Brazil and the United States in the 1910s. Sao Paulo, which welcomed 70% of Italian migrants to Brazil between 1870and 1920(Holloway, 1980: 37): 37), saw the foundation of many clubs during that decade: Palestra Italia di San Carlo, FC Italo, Juventus, Palestra Italia di Santos, Palestra Italia-Ribeiro Preto and, in particular, Società Sportiva Palestra which was to become the largest Italian association in the city (Trento, 1992). Created in 1914 after a visit by two clubs from Italy, Pro-Vercelli and Torino, Società Sportiva Palestra adopted the colours of the homeland's flag (green, white and red) and added the Savoy royal blazon and set up a team of players who were either Italians or sons of Italians (De Campos Araújo, 2000).

In the United States, the major area of Italian immigration had similar clubs. Despite abundant research on Little Italies in New York, which has the largest Italian population in the United States, there are no references to this sport. Sociological and ethnographical studies on activities by youths, many of whom were Italian, in the streets of New York in the late 19th and early 20th centuries, do not discuss soccer [START_REF] Bevilacqua | Italian-American children at play on city playgrounds and streets 1880-1930[END_REF]. But according to Carmelo Bazzano (1994: 112) «during the 1920s, many Italian soccer clubs had been established in the Italian neighborhoods of many eastern cities»; some of them in New York City. In Chicago -the third largest Italian population in USA after New York and Philadelphia (Candeloro, 2003: 151) -the first Italian sports clubs were created in 1929. St. Louis was also an important place of Italian soccer during the 1920s (Mormino, 1982). In San Francisco, where there was a substantial Italian community in North Beach, we know that the largest sports club, Circolo Recrativo Italiano Virtus was founded in 1917, followed by two more clubs in 1919: Unione Sportive Italiana and Sporting Club Italiana. In 1921, Sporting Club Italiana and Circolo Recrativo Italiano Virtus merged and the new club was renamed Italia Virtus Club before again merging, this time with Unione Sportiva italiana, and becoming Unione Sportiva Italia Virtus in 1926[START_REF] Van Rheenen | The promise of soccer in America: the open play of ethnic subcultures[END_REF].

Given that both countries had an Italian migrant population for several decades by the time these sports clubs were founded, this seems to be a relatively late phenomenon. In 1912, Fanfulla, Sao Paolo's Italian newspaper, launched an appeal:

In Sao Paulo we have… football clubs for the Germans, English, and Portuguese, an international club and even clubs for Catholics and Protestants, but nothing has [yet] been done to create a club composed entirely of Italian "sportsmen" even though we have the largest colony in the state (Bocketti, 2008: 282).

In Chicago USA, when the Maroons club was founded, there were already teams of Irish, Scottish, Swedes, Danes, German, Hungarians, Bohemians and even Jews in the local soccer championship [START_REF] Trouille | Association football to fútbol: ethnic succession and the history of Chicago area soccer, 1890-1920[END_REF].

There are several reasons for this interval between the arrival of migrants and the creation of clubs. In one of the first studies published, in 1929, on the Italo-American community in the United States, Giovanni Schiavo (1929: 81) noted that «It is well known that, as a whole, the Italian immigrant has not brought with him any tradition of national games… the only pastime for which the immigrant is noted is apparently bocce -an Italian game resembling lawn bowling».

The traditional games that attracted the most interest from migrants were Bocce and Pallone in its many forms, the most popular being Pallone col bracciale1 . In fact, Pallone was the reason behind a first attempt to organize an ethnic sports team for New York's Italian community in 1877: it was a failure (Gems, 2013: 1). The principal sport practised by Italian migrants was cycling, which was very popular in Italy (Marchesini, 2009). The Unione sportiva italiana, one of the oldest Italian sports clubs in New York, was been particularly successful (Bazzano, 1994: 108). However, the conditions for introducing football were not particularly favourable, as it did not become a popular sport in Italy until the First World War (Papa-Panico, 1993;Dietschy, 2014;Dietschy-Pivato, 2019). At the time, most migrants came from poor rural areas, and soccer was mainly seen as a «game for the young urban elite» (Lanfranchi, 1994).

Among Italian communities abroad, it was often the elite who encouraged the creation of football clubs and many other associations, including the numerous mutual support groups which, frequently organized cultural activities (Bugiardini, 2002;[START_REF] Bertagna | L'associazionismo in America Latina. In Bevilacqua -De Clementi -Franzina (a cura di)[END_REF]. More importantly, the main purpose of Italian associations was to promote solidarity amongst their members, maintain the image of their homeland, and develop cultural activities: football clubs clearly contributed to meeting these objectives. They gave the Italian migrant a community structure and were encouraged by the Italian government which, in the early 20th century, sought to "nationalize" migrants who did not have a strong feeling of belonging to a country that had only recently been unified (Choate, 2008). The Italian consulates, which maintained close links with the local ethnic elite, played a key role in this process which was reinforced by the Fascist regime's policies during the 1920s and 1930s (Franzina-Sanfilippo, 2003;Pretelli, 2010). The creation of the Italian club, the Maroons, in Chicago in 1929 appears to have been influenced by a «heightened nationalism of fascist Italy» (Candeloro, 2003). The merger of San Francisco's football clubs into the Italian Athletic Club in 1936 appears to have been part of a "mainstream" approach which, if it did not give rise to massive numbers of migrants joining the Fascist cause, did in fact strengthen their identification with Italy.

The Church also played an important role in maintaining this sentiment and in providing structure for migrants by sending out missionaries (Rosoli, 1996;Principe, 2009). In St Louis's Hill district, with a 90% Italian population, membership of football teams organized by parishes was clearly based on ethnic background in the 1920s through the presence of Italian priests seeking to reduce violence among migrant youths (Mormimo, 1982).

In many cases, the clubs took on a social function other than sport either by bringing immigrants into a larger social network or by organizing charitable and cultural activities. In Sao Paolo, Società Sportiva Palestra offers a good example of mutual support and solidarity activities offered by ethnic clubs (Bocketti, 2008: 286). Similarly, «[b]eyond their social function, many teams [from Chicago's ethnic clubs] also provided essential community services, such as helping players acquire jobs, housing and loans» (Trouille, 2009: 802). In addition, Gerald R. Gems noted (2013: 39) that:

For many Italian immigrants of the first and second generations [in the United States], pride in Italy was a matter of self-respect and social recognition. They propagated information of the Italian nation in order to gain attention and acceptance from the other ethnic groups and the mainstream population. For some Italians, economic reasons also may have played a role in their efforts to support identification with Italy and with the various Italian groups. Businessmen may have hoped that the united Italian populace in the United States would provide greater commercial opportunities and additional patrons beyond their own paesani. In particular, the early northern Italian middle-class immigrants spearheaded the efforts to unite Italians in United States by creating an 'imagined community' among them through the establishment of clubs, fraternal organizations, and commercial associations.

Ethnic clubs were far from being universally present; they were almost non-existent in Argentina, due to competition from other local and national cultures and identities in that country.

Play the Game! from Neighbourhood to Nation

Ethnic clubs could not completely avoid the process of integration that occurs as generations of migrants replace each other, nor overcome the constraints imposed by the introduction of football in the national culture of the country of destination.

A good indication is the use of Italian in the clubs: for example, by 1927, Sao Paulo's Società Sportiva Palestra wrote its documents in Portuguese (Bocketti, 2008). By the end of the 1930s, the policy of Brasilidade (Brazilianness), introduced by Vargas' Estado Novo, covered all aspects of life including football. The coup de grâce for Italian clubs came when Brazil joined the Allies during World War II: Germany and Italy were immediately treated with scorn. As a result, SC Germânia was renamed Pinheiros in 1942, Sao Paolo's Società Sportiva Palestra became Sociedade Esportiva Palmeiras and Palestra Italia, founded in Belo Horizonte in 1921, was reborn as Cruzeiro Esporte Clube (Goldblatt, 2014: 61-62). The two Italian clubs were even forced to give up the traditional colours of green, white and red from the Italian flag. During the second half of the 20th century, Palmeiras and Cruzeiro made their mark on the history of Brazilian -and South Americanfootball: Cruzeiro won the Copa Libertadores in 1976 and 1997, followed by Palmeiras in 1999. Nevertheless, their Italian origins were fading and now have almost completely disappeared.

In the United States where, before World War II, soccer was played almost exclusively by migrants, it was seen as a "foreign game" (Markovits-Hellerma, 2001;[START_REF] Wangerin | Distant corners: American Soccer's History of Missed Opportunities and Lost Causes[END_REF] placed mostly in the suburbs (David, 1997). It was many years before the game was subjected to a process of "domestication" or "americanization" (Keys, 2014) and the Italianness of clubs became less visible. This process also affected clubs created by later migrants: recent migrants from Southern Italy to New York founded Italians Brooklyn in 1949, a club still active today. Among migrants arriving in the 1990s was Baldo Savarino, a former player for the Maroons and now Chairman of Chicago Park District Soccer, who said: «Soccer is going to be a lot more popular, but I think its ethnicity will begin to diminish» (Mayer, 1991).

Although time has served to wipe out traces of Italianness in clubs initially formed on the basis of ethnic ties, those ties were not always the reason for the presence of Italian migrants in football clubs. They might also join clubs sponsored by companies or teams in their local neighbourhood. Even if this resulted in a high proportion of Italian members, because of links with their place of work or of residence, we cannot consider them to be a vector for Italianness. Certainly, they foster links within migrant communities, but their primary role was to contribute to the integration of migrants at the local and national levels. These clubs developed social and territorial identities within which Italianness played only a second role, in conjunction with the national identity of the country of destination. In Sao Paolo, Sport Club Corinthians Paulista, rival of Società Sportiva Palestra, offers an interesting example (Bocketti, 2008). Based in Bom Retiro near the city centre, where the majority of the inhabitants were migrants from Italy and other countries, Sport Club Corinthians Paulista was founded in 1910 by a group of workers, two from Spain, two from Portugal and one Italian. The presence of Italians in the initial management team and among the players was noteworthy in itself, but they did not appear to exert more influence than anyone else in the club. The club preferred to promote a working-class identity, as can be seen in its charter: «the Corinthians will be the people's team, run by the people and for the people».

In Argentina and, in particular Buenos Aires, sports clubs were the flag bearers for the Barrio (neighbourhood). By the early 20th century, when the first clubs were founded by the British elite, football was already spreading to the working classes. In 1907, there were 300 club teams in the capital «[that are] community institutions rooted in neighbourhoods and connected to the local and national political spheres; they are spaces with multiple internal forms of sociability» (Frydenberg-Daskal-Torres, 2013: 1674). Migrants, who represented 50% of the Argentinian population in 1914, played a major role in the development of these clubs. In particular, Italians (34% of the city's inhabitants in 1914) contributed to the creation of Argentina's two largest -and rivalfootball clubs in La Boca. The first club, River Plate, was created in 1901 after the merger of Las Rosales and Santa Rosa which was a club for young men from Genoa. The majority of the new club's founders were Italian or of Italian descent. Following the presidency of Leopoldo Bard, son of Jewish immigrants, they became directors of River Plate and when it was decided to move its stadium in 1923, the site chosen was the neighbourhood of Palermo with its strong Italian community. The second club from La Boca, Boca Juniors, was even more representative of the neighbourhood when it was created in 1905 by Genoese migrants and this is still the case today.

In parallel with the emergence of territorial identities, Italians were contributing to the nationalization of football [START_REF] Archetti | Masculinity and football: The formation of national identity in Argentina[END_REF] with the debate about their contribution to a specifically Argentinian style. This was important because it was thought to contribute to the definition of national identities in Latin America, in opposition to a European identity, and became particularly pertinent after World War I. Joshua H. Nadel in a recent publication stated that «[i] n soccer terms, the intermixing of immigrants from around Europe with the Argentine population supposedly gave birth to the criollo style, while the Brazilian game purportedly rested on the mulatto heritage of the nation. In both cases, what mattered was that the national style embodied the composition of the nation and was able to surpass the supposedly superior European "race"» (Nadel, 2014: 45). In Argentina, the sports magazine El Grafico described criollo as an individualistic, but undisciplined, style based on agility and, in particular, dribbling (Archetti, 1999: 60). For the journalists, it is not a specifically Italian or Spanish style: migrant children had been transformed through their contact with Argentinian culture based on the myth of the Gaucho on the Pampas. As a result, criollo was seen as a barrier to cultural transfer and football styles imported from Europe (Archetti, 1999: 69). But, according to Eduardo P. Archetti, this idea was subject to debate, since criollo could be seen as a consequence of the Argentinian melting pot (Archetti, 1999: 70). To avoid this ideological debate, let us say that this football style is one element in a sporting culture that is essential to the Argentinian national identity. This sporting culture is obviously linked to the cultural environment of football players, many of whom were imbued by their Italianness. Nevertheless, the extent to which Italian migrants influenced this style has still to be elucidated.

It is easier to observe and measure the contribution of Italian migrants to the international reputation of Argentinian, Brazilian and American football by analysing the presence of these players in their national teams.

Argentina's team certainly benefited from the presence of Italian players during the 1920s and 1930s and, in particular, in the early major international competitions. It won the Copa America for the first time in 1921, largely thanks to Julio Libonattti, originally from Genoa. At the Amsterdam Olympics in 1928, when Argentina was one of the finalists, the most successful striker was Domingo Tarasconi, a player from Boca Juniors, as was Raimondo Orsi. Argentina also played in the final of the first World Cup held in Uruguay in 1930, during which Guillermo Stabile was the leading goal-scorer. He then joined Genoa, in his parents' homeland, with his teammate, Luis Felipe Monti. Other descendants of Italian migrants include emblematic players such as Daniele Passarella, who played in Argentina's winning team in the World Cup at home in 1978, and above all the famous Diego Maradona and Lionel Messi, who both have Catalan and Italian origins.

The contribution of Italians to Brazil's national team is less noteworthy. The great football stars, Pelé and Garrincha, reflect the racial mixity that Brazil sought for developing its identity and culture. However, this does not negate the influence of Italian migrants. In the 1930 World Cup, the Brazilian team included Luis Gervasoni, whose nickname (common for Brazilian players) "Itália" clearly refers to his origins. Later, Roberto Rivelino (Rivellino), son of a family originally from Molise, was an important player in the history of the Seleção, having played alongside Pelé in the winning team at the 1970 World Cup.

The USA team's record is somewhat less glorious. It participated in the first World Cup in 1930 with a team of players mostly of Anglo-Saxon origin. At the 1934 World Cup in Italy, the team was led by its striker, Aldo Donelli from Pittsburgh (Allaway-Jose-Litterer, 2001: 80). The contribution of Italian migration was more visible, when the USA team beat England in the Brazilian World Cup in 1950, with several Italo-American players from St. Louis, including the goalkeeper Frank Borghi, the "hero" of the match against England, and Gino Pariani, who scored the only goal.

Not all Italian migrant players in the Americas were symbols of integration. Career choices led some to travel back across the Atlantic Ocean to the homeland of their parents or grand-parents and in some cases, to play with the Azzurri, Italy's national team.

Between two worlds

This migration of American footballers originally from Italy was part of a major return movement by Italian migrants (Cinel, 1991;Cerase, 2001). We do not know for certain how many left the Americas, but the number is estimated at between a third and a half of Italian emigrants. Until the mid-20th century, returning footballers selected for the Italian team were known as rimpatrati, as were other returning emigrants, but in 1950 a new term oriundi (from the Latin oriri, "to be born") was introduced and was later extended retrospectively to those returning in the 1920s and 1930s. Some 40 oriundi have been selected for the Nazionale. Other South Americans with Italian origins agreed to return through transfers to Italian clubs, usually for financial reasons, but also because South America was slow in introducing professional football (Lanfranchi-Taylor, 2001). These transfers were arranged by a network of official and unofficial agents, but were not necessarily part of a search for identity, even though they participated in a circular migratory space for Italians [START_REF] Baldassar | Ritorni e visite in Patria: la circolarità dello spazio migratorio[END_REF].

Half of the oriundi in the Italian team came from Argentina; the other are mostly from Brazil or Uruguay, another country with a large Italian migrant population. Some of these players were capped by two countries, notably in the 1920s and 1930s when this was still possible under international rules. The first of the oriundi, Julio Libonnati, won the Olympics competition for Argentina in 1928 before playing for Torino and being selected 17 times for the Italian team. Others include the tough defender Luis Felipe Monti whose parents came from Romagna and who is the only player in the history of football to have played for different countries in two World Cup finals: Argentina in 1930 and Italy in 1934. Monti's inclusion in the Nazionale occurred at a time when many oriundi were recruited for Italian clubs. The 1912 Law on Nationality made the process more favourable by introducing a jus sanguinis: losing Italian nationality could only be made by a conscious act and could be recovered after two years' residence in the homeland [START_REF] Tintori | Cittadinanza e politiche di emigrazione nell'Italia liberale e fascista. Un approfondimento storico[END_REF]. The Fascist regime reduced this residency requirement to six months and authorized acquisition of Italian nationality "in exceptional circumstances". More generally, its appeal for oriundi was part of a policy of defending and promoting Italianness among emigrants. This policy targeted Latin American countries, such as Argentina and Brazil (Mugnaini, 2008;[START_REF] Bertonha | Fascismo, antifascismo y las comunidades italianas en Brasil, Argentina y Uruguay: una perspectiva comparada[END_REF][START_REF] Aliano | Mussolini's National Project in Argentina[END_REF]Goebel, 2014) where regulations on nationality had created «a permanent situation for potential tension» since the beginning of the 20th century (Pastore, 1999). In addition, the oriundi sportsmen were useful figures for internal and international propaganda (Dietschy, 2008). Initially unimpressed by football, which he saw as a foreign sport, Mussolini was nevertheless persuaded to take advantage of the game's popularity and organize the 1934 World Cup (DietschyGastaut-Mourlane, 2006). The competition not only offered Fascism an opportunity for displaying the modernity of its ideology, but also for demonstrating Italy's superiority through the success of its national team. The Nazionale del Duce (Grimaudi, 2003), which won the Cup, included no less than four Italo-Argentinians (Enrique Guaita, Luis Felipe Monti, Raimundo Orsi and Atilio Demaria) and one Italo-Brazilian (Amphilóquio Guarisi).

Footballers playing in the Italian team created problems in Latin American countries, not only in Italy but also with the Azzurri's competitors. In France, where the national team was unable to beat the Italians, questions were raised about the nationality of the team members and people talked, not without prejudice, the supposed deception and trickery of Italians (Mourlane, 2006).

In Italy, on the other hand, their national identity was never in doubt. The Gazzetta dello Sport said that «there can be no question about [Raimundo Orsi's] right to acquire Italian citizenship. He has always been Italian» (Ghirelli, 1990: 100) However, Orsi, the player with the highest number of caps among the oriundi (35 selections), did not speak a word of Italian when he arrived in the country. Like many other oriundi, apart from Renato Cesarini (born in Senigallia in the Marche, with 11 selections between 1931 and 1935), Orsi was one of the second generation migrants born in Argentine or Brazil. Some, notably the Italo-Brazilians, did not play under the same name in the two countries. In Brazil, they adopted a football nickname while, in Italy, they used their Italian surname: for example, Amphilóquio Guarisi was known as Filó in Brazil.

Their departure was not appreciated in Argentine or Brazil. For the Argentinians, the Italianness of these players was a far memory that had dissolved into their Argentinian culture (Lanfranchi-Taylor, 2001: 73). In Brazil, they were called "renegades" (Bocketti, 2008: 294). During the 1934 World Cup, the Palestra Italia club house in Sao Paolo was damaged after the Brazilian team was beaten by Spain by people protesting against the defection of Italo-Brazilian players (Goldblatt, 2014: 81). In Italy, similar events occurred in 1935 when three players from AS Roma, including 2 Italian team players (Enrique Guaita and Alejandro Scopetti), returned to Argentina in order to avoid having to fight in the war in Ethiopia (Simon, 2004: 196). In a climate of nationalist fervour, the high point of the consensus around the Mussolini regime, his decision was fustigated and many questioned the attachment of oriundi to their parents' homeland. After this episode, the number of players travelling between South America and Italy diminished (Lanfranchi-Taylor, 2001: 81). The debate was revived after the war when the results of the Italian team were less good. Quite apart from the more restrictive conditions established by FIFA, early elimination in the 1962 World Cup and, above all, the trauma of being defeated by North Korea in the 1966 World Cup led to the Italian Federation giving priority to the training and development of national players. Forty years passed between the last selection of an Italo-Brazilian player, Angelo Sormani, in 1963, and the first selection of an Italo-Argentinian footballer, Mauro Camoranesi, in 2003. Great-grandson of Italian migrants who had left the Marche in 1876, Camoranesi is the oriundo with the most caps for Italy, with 55 selections, Thiago Motta, the ItaloBrazilian with similarly remote Italian origins, has been a key player in the Nazionale since 2011. The selection of these footballers is made possible by the Nationality Law of 1912, with minimal amendments in 1992. Today, nearly 60 million people can claim Italian nationality. Between 1998 and2007, 786,000, of whom 66.2% were Argentinian or Brazilian, took up Italian nationality during the economic crisis in those countries (Tintori, 2009;Pastore, 2004). To encourage tighter links with the diaspora of "Italians abroad", the Berlusconi government, shortly before Camoranesi's first selection, passed a law in 2001 giving oriundi the vote in general elections and the possibility of regaining the nationality of their ancestors. For the footballers, choosing Italy obviously allows them to play at a higher level, as Giuseppe Rossi, the first Italo-American oriundo, realized when he played for the Italian team. He left the USA at the age of 13 in order to finish his training in Parma, but he refused to play in the 2006 World Cup with the modest American team in order to maintain his chances of being selected for Italy.

In 2010, the Italian coach Cesare Prandelli said «The oriundi are the new Italians» (La Gazzetta dello Sport, July 13, 2010). Yet not all were thought to be particularly good at their sport, so that now, given new concerns because of the growing wave of migrants in Italy, the debate has been reopened. The selection of Camoranesi and Thiago Motta has not been unanimously popular. The journalist Greg Lea recalls that «Camoranesi was refreshingly honest when quizzed on his national identity, admitting that he considered himself Argentinian and did not know the words to the Italian anthem, but this candour did not do much to endear him to football fans of the country he had chosen to represent» (The Guardian, February 18, 2015). Cesare Maldini, the former Italy defender and manager said in an interview with Radio Incontro that the resumption of this practice was negative, a "return to the past" and referred to Thiago Motta as "the Brazilian". In 2015 when the new coach, Antonio Conte, selected the Brazilian Eder and the Argentinian Franco Vasquez, Roberto Mancini, Inter's coach, said that «The Italian national team must be Italian. Those who aren't born in Italy, but have distant relatives should not be called up» (La Gazzetta dello Sport, March 25, 2015). Seventy-four percent of La Gazzetta dello Sport readers, when asked for their opinion, shared this point of view.

Conclusion

Even today, football offers indications of the identity issues raised by Italian emigration. The question of integration crystallizes these difficulties, although it appears now that they can be overcome. It does not seem opportune to make ethnic clubs an obstacle to the integration of migrants when they are also agents for their participation in local championships. Similarly, the de-ethnicization and nationalization of football in both Latin America and North America does not prevent the preservation of a memory of migration in diverse forms.

Like Garibaldi, footballers of Italian origin in the Americas can appear to be "heros in two worlds". Their fame spreads, and is sometimes divisive, on both sides of the Atlantic. They contribute to the oftenflouted pride of migrants, notably by insisting on their contribution to the development and reputation of the country of destination. In Italy, these success stories act as a counterweight to negative representations of emigration which is often seen as a form of national dishonour.

Yet, quite apart from these divergent representations, Italian migrants have made a substantial contribution to the growth of football both in South America, where the game is a key element in the national culture, and in the USA, where it faces serious competition from other American sports. Further, for the Italian diaspora, football has led to the emergence of a transnational circular space linking both sides of the Atlantic, not only through tours by Italian teams in South America since the early 20th century, but also by the transfer of players of Italian origin back to Italy. As a result, we can see that Italian emigration, as a certain form of Italianness (Corti, 2011), varies in line with the country of destination, the period of emigration and the contemporary context and also plays a major role in the globalization of football.

Balls are struck back and forth with a wooden cylinder, called a bracciale, worn over the forearm. Pallone is often played on courts marked out on town streets or in sferisteri for championship.

Emilio (a cura di) (2002). Storia dell'emigrazione italiana, II, Arrivi. Roma: Donzelli.

- Bocketti, Gregg (2008) 1914-1998 (146158). Paris: Éditions Autrement. - Mugnaini, Marco (2008). L'America latina e Mussolini. Brasile e Argentina nella politica estera dell 'Italia (1919'Italia ( -1943)). Milano: Franco Angeli.

- Murray, Bill (1996). -Simon, Martin (2004). Football and Fascism. The National Game under Mussolini. London: Bloomsbury Academic.


Samuel Thibault

Ens Lyon

Xavier Cavin

Olivier Festor

Eric Fleury

Unreliable Transport Protocol for Commodity-Based OpenGL Distributed Visualization

This paper presents a way to use an unreliable transport protocol to perform distributed OpenGL visualization on a commodity network of computers. We present a performance evaluation of a prototype implementation based on the Chromium environment.

Introduction

Clusters of commodity computers are becoming a practical tool for visualization. However, most of the time, they are dedicated clusters with a fast (and sometimes expensive) network interconnect: gigabit Ethernet, Myrinet, etc. On these networks, transfer time and packets loss are very low. For this reason, the chosen transport protocol (IP) is often a reliable one (i.e. TCP).

In this paper, we consider alternative network interconnects, that are both cheaper and more commonly available: 100 Mbps, 10 Mbps or wireless networks. On these networks, transfer time is longer and packets loss happens more often (which requires to resent the packets and adds a delay in the transfer time). In those cases, it may be more interesting to use an unreliable transport protocol to get a higher framerate, if we accept a degradation of the final rendering quality (due to packets loss).

We show in this paper how to use an unreliable transport protocol (i.e. UDP) to perform distributed OpenGL visualization on a commodity network of computers. We present a performance evaluation of a prototype implementation based on the Chromium environment [START_REF]Chromium[END_REF].

Network Interconnect

As opposed to shared memory parallel machines -where multiple processors share a common adress space -a cluster of computers requires to specify a way for the machines to communicate between them: a protocol describes "the message formats and the rules two or more machines must follow to exchange those messages" (Douglas Comer). * Contact: mailto:Olivier.Festor@loria.fr Applications are usually built on the top of the process layer protocols. However, an application is free to bypass the defined transport layers and to directly use IP or one of the underlying networks.

The internet layer protocols are fundamentally datagramoriented and unreliable. It is the responsibility of the hostto-host and process layer protocols to enhance the quality of service to that desired by a particular application. These protocols function as an intermediary between the application and network layers. The two main Internet host-to-host layer protocols are:

• TCP is a sliding-window protocol providing reliable, stream-oriented delivery. TCP includes support for guaranteed delivery, meaning that the recipient automatically acknowledges the sender when a message is received, and the sender waits and retries in cases where the receiver does not respond in a timely way.

• UDP provides almost no additional functionality over IP. It performs fast, unreliable, datagram delivery. UDP does not implement guaranteed message delivery and provides no guarantees that the order of data delivery is preserved. UDP is called an unreliable transport protocol for this reason.

The Maximum Transmission Unit (MTU) is the biggest packet size that can be transmitted over a physical network. Different networks have different MTUs; for Ethernet, 1500 is the maximum, and recommended MTU. When a packet reaches a network whose MTU is smaller than the packet size, the Internet Protocol will "fragment" it by breaking it up into smaller ones.

TCP automatically breaks up the data to adapt to the MTU, while UDP does not: the application has to take care about the size of the packets it sends.

Distributed OpenGL

In this section, we assume that we have an OpenGL application that encodes its OpenGL commands, send them over a network to a rendering server that in turn decodes the commands and renders them on its own graphics hardware. This is the basic brick of a distributed OpenGL visualization system, where multiple applications can send OpenGL commands to multiple rendering servers.

Most implementations of a distributed OpenGL system rely on TCP for the transport protocol, because it is reliable by nature and rather efficient on a dedicated cluster of computers. All OpenGL commands sent over the network are guaranteed to be received, in the same order that they have been sent. Figure 2 shows a rendering of a crocodile model using TCP: it appears as if it was rendered locally.

UDP versus TCP

Replacing TCP by UDP is equivalent to remove the acknowledgment mechanism, so that some data may not be received, or at least not in the order they were sent. It allows to optimize the use of the network, because the sender no longer have to care about the loss of packets.

Simply replacing TCP calls by UDP calls is obviously not a viable solution. Indeed, if some important OpenGL commands are lost, the execution of the received ones (possibly in a different order) often leads to a fatal crash. There is also the case of OpenGL commands that are supposed to return a value (like the glGet* family).

One solution is to combine the use of TCP for commands that can not be lost and UDP for all other commands. However, the two communication channels have to be synchronized so that commands sent through UDP are correctly interleaved with commands sent through TCP. We so add a sequence number to UDP packets, which is incremented each time new commands are sent through TCP, as shown by Rather than exhaustively listing the OpenGL commands that can not be lost, we have chosen to incrementally choose the one that can be lost and therefore be sent through UDP.

Actually, we have first decided to only send the glVertex commands through UDP. These commands are used within glBegin/glEnd pairs to specify point, line, and polygon vertices, and may represent up to 80% to 99% of the whole commands, depending on the type of application.

Loosing packets

If the whole content of a glBegin/glEnd pair is lost, then some holes would appear in the final rendering. Unfortunately, this is not so simple: Figure 4 shows a rendering of the crocodile model using the interleaved UDP/TCP mechanism. We can notice that a lot of "bad", overlapping triangles have been rendered. Real 3D objects (such as the crocodile model of Figure 2) are composed of many more points, and it can happen that the content of a glBegin/glEnd pair becomes larger than the MTU of the network, and then gets fragmented before being sent. If a subset of the content of a glBegin/glEnd pair is lost, we may get to a case similar to the (simplified) one depicted on Figure 6 A solution to overcome this problem is to group the points by three (in the case of our simplified example) in the UDP packets, so that the loss of vertices happen with triplets of vertices, as shown on Figure 7 A similar attention must be given to other cases of glBegin/glEnd pairs:

GL_LINES, GL_LINE_STRIP, GL_LINE_LOOP, GL_TRIANGLE_STRIP, GL_TRIANGLE_FAN, GL_QUADS, GL_QUAD_STRIP and GL_POLYGON.

Figure 8 shows a better rendering of the crocodile model with the interleaved UDP/TCP mechanism and the MTU consideration. We can observe holes due to the packets loss (about 10% in this case), but the result looks much better than on Figure 4.

Experimentations

We have implemented the UDP/TCP mechanism inside the Chromium environment [START_REF]Chromium[END_REF]. Chromium is a flexible framework for scalable real-time OpenGL rendering on clusters of workstations. The communication is done through a stream packing library that takes a sequence of OpenGL commands and produces a serialized encoding of the commands and their arguments. The serialized representation of the OpenGL command is sent over the network and then decoded for rendering or further processing.

Chromium provides a point-to-point connection-based networking abstraction, that abstracts the details of the underlying transport mechanism. This library is currently im- plemented on top of TCP/IP and Myrinet. We have extended this library so that it supports both the version 6 of the Internet Protocol (IPv6) and our UDP/TCP mechanism. 1 We have experimented our implementation between two standard PC's on a local network with three different configurations and with the following applications :

• Crocodile: a triangulated surfaces OpenGL viewer, loaded with the crocodile model (see Figure 2).

• Rings: one of the Chromium's test programs that implements a simple parallel application displaying rings (see Figure 9(a)).

• Chromium B.S.U. [START_REF] Chromium | [END_REF]: a space shooter game written in OpenGL (see Figure 9(b)).

100 Mbps: our first experiment is on the 100 Mbps (100BaseTX) local network. Table 1 shows the obtained framerates with the three applications.

In this configuration, we observe a small gain with the crocodile application (22%), with few missing triangles, but the performance is the same with the rings application. Indeed, the geometry of the rings is very light, and all synchronization operations must be done through TCP. 10 Mbps: in order to reduce the available bandwidth, we have set the network card of the PC's to 10 Mbps (10BaseT). Table 2 shows the obtained framerates with the three applications.

In this case, we observe a gain with the crocodile application (28%) but also with the rings application (65%).

Wireless: finally, we have replaced the network card of the PC's with a 11 Mbps wireless card. In this configuration, 1 These modifications have been applied to the patches-1-branch of the Chromium's SourceForge CVS repository. a lot of packets loss happen, and the TCP performance goes dramatically down: for instance, the transfer rate for sending a file is about 200 KB/s. Table 3 shows the obtained framerates with the three applications. We obviously observe an important gain for the crocodile application (560%), because TCP needs to resent a lot of geometry. The gain is less important with the rings application (145%), because it uses less geometry and relies more on TCP. However, the rendering quality with UDP/TCP is very bad in both cases, because the packets loss degrading the TCP performance are directly visible on the screen.

Conclusion

This paper presents a novel way to use an unreliable transport protocol for commodity-based OpenGL distributed visualization, by interleaving UDP and TCP communications. We have implemented this mechanism inside the Chromium environment, and our preliminary experiments show that it can be adapted if the geometry represents most of the traffic, if rendering quality is not the first interest, if it is animated (the eye compensates the holes), if there are a lot of packets loss, and if network is the bottleneck. However, if the application requires a lot of synchronization, or if the network is faster than what the application can handle, TCP may be better adapted. We also believe that our current implementation could benefit from some UDP specific optimizations in order to obtain better performance.

Figure 1 :

 1 Figure 1: The four layers in the DoD model (Freesoft.org).

Figure 2 :

 2 Figure 2: TCP rendering of a crocodile.

Figure 3 :

 3 Figure 3: Interleaving UDP and TCP packets.

Figure 4 :

 4 Figure 4: UDP/TCP rendering of the crocodile.

Figure 5 :

 5 Figure 5: Rendering without loss.

  (a), where a single vertex (the third one) has been lost during the transmission. The triplets of vertices representing the individual triangles are wrong, the two last vertices are ignored, and the final rendering shown on Figure 6(b) is completely incorrect.

Figure 6 :

 6 Figure 6: Rendering with a single vertex loss.

  (a). A packet loss so leads to a coherent hole, as shown on Figure7(b).

Figure 7 :

 7 Figure 7: Rendering with a triplet of vertices loss.

Figure 8 :

 8 Figure 8: Fixed UDP/TCP rendering of the crocodile.

  Chromium B.S.U.

Figure 9 :

 9 Figure 9: Screenshots of two tested applications.

Crocodile Rings Chromium B.S.U.

	TCP	2.2fps	110fps	playable
	UDP/TCP	2.7fps	110fps	playable

Table 1 :

 1 Framerates on the 100 Mbps local network.

Crocodile Rings Chromium B.S.U.

	TCP	0.28fps	24fps	playable
	UDP/TCP	0.36fps	39fps	playable

Table 2 :

 2 Framerates on the 10 Mbps local network.

Table 3 :

 3 Framerates on the wireless network.

		Crocodile Rings Chromium B.S.U.
	TCP	0.066fps	5.5fps	quite playable
	UDP/TCP	0.37fps	8fps	non playable

Acknowledgments

The authors would like to thank Alain Filbois for his constant technical support and Bruno Levy for his help with OpenGL related problems. This material is partially based upon work supported by the AVTC under the DOE VIEWS program.