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

Pierre Ribereau
email: pierre.ribereau@univ-lyon1.fr

Manal Zeidan
email: manal.zeidan@univ-lyon1.fr

Regionalization of the extremal dependence structure using spectral clustering

Keywords: Max-stable processes, Extremal dependence, Extremal concurrence probability, Spectral clustering

come

Introduction

Constructing a statistical model for environmental extreme events, such as rainfall, temperature, and so on, is very important for understanding their behavior and accurately predicting their occurrence. Max-stable processes are natural models for spatial extremes, as they are natural extensions of the Extreme Value Theory (EVT) to spatial domains. They are powerful statistical models for extreme events in a continuous space and can assess the risk in areas that do not have stations. One common assumption used for maxstable models is that the extremal dependence is fixed across spatial domains. However, this assumption may be incorrect and can lead to the construction of meaningless models, particularly when the data sets are taken from a large region or regions with complex geographical features [START_REF] Richards | Spatial deformation for nonstationary extremal dependence[END_REF]). For instance, consider the rainfall data from the east coast of Australia, a region characterized by mountain ranges. These mountains are known to significantly influence rainfall events and can give rise to complex rainfall patterns. Similarly, when considering rainfall data in France, which features extensive coastal areas, extreme rainfall can be produced by disturbances from both the Mediterranean Sea and the Atlantic Ocean. Researching a max-stable model capable of capturing changing extremal dependence is a recent topic of study. For instance, [START_REF] Huser | Non-stationary dependence structures for spatial extremes[END_REF] developed an approach that captures non-stationary patterns in spatial extremes using covariates. They developed a non-stationary extremal-t model by combining the extremal-t model with a non-stationary correlation function. This model is satisfactory in capturing the extremal dependence. [START_REF] Huser | Non-stationary dependence structures for spatial extremes[END_REF] presented another non-stationary max-stable model without using it in their paper. This model is defined as a max-stable mixture, and the details of this model are presented in Section 5. In short, this model is a mixture of two max-stable models with a proportion of mixing that varies spatially. This proportion can be modeled as a function depending on covariates. This model could capture different extremal dependence in different spatial regions. However, it is computationally difficult and requires prior knowledge of relevant covariates. Recently, clustering was used to create regionalisations of extreme events. Clustering is an unsupervised machine learning tool that is widely used in data analysis to identify subgroups with similar features. It has a wide range of applications in fields such as computer science, statistics, biology, and climate science. In the context of spatial extremes, only a few studies use clustering to partition an entire region into homogeneous sub-regions based on similarities in extremal dependence. For instance, [START_REF] Bernard | Clustering of maxima: Spatial dependencies among heavy rainfall in france[END_REF] presented a novel clustering algorithm for maxima, using the F-madogram introduced by Cooley et al (2006). By combining the F-madogram with a partitioning around medoids (PAM) algorithm, they were able to cluster the extremes based on dependence strength. The algorithm was applied to analyze rainfall patterns over France. Afterward, [START_REF] Bador | Spatial clustering of summer temperature maxima from the cnrm-cm5 climate model ensembles & e-obs over europe[END_REF] applied this algorithm to large regions and different variables, analyzing the maxima of summer temperatures across Europe. [START_REF] Saunders | A regionalisation approach for rainfall based on extremal dependence[END_REF] demonstrated that the PAM algorithm presented by [START_REF] Bernard | Clustering of maxima: Spatial dependencies among heavy rainfall in france[END_REF] is sensitive to stations density. To address this issue, they proposed the use of hierarchical clustering with F-madogram. They applied their proposed algorithm to rainfall stations in Australia. Then, for each of the resulting regions, they fitted a stationary max-stable model. In this case, its not possible to model the dependence between two locations that belong to different clusters. However, in all of these studies, there is no clear idea about which model for the entire region can be estimated using these clusters. Our contribution in this paper has two paths. Firstly, we proposed a clustering approach for max-stable processes. We adapted spectral clustering for maxstable processes by combining it with the extremal concurrence probability introduced by [START_REF] Dombry | Probabilities of concurrent extremes[END_REF]. The extremal concurrence probability for a max-stable process is the probability that the maximum value of the process occurs at two or more locations simultaneously. This clustering algorithm aims to identify regions with similar extremal dependence. We demonstrated the applicability of this algorithm in clustering the max-stable mixture process presented by [START_REF] Huser | Non-stationary dependence structures for spatial extremes[END_REF] into k regional clusters, each with a similar extremal dependence. Additionally, we considered different sizes of blocks and discussed the effect of the block size on the clustering maps. The second path involves using the clustered regions in modeling the entire area. To the best of our knowledge, this paper is the first to address this. For this purpose, we proposed a censored pairwise likelihood based on clusters. We validated our clustering and estimation approaches through a simulation study and then applied them to two datasets. The first dataset consists of rainfall data from the east coast of Australia, while the second dataset includes rainfall data from France provided by Météo-France. The paper is organized as follows. An overview of spectral clustering is provided in Section 2. Section 3 presents Max-stable processes. Section 4 describes the adapted spectral clustering for max stable process. Section 5 presents the applicability of the adapted spectral clustering . Section 6 presents the inference using composite liklihood approach. A simulation study is presented in Section 7. Section 8 applies the methodology to data: rainfall over east coast of Australia and rainfall over France. Finally, Section 9 presents the discussion and conclusions of our study.

Spectral clustering : an overview

Spectral clustering is a technique used in machine learning and data analysis for clustering data points into groups based on the similarity between them. It is based on the concept of spectral graph theory, which is the study of the properties of graphs using linear algebra. Spectral clustering has several advantages, as it can handle high-dimensional data, which is often a limitation for other clustering algorithms. This is done by reducing the high-dimensional data to a lower-dimensional space using eigenvalue decomposition. Furthermore, it can handle different kinds of similarity measures, which makes it flexible and adaptable to different types of data. Also, it does not make any assumptions on the shape or size of clusters. Spectral clustering considers the dataset as a graph, where each data point

P i , i = 1, • • • , N represents a vertex in an undirected weighted graph. An undirected graph G = (V, E, W ) is generally defined by a set of vertices V = {v 1 , v 2 , • • • , v n }, a set of edges E = {(v i , v j )|v i , v j ∈ V }
between these vertices, and a similarity matrix W . An element w ij ∈ W represents the amount of similarity between the vertices v i ,v j and the weight that will be assigned to each edge. It is important to note that since the graph is undirected, the similarity matrix should be symmetric. If w ij = 0, this means that there is no edge between the vertices v i ,v j . Each vertex v i in the graph has a degree d i :

d i = N j=1 w ij . (1)
The degrees d 1 , • • • , d N represent the elements of a diagonal matrix called the degree matrix of the graph D. Spectral clustering aims to separate the main graph G into sub-graphs so that the weights of the edges between these sub-graphs are small, indicating dissimilarity between the clusters, while the weights of the edges connecting nodes within each sub-graph are relatively high, indicating similarity within the clusters.

Steps of spectral clustering algorithm

In general, any spectral clustering algorithm involves the following three steps.

1. Pre-processing Construct the similarity matrix W from the dataset using a measure that takes into account the aim of clustering, and then construct the similarity graph. There are different ways to do this depending on the pairwise similarity w ij . The aim is to model the neighborhood relation among the data points x 1 , • • • , x n . These ways are summarized as follows:

ε -neighborhood graph: In this graph, the vertices v i , v j will be connected by an edge if they are similar enough, i.e if w ij > ε, ε is a pre-defined non-negative real number. Usually, this graph is considered as an unweighted graph. k-nearest neighbor graphs: In this graph, the distance between each pair of vertices is computed using the Euclidean distance. Then, the vertices v i , v j are connected by an edge if v j is among the k nearest neighbors of v i or vice versa, and the edge is weighted by the similarity w ij . The neighborhood relationship among data points is controlled by a pre-defined integer number k.

The fully connected graph: In this graph, each vertex is connected to all other vertices by edges, and these edges are weighted by the similarities w ij . This type of graph is useful only if the similarity function can model the neighborhood relation among the data points. The commonly used similarity function is the Gaussian similarity function, which is defined as w ij = exp(-∥P i -P j ∥ 2 /2σ 2 ), where the neighborhood relation is controlled by σ.

For further information on similarity graphs, we refer to Von [START_REF] Luxburg | A tutorial on spectral clustering[END_REF] and Parodi (2012).

Spectral representation

Compute the Laplacian matrix of the graph, which is an essential tool to identify clusters in the data using spectral clustering. It is a matrix that characterizes the connectivity of a graph. It captures the relationships between the nodes, and can be used to identify the nodes that are most closely connected to each other. There are two different definitions for this matrix, depending on the degree matrix D and the similarity matrix S of the graph, as follows.

(a) Unnormalized Laplacian matrix

L: L = D -W . (b) Normalized Laplacian matrix L norm : L norm = D -1 2 LD -1 2
. The choice of Laplacian matrix type to use with spectral clustering depends on the application and the problem to be solved. Spectral clustering is often used to optimize two objective functions: Ratio Cut (Rcut) and Normalized Cut (Ncut). Both of these objective functions measure the quality of the partition of a graph into clusters. Let C i be a subset of vertices i.e C i ⊂ V, i = 1, • • • , k and its complement C i := V \C i , the Ratio Cut function (Rcut) [START_REF] Hagen | New spectral methods for ratio cut partitioning and clustering[END_REF]) is defined as:

Rcut(C 1 , • • • , C k ) = k i=1 cut(C i , C i ) |C i | . (2)
Where cut(C, C) := i∈C,j∈C w ij |C i | := number of vertices in C i In this function, the size of a subset C i is measured by its number of vertices. Using the unnormalized Laplacian matrix L with spectral clustering leads to minimizing the Ratio Cut function. In contrast, the Normalized cut function(Ncut) [START_REF] Shi | Normalized cuts and image segmentation[END_REF]) is defined as:

Ncut(C 1 , • • • , C k ) = k i=1 cut(C i , C i ) vol(C i ) . ( 3
)
where vol(C) := i∈C d i

In the Normalized cut function, the size of a subset C i is measured by the weights of its edges. Using the normalized Laplacian matrix L norm with spectral clustering leads to minimizing the Normalized cut function. For more details see Von [START_REF] Luxburg | A tutorial on spectral clustering[END_REF].

The matrices L and L norm have some important properties: they are symmetric and positive semi-definite matrices; the N eigenvalues λ 1 , • • • , λ N of these matrices are non negative real-valued, so 0 =

λ 1 ≤ λ 2 ≤ • • • ≤ λ N ;
the multiplicity k of the value 0 as an eigenvalue of these matrices is equal to the number of connected components C 1 , • • • , C k in the graph. (for more details, see [START_REF] Mohar | The laplacian spectrum of graphs[END_REF], [START_REF] Mohar | Some applications of Laplace eigenvalues of graphs[END_REF] and [START_REF] Chung | Spectral graph theory[END_REF]).

The eigenvalues of the graph Laplacian matrix and its associated eigenvectors are computed. Then, the eigenvectors are used to constitute a low-dimensional representation of the data, where the clusters are more separated. Typically, the k eigenvectors corresponding to the k smallest eigenvalues are used to construct a k-dimensional representation of the data, as they capture the structure of the graph and important features of the data (see [START_REF] Wierzchoń | Modern algorithms of cluster analysis[END_REF]). Reducing dimension can reveal hidden patterns in the data that may be difficult to distinguish in higher dimension.

clustering

Apply the k-means clustering algorithm to the low-dimensional representation to group the data points into k clusters.

In spectral clustering, a specific heuristic method has been proposed for choosing the number of clusters k. This method relies on the gap between two consecutive eigenvalues, with the number of clusters determined by the value of k that maximizes the eigengap δ k : [START_REF] Luxburg | A tutorial on spectral clustering[END_REF]). This method is effective in determining the number of clusters when the dataset is well separated.

δ k =| λ k+1 -λ k | , k ≥ 2 (see Von

Max-stable processes

In this section, we will provide a brief overview of max-stable processes and define the extremal concurrence probability, which is a critical tool for our approach. We essentially follow [START_REF] Dombry | Probabilities of concurrent extremes[END_REF] and refer the reader to this reference for further details.

Definition of Max-stable processes

Let Z 1 (s), Z 2 (s) • • • be a sequence of independent replications of a spatial pro- cess {Z(s), s ∈ S}, S ⊂ R d , d ≥ 1. If there exist continuous functions A n (s) > 0 and B n (s) ∈ R such that max i=1,••• ,n Z i (s) -B n (s) A n (s) d = X(s), s ∈ S, n → ∞, (4)
is non-degenerate, then {X(s), s ∈ S} is a max stable process (see De [START_REF] De Haan | Spatial extremes: Models for the stationary case[END_REF]). The univariate maxima X(s) at any location s, follows a Generalized Extreme Value distribution (GEV), i.e, for all x ∈ R,

P(X(s) ≤ x) = exp[-(1 + ξ(s) x -µ(s) σ(s) ) -1/ξ(s) ], (5)
where µ(s) ∈ R is the location parameter, σ(s) > 0 is the scale parameter and ξ(s) ∈ R is the shape parameter. These parameters are possibly different from one location to another. Setting µ(s) = σ(s) = ξ(s) = 1, leads to consider unit Fréchet distributions, i.e, P(X(s

) ≤ x) = exp[-1/x],
x > 0, and {X(s), s ∈ S} is called a simple max-stable process (see [START_REF] Ribatet | Modelling spatial extremes using max-stable processes Ribatet M, Dombry C, Oesting M[END_REF] and [START_REF] Ribatet | Modelling spatial extremes using max-stable processes Ribatet M, Dombry C, Oesting M[END_REF]). De [START_REF] De Haan | A spectral representation for max-stable processes[END_REF] provided the spectral representation for simple maxstable processes {X(s), s ∈ S} as follows:

X(s) = max i≥1 ζ i Y i (s), s ∈ S, S ⊂ R d , d ≥ 1 ( 6
)
where

{ζ i , i ≥ 1} is a Poisson point process on (0, ∞) with intensity ζ -2 dζ and Y 1 (s), Y 2 (s), • • • denote a sequence of independent replications of a non negative stochastic process {Y (s), s ∈ S} with E[Y (s)] = 1 for all s ∈ S.
Equation ( 6) may be written as follows:

X(s) = max φ∈Φ φ(s), s ∈ S (7)
where Φ = {φ i (s) = ζ i Y i (s) : s ∈ S, i ≥ 1} is a Poisson point process on C 0 , the space of non-negative continuous functions on S (see [START_REF] Ribatet | Modelling spatial extremes using max-stable processes Ribatet M, Dombry C, Oesting M[END_REF]). Let S be a set of m spatial locations : S = {s 1 , • • • , s m } ⊂ S, then the multivariate maxima distribution is given by

P{X(s 1 ) ≤ x 1 , • • • , X(s m ) ≤ x m } = exp -E max j=1,••• ,m Y (s j ) x j (8)
where {Y (s), s ∈ S} is the process appearing in Equation ( 6). The exponent function

V S (x 1 , • • • , x m ) = E max j=1,••• ,m Y (s j ) x j , (9)
is called the exponent measure. It characterizes the dependence structure of X(s 1 ), • • • , X(s m ). Since the exponent measure is homogeneous of order -1, we can obtain a useful relation by setting

x j = x for all j = 1, • • • , m such that V S (1, • • • , 1) = θ S
where θ S is the extremal coefficient that provides a summary of the dependence structure (see [START_REF] Schlather | A dependence measure for multivariate and spatial extreme values: Properties and inference[END_REF] and [START_REF] Smith | Max-stable processes and spatial extremes[END_REF]). Particularly, when S = {s 1 , s 2 } the extremal coefficient satisfies

θ S = V S (1, 1) ∈ [ 1, 2] .
The lower bound corresponds to the variables X(s 1 ) and X(s 2 ) which are completely dependent, while the upper bound corresponds to the case where they are independent.

Several models for max-stable processes have been presented based on this spectral representation, including the Brown-Resnick model (see [START_REF] Brown | Extreme values of independent stochastic processes[END_REF]), the Smith model (see [START_REF] Smith | Max-stable processes and spatial extremes[END_REF]), the Schlather model (see [START_REF] Schlather | Models for stationary max-stable random fields[END_REF]), and the Extremal-t model (see [START_REF] Opitz | Extremal t processes: Elliptical domain of attraction and a spectral representation[END_REF]).

Extremal concurrence probability

Other indices in order to measure the dependence between extremes exist in the literature. [START_REF] Dombry | Probabilities of concurrent extremes[END_REF] introduced the extremal concurrence probability for the analysis of extremal dependence, which was especially designed for max-stable processes. It has properties similar to the pairwise extremal coefficient, but it has the advantage of being a probability measure, which makes it more interpretable and axiomatic. The extremal concurrence probability focuses on the occurrence times of extremes, which means whether the record maxima occurs simultaneously, i.e., at the same time for all locations.

It can be interpreted as the chance of a single extreme event affecting all the locations and being responsible for the record maximum.

It is based on the spectral representation of the max-stable processes. The idea behind this metric can be explained as follows.

Recall the spectral representation in Equation ( 7). We say that the extremes are concurrent at locations

s 1 , • • • , s m ∈ S if X(s j ) = φ ℓ (s j ), j = 1, • • • , m (10)
for some ℓ ≥ 1. This means that the values of the process {X(s), s ∈ S} at locations s 1 , • • • , s m come from the same spectral function φ ℓ . The extremal concurrence probability is defined as

p r (s 1 , • • • , s m ) = P{for some ℓ ≥ 1 : X(s j ) = φ ℓ (s j ), j = 1, • • • , m} (11)
According to Theorem 3 in [START_REF] Dombry | Probabilities of concurrent extremes[END_REF], the bivariate extremal concurrence probability estimation coincides with Kendall's τ statistic:

pr (s 1 , s 2 ) ≡ τ = 2 n(n -1) 1≤i<j≤n sign{X i (s 1 )-X j (s 1 )}sign{X i (s 2 )-X j (s 2 )}, (12) where {X i (s), s ∈ S, i = 1, • • • , n} are n independent copies of {X(s), s ∈ S}.
The bivariate extremal concurrence probability for max-stable processes satisfies p r (s 1 , s 2 ) = 0 if and only if X(s 1 ) and X(s 2 ) are independent, and p r (s 1 , s 2 ) = 1 if and only if X(s 1 ) and X(s 2 ) are almost surely equal. These properties were stated and proved in Proposition 1 of [START_REF] Dombry | Probabilities of concurrent extremes[END_REF].

4 Adapting spectral clustering for max stable process

Let X i (s j ), s j ∈ S, S ⊂ R d , d = 2, i = 1, • • •
, n be a sequence of n independent and identically distributed max stable processes at different locations s j , j = 1, 2, • • • , m. In order to apply spectral clustering in extremal dependence context, locations s 1 , • • • , s m are considered as vertices in a fully connected graph. Each vertex is connected to all others by edges, and the weights of these edges represent the similarity values among the locations. Figure 1 represents a fully connected graph for a max-stable process consisting of 15 locations. For viewing purposes, the self-edges were not shown. Select-

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15
Fig. 1 Fully connected graph with 15 vertices. Each vertex represents a location in the max-stable process ing an appropriate metric to construct the similarity matrix is essential in the spectral clustering algorithm, especially when using a fully connected graph. It is important to choose an extremal dependence measure that can accurately model the neighborhood relations among the locations. In this study, we used the extremal concurrence probability, as introduced by Dombry et al (2018)(see Section 3.2). The similarity matrix represents the pairwise extremal concurrence probability matrix, denoted by CP ∈ R m×m , where m is the number of locations. For a pair (s, s ′ ) ∈ S × S, the element of the matrix CP is given by:

pr (s, s ′ ) = 2 n(n -1) 1≤i<j≤n sign{X i (s) -X j (s)}sign{X i (s ′ ) -X j (s ′ )} (13)
After constructing the similarity matrix CP according to Equation (13), it is used to compute the graph Laplacian matrix. Using the normalized graph Laplacian matrix L norm helps to achieve our goal of making the size of the resulting clusters dependent on the concurrence of extremes (i.e., the weights of the graph edges). The spectrum λ and the eigenvectors of L norm are then computed. The eigenvectors q 1 , • • • , q k are used to constitute a k-dimensional representation for the data. This is done by representing these eigenvectors as columns of an m × k matrix denoted Q. Each row in Q represents a location s j : [START_REF] Wierzchoń | Modern algorithms of cluster analysis[END_REF]). Normalizing each row of Q to norm 1 results in a matrix denoted Y ∈ R m×k . According to [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF], this last step improves the performance of the clustering algorithm. Instead of using kmeans, which is usually used at this step, we used a Gaussian Mixture Model (GMM) to cluster the rows of Y . GMM clusters the datapoints based on probability distribution, considering that the datapoints come from a Gaussian mixture. Each cluster has a Gaussian distribution model with parameters mean and covariance. Taking the covariance into account makes GMM more robust than k-means, which depends only on the cluster mean. For more details about GMM, see for example [START_REF] Bouveyron | Model-based clustering and classification for data science: with applications in R[END_REF]. We summarize these steps in Algorithm 1.

s j → (q j,1 , • • • , q j,k ) , j = 1, • • • , m, this is called spectral mapping (see

Algorithm 1 Adapted spectral clustering

Require: The similarity matrix CP ∈ R m×m , constructed according to Equation ( 13) . Ensure:

Clusters {C 1 , • • • , C k }. 1: Compute the normalized Laplacian matrix L norm = D -1 2 (D -CP )D -1 2 . 2:
Compute the spectrum of L norm . 3: Compute the eigengap δ k and determine the potential number of clusters which correspond to relatively large values of δ k . Then for each of them repeat the steps 4 to 7. 4: Compute the k smallest eigenvectors q 1 , q 2 , • • • , q k of L norm , and arrange these vectors in columns to be a matrix Q, where Q ∈ R m×k . 5: Normalize the rows of Q to norm 1, resulting the matrix Y ∈ R m×k :

Y jl = Q jl /( l Q 2 jl ) 1 2 , j = 1, • • • , m, l = 1, • • • , k. 6:
Consider each row of Y as a point in R k and implement Gaussian Mixture Model (GMM) to cluster them into k clusters. 7: Assign the location s j to cluster l if and only if row j of the matrix Y is assigned to cluster l.

Applicability of the adapted spectral clustering

The adapted Spectral clustering algorithm, which utilizes the extremal concurrence probability as a similarity matrix, aims to identify regions where the pairwise concurrence probability between locations is high. As a result, we obtain regions where the locations exhibit a similar extremal dependence. This can reduce the possibility that the extremal dependence will be non-stationary in the clusters.

As a preliminary step, we need to determine the spatial structure of the k clusters. This stems from the expectation that locations geographically close to each other may exhibit similar extremal dependence. We obtained the spatial structure of these k clusters by applying the Partitioning Around Medoids (PAM) clustering algorithm proposed by [START_REF] Kaufman | Finding Groups in Data: An Introduction to Cluster Analysis[END_REF] to the spatial coordinates of these locations. As a result, we obtained k clusters in which the locations are spatially closer to each other, as measured by their euclidean distances.

We illustrate the application of spectral clustering by examining a nonstationary max-stable process. We construct this process using an approach presented by [START_REF] Huser | Non-stationary dependence structures for spatial extremes[END_REF], named max-stable mixtures. Let X 1 (s) and X 2 (s) be independent max stable processes that have unit Fréchet margins defined on the same space S. Then the process X(s) = max π(s)X 1 (s), (1π(s))X 2 (s) is a simple max stable process, where π(s) is any function with 0 ≤ π(s) ≤ 1. The bivariate distribution is given by

F (x 1 , x 2 ; Ψ) = P{X(s) ≤ x 1 , X(s ′ ) ≤ x 2 } = exp -V mix S (x 1 , x 2 ) (14) and V mix S (x 1 , x 2 ) = V 1 S x 1 π(s) , x 2 π(s ′ ) +V 2 S x 1 1 -π(s) , x 2 1 -π(s ′ ) (15)
where, V 1 S and V 2 S are the exponent measure of X 1 (s) and X 2 (s), respectively, and S = {s, s ′ }. The proportion π(s) determines which of the processes X 1 (s) and X 2 (s) prevails at location s. If π(s) is constant across space, then the process X(s) is stationary. However, X(s) becomes non-stationary when π(s) varies spatially, such as if it depends on covariates. This variation may lead to capturing different extreme behaviors in different spatial regions. For clustering purposes, we will consider that each spatial cluster k has its own mixing proportion π k . To make the process more realistic, we take into account the presence of regression in the values of π k between the clusters. Figure 2 represents an example of realization of such a process by mixing a Schlather and an extremal-t process over the [0, 1] 2 . In the simulation study (Section 7), we will explore how spectral clustering can recover these clusters of locations. Assuming we have two clustered regions, depending on these clusters, we can rewrite the bivariate distribution in Equations 14 and 15 as

F (x 1 , x 2 ; Ψ) =          exp -V 1 S ( x1 π1 + x2 π1 )) + V 2 S ( x1 1-π1 + x2 1-π1 ) , if s, s ′ ∈ C 1 exp -V 1 S ( x1 π2 + x2 π2 )) + V 2 S ( x1 1-π2 + x2 1-π2 ) , if s, s ′ ∈ C 2 exp -V 1 S ( x1 π1 + x2 π2 )) + V 2 S ( x1 1-π1 + x2 1-π2 ) , if s ∈ C 1 , s ′ ∈ C 2 (
16) For simplicity we will denote the distribution function components in Equation 16by

F C1 (x 1 , x 2 ; Ψ) if s, s ′ ∈ C 1 , F C21 (x 1 , x 2 ; Ψ) if s, s ′ ∈ C 2 and F C12 (x 1 , x 2 ; Ψ) if s ∈ C 1 , s ′ ∈ C 2 .
Using the distribution in Equation 16, we can model the whole region under study depending on clustered regions. 6 Inference: composite likelihood approach

The full likelihood inference for max-stable models is computationally intractable [START_REF] Castruccio | High-order composite likelihood inference for max-stable distributions and processes[END_REF]). The most widely used approach is the pairwise composite likelihood [START_REF] Padoan | Likelihood-based inference for max-stable processes[END_REF]). Let Ψ represent the vector of unknown parameters; the pairwise composite log-likelihood can be expressed as follows:

P(Ψ) = n i=1 m-1 j=1 m j ′ =j+1 W jj ′ log L(x ij , x ij ′ ; Ψ) =: n i=1 P i (Ψ) (17)
where x ij represents the block maxima i which recorded at station j, L(x ij , x ij ′ ; Ψ) is the likelihood of the pair (x ij , x ij ′ ) and W jj ′ are non negative weights that specify the contribution of each pair. Therefore the maximum pairwise liklihood estimator is given by Ψ = arg max P(Ψ). Under some regularity conditions (see [START_REF] Padoan | Likelihood-based inference for max-stable processes[END_REF]), Ψ obtained by maximizing the pairwise composite log-likelihood in Equation ( 17) is consistent and asymptotically normally distributed, such that

Ψ ∼ N (Ψ, I(Ψ) -1 ) (18)
with

I(Ψ) = H(Ψ) J (Ψ) -1 H(Ψ) (19)
where I(Ψ) is the sandwish information matrix, H(Ψ) = E -∇ 2 P(Ψ) is the sensitivity matrix and J (Ψ) = V ∇ P(Ψ) is the variability matrix. There- fore, the variance of the estimated parameters can be assessed by estimating the matrices H(Ψ) and J (Ψ). In this context, the empirical estimates of the matrices H(Ψ) and J (Ψ) are given by

Ĥ(Ψ) = - n i=1 ∇ 2 P i ( Ψ) (20)
and

Ĵ (Ψ) = n i=1 ∇P i ( Ψ) ∇P i ( Ψ) T (21)
In practice, Ĥ(Ψ) can be obtained directly from the optimization algorithm since it equals the negative of the Hessian matrix.

In this paper, our interest is to use the clustered regions in modeling the whole region. We define the censored pairwise likelihood contribution

L CL (x ij , x ij ′ ; Ψ) for a pair (x ij , x ij ′ ) as follows L CL (x ij , x ij ′ ; Ψ) =      ∂ 2 12 F C1 (x ij , x ij ′ ), if s j , s j ′ ∈ C 1 ∂ 2 12 F C2 (x ij , x ij ′ ), if s j , s j ′ ∈ C 2 ∂ 2 12 F C12 (x ij , x ij ′ ), if s j ∈ C 1 , s j ′ ∈ C 2 (22)
where ∂ i is the differentiation with respect to variable x i .

Finally, selecting the best-fitted model under the composite likelihood approach is performed using the Composite Likelihood Information Criterion (CLIC) [START_REF] Varin | A note on composite likelihood inference and model selection[END_REF]), which is defined as CLIC = -2 P( Ψ) + 2 tr( Ĥ( Ψ) -1 Ĵ ( Ψ)). The lowest value of CLIC correspond to best fit model.

Simulation study

In order to assess the accuracy of our algorithm, we tested it on two simulation cases. In each cases, the coordinates of the locations were generated randomly and uniformly in [0, 1] 2 . In practice, we usually do not have two completely separated and independent clusters, some smoothness and dependence from one to the other may be present, this is why we will use the max-stable mixture model in two different simulation case.

Case 1: In this case, we assumed the existence of two clusters, one in the north and the other in the south. To identify the spatial structure of these clusters, we applied the PAM algorithm to the coordinates of 80 locations. The left panel in Figure 3 displays the two clusters, with the first cluster in the north (depicted by red points) and the second in the south (depicted by blue points). We then established the boundary area between the clusters arbitrarily, represented by the gray points located between two black lines (latitude ranging from 0.4 to 0.55), as shown in the right panel of Figure 3. In this context, we have named the cluster in the north as A, the cluster in the south as B, and the boundary area as middle. Then, we simulated a nonstationary process, as described in Section 5. Denoted by M 1 , the model represents a mixture of the Schlather model with a powered exponential

π(s) =      0.3, if s ∈ A 53 30 -8 3 × latitude, if s ∈ Middle 0.7, if s ∈ B ( 23
)
in order to get continuous realizations of the underlying process. With this choice, we get two different areas (A and B) where the underlying process is stationary; moreover the areas A and B are dependent leading to a more realistic model for environmental phenomena. Afterward, we applied adapted spectral clustering to this process. The eigengap heuristic indicated that the number of clusters equals 2, as illustrated in the left panel of Figure 4. In order to further confirm this choice, we employed the widely used Silhouette Index [START_REF] Rousseeuw | Silhouettes: a graphical aid to the interpretation and validation of cluster analysis[END_REF]) for clustering validation. The Silhouette Index is a valuable metric for assessing the separation of clusters, with values ranging from -1 to 1. The highest Silhouette Index value corresponds to the most reliable clustering results. We considered potential numbers of clusters, including 2, 3, and 4, which corresponded to relatively large eigengap values, the right panel of Figure 4 clearly shows that the optimal number of clusters is 2. The spectral clustering result for k equals 2 is depicted in the left panel of Figure 5, with red points representing cluster A, blue points representing cluster B, and black triangles representing the points that lie in middle. The pairwise extremal concurrence probability matrix for the clustering result is displayed in the right panel of Figure 5. Since we know the true clusters A, Middle and B, We can calculate the precision of the spectral clustering algorithm for recovering the true clusters by determining the ratio of locations in the true cluster shared in common with the cluster resulting from spectral clustering. We will denote this ratio as A Ct,Csp for simplicity: above. The mean of A Ct,Csp for the 10 simulations is shown in Table 2.

A Ct,Csp = |C t ∩ C sp | |C t | (
From Table 2, we can observe that a higher number of observations leads to better recovery of the true clusters. However, the algorithm performs well even when the number of observations is relatively small. Its worth noting that for the 40 simulations in Table 2, the accuracy of determining the number of clusters (k = 2) using the eigengap heuristic is 0.8. However, the eigengap is still useful in providing us with the potential number of clusters, which corresponds to the relatively largest eigengap.

When we calculate the Silhouette Index only for this potential number of k, we achieve an accuracy of 1 in determining the correct number of clusters. Now, we will assess the ability of the proposed censored pairwise likelihood, as defined in Section 6, Equation 22, to estimate the parameters of the whole region based on clusters. We simulate N = 100 max stable mixture processes (model M 1 ) with the same conditions as explained previously, with the number of observations fixed at 100. Then, we use Equation 22to obtain the estimator Ψ. After that, we will create a boxplot for the estimated parameters and calculate the mean estimate and the root mean square error (RMSE) for each estimated parameters. Let Ψi denoted the ith estimation, then

RMSE = 1 N N i=1 ( Ψi -Ψ) 2 1/2 (25)
The boxplots of the errors of the estimated parameters, i.e., ( Ψ -Ψ), are displayed in the left panel of Figure 6, and the RMSE values are shown in the right panel of Figure 6. The subscript index 1 (2) refers to the first (second) mixing model. The mean estimates of the parameters are π1 = 0.29, π2 = 0.67, φ1 = 0.28, α1 = 0.98, φ2 = 0.66 and α2 = 1.23. Generally, the

-1.0 -0.5 0.0 0.5 1.0 π ^1 -π1 π ^2 -π2 φ ^1 -φ1 α ^1 -α1 φ ^2 -φ2 α ^2 -α2 0.00 0.05 0.10 0.15 π ^1 π ^2 φ ^1 α ^1 φ ^2 α ^2
Fig. 6 The left panel shows the boxplots displaying ( Ψ -Ψ) of the estimated parameters using censored pairwise likelihood Equation 22based on 100 simulations of model proposed censored pairwise likelihood appears to work well in estimating the parameters for the whole region depending on clustered regions. Case 2: Here, similar to case 1, we assumed the existence of two clusters, one in the west and the other in the east. The spatial structure of these clusters is depicted in the left panel of Figure 7 after applying the PAM algorithm to the coordinates of 40 locations. The red points represent the west cluster, while the blue points represent the east cluster. We then established the boundary area between the clusters arbitrarily, represented by the gray points located between two black lines (longitude ranging from 0.4 to 0.6), as shown in the right panel of was defined as follows:

M 1 with parameters π 1 = 0.3, π 2 = 0.7, ϕ 1 = 0.3, α 1 = 1, ϕ 2 = 0.
π(s) =      0.2, if s ∈ A -1 + 3 × longitude, if s ∈ Middle 0.8, if s ∈ B (26)
Then, we applied adapted spectral clustering to this process. The eigengap heuristic and Silhouette Index indicate that the number of clusters equals 2, as illustrated in the left and right panels of Figure 8 respectively. The spectral clustering result for k equals 2 is depicted in the left panel of Figure 9, with red points representing the west cluster A, blue points representing the east cluster B, and black triangles representing the points that lie in the boundary area. The pairwise extremal concurrence probability matrix for the clustering result is displayed in the right panel of Figure 9. Table 3 4 presents the algorithms performance for different numbers of observations: n = 500, 200, 100, and 50. For each number of observations, we simulated 10 processes on the same coordinates with the same parameters as mentioned above. The table displays the mean of A Ct,Csp for the 10 simulations. From Table 4, its clear that the algorithm successfully recovers the true clusters in all the tested numbers of observations, also, the number of observations does not affect the clustering of points that lie in the boundary area. Its worth noting that, in the 40 simulations presented in Table 4 the accuracy of determining the number of clusters (k = 2) using the eigengap heuristic and the Silhouette Index is equal to 1.

In order to asses the performance of the the proposed censored pairwise likelihood, we simulate N = 100 models the same conditions as M 2 and with the number of observations fixed at 100. Afterwards, we estimate the parameters use Equation 22. underfitting for some parameters, the proposed censored pairwise likelihood works well.

Since the results of the simulations appear satisfactory, we can use this technique to clustering and modeling the maxima for rainfall in East Australia and France.

Application on data

This section is devoted to two data applications: one on rainfall in Australia's east coast, and the other on rainfall in France.

Rainfall over east coast of Australia

We will begin with a brief description of the data, followed by the application of our clustering method and a discussion of the results. 8.1.2 Regionalizing the maxima for rainfall over east coast of Australia

Description of the data

We will apply the adapted spectral clustering algorithm, described in Section 4 Algorithm 1, to clustering the extremal dependence. To demonstrate the effect of the block size on the resulting clusters, we will use different sizes of block. Specifically, we will test block sizes of 183 days, 30 days, 15 days and 10 days.

Figure 12 shows the results of spectral clustering for Australia's rainfall data for each block size. We can observe from the first column of Figure 12 that the eigengap heuristic indicates that the number of clusters is 2 for all the tested block sizes. To reinforce this result, we calculated the Silhouette Index for the potential number of clusters, and the results confirm that the best number of clusters is 2 for all the tested block sizes, as shown in the middle column of Figure 12. In the right column of Figure 12 it is evident that the spectral clustering result is the same for all the tested block sizes, with the exception of one location in a block size of 10 days. This variation is expected since it is located in the boundary area between the clusters. For this data, we can say that the size of the block does not affect the clustering map. The first (second, third, and fourth) row is for a block size of 183 days (30 days, 15 days, and 10 days) respectively.

Modelling the rainfall over east coast of Australia

In order to determine whether the clustering of extremal dependence is significant in modeling the maxima of rainfall over the east coast of Australia, we will assess this by modeling with a 30 day block size. We consider nine arbitrary models that belong to three classes: max-stable, stationary max-stable mixture (constant mixing proportion), and non-stationary max-stable mixture (different mixing proportions for each cluster). The descriptions of the models are as follows.

Class I: consists of max stable models I1-I3.

I1: Class III: consists models III1-III3 which are the non stationary max stable mixture models as defined in Equation 16for models II1-II3. For all the models above, the univariate must be unit Fréchet. Therefore, we use the empirical distribution function to transform the data to a unit Fréchet distribution. After that, we estimate the dependence parameters using the composite likelihood approach, assuming equal weights. For model selection, we calculate CLIC. The results are summarized in Table 5, where ϕ, α, df and π represent the range, smooth, degree of freedom and spatial mixing proportion parameters. The subscript index 1 (2) refers to the first (second) mixing model.

Regarding the results in Table 5, the best fitting model for the data is the max stable mixture with different spatial proportions for each cluster.

Rainfall over France

This subsection is devoted to the study of rainfall data in France.

Description of the data

This data is provided by Météo-France and represents the hourly precipitation recorded at 80 French monitoring stations. The data was measured during the fall season (September, October and November) over the period 1993 -2021. Each station has 24 * 91 * 29 = 63336 observations. The distance between the stations ranges approximately from 12 km to 1248 km. The geographic locations of these stations were chosen to cover all the French metropolitan

Regionalizing the maxima for rainfall over France

In order to clustering the rainfall extremal dependence, we implemented the adapted spectral clustering on the data. Additionally, we studied the effect of the block size on the clustering map of extremal dependence. The block sizes we considered are monthly, 2 weeks, and weekly. The results obtained with the spectral clustering on this data for each block size are shown in Figure 14.

The eigengap heuristic indicated that the number of clusters is 2 for all tested block sizes, as shown in the left panels of Figure 14. This was reinforced by the Silhouette Index, as is evident in the middle panels of Figure 14.

We will begin our discussion with the weekly block size since it was studied by [START_REF] Bernard | Clustering of maxima: Spatial dependencies among heavy rainfall in france[END_REF]. Implementing spectral clustering on the data with this block size divides France into two regional areas, north and south, along the Loire valley line. Each of these regions has a different extremal dependence, as explained in the right panel of the third row of Figure 14. This can be interpreted easily. The extreme rainfall in the north of France is produced by disturbances from the Atlantic, which affect Brittany, Paris and other areas in the north of France. In contrast, the extreme rainfall in the south of France is caused by the Mediterranean sea, which affects the coastal areas, particularly Cévennes and the Montagne Noire. The results are similar to those obtained by [START_REF] Bernard | Clustering of maxima: Spatial dependencies among heavy rainfall in france[END_REF], where the selection criterion for the number of clusters indicated that k = 2. The locations clustering result is relatively close to [START_REF] Bernard | Clustering of maxima: Spatial dependencies among heavy rainfall in france[END_REF], where France was divided into north and south regions. This indicates that the behavior of the data has not changed since 2011. The right panels of Figure 14 show the location clustering results for all tested block sizes. It is clear that the south region becomes smaller as the block size becomes larger. For this data, we can say that the size of the block may affect the clustering map.

Modelling the rainfall over France

Once again, we will check if the clustering of extremal dependence is valuable in modeling the maxima of rainfall over France. We will use the weekly block size and consider the same models proposed for Australian data, following the same modeling steps as in section 8.1.3. The results are summarized in Table 6.

The lowest value for CLIC in Table 6 indicates that the best-fitting model for the France data is the max-stable mixture with different spatial proportions for each cluster. Although the difference in the value of π between the clusters in the north and in the south is not too large, it appears significant in modeling.

Discussion and Conclusion

Modeling environmental extreme events requires an understanding of the extremal dependence structure. In many studies, it is assumed that extremal dependence is fixed. However, this assumption may be incorrect, especially in large regions or areas with complex geographical or climatic patterns. Therefore, finding a method that can detect regions with similar extremal dependence is valuable.

In this study, we combined spectral clustering with extremal concurrence probability to develop a simple clustering method for max-stable processes. Additionally, we modeled the entire region based on the regions resulting from spectral clustering. We validated our approach through a simulation study based on a non-stationary max-stable process. Then, we applied it to two environmental datasets.

The first dataset consists of daily rainfall data over the east coast of Australia. Spectral clustering divided the east coast area with respect to rainfall maxima into two regional clusters, where assuming fixed extremal dependence for each cluster is reasonable. We found that, for this data, the block size does not affect the clustering map.

The second dataset consists of hourly precipitation data over France. Spectral clustering divided the France region into two regional clusters: one in the north and the other in the south, each exhibiting similar extremal dependence. However, for this data, the clustering maps are not the same for all the tested block sizes. Therefore, we conclude that the block size can result in changes in regional clusters and should be taken into consideration. Modeling these two datasets indicates that the best-fitted model for each of them is the max-stable mixture model with different spatial mixing proportions for each clustered region. This leads us to consider regionalizing extremal dependence to help model the non-stationary max-stable mixture model, instead of relying on covariates that are not always provided to model spatial proportions. Finally, despite the simplicity of the adapted clustering algorithm, it is powerful. As a future direction of this study, one can study other variables like temperature. Another direction is to test the efficiency of this clustering algorithm when applied to a larger region, such as the whole of Australia or the continent of Europe.

Fig. 2

 2 Fig.2Example of realization of a max-stable mixtures process on the logarithm scale. Left panel: realization of an Extremal-t process with whittle-matérn correlation function ρ(h) = (2 1-c 1 /Γ(c 1 ))(∥h∥/c 2 ) c 1 Kc 1 (∥h∥/c2 ) with degree of freedom equal to 5, range c 2 = 0.7 and smooth c 1 = 1. Middle panel: realization of a Schlather process with exponential correlation function ρ(h) = exp(-∥h∥/ϕ), with range ϕ = 0.7. Right panel: corresponding realization of the max-stable mixture with π(s) = 3x -1 if x ∈ [0.4, 0.6], if x < 0.4 π(s) = 0.2 and π(s) = 0.8 if x > 0.6, where x represents the value of the X coordinate corresponding to location s.

Fig. 3

 3 Fig. 3 The left panel shows the spatial structure of the two clusters, while the right panel displays cluster A in the north, cluster B in the south, and the middle, which is the boundary area between clusters A and B. correlation function ρ(h) = exp(-(∥h∥/ϕ) α ), and the Brown-Resnick model with a semivariogram γ(h) = (∥h∥/ϕ) α . The parameters of the Schlather model are set to ϕ 1 = 0.3 and α 1 = 1, where the subscript index 1 refers to the first mixing component. The parameters of the Brown-Resnick model are set to ϕ 2 = 0.7 and α 2 = 1.2, where the subscript index 2 refers to the second mixing component. The number of observations is fixed at 500, and the spatially varying proportion was set as follows:

Fig. 4 Fig. 5

 45 Fig. 4 The left panel shows the eigengap values against number of clusters k, while the right panel displays Silhouette Index values against number of clusters k.

  Fig.6The left panel shows the boxplots displaying ( Ψ -Ψ) of the estimated parameters using censored pairwise likelihood Equation22based on 100 simulations of model M 1 with parameters π 1 = 0.3, π 2 = 0.7, ϕ 1 = 0.3, α 1 = 1, ϕ 2 = 0.7 and α 2 = 1.2. The number of observations fixed at 100. The red horizontal line represents the zero value. Meanwhile, the right panel displays the barplots for the RMSE of Ψ = {π 1 , π2 , φ1 , α1 , φ2 , α2 } for the same simulations.

Figure 7 .Fig. 7

 77 Fig.7The left panel shows the spatial structure of the two clusters, while the right panel displays cluster A in the west, cluster B in the east, and the middle, which is the boundary area between clusters A and B.

Fig. 8

 8 Fig. 8 The left panel shows the eigengap values against number of clusters k, while the right panel displays Silhouette Index values against number of clusters k.

Fig. 9

 9 Fig.9The left panel shows the spectral clustering result for k=2 where the red points represent cluster A, blue points represent cluster B and the black traingles represent the points that lie in the middle, while the right panel shows the pairwise extremal concurrence probability matrix for the clustering result.

Fig. 10

 10 Fig.10The left panel shows the boxplots displaying ( Ψ -Ψ) of the estimated parameters using censored pairwise likelihood Equation 22 based on 100 simulations of model M 2 with parameters π 1 = 0.2, π 2 = 0.8, ϕ 1 = 0.3, α 1 = 1 and ϕ 2 = 0.5. The number of observations fixed at 100. The red horizontal line represents the zero value. Meanwhile, the right panel displays the barplots for the RMSE of Ψ = {π 1 , π2 , φ1 , α1 , φ2 } for the same simulations.

Fig. 11

 11 Fig. 11 Geographic locations of 40 stations on the east coast of Australia.

Fig. 12

 12 Fig.12The left column displays Eigengap values, and the middle column shows the Silhouette Index against the number of clusters k, while the right column represents the spectral clustering result for the east coast of Australia rainfall data over the period of 1972-2019. The first (second, third, and fourth) row is for a block size of 183 days (30 days, 15 days, and 10 days) respectively.

Fig. 13

 13 Fig. 13 Geographic locations of 80 stations over France region

Fig. 14

 14 Fig.14The left column displays Eigengap values, the middle column shows the Silhouette Index against the number of clusters k, and the right column represents the spectral clustering result for the France rainfall data over the period of 1993-2021. The first, second, and third rows are for a monthly block size, a two-week block size, and a weekly block size, respectively.

Table 1

 1 24)where C t represents the true cluster, C sp represents the cluster resulting from spectral clustering, and |.| denotes the number of elements in a set. The values of A Ct,Csp for all combinations between the true clusters and the spectral clustering clusters are presented in Table.1. Its clear that the true clusters A and B are correctly recovered by spectral clustering. Regarding the Middle cluster, 4 locations are clustered with A which has π(s) values closer to 0.3 and 6 locations are clustered with B which has π(s) values closer to 0.7. To assess whether the number of observations affects the algorithm performance, we implemented it with different numbers of observations: n = 500, 200, 100, and 50. For each number of observations, we simulated 10 processes on the same coordinates with the same parameters as mentioned The values of A C t ,Csp for one simulation of case 1 with the number of observation equals 500

	Spectral clustering clusters

Table 2

 2 The mean values of A C t ,Csp for 10 simulations of case 1 with the number of observations set to n = 500, 200, 100, and 50.

				Spectral clustering clusters
		n = 500	n = 200	n = 100	n = 50
	True
	clusters	A	B	A	B	A	B	A	B
	A	1	0	1	0	1	0	0.98	0.02
	Middle	0.41	0.59	0.43	0.57	0.39	0.61	0.51	0.49
	B	0	1	0	1	0	1	0.07	0.93

Table

Table 3

 3 The values of A C t ,Csp for one simulation of case 2 with the number of observation equals 500

	Spectral clustering clusters

Table 4

 4 The mean values of A C t ,Csp for 10 simulations of case 2 with the number of observations set to n = 500, 200, 100, and 50.

				Spectral clustering clusters
		n = 500	n = 200	n = 100	n = 50
	True
	clusters	A	B	A	B	A	B	A	B
	A	1	0	1	0	1	0	1	0
	Middle	0.46	0.54	0.46	0.54	0.46	0.54	0.46	0.54
	B	0	1	0	1	0	1	0	1

  This data represents the daily rainfall totals (in millimeters) measured over a 24-hour period at 40 stations on the east coast of Australia during the winter season (April to September) from 1972 to 2019, resulting in a total of 183 * 48 = 8,784 observations at each station. The altitude of these stations ranges from 2 to 540 meters. The distance between the stations ranges approximately from 11 km to 2058 km. The geographic locations of the 40 stations are illustrated in the left panel of Figure11. More information about this data can be found in references such as[START_REF] Ahmed | Regionalization of the extremal dependence structure using spectral clustering[END_REF],[START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF],[START_REF] Ahmed | Regionalization of the extremal dependence structure using spectral clustering[END_REF][START_REF] Abu-Awwad | Fitting spatial max-mixture processes with unknown extremal dependence class: an exploratory analysis tool[END_REF]. The data is freely available on the website http://www.bom.gov.au.

Table 5

 5 Summary of the fitted models based on the 30-day maxima of rainfall over the east coast of Australia. The bold number for CLIC indicates the lowest value.

	Class I	ϕ	α	df					CLIC
	I1	169.11	1.06	-					1871755
	I2	134.23	0.63	-					1866939
	I3	661.62	-	3.23					1865962
	Class II	ϕ 1	α 1	df 1	π		ϕ 2	α 2	CLIC
	II1	380.17	1.37	-	0.25		115.73	0.61	1865515
	II2	1716.5	-	4.66	0.54		48.66	0.54	1865280
	II3	224.04	0.77	-	0.88		28.74	-	1866104
	Class III	ϕ 1	α 1	df 1	π 1	π 2	ϕ 2	α 2	CLIC
	III1	100.02	0.34	-	0.11	0.53	90.11	0.44	1866179
	III2	604.90	-	2.44	0.41	0.88	125.31	0.48	1864845
	III3	224.77	0.77	-	0.89	0.87	53.26	-	1865982
	regions. Figure 13 illustrates the geographic locations of the 80 stations. This
	data was studied by Bernard et al (2013) during the period 1993 -2011.

Table 6

 6 Summary of the fitted models based on the weekly maxima of rainfall over France. The bold number for CLIC indicates the lowest value.

	Class I	ϕ	α	df					CLIC
	I1	164.89	0.96	-					9900596
	I2	86.58	0.55	-					9918058
	I3	344.08	-	1.88					9890874
	Class II	ϕ 1	α 1	df 1	π		ϕ 2	α 2	CLIC
	II1	408.38	1.05	-	0.48		56.89	0.55	9886465
	II2	100.28	-	1	0.53		90.15	0.47	9894430
	II3	100.06	0.42	-	0.87		90	-	9909144
	Class III	ϕ 1	α 1	df 1	π 1	π 2	ϕ 2	α 2	CLIC
	III1	426.87	1.02	-	0.50	0.43	57.05	0.56	9885185
	III2	840.75	-	1.90	0.59	0.49	30.38	0.55	9884561
	III3	331.47	0.87	-	0.76	0.66	316.06	-	9895379

Acknowledgments We acknowledge partial support from the PAUSE program, which is operated by the Collège de France. Also, we would like to thank the French meteorological service (Météo-France) for providing us with the data.


est destinée au dépôt

  Tânia C. Gonçalves, Rachid Boukaiba, Jordi Molgó, Muriel Amar, Michel Partiseti, Denis Servent, Evelyne Benoit To cite this version: Tânia C. Gonçalves, Rachid Boukaiba, Jordi Molgó, Muriel Amar, Michel Partiseti, et al.. Multiscale functional approaches to evidence high affinity blockade of Nav1.6 channel subtype by huwentoxin-IV spider peptide. 29th Ion Channel Meeting, Sep 2018, Sète, France. hal-04466966


Eric Fossat
email: eric.fossat@unice.fr

Visible Astronomy as well? Why not!

Keywords:

It is quite clear that the dry and cold astronomical site of dome C has to be one of the best places on Earth, if not THE best, for infrared and sub millimetric astronomical observations. In comparison with usual observatories located in more usual latitudes, the benefit to be expected in the visible range is a little less obvious. But this benefit does exist indeed. The very long days and nights of the Antarctic plateau provide a unique opportunity for extremely long and nearly uninterrupted observations. Added to the expected reduced amplitude of the stellar scintillation, this makes possible to consider the feasibility of asteroseismic observations that were envisioned only in space so far. Such night time astronomical programmes can be developed very quickly, with moderately sized telescopes, and will benefit of the first winter-over seasons at Concordia. When thinking beyond the first few winters, the unique seeing properties of the site look extremely promising for the combination of interferometry, high performance adaptative optics and stellar coronography, or very high contrast imaging, so that the direct observations of exoplanets must be considered as a possibility for future observations on this site. I briefly review here the first projects and ideas in these directions. As many of these ideas are presented during this "visible" session, you will find details, regarding most of them, in other papers of these proceedings.

Introduction

Historically speaking, the first successful astronomical observation on the Antarctica plateau was done at South Pole during the 1979/80 summer season. Thanks to the incredible perseverance of Martin Pomerantz (second from left on Figure 1), from Bartol, Gerard Grec and I had the chance of access-ing the Amundsen-Scott South Pole station with a very first prototype of the future full disk helioseismic instruments developed for the IRIS network and then for the GOLF space instrument on SOHO. The first uninterrupted measurement of global solar oscillations much longer than one day (6 days non stop, in fact) was obtained during the first week of 1980, and it has since then been regarded as the birthdate of helioseismology, the word itself being used only a year later [START_REF] Grec | These proceedings Grec[END_REF] (Grec et Duvall (Harvey et al 1982).

al

However, helioseismology was more and more demanding in the extremely long duration of as uninterrupted as possible time series, so that it started to spread around the planet by means of ground based networks first, and of spacecrafts later on. The South Pole station was not able to provide many consecutive months of continuous data, and it became less demanded for this scientific chapter. However, the astronomical properties of the site were studied in much detail and an important night time IR observatory was logically the next step (Roche 2003).

The Concordiastro programme

To-day, the South Pole site is certainly one of the best known on Earth for its astronomical properties [START_REF] Storey | These proceedings Travouillon[END_REF]. However, if it has indeed proved to be an exceptional site in the infrared and submm ranges, it has also been a little disappointing regarding the seeing in the visible range (Marks et al. 2000). The reason is a 300 to 400 m thick layer of turbulent air due to the fact that the pole is standing on a slope of the polar plateau, even if this slope cannot be felt when looking around at the apparently absolutely horizontal snow desert. This slope is still enough to permit the presence of a katabatic wind coming down from the dome A area, and producing enough turbulence to damage the optical seeing. The dome C site is a dome, and is consequently supposed to be free from katabatic wind. Indeed, the surface wind statistical values are of the order of 6 m/s at South Pole against only 2.6 m/s at dome C. The difference has a very significant impact on the amount of flying snow, and even more important on the amount of turbulence disturbing astronomical images. Another important issue for astronomers is the frequent absence of high speed wind even at high altitude (see Fig. 2). If this can be confirmed at night on a long term basis, it will have really tremendous conse-quences for optical and near infrared astronomy: indeed, it could imply the absence of any turbulent layers other than the moderate one very near the snow surface. On one hand multiconjugate adaptative optics would then not be required, and on the other hand the stellar scintillation produced by high altitude turbulence would be reduced, possibly enough for making feasible asteroseismic measurements of very faint stellar oscillations. These are generally regarded as inaccessible from the ground because of the scintillation limitation.

Along these ideas, the Concordiastro programme was proposed to both French and Italian polar Institutes in 1999, when it became clear that the Concordia station was really being erected and would become available within only a few more years. This programme aims at measuring the atmospheric seeing quality of the site by many different ways, including in situ radiosoundings and direct optical measurements of quality of the stellar image provided by a telescope. The programme was first funded in 2000, and at the time of the Capri meeting, three summer campaigns have already been organized. They were mostly devoted to daily radio-soundings by means of meteorological stratospheric balloons, and during one month of the last campaign, to the first DIMM measurements ever made at dome C and ever made in permanent sunlight on the star Canopus [START_REF] Aristidi | These proceedings A. Baglin and the COROT Team IAU Symposium[END_REF]. During the first accessible winters, a DIMM first and a GSM then (Martin et al. 1994) will be operated to provide as many as possible seeing parameters all night long. Stratospheric radio-soundings will also be done if (financially) possible on at least a weekly basis, and the possibility of organizing a Scidar (Vernin et al. 1973) campaign is also being considered. The solar part of this site testing efforts is in the hands of the OAC group at Napoli, and will include high resolution full disk solar imaging (Moretti et al. 2003).

Another important parameter of the site qualification is the cloud cover statistics, that is not outstanding at South Pole, the sky being covered by more or less thin cirrus clouds nearly 50 percent of time. The precise statistics is not known yet at dome C, but it begins to appear that it will be much better, and even possibly outstanding. The Aastino (Lawrence 2003) and Icecam Australian programmes (Travouillon 2003) are now measuring it at night. Of course, if the seeing night time is really exceptional thanks to the slow winds, with an outstanding cloud cover statistics, the site can then be regarded as the best in Fig. 3. This wooden platform is the first of two that will support, at 6m above the ice surface, the series of small telescopes of the Concordiastro programme, making site testing on one of them and astroseismology on the other one. Photo K. Agabi the world not only in infrared but also for visible astronomy as well.

Asteroseismology, stars and giant planets

The first night time astronomy programme that comes to mind in the visible range is the stellar continuation of the pioneer work made with Pomerantz 24 years ago. Helioseismology has become to-day a mature science, while asteroseismology, that intends to do the same on distant solar-type stars, is still at its very beginning. You can read more details on this topic in another paper (Kurtz 2003). Along this line, the Concordiastro programme contains the first step of asteroseismic measurements to be conducted at Concordia, namely the double star Alpha Centauri [START_REF] Grec | These proceedings Grec[END_REF]. Solartype oscillations have recently been detected in Doppler shift measurements with a large telescope on the brightest component of this double star [START_REF] Bouchy | [END_REF]. As I already mentioned, asteroseismology is expected to be extremely difficult for ground based observations made in photometry, so that at least (Favata et al. 2000). These space missions intend to measure as many stars as possible, so that they will avoid the brightest of them for the simple reason that they are too bright and an instrument cannot be tuned for both bright and faint stars. As generally speaking, the brightest stars are also our nearest neighbors, they are extremely interesting, and they represent an ideal target for the dome C ground based site, provided the scintillation level confirms that it is indeed significantly lower than anywhere else. Alpha Centauri will be measured during the first accessible night, in 2005.

However, the stars are not the only night time objects that can show pulsations. In parallel to the selection of a few bright stars, the asteroseismological pro-gramme that is planned for the first winterover campaigns at Concordia includes the planet Jupiter. Jupiter is not a star, but it is also a non standard planet, as it has no solid ground, being made mostly of Hydrogen and Helium, as the Sun and most stars. As it is visibly extremely turbulent (see photo), it has been shown already that these noisy subsurface layers must generate enough acoustic energy for producing resonant acoustic vibrations, exactly as the subsurface convection is doing the same in the Sun [START_REF] Schmider | Possible detection of Jovian global oscillations[END_REF]. Consequently, an asteroseismic study of the deep internal structure of Jupiter is also possible, under the name of Jovian seismology (Voronstov et al. 1976), (Mosser et al. 1991). As this deep internal structure is poorly known in fact, this research appears to be very important for the general understanding of the planets and planetary systems formation (Guillot 1998). For this purpose, a specific instrument has been developed at Nice. This instrument is named "Sympa", for SIsmometre IMageur a Prismes Accoles (Schmider et al. 2002). It is a very compact Mach-Zender interferometer, that makes interferometric images of the planet. They give access to acoustic resonant modes of degrees comprised between zero and 20 to 30 depending on the atmospheric turbulence. This range should be adequate for obtaining a good enough vertical resolution in the inversion of frequencies for reconstructing the Jovian interior model. Now, in 2003, Jupiter is standing relatively high in the Northern hemisphere sky. Its continuous observation is only possible by means of coordinating several observations, using then several instruments at different longitudes. Then, Jupiter will quickly move towards the Southern hemisphere, and for a few years starting in 2006, it will be permanently visible from the dome C. That year will correspond to the second winter-over at the Concordia station, so that the Jupiter programme appears to be very well suited for an optimal deployment at Concordia.

Then, what's next?

Clearly, what will happen then, after the first asteroseismic night time campaigns, will depend on the final site testing results, that will will make the site more or less strongly attractive to visible light astronomers, for counterbalancing the difficulty inherent to the geographical situation. The first priority during these first few winter-over campaigns, then, remains without doubt the site testing that must provide extensive, robust and convincing results.

Then, if one assumes that the seeing properties will be consistent with their present expectation in term of Fried parameter, isoplanetic angle and wavefront coherence time, one cannot avoid thinking of large scale astronomical projects involving either very large apertures or long distance interferometry, or both, and with high performance adaptative optics operated in near IR. If you join the IR site quality, with its prospects in high angular resolution and high contrast imaging, the search for direct exoplanet observations is obviously the scientific target that comes to mind. The scientific community involved in this direction has been participating for the first time to this Concordia workshop, thus significantly raising the general optimism on the future of astronomy on the site. The Concordia potentialities in this direction are developed in another session of these proceedings. Now, between the first night time campaigns with modestly sized telescopes and the very ambitious prospects regarding possible ELT (Extremely Large Telescopes), there is some room and some need for intermediate size projects. We have a few of such projects presented in these proceedings, such as a wide field visible and near IR telescope (Viotti et al. 2003) or a very light robotic 1 to 1.5 m telescope [START_REF] Boer | [END_REF]). An interesting idea, with a scientific scope that goes beyond astronomy, is the measurement of the moon Earthshine for monitoring the cli-mate change on Earth through the albedo long term variation (Chou 2003).

Conclusions

This meeting at Capri was not the first one organized around the astronomical prospects at Concordia. However, it has clearly been the broadest, the most international, and all participants have felt that it indicated a very sharp step upwards in these prospects. So I will not include any conclusion in this paper, as any such attempt will be made obsolete very soon, as we all hope.

Fig. 1 .

 1 Fig. 1. From left to right, Eric Fossat, Martin Pomerantz, Lyman Page and Gerard Grec with the first solar both azimuthal and equatorial telescope that was successfully operated at the geographical South Pole in 1979/1980, providing 6 days of uninterrupted full disk oscillation data

Fig. 2 .

 2 Fig. 2. Left panel: An example of the wind speed profile above the dome C site, up to about 27 km in this case (left curve), with also the wind direction on the right curve. It shows that the maximum wind speed at any altitude above the site is of the order of 10 m/s. Right panel: For comparison, the same wind speed profile measured above the Chilean site of Cerro Paranal, shows the standard jet stream between altitudes of 10 and 15 kms. Courtesy M. Azouit

Fig. 4 .

 4 Fig. 4. The surface of Jupiter clearly shows the presence of turbulent motions (including the great red spot) that can be noisy enough, in the real acoustic sense of the word, to imply the presence of resonant acoustic vibrations of all the planet body. Photo Voyager, NASA

Acknowledgements. Thanks to the editors for their patience...