Omar Gaci
email: omar.gaci@univ-lehavre.fr

Stefan Balev

Reconstructing Amino Acid Interaction Networks by an Ant Colony Approach

Keywords: ant algorithm, interaction network, protein structure

In this paper we introduce the notion of protein interaction network. This is a graph whose vertices are the proteins amino acids and whose edges are the interactions between them. We consider the problem of reconstructing protein's interaction network from its amino acid sequence. We rely on a probability that two amino acids interact as a function of their physico-chemical properties coupled to an ant colony system to solve this problem.

Introduction

Proteins are biological macromolecules participating in the large majority of processes which govern organisms. The roles played by proteins are varied and complex. Certain proteins, called enzymes, act as catalysts and increase several orders of magnitude, with a remarkable specificity, the speed of multiple chemical reactions essential to the organism survival. Proteins are also used for storage and transport of small molecules or ions, control the passage of molecules through the cell membranes, etc. Hormones, which transmit information and allow the regulation of complex cellular processes, are also proteins.

Genome sequencing projects generate an ever increasing number of protein sequences. For example, the Human Genome Project has identified over 30,000 genes which may encode about 100,000 proteins. One of the first tasks when annotating a new genome, is to assign functions to the proteins produced by the genes. To fully understand the biological functions of proteins, the knowledge of their structure is essential.

In their natural environment, proteins adopt a native compact three-dimensional form. This process is called folding and is not fully understood. The process is a result of interactions between the protein's amino acids which form chemical bonds. In this chapter we consider the problem of reconstructing the network of amino acid interactions when only the protein sequence and its structural family are known.

In this study, we treat proteins as networks of interacting amino acid pairs [START_REF] Dokholyan | Topological determinants of protein folding[END_REF]. In particular, we consider the subgraph induced by the set of amino acids participating in the secondary structure also called Secondary Structure Elements (SSE). We call this graph SSE interaction network (SSE-IN). We carry out a study to identify the interactions involved between the amino acids in the folded protein. First, we build an interaction probability between two amino acids as a function of their physicochemical properties to predict the graph of the folded protein. Second, we develop an ant colony algorithm to fold or also to reconstruct a sequence SSE-IN.

Protein structure

Unlike other biological macromolecules (e.g., DNA), proteins have complex, irregular structures. They are built up by amino acids that are linked by peptide bonds to form a polypeptide chain. We distinguish four levels of protein structure:

• The amino acid sequence of a protein's polypeptide chain is called its primary or one-dimensional (1D) structure. It can be considered as a word over the 20-letter amino acid alphabet. By formation of such tertiary and quaternary structure, amino acids far apart in the sequence are brought close together to form functional regions (active sites). The reader can find more on protein structure in [START_REF] Branden | Introduction to protein structure[END_REF].

One of the general principles of protein structure is that hydrophobic residues prefer to be inside the protein contributing to form a hydrophobic core and a hydrophilic surface. To maintain a high residue density in the hydrophobic core, proteins adopt regular secondary structures that allow non covalent hydrogen-bond and hold a rigid and stable framework. There are two main classes of secondary structure elements (SSE), α-helices and β-sheets (see Fig. 1). A β-sheet is build up from a combination of several regions of the polypeptide chain where hydrogen bonds can form between C'=O groups of one β strand and another NH group parallel to the first strand. There are two kinds of β-sheet formations, anti-parallel β-sheets (in which the two strands run in opposite directions) and parallel sheets (in which the two strands run in the same direction).

Amino Acid Interaction Networks

The 3D structure of a protein is determined by the coordinates of its atoms. This information is available in Protein Data Bank (PDB) [START_REF] Berman | The protein data bank[END_REF], which regroups all experimentally solved protein structures. Using the coordinates of two atoms, one can compute the distance between them. We define the distance between two amino acids as the distance between their C α atoms. Considering the C α atom as a "center" of the amino acid is an approximation, but it works well enough for our purposes. Let us denote by N the number of amino acids in the protein. A contact map matrix is a N × N 0-1 matrix, whose element (i, j) is one if there is a contact between amino acids i and j and zero otherwise. It provides useful information about the protein. For example, the secondary structure elements can be identified using this matrix. Indeed, α-helices spread along the main diagonal, while β-sheets appear as bands parallel or perpendicular to the main diagonal [START_REF] Ghosh | Dynamics of lysozyme structure network: probing the process of unfolding[END_REF]. There are different ways to define the contact between two amino acids. Our notion is based on spacial proximity, so that the contact map can consider non-covalent interactions. We say that two amino acids are in contact iff the distance between them is below a given threshold. A commonly used threshold is 7 Å [START_REF] Atilgan | Small-world communication of residues and significance for protein dynamics[END_REF] and this is the value we use.

Consider a graph with N vertices (each vertex corresponds to an amino acid) and the contact map matrix as incidence matrix. It is called contact map graph. The contact map graph is an abstract description of the protein structure taking into account only the interactions between the amino acids. Now let us consider the subgraph induced by the set of amino acids participating in SSE. We call this graph SSE interaction network (SSE-IN) and this is the object we study in the present paper. The reason of ignoring the amino acids not participating in SSE is simple. Evolution tends to preserve the structural core of proteins composed from SSE. In the other hand, the loops (regions between SSE) are not so important to the structure and hence, are subject to more mutations. That is why homologous proteins tend to have relatively preserved structural cores and variable loop regions. Thus, the structure determining interactions are those between amino acids belonging to the same SSE on local level and between different SSEs on global level. Fig. 2 gives an example of a protein and its SSE-IN.

In [START_REF] Muppirala | A simple approach for protein structure discrimination based on the network pattern of conserved hydrophobic residues[END_REF][START_REF] Brinda | A network representation of protein structures: implications for protein stability[END_REF] the authors rely on similar models of amino acid interaction networks to study some of their properties, in particular concerning the role played by certain nodes or comparing the graph to general interaction networks models. Thanks to this point of view the protein folding problem can be tackled by graph theory approaches.

A Comparaison of Amino Acid Interactions Prediction

In previous works [START_REF] Gaci | Proteins: From structural classification to amino acid interaction networks[END_REF][START_REF] Gaci | Hubs identification in amino acids interaction networks[END_REF][START_REF] Gaci | The small-world model for amino acid interaction networks[END_REF], we have studied the protein SSE-IN. We have identified notably some of their properties like the degree distribution or also the way in which the amino acids interact. These works have allowed us to determine criteria discriminating the different structural families. We have established a parallel between structural families and topological metrics describing the protein SSE-IN.

Using these results, in [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF] we have proposed a method of predicting the structural family of a protein. Then, a protein or an unknown sequence hasn't a family in the SCOP v1.73 classification. It is defined by its sequence in which the amino acids participating in the secondary structure. This preliminary step is usually ensured by threading methods [START_REF] Mirny | Protein structure prediction by threading: Why it works and why it does not[END_REF] or also by hidden Markov models [START_REF] Asai | Prediction of protein secondary structure by the hidden markov model[END_REF].

In [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF], we consider that the motifs formed by the SSEs in an unknown sequence can be represented as an adjacency matrix where 1 means that the two SSEs are in contact. Then, we use a comparative model built with template proteins from the predicted family and we rely on a genetic algorithm (GA) to predict the unknown sequence matrix of motifs, see Fig. 3.

Fig. 3. 2OUF SS-IN (left) and its associated incidence matrix (right). The vertices represent the different α-helices and an edge exists when two amino acids interact.

To fold a SSE-IN or to reconstruct it, we rely on the Levinthal hypothesis also called the kinetic hypothesis. Thus, the folding process is oriented and the proteins don't explore their entire conformational space. We use the same approach: to fold a SSE-IN we limit the topological space by associating a structural family to a sequence [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF]. Since the structural motifs which describe a structural family are limited, we propose a GA to enumerate all possibilities. Now, in this work, we want to identify the interactions involved between the amino acids in the folded protein, we proceed in two steps.

In section 2, we elaborate a method to predict the interactions between amino acids in the tertiary structure. We start by a disconnected graph from which we want to add edges between amino acids belonging to different SSEs. Relying on a comparative model we evaluate the quantity of edges to add and we build an interaction probability between amino acids considering their chemical properties. Our goal is to identify which parts are in interaction between two SSEs. To measure the performance of our method, we are interested in the contact zones correctly identified.

In section 3, we want to identify which are the edges which link two SSEs, see Fig. 4. considering that the predicted matrix of motif is correct, see Fig. 3. To do that, we start by isolating two SSEs which are in contact and we link each node from the first SSE with each node from the second SSE. Then, it remains to select the suitable edges by using an ant colony approach. When the edges have been collected for each couple of SSE, we consider the disconnected SSE-IN (that is there are none edges inter-SSE yet) in which we add the edges identified during the previous approach. Finally, we select again the global suitable edges using an global ant colony approach.

Further, we use a dataset composed by proteins which have not fold families in the SCOP v1.73 classification and for which we have predicted a family in [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF].

A Stochastic Interaction Prediction

Here, we want to identify which zones are in contact between two SSEs assuming that the predicted matrix of motif is correct. We start by considering that the sequence SSE-IN is a disconnected graph with no edges between the amino acids. Our goal is to add edges between amino acids belonging to different SSE to rebuild the sequence SSE-IN. Thus, to predict the sequence SSE-IN we have to predict the quantity of edges to add and we have to identify the nodes which will be connected.

Prediction of the edge quantity

In [START_REF] Gaci | The small-world model for amino acid interaction networks[END_REF] we have studied the quantity of edges between SSEs, also called shortcuts, and we have shown that it is bounded mainly due to the excluded volume effect. Nevertheless, even if the shortcut number can be estimated knowing its upper bound, the distribution of shortcuts between the different SSEs composing the sequence stays difficult to predict (actually it is too variable from one protein SSE-IN to another).

To predict the shortcut edge rate of the sequence, we rely on the template proteins. We present a protein as an array with elements corresponding to the SSEs, Each cell represents a SSE notably considering its size that is the number of amino acids which compose it. The size is normalized contributing to produce arrays whose cells describe a value between 0 and 100, see Fig. 5.

From the predicted family, we choose the proteins with the same number of SSEs as the unclassified protein and we construct their arrays. Let s = (s 1 , . . . , s n ) be the sequence array and t = (t1, . . . , tn) be the template protein array. We use the standard

L 1 distance d(s, t) = n i=1 |s i -t i |. We admit 20% of difference, if d(s, t) < 0.2 n i=1
si, we consider that the template protein is near the sequence. Finally, we compute the average shortcut rate of the near proteins to predict the quantity of edges to add in the initial disconnected graph of the sequence. If no near proteins are found, we compute the edge quantity to add as the average of the template proteins shortcut rate.

Contact zones between the nodes to be connected

Our goal is to predict the motifs involved in the unknown sequence, we want to identify which parts are in interaction between two SSEs, that is why we are interested in predicting the zones of contact. Thus, when we add an edge in the disconnected sequence SSE-IN, what is important is not the two nodes linked but rather the zones which the nodes can cover. When measuring the performance of our method, we will be in the contact zones correctly identified. We define the notion of a zone for a helix or a sheet as follows. Fig. 6. Contact zones for a node p belonging to a helix For a node belonging to a helix in position p the contact zones in which this node participates are the quadrilaterals we can form with its spatial neighbours, namely (p, p + 1, p + 5, p + 4), (p, p + 4, p + 3, p -1), (p, p -1, p -5, p -4) and (p, p -4, p -3, p + 1), see Fig. 6.

For a node belonging to a strand in position p, the contact zone contains the node and its two neighbours in the chain, (p -1, p, p + 1).

Prediction of the nodes to be connected

The matrix of motifs determined by the GA (in [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF]) indicates which SSEs are in contact and it remains to predict which zones will be in contact. First, we choose to add the same quantity of edges between SSEs in contact, then the number of edges between two SSEs is equal to the number of shortcut predicted previously divided by the number of SSEs in contact. Second, to add an edge between two SSEs we use a probability built from the predicted family. Indeed, we consider the occurrence of a shortcut as a function of the amino acids it links. Fig. 7 shows the shortcut occurrence matrix of the families All alpha 47472 and All beta 50813. We remark that shortcuts between amino acids Leucine are the most frequent.

The problem of deciding how to add edges between two SSEs can be stated as follows. Let A = a 1 . . . a s and B = b 1 . . . b 1 . . . b t be two SSEs.

We want to add k edges among the s × t possibilities. For i ∈ [1, s] and j ∈ [1, t] the probability to link the amino acids ai and bj will be proportional to the number of occurrences of the shortcut (a i , b j ) denoted S(a i , b j ):

p ij = kS(a i , b j ) s i=1 t j=1 S(a i , b j )

Measures and Reliability

In [START_REF] Gaci | Characterization of amino acid interaction networks in proteins[END_REF][START_REF] Gaci | Proteins: From structural classification to amino acid interaction networks[END_REF] we have shown that the proteins SSE-IN can be described by their topological properties. We have shown that there exists a parallel between structural biological classification (notably the SCOP fold family level) and the SSE-IN topological properties. Here we use these properties to exclude the incompatible SSE-IN built by adding of edges using the probabilities pij, defined above. The principle is the following, we have predicted a fold family for the sequence from which we extract the template proteins. Then, we compute the diameter, the characteristic path length and the mean degree of the extracted proteins to evaluate the average topological properties of the family for the particular SSE number. Then, after we have built the sequence SSE-IN, we compare its topological properties to the average template properties. We admit an error rate up to 20% to accept the built sequence SSE-IN. We also compare of the edge quantity added the real number of edges. Let m P be the predicted number of edges and m R be the real one. Then

precision = 1 - |m R -m P | m P

Results

We have tested our method according to the predicted family because the probability of adding edge is determined by the family occurrence matrix. We have used the same dataset of sequences whose family has been predicted (in [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF]). For each protein, we have launched 60 simulations and we accept the built SSE-IN only when its topological properties are compatible to the template properties. The results are presented in Tab. 1. The score is the percentage of correctly predicted contact zones. The scores evolve around 65% meaning that our probabilistic method is relatively reliable. Moreover, we measure the accuracy of the shortcut quantity prediction to see that our prediction stays close to the reality, see Fig. 8. Consequently, our results validate the prediction done in [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF]. Nevertheless, the presented results are average and there are fluctuations as indicated by the standard deviation values. One can easily understand that the intersection score depends before all on the number of edges we decided to add. If the prediction of the edge quantity to add is precise, then the score is significantly improved. To show this correlation, we have plotted the scores as a function of the shortcut prediction error rate, see Fig. 9. It appears that when the shortcut prediction error rate is less than 25%, the score of contact zone reaches 70%.

Interaction Prediction by Ant Colony Algorithm

Here, we continue the work done in the previous section since we want to predict the nodes which are involved when two SSEs are in contact. We propose an ant colony algorithm to perform this last step of folding a SSE-IN.

In this section we consider that the matrix of motifs is already known. Our goal is to add inter-SSE edges and to reconstruct the SSE-IN, see Fig. 4. We use a two-step approach following the hierarchical structure of the network. The first step consists in considering separately each pair of SSEs in interaction. We use an ant algorithm to identify the suitable interactions between amino acids belonging to these SSEs. When this step is done for all pairs of SSEs in interaction, we go back to global level and use another ant algorithm to select the final set of inter-SSE edges.

Ant systems

The studies of social insects have shown that their behavior is self-organized [START_REF] Deneubourg | Probabilistic behaviour in ants : a strategy of errors ?[END_REF] notably concerning the food search. The self-organisation is the process during which structures emerge at collective level resulting from the multiple but simple interactions between the individuals of a population.

The ants have the particularity to communicate using volatile chemical substances called pheromones. The ants drop them to form pheromone trails which lead the other ants and allow for example to find shortest path between their nest and a food source [START_REF] Goss | Self-organized shortcuts in the argentine ant[END_REF][START_REF] Beckers | Trails and u-turns in the selection of a path by the ant lasius niger[END_REF] although the ants do not have a global vision of their environment.

The general concept of the ant systems relies on agents collaborating to search a solution using indirect communication mechanism inspired by the pheromone trails. The artificial ants have the following characteristics:

• an ant is able to perceive locally its environment.

• it has a memory to save the solution it is building.

• it can adapt the pheromone quantity to drop as a function of the solution it is building.

The local perception given to ants is also called heuristic value. Together with the pheromone quantity, it influences their choices.

Formally, at each point where a choice is needed, the artificial ants use probabilistic rules to determine their moves. Let τ ij be the quantity of pheromone present on a path between the points i and j and w ij be the heuristic value associated to this path. Then, the probability pij that an ant reaches the point j from the point i is given by the next formula:

pij = [τ ij ] α . [w ij ] β k∈V (i) [τ ik ] α . [η ik ] β (1)
where V (i) is the neighborhood of point i. The parameters α and β control the relative influence of the pheromone and the heuristic values.

In nature, as in computer models, the pheromone evaporates. Let ρ be the pheromone persistence factor. Then 1 -ρ is the evaporation rate. The pheromone quantity is updated according to this rate:

τij = (1 -ρ) τij + ∆τij
where ∆τ ij is the pheromone dropped by the ants on the path between i and j.

The local choices made by the ants will influence the global behavior of the colony. A global behavior emerges from the interactions between the ants without being specified or planned at insect level. In this way, the ant systems allow to find solutions of different problems relying on the population local interactions.

Local algorithm

This algorithm is used to identify the suitable shortcuts between a pair of SSEs in interaction. We start with a graph in which each node form the first SSE is connected to each node from the second SSE. The edges have weights w ij corresponding to the relative frequency of each kind of shortcut in the family. The edge weights are computed as explained in section 2.3. Each ant walks on the graph dropping pheromone on its way. At the end, we will keep the edges with highest pheromone rate and use them for the next phase. An overall description is given as algorithm 1.

The number of ants in our colony is equal to the total number of nodes. The initial positions of the ants are chosen randomly, so at the beginning more ants may be at the same node. Each ant chooses the next node to move randomly, with probabilities defined by eq. 1. The weights of the inter-SSE edges are computed as explained in section 2.3: Algorithm 1: Algorithm to the emergence of shortcut edges between two SSEs using an ant colony. begin create n ants while stop condition do for each ant a do moveAnt(a) updatePheromone() selectShortcut(phero min ) end

w ij = kS(ai, bj) s i=1 t j=1 S(a i , b j )
For the edges connecting nodes from the same SSE, we take as weight the average of the weights of inter-SSE edges:

w = 1 st s i=1 j = 1 t wij
After each move we update the pheromone quantity on the inter-SSE edges using the formula

xτ ij ← (1 -ρ)τ ij + a ij ∆τ
where a ij is the number of ants that moved on the edge (i, j) and ∆τ is the quantity of pheromone dropped by each ant. As far as the edges belonging to the same SSE are concerned, we keep the pheromone rate on them equal to the average pheromone rate on the inter-SSE edges

τ = 1 st s i=1 j = 1 t τ ij
In this way ants move inside an SSE completely randomly, while if they decide to change the SSE they are guided by the edge weights ant the pheromone rates.

The algorithm stops when a maximum number of iterations is reached. It can also stop earlier if the maximum pheromone rate is k times bigger than the average pheromone rate on the edges. In this case we consider that the reinforcement is sufficient. At the end, we keep only edges with pheromone quantity exceeding a threshold phero min .

Global algorithm

Once the local algorithm has provided inter-SSE edges, we build the SSE-in composed by these edges. Now our goal is to keep only the most appropriate among them. The number of edges to keep, m P , we choose as in section 2.1. Global algorithm works similarly to the local one, but this time the ants move on the entire SSE-IN graph. Their number is equal to the total number of nodes. After the end of this algorithm we sort the edges by decreasing pheromone rate and keep only the first m P of them.

Finally, we compare the topological properties of the obtained graph to the average properties of the corresponding family and accept or reject it as explained in section 2.4.

Results

The experimental settings used in this section are the same as in section 2.5. Tab. 2 summarizes the results. In most of the cases the score is greater than 70% and better than the score obtained by the stochastic approach (compare to Tab. 1). The average score decreases for big proteins (of size more than 200). For these proteins there are many pairs of SSE in contact and the errors made by the local algorithm accumulate to global level.

Table 2. Results of the colony approach. The score measures the intersection between the effective and predicted shortcut edges. The algorithm parameter values are: α = 25, β = 12, ρ = 0.7, ∆τ = 4000, k = 2, phero threshold = 0.8.

Class

Conclusion

We propose in this paper a means to fold a protein SSE-IN. Thus, we rely on the Levinthal hypothesis, since the folding process is oriented, we observe four steps to fold a SSE-IN. First, we limit the topological space by associating to a sequence a structural family [START_REF] Gaci | Prediction of protein families and motifs by topological inference[END_REF].

Second, we propose a genetic algorithm trying to construct the interaction network of SSEs. It allows enumerating the possible structural motifs.

Third, we develop a method to identify the motifs involved in a sequence SSE-IN. Then, we build a probability of node interaction between different SSEs and we use a comparative model to predict the quantity of inter-SSE edges to add. The score obtained measure the percentage of correctly predicted contact zones. When the inter-SSE edges prediction error rate is less than 25%, the score of contact zone reaches 70%.

Fourth, we use an ant colony system to predict the nodes involved when two SSEs are in contact. When two SSEs are in contact we predict the inter-SSE edges by an ant colony approach and we repeat the same process at the global level. By measuring the resulting SSE-IN, we remark that their topology depends on the local research. If, during the local folding, we collect the right edges (more than 75%) then the global score stays close to 76% of the real edges inter-SSE.

The characterization we propose constitutes a new approach to the protein folding problem. Indeed, we propose to fold a protein SSE-IN relying on topological properties. We use these properties to guide a folding simulation in the topological pathway from unfolded to folded state.

Fig. 1 .

 1 Fig. 1. Left: an α-helix illustrated as ribbon diagram, there are 3.6 residues per turn corresponding to 5.4 Å. Right: A β-sheet composed by three strands.

Fig. 2 .

 2 Fig. 2. Protein 1DTP (left) and its SSE-IN (right).

Fig. 4 .

 4 Fig. 4. Protein 1DTP SSE-IN. Interaction we want to predict by an ant system are plotted in green.

Fig. 5 .

 5 Fig. 5. Building the array representation for the unclassified protein 2OUF.

Fig. 7 .

 7 Fig. 7. Occurrence matrix of inter-SSE edges in the family All alpha 47472 (left) and All beta 50813 (right).

p

  ij ∼ S(a i , b j )In order to add approximately k edges, we need

Fig. 8 .

 8 Fig. 8. Precision of the edge quantity added. The most of the predictions are close the real value.

Fig. 9 .

 9 Fig. 9. Score as a function of the precision of the predicted edge quantity

•

  Different elements of the sequence form local regular secondary (2D) structures, such as α-helices or β-strands.• The tertiary (3D) structure is formed by packing such structural elements into one or several compact globular units called domains. • The final protein may contain several polypeptide chains arranged in a quaternary structure.

Table 1 .

 1 Experimental results by family. The score measures the intersection between the effective and the predicted contact zones.

	Class	SCOP	Protein	Protein	Score	Standard
		Family	Number	Size		Deviation
	All α	46688	17	27-46	68.65	14.89
		47472	10	98-125	64.25	6.45
		46457	25	129-135	64.75	13.37
		48112	11	194-200	76.87	18.80
		48507	18	203-214	60.08	10.19
		46457	16	241-281	69.40	5.53
		48507	20	387-422	65.27	16.19
	All β	50629	6	54-66	75.71	7.54
		50813	11	90-111	69.13	8.28
		48725	24	120-124	63.17	16.87
		50629	13	124-128	65.22	12.65
		50875	14	133-224	66.36	14.97


Romina Gonella

Mathias Bourel

Liliane Bel
email: liliane.bel@agroparistech.fr

Facing spatial massive data in science and society : variable selection for spatial models

Keywords: LASSO, Variable Selection, Spatial Statistics

This work focuses on variable selection for spatial regression models, with locations on irregular lattices and errors according to Conditional or Simultaneous Auto-Regressive (CAR or SAR) models. The strategy is to whiten the residuals by estimating their spatial covariance matrix and then proceed by performing the standard L1-penalized regression LASSO for independent data on the transformed model. A result is stated that proves the sign consistency for general dependent errors provided that the transformed design matrix fulfills standard assumptions for the LASSO procedure and that the estimate of the residual covariance matrix is consistent. Then sufficient conditions on the weight matrix of the SAR or CAR model are given that ensure those conditions hold. A simulation study is driven that shows this method gives good result in terms of variables selection, while some underestimation of the coefficients is noted.

It is compared to a strategy that estimates both the regression and the covariance parameters in a LARS procedure. Coefficient are better estimated with the Least Angle Regression (LARS) procedure but it gives in some cases much more false positive in the variable selection. The application is on the regression of income data in rural area of Uruguay on a set of covariates describing socio-economic characteristics of the households.

Introduction

Thanks to new technologies, data acquisition is nowadays much easier and the challenge is now to process this mass of data. Variable selection is one of the aspects of the question, the number of covariates is often so large that it must be reduced in order to avoid numerical problems and gaining interpretability. The LASSO (Least Absolute Shrinkage and Selection Operator, [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]) method which cancels the small coefficients of the regression thanks to a 1 penalty, has become very popular and has experienced many theoretical and practical developments : [START_REF] Tibshirani | Regression shrinkage and selection via the lasso: a retrospective[END_REF], [START_REF] Hastie | Statistical Learning with Sparsity. The Lasso and Generalizations[END_REF] state the basics of the method, [START_REF] Knight | Asymptotics for Lasso-Type Estimators[END_REF] establish the asymptotics, [START_REF] Zou | The Adaptive Lasso and Its Oracle Properties[END_REF] the oracle properties, [START_REF] Zhao | On Model Selection Consistency of Lasso[END_REF] a consistency result, [START_REF] Simon | A Sparse-Group Lasso[END_REF] extend to group-LASSO amongst many others.

In many domains, environment, climate, econometric, agronomy, data are georeferenced and their modeling includes an error term with a spatial dependence.

It is well known that this dependence must be taken into account to avoid making errors, whether in the estimation of parameters or their significance. In these domains, the increase in the size of the data is also a reality, even if it does not reach the magnitudes that can be observed in biology for example.

Variable selection by LASSO in spatial models has been studied by several authors : [START_REF] Nandy | Additive model building for spatial regression[END_REF] develop an additive model, [START_REF] Chu | Penalized maximum likelihood estimation and variable selection in Geostatistics[END_REF], [START_REF] Huang | Optimal Geostatistical Model Selection[END_REF], [START_REF] Wang | Variable selection in spatial regression via penalized least squares[END_REF] consider the geostatistical framework, while [START_REF] Reyes | Selection of spatial-temporal lattice models: assessing the impact of climate conditions on a mountain pine beetle outbreak[END_REF], [START_REF] Zhu | On selection of spatial linear models for lattice data[END_REF] and [START_REF] Cai | Variable selection and estimation for high-dimensional spatial autoregressive models[END_REF] consider lattice models. Depending on the case different contexts either on the form of the spatial dependence, or on the approach considered, by maximum likelihood or by minimizing a least squares criterion are handled. We show in this work a sign consistency result, which guarantees to recover asymptotically the true support of the regression parameters, for dependent data. We give necessary conditions on the weight matrix when the spatial model incorporates CAR or SAR errors for the conditions of the theorem to be verified. These two results are presented in section 2 along with the procedure for implementing the selection. In section 3 we compare this method with the method in [START_REF] Zhu | On selection of spatial linear models for lattice data[END_REF] which regularizes the full likelihood on the same models, on simulated data in several situations. Knowledge of household income is essential for setting public policy. In developping countries such as Uruguay it may be difficult to collect direct data on income especially for some specific population such as in rural areas and it may be convenient estimating it from covariates easier to collect describing the socio-economic characteristics of the household as well as the comfort level and the satisfaction of their basic 35 needs. In section 4 we compare the two approaches on a survey data in order to explain the per capita income of households from Uruguayan rural areas by covariates A discussion and conclusions are given in section 5.

Variable selection for dependent errors, the spatial case

Let us consider a linear model

Y = Xβ + ε (1)
where Y is a n × 1 vector of outputs, X is the n × p design matrix of covariates, with p that can be very large, β is a p × 1 vector of unknown parameters and ε is a n × 1 random vector with E(ε) = O and V ar(ε) = Σ.

Without loss of generality, it can be assumed that X and Y are standardized with 0 mean.

One is willing that the fitted model fulfills two properties : it has good predictive ability, it is interpretable that is if the number of covariate is large only a few of the associated coefficients are significant. The LASSO (Least Absolute Shrinkage and Selection Operator, [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]) procedure minimizes the following penalized criterion

L (β) = Y -Xβ 2 2 + λ β 1 for some λ > 0 (2)
The penality λ β 1 as being a L γ norm with γ ≤ 1 has the nice feature to cancel parameters when λ is growing, and selects by this means variables. When γ = 1, the minimization problem is convex, hence it is computationally tractable. Regarding the predictive performance of LASSO, it has been shown that it is equivalent to other penalized methods such as ridge or bridge regression ( [START_REF] Tibshirani | Regression shrinkage and selection via the lasso: a retrospective[END_REF]) and these two features good performance prediction and providing sparse models make the LASSO procedure very popular and efficient for fitting models especially for large number of covariates, even larger than the observation number.

We assume that the true support of the vector β, denoted A * = {j :

β j = 0, 1 ≤ j ≤ p} is size |A * | = p * , 0 < p * ≤ p,
and we denote A * c the A * complementary,

|A * c | = p -p * .
The aim of variable selection is to determine the set A * and estimate the corresponding parameters {β j , j ∈ A * }.

Theoretical results for the LASSO procedure are mainly stated for models with independent errors (Σ = σ 2 Id), a way to deal with correlated errors is to transform the model in order to remove the correlation :

Σ -1/2 Y Y = Σ -1/2 X X β + Σ -1/2 ε ε (3)
with ε ∼ N (0, Id) and the procedure may be driven on the transformed model.

When λ = 0 the estimator of β is then the GLS estimator β = ( X X) -1 X Y = (X Σ -1 X) -1 X Σ -1 Y . When Σ is not known but may be estimated by Σ, a plug-in approach leads to consider the model

Σ -1/2 Y Y = Σ -1/2 X X β + Σ -1/2 ε ε (4)
In the following we state conditions under which the estimator of β is sign consistent as defined in [START_REF] Zhao | On Model Selection Consistency of Lasso[END_REF], that is:

lim n→∞ P (sign( β) = sign(β)) = 1
Sign consistency insures the support of the β estimate is asymptotically the true support.

Without loss of generality , model (1) can written:

Y = [X A * X A * c ]   β A * β A * c   + ε (5)
where

β A * = {β j , j ∈ A * }, X A * = ((X •j )) j∈A * , where X •j is the jth column of X.
The following theorem is an adaptation to model [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] with correlated errors of the results stated in [START_REF] Zhao | On Model Selection Consistency of Lasso[END_REF] for models with independent errors.

Theorem 1 (Condition for strong sign consistency when errors are dependent).

Considering the model ( 4), and β the LASSO estimator for this model, if there

exist positive constants M 1 , M 2 , M 3 , M 4 , M 5 , M 6 and M 7 and c 1 , c 2 with 0 ≤ c 1 ≤ c 2 ≤
1/2 such that the following conditions are:

1. 1 n X •j Σ -1 X •j ≤ M 1 ∀ j 2. ρ min ( 1 n X A * Σ -1 X A * ) ≥ M 2 3. p * = O(n c1/2 ) 4. n (1-c2)/2 min 1≤j≤p * (|β j |) ≥ M 3 5. it exists a constant positive vector δ such that | 1 n X A * c Σ -1 X A * ( 1 n X A * Σ -1 X A * ) -1 sign(β A * )| ≤ 1 -δ
where 1 is a p -p * vector of 1, and the inequality holds element-wise.

6. λ √ n → n→∞ ∞ and λ √ n = o(n (c2-c1)/2 ) 7. (X X)/n ∞ ≤ M 4 8. ρ max (Σ -1 ) ≤ M 6 9. ρ min (Σ -1 ) ≥ M 7 10. Σ -1 -Σ-1 ∞ = O P ( 1 √ n ) when n → ∞
where ρ(X) is the vector of eigenvalues of X, ρ min (X) and ρ max (X) are respectively the minimum and maximum eigenvalues,

|||X||| ∞ = max 1≤i≤n n j=1 |X ij | .
Then the strong sign consistency property is achieved for the LASSO estimator

β, that is, lim n→∞ P (sign( β) = sign(β)) = 1.
Condition 5 is the so-called [START_REF] Zhao | On Model Selection Consistency of Lasso[END_REF] strong irrepresentable condition for matrix X in model (3).

Conditions 3, 4 and 6 are present in the result for the independent errors case, while conditions 1, 2 and 5 are analog but for X of model (3) instead of X.

Conditions 8 and 9 indicate that eigenvalues of Σ are positive and bounded and condition 10 states that Σ is a good estimator for Σ.

The proof of this Theorem is highly related to that of the one presented in 70 [START_REF] Perrot-Dockès | Variable selection in multivariate linear models with high-dimensional covariance matrix estimation[END_REF] established for the multivariate responses case, when the error vector for one observation are dependent, but the errors for different observations are independent. The proof is available in the supplementary material.

Spatial case

We focus now on spatial models and investigate conditions on the error covariance matrix in order to fulfill the conditions for strong sign-consistency in Theorem 1 . Specifically we deal with models on lattice, focusing on autoregressive CAR or SAR models, with weights matrix W . More precisely the CAR (Conditional Auto Regressive) model is written

E(Z i |Z j : j = i) = n j=1 c ij Z j V ar(Z i |Z j : j = i) = σ 2
The Z covariance matrix is

Σ CAR = (Id -C) -1 V C = θW , |θ| < 1, V = diag(σ 2 ).
The SAR (Simultaneaous Auto Regressive model) model is written

Z = CZ + ν
where ν is a iid random vector with 0 mean and variance σ 2 .

The Z covariance matrix is

Σ SAR = (Id -C) -1 V(Id -C ) -1
with same as for the CAR model.

75

We seek for conditions on W that guarantee conditions 8, 9 and 10 of Theorem 1 .

To bound ρ max (Σ -1 ) using inequalities results in ( [START_REF] Horn | Matrix Analysis[END_REF]) for positive definite matrices

ρ max (Σ -1 ) ≤ γ(Σ -1 ) ≤ Σ -1 ∞ (6)
where γ(X) is the spectral radius of X and decomposing Σ -1 in a product of matrices and bounding each term we obtain for both Σ CAR and To bound ρ min (Σ -1 ) using the same arguments we obtain for both Σ CAR and

Σ SAR Σ -1 ∞ ≤ 1 σ 2   1 + |θ| max
Σ SAR ρ min (Σ -1 ) ≥ 1 -|θ| max 1≤i≤n n j=1 |w ij | 2 σ 2
and condition 9 holds if max ) for the CAR model we have

Σ -1 CAR -Σ-1 CAR ∞ = 1 σ 2 - 1 σ2 + θ σ2 - θ σ 2 max 1≤i≤n   n j=1 |w ij |   If √ n( θ -θ) = O P (1) and √ n 1 σ2 -1 σ 2
= O P (1), and max For the SAR model Let B a binary weight matrix, such that each location has a bounded number of neighbours (for instance stemming from a triangulation neighbour structure), then the following W weight matrix fulfills the 3 conditions of Table 1:

Σ -1 SAR -Σ-1 SAR ∞ = max 1≤i≤n n k=1 w 2 ik θ 2 σ 2 - θ2 σ2 + 1 σ 2 - 1 σ2 ≤ 1 σ 2 - 1 σ2 + θ 2 σ 2 - θ2 σ2 max 1≤i≤n     n j=1 j =i n k=1 |w ik w kj |     + 2 θ σ2 - 2θ σ 2 max 1≤i≤n   n j=1 |w ij |   If √ n( θ -θ) = O P (1), √ n 1 σ2 -1 σ 2 = O P (
w ij =    min 1 |N (i)| , 1 |N (j)| if j ∈ N (i) (equiv. i ∈ N (j))
0 in other case [START_REF] Simon | A Sparse-Group Lasso[END_REF] where

N (i) = {j : B ij = 1}
This can be verified bounding max We denote N 2 (i) the neighbours of order two of i, and

|N 2 (i)| = j∈N (i) |N (j) -{i}|. We have max 1≤i≤n     n j=1 j =i n k=1 |w ik w kj |     ≤ max 1≤i≤n |N 2 (i)| |N (i)| |N 2 (i)|
|N (i)| can be interpreted as the mean quantity of neighbours that have a ineighbour (without i), and we expect that max

1≤i≤n |N 2 (i)| |N (i)|
is bounded for a variety of neighbourhood structures, such as the triangulation one for instance.

A simulation study showed empirically that for n ∈ {100, . . . , 40000}, 100 replicates for each n, and a triangulation neighbourhood structure the quantity

max 1≤i≤n |N 2 (i)| |N (i)|
although increasing slowly with n remains less than 9 and we may consider that condition 3 holds. Parameters θ and σ are estimated from the residuals of model without correlation. They can be estimated by maximum likelihood [START_REF] Gaetan | Spatial Statistics and Modeling[END_REF], or by the generalized moments procedure. In the latter case [START_REF] Kelejian | A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF] establish the consistency of the estimator so the condition 10 of Theorem 1 holds. To summarize, the procedure to select and estimate the parameters for model 5 is the following:

• Step 1: Fit a LASSO model (Standard LASSO) as if there was no spatial dependence.

• Step 2: Test the spatial autocorrelation of the residuals, using for instance the Moran or Geary index.

• Step 3: If the null hypothesis of no correlation is rejected, consider CAR and SAR models, estimate parameters θ and σ 2 by maximum likelihood or generalized moments and plug them in ΣCAR or ΣSAR

• Step 4: Eliminate the spatial dependence using the estimated matrix of the previous step applying 4.

• Step 5: Estimate the LASSO model to the transformed data, selecting the parameter λ by cross validation. This model is named Spatial LASSO.

Simulations

In order to investigate how performs the procedure designed in the previous section we proceed to simulations according to several scenarios. n locations irregularly spaced are randomly drawn, p = 20 random variables

N (0, 1) are simulated forming the n × p design matrix X. The output Y is set as Y = X 1 + X 2 + X 3 + X 4 + ε, with ε ∼ N (O, Σ), that is p * = 4.
We consider 2 distinct situations, namely

• Problem 1 : random variables (X j , 1 ≤ j ≤ 20) are independent • Problem 2 : random variables (X j , 1 ≤ j ≤ 20) are correlated with cor(X i , X j ) = 0.9 |i-j| .
In each case we check that the conditions on the matrix X in Theorem 1, that is conditions 1, 2, 5 and 7 hold, and to investigate the asymptotic side of the consistency result we vary n to large numbers. The distinct scenarios correspond to a specific value of n, σ 2 , model error (CAR or SAR), the parameter θ is fixed.

Values for n, σ 2 and θ are n ∈ {100, 200, 400, 800}, σ 2 ∈ {0.5, 1, 1.5, 2, 5, 10}, θ = 0.9. For each scenario 100 replicates are drawn.

The procedure described in the previous section named "Spatial LASSO" is compared to the "Standard LASSO" procedure for independent errors and to a procedure named "LARS m " described in [START_REF] Zhu | On selection of spatial linear models for lattice data[END_REF] which selects the significant variables by regularizing the complete likelihood of the model, that is including parameters β, θ and σ 2 .

Simulations were driven in R using packages glmnet, lars, MASS, spdep, expm and TruncatedNormal ( [START_REF] Core | R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing[END_REF], [START_REF] Friedman | Regularization Paths for Generalized Linear Models via Coordinate Descent[END_REF], [START_REF] Hastie | lars: Least Angle Regression, Lasso and Forward Stagewise, R package version 1[END_REF], [START_REF] Venables | Modern Applied Statistics with S, 4th Edition[END_REF], [START_REF] Bivand | Applied spatial data analysis with R, 2nd Edition[END_REF], [START_REF] Goulet | Matrix Exponential, Log, "etc", R package version 0[END_REF] and [START_REF] Botev | TruncatedNormal: Truncated Multivariate Normal and Student Distributions, R package version 2[END_REF]). The neighbourhood structure was designed following reference [START_REF] Bivand | Applied Spatial Data Analysis with R, 1st Edition[END_REF]. ters θ and σ 2 they are better estimated by the LARS m procedure than by the Spatial LASSO one.

As a conclusion on this simulation study we can assert that the procedure Spatial LASSO recovers better the parameters β support while LARS m gives better estimations of all the parameters at the cost of a greater number of False Positive. More figures and results are available in the supplementary material.

Real case : income in rural areas in Uruguay

Knowing the income level of a household is fundamental to determine when it can apply for the different public policies in a country, such as access to housing, food allowances, etc. In the simplest situation, the household's income is determined by taking into account all the regular income of each member of the household, through the documentation requested. However, in developing countries, as is the case in Uruguay, and particularly for some specific populations such as rural areas, it is not at all simple to collect this data, since many households receive informal and/or very irregular income, so that in many cases there is no formal documentation to prove the declared income. In these cases, indirect methods can be used to approximate income, such as the level of consumption or the degree of comfort of the household. The characteristics of the territory, the political and economic reality and the moment also influence the determination of income, and identifying the most important variables in the modeling carried out in a spatial analysis framework makes sense, because of their correlations.

We apply the three methods described in the previous sections to an Uruguayan socio-economic dataset including per capita income in rural areas, variables describing labour market (activity, employment, unemployement) and comfort items. The data are from the survey ECH (Encuesta Continua de Hogares) in Uruguay conducted by INE (Instituto Nacional de Estadística).

We use a subset of the survey corresponding to year 2018, including 933 households in rural areas which are not owning their housing and have at most one basic need 1 not satisfied. The microdata of the ECH are freely accessible and are available on the INE website; however, for confidentiality reasons, the georeferencing of the surveyed households is not available. In order to proceed a spatial analysis, and for academic purposes, it was decided to impute to each household a random location drawn within the census tract area to which each household belongs. The spatial location of the households considered, identifying the quartile of per capita income to which they belong, is presented in With the information available in the survey we seek to explain the per capita household income (without rental value and without domestic service), expressed in RU (Readjustable Units2 ), according to a set of quantitative variables, reflecting the main socio-economic characteristics of the household as well as its level of comfort. The 29 covariates are listed in Table 3, they include information on the household members (number, gender, education, occupation)

and comfort elements (cars, laptop, mobile phone ...). On real data it is not possible to check that the conditions in Theorem 1 hold as the real covariance matrix Σ is unknown. In particular in our case as the covariates are highly correlated it is doubtful that condition 5. (the irrepresentability condition) which roughly states that variables in the support are weakly correlated to variables out of the support, is fulfilled. Nevertheless it is interesting using this variable selection procedure having in mind that likely we are not in the good framework

to investigate what could be the relevant covariates. The weight matrix W is driven from a triangulation neighborhood structure and weights are set according to 7. In addition to the packages already mentioned, we use shapefiles, sp and rgdal ( [27], [START_REF] Pebesma | Classes and methods for spatial data in R[END_REF] and [START_REF] Bivand | rgdal: Bindings for the "Geospatial[END_REF]).

According to this structure each household has a number of neighbours in the 220 range [START_REF] Hastie | Statistical Learning with Sparsity. The Lasso and Generalizations[END_REF][START_REF] Reyes | Selection of spatial-temporal lattice models: assessing the impact of climate conditions on a mountain pine beetle outbreak[END_REF] the average number is 5.94. The optimal parameter λ calculated by leave one out procedure is equal to 0.575 for Standard LASSO and 0.082 for Spatial LASSO. 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Observed values

Standard LASSO q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ρ = 0.67 ρ = 0.67 ρ = 0.67 ρ = 0.67

Observed values

Spatial LASSO q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ρ = 0.63 ρ = 0.63 ρ = 0.63 ρ = 0.63

Discussion and conclusion

In this work we established a result giving conditions to ensure the sign consistency of the LASSO estimator for dependent data. This result guarantees to recover asymptotically the true support of the regression parameters. The conditions involved in this result relate to the design matrix and the error covariance matrix and may be difficult to check in some cases. We focus on the particular case of spatially dependent data on irregular lattice according to a CAR or a SAR model and we give sufficient conditions on the weight matrix for some of these conditions to be satisfied. These conditions are obtained by straightforward calculations, some lighter conditions might be obtained by more precise methods. In a geostatistical framework, properties of the most popular covariance functions could be used to derive similar conditions to the error covariance matrix satisfy the theorem assumptions.

A simulation study shows for the situations investigated that the support is well recovered for sufficient sample sizes and reasonable noise levels. This simulation study also shows that the recommended method: estimate first the parameters in an independent setting and identification of the covariance structure on the residuals and then implementation of the procedure for dependent data using the estimated covariance matrix is more efficient than the method consisting in regularizing the likelihood involving both the regression parameters and the covariance parameters. It would be worthwhile to study more extensively various situations of dependence in errors to determine to what extent the exact support of the parameters can be recovered. Other error models can be considered with higher autoregressive order or for data in a continuous domain instead of irregular lattice.

The application of these methods to a socio-economic dataset, with the objective of explaining the per capita income of rural households in Uruguay, showed that these two approaches give significantly different results nevertheless the usual goodness-of-fit criteria give the advantage to the plug-in method which selects a larger number of variables than the maximum likelihood method. It should be

  ij | does no depend on n.

  ij | does not depend on n and |θ| max 1≤i≤n n j=1 |w ij | < 1. Let C = θW , and V = diag( σ 2

  10 holds for the case CAR.

  80

  Considering different situations according to the relative sizes of N (i) and N (j), we obtain max 1≤i≤n verified. Condition 2 is verified because |θ| < 1.

Figure 1 forFigure 1 :

 11 Figure 1 for Problem 2 and SAR error model. Regarding the spatial parame-

Figure 2 . 1 Figure 2 :

 212 Figure 2.

225

  Standard LASSO and Spatial LASSO (both CAR and SAR errors model) select the same 11 variables from the 29, they are mostly related to the income earners, their education, their employment and some comfort elements. LARS m method, selects 5 with the CAR model and 8 variables with the SAR model. The variables selected included in the set of 11 variables selected by standard and Spatial LASSO. Both Spatial LASSO and LARS m results depend on initial values of the parameters. These values are drawn at random and introduce randomness in the results. To evaluate the sensitivity to the initial parameters we draw for each method 100 initial values and examine how vary the estimates. While β estimates obtained with Spatial LASSO do not vary much, those obtained with LARS m vary greatly, some of them even change signs. Even the number of selected variables may change either with the CAR or the SAR error model. On the other hand θ and σ 2 estimates don not vary much for either Spatial LASSO and LARS m .

Table 1 :

 1 Conditions on weights w ij of matrix W.

		n
	1), and both max 1≤i≤n	j=1	|w ij | ,

Table 2

 2 displays the average number of selected variables (Vars), the True Positive and False Positive for each method, according to different parameters of the scenarios. All the detailed results are available in the supplementary material.

			Vars			True Positive		False Positive
	scenario
		St LS Sp LS LARS m	St LS Sp LS LARS m	St LS Sp LS LARS m
	CAR	4.26	4.28	4.83	3.87	3.89	3.89	0.40	0.39	0.94
	SAR	4.13	4.28	4.87	3.72	3.91	3.87	0.41	0.37	0.99
	Problem 1	4.31	4.38	5.21	3.79	3.90	3.92	0.52	0.48	1.29
	Problem 2	4.09	4.17	4.49	3.80	3.89	3.85	0.29	0.28	0.64
	n = 100	4.40	4.38	5.17	3.88	3.89	3.94	0.52	0.49	1.24
	n = 200	4.21	4.38	5.24	3.70	3.92	3.90	0.52	0.47	1.34
	σ 2 = 1	4.55	4.52	5.20	4.00	4.00	4.00	0.55	0.52	1.20
	σ 2 = 2	4.52	4.44	5.11	3.98	4.00	4.00	0.54	0.44	1.11
	All simulations	4.20	4.28	4.85	3.80	3.90	3.88	0.40	0.38	0.97

Table 2 :

 2 These results are not really different according to the problem or the n value. When the error model is CAR, Spatial LASSO and LARS m perform similarly, when the error model is SAR, Spatial LASSO is much better. Spatial LASSO performs better for intermediate values of σ 2 , LARS m for high values of σ 2 .Estimations of the parameters β j are as expected underestimated by the Stan-dard LASSO and Spatial LASSO, the bias decreases when n increases. LARS m gives better estimations for the first 4 coefficients with value 1, but not for the null coefficients as it produces much more False Positive. This is illustrated in

Average number selected variables(Vars), True positive, False Positive for Standard LASSO (StLS), Spatial LASSO (SpLS) and LARSm, according to different scenarios The average number of selected variables is slightly overestimated for the three methods, but more for the LARS m one. The True Positive are well recovered for Spatial LASSO and LARS m , Standard LASSO is lower. False Positive are much higher for LARS m , almost twice than Standard LASSO and Spatial LASSO, the latter being slightly lower. Results for CAR and SAR error models are not very different and none give systematically better results. Results are systematically better for Problem 2 than for Problem 1 regardless the method, this kind of decreasing correlation in the covariates help selecting the true support. As expected when n increases or σ 2 decreases the three indicators improve.

Let us consider the euclidian distance between the pair (TP,FP) and the optimal situation (4,0) for each scenario in order to rank the three procedures.

According to this criterion Spatial LASSO ranks first with 46% scenarios, then LARS m with 33% and standard LASSO with the remaining 21%.

Table 3

 3 gives the β j selected and estimated for Standard LASSO, Spatial LASSO and LARS m .

				CAR	SAR
			Standard
	Variable	Parameter		Spatial		Spatial
			LASSO		LARSm	LARSm
				LASSO		LASSO
	intercept	β0	6.14	0.64	-0.91	0.65	5.80
	number of Household Members (HM)	β1	-0.90	-0.80	-	-0.81	-0.31
	proportion of Income Earners (IE)	β2	8.57	9.00	10.15	8.99	4.33
	proportion of male IE	β3	-	-	-	-	0.17
	average age of not IE	β4	-	-	-	-	-
	average age of male IE	β5	-	-	-	-	0.01
	average age of female IE	β6	-	-	-	-	-
	average years formal education IE	β7	0.01	0.005	-	0.01	-
	average years formal education male IE	β8	-	-	-	-	-
	average years formal education female IE	β9	-	-	-	-	-
	proportion HM receiving social benefits	β10	-0.21	-0.19	-	-0.19	-
	average age HM receiving social benefits	β11	-	-	-	-	-
	proportion of employed (E)	β12	0.51	0.36	6.10	0.37	5.46
	average age male E	β13	-	-	-	-	-
	average age female E	β14	-	-	-	-	-
	proportion male E	β15	-	-	-	-	-
	average years formal education E	β16	-	-	-	-	-
	average years formal education male E	β17	-	-	-	-	-
	average years formal education female E	β18	-	-	-	-	-
	average number hours worked among E	β19	0.09	0.09	-	0.09	-
	average hours worked by male E	β20	-	-	-	-	-
	average hours worked by female E	β21	-	-	-	-	-
	number unsatisfied basic needs	β22	-0.39	-0.35	-	-0.35	-
	number comfort elements	β23	0.06	0.06	-	0.06	-
	number cars	β24	2.18	2.08	2.02	2.08	0.43
	number motorcycles	β25	-	-	-	-	-
	number laptops	β26	2.06	1.96	1.30	1.96	1.50
	number air conditioners	β27	-	-	-	-	-
	number colour televisions	β28	-	-	-	-	-
	proportion HM with mobile phone	β29	1.20	1.32	5.86	1.32	3.65
		θ	-	0.41	0.36	0.21	0.22
		σ2	-	48.02	55.05	48.29	57.17

Table 3 :

 3 Estimated parameters by Standard LASSO, Spatial LASSO and LARSm, CAR and SAR errors.

It is a unit of measurement, its value is periodically adjusted according to the Average Salary Index, quantifying the variations in the previous twelve months

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The LASSO method (both Standard and Spatial by both error models) selected one third of the available variables, while the LARS m method selected 5 variables with the SAR model and 8 variables with the CAR model. Figure 3 shows the fitted values versus the observed values according to the 3 models.

Except for specific cases, in general there is no substantial change in the point 250 estimation of the parameters by the three methods. Comparison according to indicators such as the pseudo R 2 , AIC and BIC suggests that the Spatial LASSO has a better fit than the others. While all variables a priori could be eligible to explain per capita household income in the extended rural environment, it is understood that those selected by LASSO form a reasonable subset of those 255 with the greatest explanatory power, even in a context of variables that are not independent of each other. The LARS m method further narrows the set of variables selected by LASSO, but the drawback is that the results are sensitive to the initial values, the tolerance and the maximum number of steps considered.

noted that this dataset, as the covariates are highly correlated, likely does not fulfill the conditions of the theorem, which may explain this discrepancy. Methods taking into account correlation or other kind of proximity in the covariates such as Elastic-Net ( [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]), Fused-LASSO ( [START_REF] Tibshirani | parsity and smoothness via the fused lasso[END_REF]) or Group-LASSO ( [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF]), should be investigated considering models for data with spatial dependence.

Supplementary material can be download at https://www6.inrae.fr/mia-paris/Equipes/Membres/Liliane-Bel/SM


Marwa Dakik

Ahmed Rouabhi

Hedi Sellami

Isabelle Thenevin

Kathy Bru

Yannick Ménard

Fragmen- Rock

Rock Fragmentation: High-Voltage Electrical Pulse Effects Investigated Through Electrical Modeling

come

Rock Fragmentation: HighVoltage Electrical Pulse Effects Investigated Through Electrical Modeling M. DAKIK 1 , A. ROUABHI 1 , H. SELLAMI 1 , I. THENEVIN 1 , K. BRU 2 , Y. MENARD 2 1 Mines Paris, PSL University, Center for geosciences and geoengineering, 77300 Fontainebleau, France 2 BRGM, F45060 Orléans, France

The electric problem of HVPF is described by the equation of conservation of charge in which the unknown variable is the electric potential V.

∇. (𝜎∇𝑉) +

𝜕 𝜕𝑡 ∇. 𝜀𝜀 0 ∇𝑉 = 0 σ is the electric conductivity, and ε is the relative electric permittivity. These two parameters control the dielectric breakdown process and are empirical function of the electric field norm.

The distribution of σ (S/m) at the end of the simulation is our point of interest.

[1] Xiaohua Zhu, Mengqiu Chen, Weiji Liu, Yunxu Luo, and Hai Hu. The fragmentation mechanism of heterogeneous granite by highvoltage electrical pulses. Rock Mechanics andRock Engineering, 55(7):4351-4372, 2022 Modeling electrical fragmentation of rocks to understand its underlying mechanisms and potential applications in various fields.

Rock fragmentation is a main process in mining, quarrying, and civil engineering. In recent years, highvoltage pulse fragmentation (HVPF) has emerged as a promising alternative to rock fragmentation using traditional mechanical tools due to its high energy efficiency.

HVPF uses a highpulse voltage to initiate electrical breakdown inside a dielectric material immersed in water, generating its disintegration. In HVPF, a discharge channel is formed inside the rock after applying a highvoltage pulse resulting in a high initial current flow. Consequently, the rock, a nonconductive material, appears to be locally transformed into a conductive material [1], and the electrical energy is transformed into heat and mechanical energy.

The highvoltage rock destruction may be considered a dynamic multiphysical problem consisting of three main physical problems: electric, thermal, and mechanical problems. In this study, we present a focused investigation of the electrical problem only aiming to gain deeper insights into the underlying mechanisms of electrical fragmentation. The rock is considered an insulator with constant 𝜎 0 at 𝑡 = 0 (before Breakdown). The rock underwent a state transition and became a conductor with varying conductivity levels after breakdown.

Introduction and Goals

Methodology

Figure 2a highlights regions characterized by 𝜎 𝑚𝑎𝑥 values, denoted as "electrical damage zones," which suggest a potential for mechanical damage in these areas. The dimensions (length, depth, width) of these zones can be determined numerically, which can then be compared with experimental data to evaluate the model's performance and accuracy.

FIGURE 1 .

 1 FIGURE 1. (a): HVPF Schematic Diagram in COMSOL (b,c): Macroscopic results of sandstone tested with 90 kV electric pulse (d,e): Microscopic results by Xray tomography of the tested sandstone

Fig

  Fig. 2b revealed that these electrical damage zones are interconnected by a channel with σ < 𝜎 𝑚𝑎𝑥 , suggesting an important role in the overall breakdown mechanism of the rock.

Results

FIGURE 2 :

 2 FIGURE 2: Electric Conductivity (S/m) distribution at 𝑡 𝑓 (s). (a): Region 𝜎 𝑚𝑎𝑥 (b): Potential channel