Annabelle Gillet
email: annabelle.gillet@u-bourgogne.fr

Éric Leclercq
email: eric.leclercq@u-bourgogne.fr

Lucile Sautot
email: lucile.sautot@agroparistech.fr

The Tucker tensor decomposition for data analysis: capabilities and advantages

Keywords: Information systems → Information extraction; Clustering and classification data analysis, tensor, tensor decomposition, Tucker

Tensors are powerful multi-dimensional mathematical objects, that easily embed various data models such as relational, graph, time series, etc. Furthermore, tensor decomposition operators are of great utility to reveal hidden patterns and complex relationships in data. In this article, we propose to study the analytical capabilities of the Tucker decomposition, as well as the differences brought by its major algorithms. We demonstrate these differences through practical examples on several datasets having a ground truth. It is a preliminary work to add the Tucker decomposition to the Tensor Data Model, a model aiming to make tensors data-centric, and to optimize operators in order to enable the manipulation of large tensors.

INTRODUCTION

From a general point of view, data analysis is a complex process that requires the selection and retrieval of data as well as the application of one or several algorithms. This leads to two major issues: data must be transformed to fit the selected algorithms, and the output results must be interpretable. Furthermore, data analysis often requires to take into account multiple characteristics or features in algorithms to fully reveal and understand hidden value in data. For example, the study of community structures in social networks needs the possibility to discover overlapping communities as well as adding a temporal context for each community, i.e., over which period of time the community is active. Unfortunately, most algorithms do not take into consideration this kind of needs, and integrating them might require heavy modifications to the algorithm and/or produce a limited solution. To continue with the previous example, when using snapshots to detect communities their fixed size does not necessary correspond to the temporal activity of all the communities, and thus this can hide some information.

In this context, tensors are a valuable solution [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multiway data analysis and blind source separation[END_REF]. Indeed, their multi-dimensional nature allows to embed easily different data models. Graphs can be represented as adjacency matrices, that are 2-order tensors. In addition, when dealing with labelled graphs, the 2-order tensor can be extended to a (2 + 𝑛)-order tensor, with 𝑛 the number of types of labels. This includes temporal graphs, in which the temporality can be seen as a specific label, and they can be used for example to represent social network data [START_REF] Araujo | Com2: fast automatic discovery of temporal ('comet') communities[END_REF]. Time series can be modeled as vectors in which each entry is the value at the time 𝑡. Vectors are 1-order tensors, so time series are directly embedded into tensors. Dimensions can be added to include more contextual information, as for example, in an IoT application, the sensor that took the measures and its localization. Other models such as relational, OLAP data cubes or key-value can also be represented by tensors.

Furthermore, tensors have powerful analytical operators: the decompositions. Tensor decompositions are used for various purposes such as dimensionality reduction, noise elimination, identification of latent factors, pattern discovery, ranking, recommendation or data completion. They are applied in a wide range of applications, including genomics [START_REF] Hore | Tensor decomposition for multiple-tissue gene expression experiments[END_REF], analysis of health records [START_REF] Yang | TaGiTeD: Predictive task guided tensor decomposition for representation learning from electronic health records[END_REF], graph mining [START_REF] Sun | Beyond streams and graphs: dynamic tensor analysis[END_REF] and identification and evolution of communities in social networks [START_REF] Araujo | Com2: fast automatic discovery of temporal ('comet') communities[END_REF][START_REF] Evangelos E Papalexakis | Do more views of a graph help? community detection and clustering in multi-graphs[END_REF]. Papalexakis et al. in [START_REF] Evangelos E Papalexakis | Tensors for data mining and data fusion: Models, applications, and scalable algorithms[END_REF] review major usages of tensor decompositions in data mining applications. One of these decomposition is Tucker, that factorizes a tensor with 𝑁 dimensions into a smaller core tensor and a set of 𝑁 factor matrices, i.e., one for each dimension. However, the results of the Tucker decompositon can be tricky to interpret, especially compared to more straightforward decompositions such as the CANDE-COMP/PARAFAC [START_REF] Nicholas D Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF]. Nevertheless, the core tensor, missing from the CANDECOMP/PARAFAC decomposition, provides additional insights concerning the relationships among the dimensions.

Our contributions are the following. We propose to study the Tucker decomposition, its two major algorithms and its uses in data analysis. We also identify the key points that can impact or reduce the processing of massive data, and conduct an experimentation of the algorithms on two real datasets with ground truth to show their capabilities and interpretability. It is a preliminary work aiming to incorporate the Tucker decomposition into the Tensor Data Model [START_REF] Gillet | Empowering big data analytics with polystore and strongly typed functional queries[END_REF][START_REF] Leclercq | Polystore and Tensor Data Model for Logical Data Independence and Impedance Mismatch in Big Data Analytics[END_REF]. TDM adds the notion of schema and data manipulation operators to tensors, in order to make them data-centric and to avoid technical and functional errors brought by the manipulation of dimensions and elements of dimensions solely through integer indexes [START_REF] Rush | Tensor Considered Harmful[END_REF]. It also uses optimization techniques to allow the execution of operators, including the decompositions, on large-scale data [START_REF] Gillet | MuLOT: Multi-level Optimization of the Canonical Polyadic Tensor Decomposition at Large-Scale[END_REF].

The remaining of this article is organized as follow: section 2 gives an overview of tensors and of some main operators, section 3 presents the Tucker decomposition and two major algorithms to compute it, section 4 relates of experiments that illustrate the different uses of the Tucker decomposition and section 5 concludes the article and presents perspectives of future works.

BACKGROUND OF TENSORS

Tensors are general abstract mathematical objects which can be considered according to various points of view such as a multilinear application, or as the generalization of matrices to multiple dimensions. We will use the definition of a tensor as an element of the set of the functions from the product of 𝑁 sets 𝐼 𝑗 , 𝑗 = 1, . . . , 𝑁 to R :

𝒳 ∈ R 𝐼 1 ×𝐼 2 ו••×𝐼 𝑁 ,
where 𝑁 is the number of dimensions of the tensor or its order or its mode. Table 1 summarizes the notations used in this article.

Tensor operators, analogy with operations on matrices and vectors, are multiplications, transpositions, matricizations (or unfolding) and decompositions (also named factorizations). We only highlight the most significant operators on tensors and matrices which are used in Tucker decomposition algorithms. The reader can consult [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multiway data analysis and blind source separation[END_REF][START_REF] Tamara | Tensor decompositions and applications[END_REF] for an overview of the major operators.

The outer product between a tensor The mode-𝑛 product allows to multiply a tensor by a matrix or a vector. For a tensor

𝒴 ∈ R 𝐼 1 ×𝐼 2 ו••×𝐼 𝑁 and another tensor 𝒳 ∈ R 𝐽 1 ×𝐽 2 ו••×𝐽 𝑀 noted 𝒴 • 𝒳 produces a tensor 𝒵 ∈ R 𝐼 1 ×𝐼 2 ו••×𝐼 𝑁 ×𝐽 1 ×𝐽 2 ו
𝒳 ∈ R 𝐽 1 ×𝐽 2 ו••×𝐽 𝑛 ו••×𝐽 𝑁 and a matrix M ∈ R 𝐼 𝑛 ×𝐽 𝑛 , the result of the mode-𝑛 product noted 𝒳 × 𝑛 M is a new tensor 𝒴 ∈ R 𝐽 1 ×𝐽 2 ו••×𝐼 𝑛 ו
∈ R 𝐼 1 ו••×𝐼 𝑛-1 ×𝐼 𝑛 ×𝐼 𝑛+1 ו••×𝐼 𝑁 and a vector v ∈ R 𝐼 𝑛 noted 𝒳 × 𝑛 v produces a tensor 𝒴 ∈ R 𝐼 1 ו••×𝐼 𝑛-1 ×𝐼 𝑛+1 ו••×𝐼 𝑁 where: 𝑦 𝑖 1 ,...,𝑖 𝑛-1 ,𝑖 𝑛+1 ,...,𝑖 𝑁 = 𝐼 𝑛 ∑ 𝑖 𝑛 =1 𝑥 𝑖 1 ,...,𝑖 𝑛-1 ,𝑖 𝑛 ,𝑖 𝑛+1 ,...,𝑖 𝑁 v 𝑖 𝑛 The mode-n matricization of a tensor 𝒳 ∈ R 𝐼 1 ×𝐼 2 ו••×𝐼 𝑁 noted 𝒳 (𝑛) produces a matrix M ∈ R 𝐼 𝑛 ×Π 𝑗 ≠𝑛 𝐼 𝑗 , where: 𝑚 𝑖 𝑛 ,𝑗 = 𝑥 𝑖 1 ,...,𝑖 𝑛 ,...,𝑖 𝑁 with 𝑗 = 1 + 𝑁 ∑ 𝑘=1 𝑘≠𝑛 (𝑖 𝑘 -1) 𝑘-1 𝑚=1 𝑚≠𝑛 𝐼 𝑚
The Kronecker product between two matrices A ∈ R 𝐼 ×𝐽 and B ∈ R 𝐾×𝐿 noted A ⊗ B produces a matrix C ∈ R (𝐼𝐾)×( 𝐽 𝐿) , in which every elements of A are multiplied by the matrix B: 𝑐 𝑚,𝑛 = 𝑎 𝑖,𝑗 𝑏 𝑘,𝑙 where 𝑚 = 𝑘 + (𝑖 -1)𝐾 and 𝑛 = 𝑙 + ( 𝑗 -1)𝐿

TUCKER DECOMPOSITION

The Tucker decomposition [START_REF] Ledyard | Some mathematical notes on three-mode factor analysis[END_REF] factorizes a 𝑁 -order tensor 𝒳 1 for a representation of the Tucker decomposition on a 3-order tensor). The ranks 𝑅 1 , . . . , 𝑅 𝑁 are input parameters that determine the number of vectors (that can be seen as different features) for each factor matrix. The input tensor can be approximated with:

∈ R 𝐼 1 ו••×𝐼 𝑁 into a core tensor 𝒢 ∈ R 𝑅 1 ו••×𝑅 𝑁 and 𝑁 factor matrices A (𝑛) ∈ R 𝐼 𝑛 ×𝑅 𝑛 (see for example figure
𝒳 ≃ 𝒢 × 1 A (1) • • • × 𝑁 A (𝑁 ) 𝒳 𝐼 1 𝐼 2 𝐼 3 ≃ 𝒢 A (1)𝑇 A (2) A (3) Figure 1: The Tucker decomposition, with 𝒳 ∈ R 𝐼 1 ×𝐼 2 ×𝐼 3 the input tensor, 𝒢 ∈ R 𝑅 1 ×𝑅 2 ×𝑅 3 the core tensor, and A (1) ∈ R 𝐼 1 ×𝑅 1 , A (2) ∈ R 𝐼 2 ×𝑅 2 and A (3) ∈ R 𝐼 3 ×𝑅 3 the

factor matrices

The order of the elements of the dimensions of the input tensor does not impact the result of the decomposition. Indeed, changing it would only reorder the line vectors of the factor matrices, as each line vector stores the result of the decomposition for a given element on the dimension corresponding to the factor matrix.

To compute the Tucker decomposition, several algorithms have been proposed. Each has some advantages, as for example imposing more easily the orthogonality constraint (that allows a good clusterization of elements) or the non-negativity constraint (that provides more interpretable results). Two major algorithms are presented in this section: the Higher-Order Orthogonal Iteration (HOOI) and the Hierarchical Alternating Least Squares Non-negative Tucker Decomposition (HALS-NTD).

Higher-Order Orthogonal Iteration algorithm

The HOOI algorithm [START_REF] De Lathauwer | A multilinear singular value decomposition[END_REF] is the most famous one to compute the Tucker decomposition (algorithm 1). It depends primarily on the Singular Value Decomposition (SVD), that it extends to cope with multiple dimensions.

The HOOI starts by initializing the factor matrices, by matricizing the original tensor on each dimension in order to apply the SVD and to use the 𝑅 𝑛 left singular vectors (matrix U of the result of the SVD truncated at the 𝑅 𝑡ℎ 𝑛 column) as factor matrices. During the iterative phase (lines 2 to 7), each factor matrix is improved. To do so, a partial core tensor

𝒴 ∈ R 𝑅 1 ו••×𝐼 𝑛 ו••×𝑅 𝑁
is computed by performing the mode-𝑛 product on the original tensor and all the factor matrices except the one being improved. This partial core tensor is then matricized on the mode corresponding to the concerned dimension, and the SVD is executed on it. As for the

Symbol Definition Symbol Definition 𝒳 A tensor • Outer product X (𝑛)
Matricization of a tensor 𝒳 on mode-𝑛 ⊗ Kronecker product

𝑎 A scalar A ⊗ -𝑛 A (𝑁 ) ⊗ • • • ⊗ A (𝑛+1) ⊗ A (𝑛-1) • • • ⊗ A (1) v A column vector [M] +
Replace negative elements by 0 or small positive value

M A matrix × 𝑛 Mode-n product 𝒳 × -𝑛 {A} 𝒳 × 1 A (1) • • • × 𝑛-1 A (𝑛-1) × 𝑛+1 A (𝑛+1) • • • × 𝑁 A (𝑁 )
Table 1: Symbols and operators used initialization step, the 𝑅 𝑛 left singular vectors are used as the new factor matrix. The iterative phase allows to refine the result, as the partial core tensor takes into consideration the other factor matrices. So, each factor matrix is improved depending on the other factor matrices, thus reinforcing the discovering of relationships among elements of dimensions. When a convergence criteria is met, the final core tensor is computed from the original core tensor and all the factor matrices (line 8).

Algorithm 1 Higher-Order Orthogonal Iteration (HOOI)

Require: Tensor 𝒳 ∈ R 𝐼 1 ×𝐼 2 ו••×𝐼 𝑁 and target ranks 𝑅 1 , . . . , 𝑅 𝑁 Ensure: Core tensor 𝒢 ∈ R 𝑅 1 ×𝑅 2 ו••×𝑅 𝑁 and factor matrices U (1) , . . . , U (𝑁 ) with U (𝑛) ∈ R 𝐼 𝑛 ×𝑅 𝑛 1: Initialize U (1) , . . . , U (𝑁 ) , with U (𝑛) ∈ R 𝐼 𝑛 ×𝑅 𝑛 , U (𝑛) ← 𝑆𝑉 𝐷 (X (𝑛) ).U(:, 1 : 𝑅 𝑛 ) 2: repeat 3: for 𝑛 = 1, . . . , 𝑁 do 4: 𝒴 ← 𝒳 × 𝑁 U (𝑁 )𝑇 × 𝑛+1 U (𝑛+1)𝑇 × 𝑛-1 U (𝑛-1)𝑇 • • •× 1 U (1)
end for 7: until < convergence > 8: 𝒢 ← 𝒳 × 𝑁 U (𝑁 )𝑇 • • • × 1 U (1)𝑇
A simpler version of the HOOI algorithm exists, the Higher-Order Singular Value Decomposition (HOSVD), that removes the iterative part of the HOOI algorithm (lines 2 to 7). It is less precise, as the iterative part allows to refine the result until a convergence is met.

The HOOI algorithm inherits from the orthogonality constraint of the SVD for the computation of the factor matrices. Thus, it works pretty well to cluster elements of a dimension depending on their behavior on the other dimensions. However, as the SVD produces matrices with positive and negative values, the HOOI is not well suited to impose the non-negativity constraint on factor matrices, as some negative values will be found (and must be removed) at each iteration.

An advantage of this algorithm is that it can easily be implemented on large tensors. The most costly operation is the computation of the SVD, that is found at the initialization (line 1) and during the iteration phase (line 5). During the iteration, as the SVD is executed on the mode-𝑛 matricized partial core tensor, that is relatively small compared to the matricized original tensor, the time and space complexity is reduced. At the initialization of the algorithm, it can be replaced with a random one to avoid the computation of the SVD on a too large matrix.

Hierarchical Alternating Least Squares algorithm

The HALS-NTD algorithm [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multiway data analysis and blind source separation[END_REF] uses a different approach than the HOOI algorithm (algorithm 2), even if the initialization step (line 1) can be done by using the HOSVD. An alternative version of the HALS-NTD was later proposed [START_REF] Huy | Extended HALS algorithm for nonnegative Tucker decomposition and its applications for multiway analysis and classification[END_REF].

Algorithm 2 Hierarchical Alternating Least Squares (HALS-NTD)

Require:

Tensor 𝒳 ∈ R 𝐼 1 ×𝐼 2 ו••×𝐼 𝑁 and target ranks 𝑅 1 , . . . , 𝑅 𝑁 Ensure: Core tensor 𝒢 ∈ R 𝑅 1 ×𝑅 2 ו•
•×𝑅 𝑁 and factor matrices A (1) , . . . , A (𝑁 ) with A (𝑛) ∈ R 𝐼 𝑛 ×𝑅 𝑛 1: Initialize A (1) , . . . , A (𝑁 ) with non-negativity constraint

2: ℰ ← 𝒳 -𝒢 × 1 A (1) • • • × 𝑁 A (𝑁 ) 3: repeat 4:
for 𝑛 = 1, . . . , 𝑁 do 5:

for 𝑟 = 1, . . . , 𝑅 𝑛 do 6: for 𝑟 1 = 1, . . . , 𝑅 1 , . . . , 𝑟 𝑁 = 1, . . . , 𝑅 𝑁 do 13:

X (𝑟 ) (𝑛) = E (𝑛) + a (𝑛) 𝑟 G (𝑛) 𝑟 A ⊗ -𝑛
𝑔 𝑟 1 ,...,𝑟 𝑁 ← 𝑔 𝑟 1 ,...,𝑟 𝑁 + ℰ × 1 a

(1)

𝑟 1 • • • × 𝑁 a (𝑁 ) 𝑟 𝑁 14: ℰ ← ℰ + Δ 𝑔 𝑟 1 ,...,𝑟 𝑁 a (1) 𝑟 1 • • • • • a (𝑁 ) 𝑟 𝑁 15:
end for 16: until < convergence >

The HALS-NTD starts also by initializing the factor matrices (line 1), but adds a non-negativity constraint to manipulate only positive values in the remaining of the algorithm. An error tensor

ℰ ∈ R 𝐼 1 ו••×𝐼 𝑁 (noted E (𝑛)
when it is matricized on dimension n), that stores the difference between the original tensor and the reconstructed tensor from the core tensor and the factor matrices, is computed (line 2). The iteration phase (lines 3 to 16) is more complex than the one of the HOOI algorithm. Rather than improving a whole factor matrix at a time, it improves a vector of a factor matrix at a time. To do so, at the line 6, the current vector (the one being improved) is put in relation with all the other factor matrices associated with the part of the core tensor representing the strength of the relationships of the current vector regarding the vectors of the other factor matrices. This result is added to the error tensor, and stored in X (𝑟 ) (𝑛) , that can be seen as a matricized tensor representing the contribution of the current vector to the global result combined to the error tensor. At line 7, the current vector is improved by multiplying X (𝑟 ) (𝑛) with the part of the reconstructed tensor corresponding to the current vector. The current vector is then normalized with a 𝑙 2 norm (line 8), and the error tensor is updated (line 9). Once all the vectors of the factor matrices have been improved, the core tensor is updated from the previous core tensor, the error tensor and the new vectors of the factor matrices (line 13), and finally the error tensor is updated to integrate the changes in the core tensor (line 14).

The major advantage of the HALS-NTD is that it enforces the non-negativity constraint by imposing it during the initialization step, and then by improving the result without obtaining (and without having to eliminate) negative values during the iterative part (line 3 to 12). Thus, it eases the direct interpretation of the factor matrices as well as the core tensor.

However, as this algorithm computes the decomposition column vector by column vector for each factor matrices, it is computationally demanding, and harder to optimize than the HOOI one. Furthermore, there is almost no implementation of the HALS-NTD alogrithm. To the best of our knowledge, only Cichocki and Phan have provided a Matlab implementation in [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multiway data analysis and blind source separation[END_REF].

Related work

The Tucker decomposition has been used in several applications. Due to the lack of implementation for the HALS-NTD algorithm, the following articles are related mainly to the HOOI algorithm.

Cichocki in his book [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multiway data analysis and blind source separation[END_REF] shows several application of various decompositions on small tensors, mainly for image analysis. Romeo et al. [START_REF] Romeo | Semantic-based multilingual document clustering via tensor modeling[END_REF] used the Tucker decomposition to cluster documents. Thanks to this decomposition, they were able to process documents in several languages in the same tensor, in order to find similarities in the whole dataset. Sun et al. [START_REF] Sun | Multivis: Content-based social network exploration through multi-way visual analysis[END_REF] used Tucker on social network data in order to find clusters. They applied it on the Enron dataset. They also propose visualization techniques based on graph to display the result of the decompositon. Huang et al. [START_REF] Huang | Simultaneous tensor subspace selection and clustering: the equivalence of high order svd and k-means clustering[END_REF] compare the Tucker decomposition to the PCA and SVD associated to k-means. They run experiments on three datasets of images to show the similarities among these algorithms. Zhou et al. [START_REF] Zhou | Efficient nonnegative tucker decompositions: Algorithms and uniqueness[END_REF] took a different approach and used the Tucker decomposition as a supervised learning algorithm. They obtained promising results to cluster images.

As these works only use the HOOI algorithm, they only benefit from a part of the Tucker decomposition capabilities. They aim to cluster data, and do not rely on the direct interpretability of the factor matrices and the core tensor even if it brings valuable insights.

EXPERIMENTS

To better understand the capabilities of the Tucker decomposition regarding data analysis, we perform some experiments on two datasets: the database of faces (AT&T) [START_REF] Ferdinando | Parameterisation of a stochastic model for human face identification[END_REF] and the primary school dataset [START_REF] Stehlé | High-resolution measurements of face-to-face contact patterns in a primary school[END_REF]. The full code of the experiments is available online 1 .

1 https://github.com/AnnabelleGillet/Tucker-experiments

Database of faces experiment

This dataset contains greyscale images of the faces of 40 persons. Each person has 10 images, so the dataset is composed of 400 images. The dimensions of the images are 92 × 112. The ground truth is the person corresponding to an image.

The goal of this experiment is to evaluate the capabilities of the Tucker decomposition to find clusters in data. To do so, we create a 3-order tensor of size 92 × 112 × 400, with one dimension for the vertical pixels, one dimension for the horizontal pixels and one dimension for the images. The values of the tensor are these of the corresponding pixel in the images. We then run the Tucker decomposition with the ranks 3, 4 and 7, with both the HOOI and the HALS-NTD algorithms.

To cluster the images, we apply a k-means on the factor matrix corresponding to the dimension of the images. As we know that there are 40 subjects, we choose 𝑘 = 40. To evaluate the result, we compute its accuracy with:

𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇 𝑃 + 𝑇 𝑁 𝑇 𝑃 + 𝑇 𝑁 + 𝐹 𝑃 + 𝐹 𝑁 with 𝑇 𝑃 = True Positive, 𝑇 𝑁 = True Negative, 𝐹 𝑃 = False Positive and 𝐹 𝑁 = False Negative.
As the k-means selects random seeds to perform its computations, we repeat 100 times its execution and summarize in table 2 the results obtained. The confusion matrices of the results can be found on the Github of the experiments. The HOOI algorithm presents an accuracy score almost twice as good as the HALS-NTD one. With a mean of 61% as accuracy score, the HOOI algorithm is promising as an unsupervised classification tool. Indeed, this kind of task is often realized with supervised methods in which the algorithm is first trained on known images, but it comes at the cost of the necessity of having ground truth data to train the algorithm that also induces a risk of overfitting. With no training step, the Tucker decomposition avoids this risk.

Measure

This experiment highlights the power of the Tucker decomposition principle, which captures the relationships among dimensions in factor matrices. Here we used only one factor matrix. Thus, compared to the size of the original tensor, only few information have to be used to obtain satisfying results without any preprocessing nor using a image-specialized algorithm.

Primary school experiment

This dataset contains the interactions among students in a primary school, along with their teachers. There are 242 persons (including 10 teachers) split into 10 classes. An interaction is recorded if it lasts at least 20 seconds. The recording takes the form (person1, person2, timestamp in second). The class of each student is the ground truth of this dataset.

In this experiment, we seek to highlight the capabilities of the Tucker decomposition to provide meaningful factor matrices. To do so, we build a 3-order tensor of size 242 × 242 × 208, with two symmetric dimensions used to represent the persons, and the third dimension to represent the time, with a granularity of 5 minutes. If a person has been in contact with another person at a time 𝑡, then the value in the tensor indexed by the corresponding dimension values is 1.

As for the previous experiment, we run the Tucker decomposition with both algorithms in order to compare the results, with the empirically chosen ranks 13 (for the first person dimension), 13 (for the second person dimension) and 4 (for the time dimension).

The factor matrices for the first dimension are shown in figure 2. Each line represents a rank, and the columns are the persons. The students are ordered by their class: the first columns are the students of the class 1A, then 1B, and so on until 5B, and the 10 teachers are the last 10 columns. The natural non-negativity constraint of the HALS-NTD is of great help to understand the result. Indeed, we can distinguish 10 ranks in which each class appears distinctly, and three heterogeneous ranks. Conversely, the results of the HOOI algorithm are harder to interpret, and the classes do not appear as precisely as with the HALS-NTD algorithm.

In this experiment, the role of the core tensor is important: it gives insights regarding the strength of the relationships of the ranks among dimensions. For example, the 𝑔 1,1,1 value indicates if the vectors a

1 and a

(3) 1 are strongly related or not. To illustrate the usefulness of the core tensor, we can focus on a particular rank of the first dimension and see how it is related to the ranks of the other dimensions. The figure 3 represents this mechanism when fixing the rank of the first dimension to the one corresponding to the class 1A in the result of the Tucker decomposition performed with the HALS-NTD algorithm.

This figure shows some interesting results. The 1 𝑠𝑡 strongest value of the core tensor indicates that the class 1A has strong ties with itself, mainly at the break times and before and after the lunch break (figure 3a). It makes sense because at the breaks the students move from their classroom and go outside, so it creates more interactions among students. The 2 𝑛𝑑 strongest value of the core tensor shows again a relationship of the class 1A with itself, but this time during the class hours (figure 3b). The 3 𝑟𝑑 strongest value indicates a relationship between the class 1A and 1B during the breaks, including the lunch one (figure 3c). As the students of these two classes are of the same age, it is logical that they have more ties. Finally, the 4 𝑡ℎ value of the core tensor shows a relationship between the class 1A and a heterogeneous cluster that gathers students from grades 1, 2 and 3, during the breaks (figure 3d).

The advantages of the high interpretable capability of the HALS-NTD algorithm are twofold: 1) the vectors of the factor matrices give insights regarding the elements that contribute to the rank; and 2) the core tensor allows to link the ranks of one factor matrix to the ranks of the other factor matrices, and thus it gives more context to the result, as for example in figure 3 where we have the temporal activity of the different communities.

CONCLUSION

To conclude, the Tucker decomposition is a useful data analysis technique. Its two main algorithms, the HOOI and the HALS-NTD, have both advantages over the other: the HOOI algorithm shows promising results to cluster elements of a dimension, while the HALS-NTD provides interpretable insights about the relationships of the elements among dimensions.

However, the HALS-NTD algorithm is less known than the HOOI one, and in consequence it has almost never been implemented. We plan to integrate these Tucker algorithms to the Tensor Data Model, and to optimize them in order to allow their execution on large tensors, as we did for the CANDECOMP/PARAFAC decomposition [START_REF] Gillet | MuLOT: Multi-level Optimization of the Canonical Polyadic Tensor Decomposition at Large-Scale[END_REF]. Indeed, real data can create such tensors, that emphasis the need for optimized algorithms regarding the space and the execution time.

From a more applied point of view, we plan to use the Tucker decomposition to find patterns among birds migration, in order to prevent the spread of the avian influenza, in the context of the MOOD project2 .

  𝑇

5 :U

 5 (𝑛) ← 𝑆𝑉 𝐷 (Y (𝑛) ).U(:, 1 : 𝑅 𝑛 ) 6:

  𝑇

Figure 2 :

 2 Figure 2: Factor matrices for the dimension representing the persons in the primary school experiment

Figure 3 :

 3 Figure 3: Ranks from each factor matrix that are strongly related to each other when fixing the rank of the first dimension to the one representing the class 1A

  ••×𝐽 𝑀 in which the elements are equal to: 𝑧 𝑖 1 ,𝑖 2 ,...,𝑖 𝑁 ,𝑗 1 ,𝑗 2 ,...,𝑗 𝑀 = 𝑦 𝑖 1 ,𝑖 2 ,...,𝑖 𝑁 𝑥 𝑗 1 ,𝑗 2 ,...,𝑗 𝑀

••×𝐽 𝑁 where: 𝑦 𝑗 1 ,...,𝑗 𝑛-1 ,𝑖 𝑛 ,𝑗 𝑛+1 ,...,𝑗 𝑁 = 𝐽 𝑛 ∑ 𝑗 𝑛 =1 𝑥 𝑗 1 ,...,𝑗 𝑛-1 ,𝑗 𝑛 ,𝑗 𝑛+1 ,...,𝑗 𝑁 𝑚 𝑖 𝑛 ,𝑗 𝑛 The mode-𝑛 product between a tensor 𝒳

Table 2 :

 2 Accuracy of the result of the clustering of the images of the database of faces for each algorithm of the Tucker decomposition

		HOOI HALS-NTD
	Minimum	0.5425	0.2425
	Maximum	0.7175	0.38
	Mean	0.6122	0.3184
	Median	0.61	0.32
	Standard deviation 0.0346	0.0209
	Variance	0.0012	0.0004

BDA'22, October 24-27, 2022, Aubière, France Gillet et al.

https://mood-h2020.eu/

ACKNOWLEDGMENTS

This study was partially funded by EU grant 874850 MOOD and by ISITE-BFC (ANR-15-IDEX-0003).


Bernard Dolez

Annie Laurent

A Legislative Majority for the Future President? Revisiting the Phenomenon of Honeymoon Elections Under the Fifth Republic

Since the French presidential term was shortened from 7 years to 5 in 2002, the president and representatives in France's lower house of parliament (the National Assembly) have each been elected to five-year terms, and legislative elections are now scheduled "just after" the presidential election. In the past two decades, the party of the new or reelected president has systematically triumphed in the legislative elections held a few weeks later. It is as though these back-to-back two-round elections have coalesced to form a single four-round electoral sequence, with the result of each round determining the outcome of the one that follows. The winner of the presidential elections has thus enjoyed a certain "state of grace" that has enabled him to emerge victorious in the ensuing legislative race. Is this "honeymoon" election scenario destined to repeat itself in the presidential and legislatives elections scheduled for April 10 and 24 and June 12 and 19, 2022?

An Electoral Cycle Designed to Avoid Divided Government

France's Fifth Republic has been a semi-presidential system since the Constitution was modified in 1962 at the initiative of Charles de Gaulle to allow the President to be elected by direct universal suffrage [START_REF] Elgie | Semi-presidentialism in Europe[END_REF]. Until 2002, the President was elected for seven-year terms and members of the National Assembly for five-year terms. In the first two decades of the Republic, this had no institutional impact whatsoever. The French political game of the era was essentially a very simple one: each election pitted the Left against the Right, and in the end, the Right always won. Until 1981, the Right held on to the presidency while also systematically sealing a majority in the National Assembly, thereby facilitating political coherence between the President, the Prime Minister, and the National Assembly. Upon assuming the presidency in 1981, François Mitterrand immediately dissolved the National Assembly to secure a parliamentary majority and enable his administration to govern. The socialist party then successfully won an absolute majority in the National Assembly, marking France's first "honeymoon" elections. Thus, from 1962 to 1986, the French President systematically enjoyed a parliamentary majority that allowed him to govern the country.

Yet François Mitterrand's election signaled the start of a new era for the Fifth Republic, which would be characterized by the systematic defeat of the parliamentary majority in place in the following elections. Between 1981 and 2002, power systematically shifted with each parliamentary election cycle. Each of the three seven-year presidential terms in the 1980s and 1990s systematically included a period of joint majority rule by the president and the parliament and another period of divided government (cohabitation): 7 = 5+2 (for each of François Mitterrand's seven-year terms) or 2+5 = 7 (for Jacques Chirac's first seven-year term).

The extended duration of the third period of divided government-spanning from 1997 to 2002-showed that it was no longer a simple parenthesis until the next presidential elections. Instead, the risk was real that it might become the norm, spelling the demise of the constitutional framework instituted by Charles de Gaulle. The peril was made all the more dire by the fact that the next legislative elections were slated for March 2002, just weeks ahead of the presidential vote in April. Two major successive reforms have been advocated and adopted to prevent this risk: In 2000, former President Giscard d'Estaing and then socialist Prime Minister Lionel Jospin, with support from influential pundits, succeeded in forcing a politically weakened Jacques Chirac to assent to a shift to a five-year presidential term meant to "modernize" French politics. Next, the electoral calendar was inverted to avoid any "risk" of cohabitation, restoring the "spirit" of the Fifth Republic.

The two rounds of the presidential elections and the two rounds of the legislative elections now coalesced into a single four-round electoral sequence [START_REF] Parodi | L'ancrage d'une curiosité française : l'élection « exécutive » à quatre tours[END_REF][START_REF] Dupoirier | Four rounds in a row: the impact of presidential election outcomes on legislative elections in France[END_REF]. Following the presidential election, the "momentum" was now such that the President's party was virtually guaranteed to win the legislative elections. This was a foreseeable outcome: it had already occurred in the dissolutions of 1981 and-albeit less spectacularly-1988 (but it is true that François Mitterand's statement that, "It is not a good thing for a single party to govern," made during his presidential campaign in which he advocated for a "united France," had not encourage turnout from socialist voters). On April 21, 2002, on the eve of the first round of the presidential elections, Lionel Jospin was eliminated. Jacques Chirac emerged ahead of the pack in the first round with under 20% of the votes cast. Two weeks later, in the first sequential elections, he was re-elected against Jean-Marie Le Pen in the second round with upwards of 80% of the vote. In June, his UMP (Union pour la Majorité Présidentielle -"Union for the Presidential Majority") party and its allies won 365 legislative seats (63.3%).

The Mechanism Behind French Honeymoon Elections

This scenario is not unique to France and has now been well documented. Using data from approximately thirty recent elections in "pure" and "semi-" presidential systems, Matthew [START_REF] Shugart | Votes from Seats: Logical Models of Electoral Systems[END_REF] have shown that the "honeymoon surge" is dependent on the length of time between the two elections, empirically devising the following equation:

Rp = 1.20 -0.725E (1)
where Rp is the vote share of the president's party in the legislative elections divided by the president's vote share in the first round of the presidential election; and E is the number of months between presidential and legislative elections (0 means simultaneous elections-on the same day).

Stated another way, based on the data collected by Shugart and Taagepera, whenever E is close to zero (i.e., when legislative voting takes place "just after" the presidential elections, as it does in France), legislative elections theoretically enable the President's party to achieve a score that is 20% higher on average than the new President's score in the first round of the presidential elections. In France, however, this surge is considerably higher. The presidential party scores an average of 9 points higher than the President himself-an increase of over one third of the vote share. From 1981, it became apparent that the presidential party's victory in the legislative elections is less the result of the defection-temporary or otherwise-of the voters who supported defeated candidates in the presidential race, than the consequence of differential turnout, with supporters of vanquished presidential candidates heading to the polls in far lower numbers [START_REF] Goguel | Les élections législatives des 14-21 juin 1981[END_REF]. This phenomenon was also observed in 2017, when an Ifop survey found that a mere 45% of Marine Le Pen's and Jean-Luc Mélenchon's voters turned out for the first round of legislative voting, compared to 68% of Emmanuel Macron's supporters.

It is as though the drama of the presidential race has quenched their thirst for politics for the next five years, leaving them weary until the next presidential race. Yet there is more to it than that: voters also have the impression that the outcome of the elections is all but a foregone conclusion. As Shugart (2017) notes, "The purpose of the upcoming election is to ratify the new executive's direction, not to be a second chance for an alternative vision". This feeling has only intensified over time. The gap in turnout levels between the two elections, which stood at just over 10 points in 1981, has now regularly topped 20 points since the late 2000s, thus mechanically favoring the victory of the President's party (Table 2). In addition, France's legislative election system greatly contributes to forging a parliamentary majority. In practice, the two-round electoral system used in France is even more brutal than the single-round voting system used in the United Kingdom (country of the cube law), in which, according to [START_REF] Taagepera | Seats and Votes: A Generalization of the Cube Law of Elections[END_REF] But in France, an analysis based on the results of the six legislative elections held under the two-round majority system between 1978 and 2002 shows that the seat-vote equation did not follow the "cube law," but instead generally obeyed the "fourth exponent law," meaning that during this period the amplification effect was stronger in France than in the U.K. [START_REF] Dolez | The Seat-Vote Equation in the French Legislative Elections (1978-2002)[END_REF].

Two factors explain this phenomenon.

The first, strategic voting, is not unique to France. It contributes to marginalizing candidates, district by district, with ostensibly low chances of being elected, or at least of making it to the second round (the threshold for qualification in a district being set at 12.5% of registered voters). Competition thus tends to focus on one or two major political parties, or on one or two major alliances in which electoral coalitions have formed (Di [START_REF] Virgilio | Systèmes électoraux 'complexes', coordination pré-électorale complexe. Une comparaison France-Italie[END_REF]. The result was that until 2012, in the vast majority of cases, the second round consisted of a duel between the Left and the Right, ordinarily between one candidate from the PS and another from the UMP.

The second factor, however, is unique to France, clearly distinguishing it in this respect from the United Kingdom (and even more so from the United States). In France, legislative races are not particularly "local" in nature, which concretely means two things:

(1) The scores for political parties in each constituency mirror their national averages relatively closely. In other words, for a given political party, variations between different districts have been the result of the local specificities of the French election map and the quality of the local foothold of incumbent contenders [START_REF] Loonis | Les déterminants de la réélection des députes français de 1871 à 2002[END_REF], especially important in the case of multiple office holders-a particularity of the French system, which, until recently, allowed mayors of large cities to also simultaneously serve in the National Assembly [START_REF] Foucault | How useful is the cumul des mandats for being re-elected? Empirical evidence from the 1997 French legislative elections[END_REF]. Throughout the 2000s and into the early 2010s, these two phenomena helped consolidate the power of the two major governing parties-the UMP and the PS-over French politics;

(2) The same shifts in opinion have been observed in districts throughout the country: when the Left has gained support, it has gained support everywhere, and the same is true for the Right.

Thus, at each sequential legislative election, the President's party has systematically succeeded in achieving an appreciable lead not only nationally, but also across most of the country's electoral districts, handily carrying most of them in either the first or second round.

The dynamics of this lead can be demonstrated using the equation devised by [START_REF] Taagepera | Predicting Party Sizes: The Logic of Simple Electoral Systems[END_REF]:

s1 = 1 / (MS) 1/8 = 0.45 (3)
where s1 corresponds to the share of seats that the presidential party could have hoped to win given the district magnitude (M) and the number of districts in the election (S = 577 since 1988).

As Table 3 shows, the proportion of seats won is significantly higher than the proportion of seats that could be expected in both 1981 and 1988 and from 2002 to 2012.

2017: The "Honeymoon" Election Model Applied to Emmanuel Macron's Election

Would the victory of an outsider in the 2017 presidential race call this model into question?

Two factors seemed to coalesce to complicate the task facing the new President, a fact pundits were all too happy to underscore. Some were betting on the resiliency of the two major parties, which have structured French political life for ages, as well as on their ability to immediately bounce back from their defeat in the presidential race, in part due to the "localness" of the legislative elections and the strong foothold of their incumbent candidates. Others, in parallel, pointed to the difficulty that La République En Marche (LREM)-founded just one year priorwould face in finding candidates capable of securing a seat in each of the country's 577 electoral districts. Indeed, only twenty-nine incumbents had chosen to defect to the LREM, and in most districts, the party had selected novice politicians without an ounce of political experience.

But the uncompromising logic of sequential elections ultimately won out [START_REF] Dolez | Les législatives de 2017 ou la logique implacable des élections séquentielles[END_REF]. The political landscape was decimated by the presidential elections, and the shockwave that it generated extended to the ensuing legislative race. The opprobrium cast upon the socialist party in the wake of François Hollande's term and the chaos that reigned among the country's right-wing candidates-now deprived of a clear leader-facilitated the LREM's victory. Turnout plummeted by 29 points compared to the first round of the presidential elections, due in part to very low participation from voters who had backed Marine Le Pen and Jean-Luc Mélenchon. While candidates for the LREM and its allies earned 1.3 million fewer votes than Emmanuel Macron had just two months earlier, they nevertheless received 32.3% of the votes cast, an 8% gain compared to President Emmanuel Macron's score (24%).

Two new factors even worked in favor of an LREM victory. The first relates to the ban on holding multiple elected offices. Most members of parliament then also serving as mayors opted to retain their mayoral seats. Over 200 representatives (around 40% of incumbents) did not run for reelection, thus offering inroads for candidates from the LREM [START_REF] Evans | An atypical 'honeymoon' election? Contextual and strategic opportunities in the 2017 French legislative elections[END_REF]. A second element favored LREM in the second round of voting: its central position on the political spectrum, which allowed it to win in the second round both in face-offs against the Right (273 cases), through support from left-wing voters, and in face-offs against the Left (135 cases), through support from right-wing voters [START_REF] Dolez | From votes to seats. The 2017 French legislative elections[END_REF]. As a result, the LREM and its allies succeeded in securing an absolute majority of the seats in the National Assembly.

The electoral outcome in 2017 thus shows that a candidate capable of rallying over half of the vote in the second round of the presidential elections-regardless of the strength of his or her local party's local presence-has the ability to win a legislative majority by taking advantage of the "honeymoon" election surge.

The Outlook for 2022

In light of these factors, all signs point to a scenario in which the next President will again benefit from "honeymoon" elections next June. The last six sequential elections have demonstrated that:

1. Between the presidential and legislative elections, the presidential party gathers substantial momentum-surging by an average of 9 points thanks to the differential abstention of voters supporting the candidates defeated two months prior. The newly elected President's party systematically scores over 30% higher in the first round of the legislative elections, and this number often exceeds 35%.

2. In the second round of voting, votes from supporters of candidates eliminated in the first round and the inherent effects of the two-round system have offered the President's party an absolute legislative majority five out of six times. The share of seats secured by the presidential party in the second round has been far greater than the share of the vote received in the first round. The difference has never been less than 12 points. The average difference has been 20.5 points.

Given this information, the seat share (SL) obtained by the presidential party depends on the vote share (VL) it obtains in the first round, itself dependent on the vote share (VP) received by the future president in the first round of presidential voting. Thus:

SL = f (VP) (4)
Yet this is not enough to produce reliable electoral forecasts.

These will only make sense after the presidential election. However, it is currently possible to imagine three potential scenarios. Emmanuel Macron's reelection seems to be the most likely outcome. This would therefore be his second term. The LREM would, of course, be able to count on the local presence of its incumbent contenders. Above all, the presidential party, which benefited from the collapse of the left in 2017, now stands to benefit from the fragmentation of the Right, caught in a stranglehold between Emmanuel Macron and candidates on the Far Right. Nevertheless, it is important to remember that in 1988, when François Mitterrand was reelected, the socialists were only able to win a plurality in the National Assembly, a far cry from the "Pink Wave" seen in 1981. Likewise, in 2007, the Right had been in power for five years, and, although they were able to hang on to a majority in the wake of Nicolas Sarkozy's election, they ultimately surrendered 45 seats in the National Assembly. If Macron were to be re-elected, it is likely that a similar scenario will play out, and in particular, that LREM candidates in the second round will benefit less from their central position than they did in 2017, whether they are facing the Left, the Right, or the Far Right.

If Valérie Pécresse were elected, however, it would be her first term. She would benefit from the local presence of LR candidates, and would also be the first beneficiary of the reclassification of Emmanuel Macron's voters, whose champion would had been defeated a few weeks. She would certainly secure a large parliamentary majority.

Lastly, there is the possibility that Marine Le Pen or Jean-Luc Mélenchon might also win. For the time being, this seems unlikely, but were it to occur, they would doubtless find securing a legislative majority to be a more complicated endeavor, due both to their respective places on the political spectrum and to the limited local foothold of their parties-and particularly the low number of incumbents already in office. Yet, the shockwave created by the presidential elections should not be underestimated. Who, in January 1981, would have imagined the "Pink Wave" later that year in June? Likewise, who, in January 2017, would have imagined that Emmanuel Macron would win the election and boast the support of 350 representatives just six months later? While the likelihood that we will see new honeymoon elections next June currently seems quite high, the actual extent of future parliamentary majority will be contingent on factors that will only become clear once the presidential election has been decided.

As such, we recommend that legislative election forecasts made after the presidential election account for whether the president is being elected for the first time or the second time (dummy variable) and his or her place on the political spectrum (center or non-center). Leveraging data gathered from each individual electoral district is also a good way to improve the quality of these forecasts. In particular, at the electoral district level, we suggest taking stock of (1) the score of the newly elected president in the first round of presidential elections; and (2) whether or not the incumbent is running for reelection.

  , amplification long took the form of an exponent of 3 or: S(a)/S(b) = (Va/Vb) 3 (2) where V(a) and S(a) respectively correspond to the share of votes and seats won by party (a) and V(b) and S(b) to those won by party (b).

Table 1 :

 1 Presidential and legislative elections results in the first round since 1981 (in % of votes cast -France total) Laurent de Boissieu, https://www.france-politique.fr/elections-legislatives.htm The legislative elections do not only confirm the results of the presidential race. Rather, they profoundly amplify the outcome in terms of both votes and seats. But what are the mechanisms at work?

	Presidential	Legislative	Difference	Difference
	election,		elections,	in % points	in %
	Round 1		Round 1
	1981: Mitterrand/PS and	25.8	37.4
	allies			+11.6	+45%
	1988: Mitterrand/PS and	34.1	35.9
	allies			+1.8	+5%
	2002: Chirac/UMP and allies	19.9	33.8	+13.9	+70%
	2007: Sarkozy/UMP and	31.2	41.9
	allies			+10.7	+34%
	2012: Hollande/PS and allies	28.6	36.4	+7.8	+42%
	2017: Macron/LREM and	24.0	32.3
	allies			+8.3	+35%
	Mean	27.3	36.3	+9.0	+39%
	Source:

Table 2 :

 2 Abstention in legislative elections from 1978 to 2017 (France total) *Only the 1986 legislative elections featured a party-list proportional representation race held at the level of each département. Source: Laurent de Boissieu, https://www.france-politique.fr/elections-legislatives.htm

	Presidential election,	Legislative elections,	Difference in % points
	Round 1		Round 1
	1978		17.2
	1981	18.9	29.7	+10.8
	1986*		22.0
	1988	18.6	34.2	+15.6
	1993		32.3
	1997		28.5
	2002	28.4	35.6	+7.2
	2007	16.3	39.6	+23.3
	2012	20.5	42.8	+22.3
	2017	22.3	51.3	+29.0
	Mean	20.8	33.3	+18.0

Table 3 :

 3 Vote share and seat share of the President's party in legislative elections since 1981 (France total) Source: Laurent de Boissieu, https://www.france-politique.fr/elections-legislatives.htm

	Legislative	Legislative	Difference
	elections,	elections	in % points
	Round 1	Seat share
	Vote share
	1981: PS and allies	37.4	58.0	20.6
	1988: PS and allies	35.9	48.0	12.1
	2002: UMP and allies	33.8	63.3	29.5
	2007: UMP and allies	41.9	57.1	15.2
	2012: PS and allies	36.4	53.5	17.1
	2017: LREM and allies	32.3	60.6	28.3
	Mean	36.3	56.8	20.5

Acknowledgments

We thank the reviewers for their many helpful comments.

CONFLICTS OF INTEREST

The authors declare no ethical issues or conflicts of interest in this research.


Steven Araujo

Laurent Delbreilh

Laurent Laguerre

Hervé Dumont

Éric Dargent

Cyrille Fauchard
email: cyrille.fauchard@cerema.fr

Rock permittivity characterization and application of electromagnetic mixing models for density/compactness assessment of HMA by means of Step Frequency Radar

Keywords: Permittivity, GPR, density, compactness, Step Frequency Radar, electromagnetic mixing model

This work aims to determine the compactness/density of Hot Mix Asphalt (HMA) by measuring its permittivity by means of Step Frequency Radar (SFR). As HMA is mainly made of rocks, their dielectric properties are measured in the frequency range applied with SFR [0.5-4 GHz] using cylindrical cavities. The results show that the rocks can be considered as lowloss dielectric. As electromagnetic mixing models are required to switch from the measured permittivity to the compactness, Power Law (PL) models and Unified Mixing Rules (UMR) are applied to laboratory experiment. Slabs permittivity of various compactness are determined with the help of the SFR system. This study shows that (i) the selection of the electromagnetic mixing model has a critical impact on the accuracy of the calculated compactness, (ii) the choice of the host matrix for UMR family has huge consequences and (iii) the best assessment of compactness/density is given by CRIM, Rayleigh and Böttcher models with aggregate matrix.

Introduction

The dielectric characterization of rocks is of great interest for exploration in geophysics. The electromagnetic (EM) methods for soil prospection require a good comprehension of EM waves interaction with soil, as well as an accurate knowledge of the dielectric properties of the observed materials. Dielectric measurements can help in various areas such as volcanic rock study [START_REF] Gomaa | Thermal effect of magma intrusion on the electrical properties of magnetic rocks from hammamat sediments, Cairo, Egypt[END_REF], estimation of rock porosity in petrolum reservoirs [START_REF] Tong | Permeability estimating from complex resistivity measurement of shaly sand reservoir[END_REF], new methods for oil shale extraction [START_REF] Lopatin | Investigation of the dielectric properties of oil shale[END_REF], rock wettability and grain size, water and clay influences in soil [START_REF] Bona | Advances in understanding the relationship between rock wettability and high-frequency dielectric response[END_REF][START_REF] Dobson | Microwave dielectric behavior of wet soil-part ii: Dielectric mixing models[END_REF][START_REF] Hallikainen | Microwave dielectric behavior of wet soil-part 1: Empirical models and experimental observations[END_REF][START_REF] Lesmes | Dielectric spectroscopy of sedimentary rocks[END_REF], water-bearing rocks (Chelidze and Gueguen, 1999;Chelidze et al., 1999) or dry geological materials [START_REF] Saint-Amant | Dielectric properties of dry, geological materials[END_REF]. In civil engineering, the widespread use of Ground Penetrating Radar (GPR) methods for non-destructive purposes [START_REF] Al-Qadi | Use of gpr for thickness measurement and quality control of flexible pavements (with discussion)[END_REF][START_REF] Benedetto | Inferring Bearing Ratio of Unbound Materials from Dielectric Properties Using GPR: The Case of Runaway Safety Areas[END_REF] has led to the study of man-made materials dielectric properties. [START_REF] Robert | Dielectric permittivity of concrete between 50 mhz and 1 ghz and gpr measurements for building materials evaluation[END_REF] studied the dielectric behaviour of concrete between 50 MHz and 1 GHz. [START_REF] Maierhofer | Investigation of dielectric properties of brick materials as a function of moisture and salt content using a microwave impulse technique at very high frequencies[END_REF] assessed the dielectric properties of brick materials and porous concrete as a function of moisture and salt content at 7 GHz. [START_REF] Hasar | Non-destructive testing of hardened cement specimens at microwave frequencies using a simple free-space method[END_REF] measured the hardened mortar and cement with various water-content in the frequency band [8][9][10][11][12].

For dry civil engineering materials such as HMA, the rock proportion generally represents as much as 88 to 96 % of the mix. Hence the knowledge of the rock dielectric properties is of major importance and represents a consistent a-priori information for thickness or density controls performed on site with GPR or SFR systems. As a matter of fact, the dielectric characterization of dry rocks has already been undertaken but for different purposes: [START_REF] Ulaby | Microwave dielectric properties of dry rocks[END_REF] in support of radar investigation to Mars, studied 80 samples of igneous and sedimentary rocks in the microwave region . [START_REF] Campbell | Electrical properties of rocks and their significance for lunar radar observations[END_REF] conducted similar measurements on lunar samples at 450 MHz and 35 GHz.

In the first section of this work, the complex permittivity of 24 magmatic, sedimentary and metamorphic rocks are measured in cylindrical cavities at 6 frequencies in the microwave region. They have mainly been extracted from quarries providing road manufacturers in building materials. These measurements complete a previous study carried out on 11 rock types [START_REF] Fauchard | Determination of the compaction of hot mix asphalt using highfrequency electromagnetic methods[END_REF] and a discussion is proposed on the measured values and their dependency to the rock bulk density and their mineral composition.

The HMA density (and its equivalent compactness, or indice of compaction) is one of the key parameters to control for the reception of newly-paved roads. Nuclear tests based on gamma ray interaction with matter (ASTM D2950 / D2950M, 2011; ASTM D6938 -10, 2010) can be implemented directly on site and have been considered since the 70's as the only non-destructive method for assessing in-place density. Nevertheless, nuclear methods tend to disappear because of the cost induced by their transportation and storage. Their radiations also threaten the life users. Nowadays, core testing in the lab remains the only reference method valuable for density evaluation. Two main core methods are suitable and standard: the paraffin coated test samples (NF EN 12697-6, 2012) and the nuclear bench (NF EN 12697-7, 2003). The challenge to take up today is to develop non-destructive on site testing, without the constraints of nuclear devices. SFR system allows getting the HMA permittivity which is relied to the density by electromagnetic models. In that purpose, the basic principle of mixing rules are mainly taken from the work of [START_REF] Sihvola | Electromagnetic mixing formulas and applications[END_REF] and the results are widely compared with [START_REF] Al-Qadi | In-place hot-mix asphalt density estimation using ground-penetrating radar[END_REF], [START_REF] Leng | Development and validation for in situ asphalt mixture density prediction models[END_REF][START_REF] Fauchard | Non-destructive assessment of hot mix asphalt compaction/ density with a step-frequency radar: case study on a newly paved road[END_REF]. In the second section of this work, these models are applied and discussed to a laboratory investigation.

Rock permittivity measurements

Lab experiments and results

As a HMA mainly consists of 88% to 96% of rocks, the dielectric characterization of rocks is a major issue for the comprehension of HMA electromagnetic behaviour. Hence, the knowledge of dielectric properties of rocks in the frequency band used in civil engineering application has been undertaken. This study updates some results of a previous work [START_REF] Fauchard | Determination of the compaction of hot mix asphalt using highfrequency electromagnetic methods[END_REF] and intends to test dielectric mixture models selected in the litterature. In order to evaluate the HMA density with radar methods, rocks extracted from various quarries for road manufacturing have been studied. The table 1 reports the name, the measurement date (2013 and 2015), the origin and the rock feature. Furthermore, other rocks that are not used for road manufacturing such as the volcanic basalt of the Piton de la Fournaise, or the limestone from Caen, are reported for knowledge purpose.

A portable network analyser (PNA, Agilent E8362B) and two cylindrical cavities (φ=400 mm and φ=200 mm) allow the complex permittivity measurement of 4-5 cm-diameter and 2.52 cm-height cylindrical rock samples at six frequencies (0.57 GHz,1.32 GHz,2.06 GHz and 1.15 GHz,2.63 GHz and 4.13 GHz,respectively). A previous study carried out on 11 different rocks has reported the good repetability of this method [START_REF] Fauchard | Determination of the compaction of hot mix asphalt using highfrequency electromagnetic methods[END_REF]. Indeed, the permittivity of each tested rocks was averaged on 10 to 15 cores for one single type of rock at six frequencies. However a substantial heterogeneity were observed in a given set of cores extracted from a single vein. In this study, 24 rocks are tested, including the aformentioned rocks. Herein, just one cylindrical core by rock type is extracted and measured twice (side and upside down) at six frequencies. Hence, the standard deviation of the 2015 measurements is calculated from 12 permittivity values. The purpose is to record a high frequency reference permittivity that will be used for further works in the low frequency domain. Rocks were dried 48h at 115 • C in oven before measurement.

The real ( r ) and imaginary dielectric ( " r ) constants, the loss tangent (tanδ) and the density (ρ1 ) are reported in the table 1. The real and imaginary part of the permittivity and the loss tangent are respectively shown in the Fig. 1(a), 1(b) and 1(c).

Discussion on the measured permittivity

The real dielectric permittivity ranges from 4 to 9 and these results are in accordance with the studies of [START_REF] Ulaby | Microwave dielectric properties of dry rocks[END_REF] and [START_REF] Parkhomenko | Electrical Properties of Rocks[END_REF]. There is no significant evidence of the values distribution of permittivity as a function of rock formation (sedimentary, magmatic of metamorphic). The imaginary part of the permittivity exhibits strong standard deviation related, first, to the experimental system, and second, to the losses. According to [START_REF] Li | Precise calculations and measurements on the complex dielectric constant of lossy materials using tm010 cavity perturbation techniques[END_REF], cylindrical cavity allows an accuracy of 1% on the real part of the permittivity and 2% for the imaginary part.

This uncertainty may increase respectively from 1% to 2% and from 2% to 5% by taking into account the uncertainty measurement of sample dimensions. The measurements highlight a slight dispersion on the real part of the permittivity which is negligible over the 6 used frequencies (less than 1%). However, a stronger dispersion is noticeable on the imaginary part. First, this could be due to water molecules which were not evaporated during the annealing (115 • C for 48h). This water (probably bounded) induces an interfacial polarization (Maxwell-Wagner-Sillar) ( [START_REF] Benedetto | Water content evaluation in unsaturated soil using gpr signal analysis in the frequency domain[END_REF][START_REF] Kaden | Low-frequency dielectric properties of three bentonites at different adsorbed water states[END_REF][START_REF] Saltas | Dielectric properties of non-swelling bentonite: The effect of temperature and water saturation[END_REF]) which is basically observable about 0.5 to 1 GHz and at lower frequencies as well. Second, another phenomenon such as interfacial polarization between mineral structures should also be considered and will be undertaken in the future through petrographic and chemical studies. By covering the frequency band from 0.5 to 4 GHz, relaxation phenomena may occur and might explain such high dispersion. However, despite the variation of the imaginary part, the loss tangent ranges between 0.1 to 0.01, in such a manner that tested rocks can be considered as low-loss dielectrics.

Overall, the differences betweeen the 2013 and 2015 measurements are about 0.2 for the real part of the permittivity which is close to the standard deviation of the measurement for a given rock. However four of them (rhyolite, gneiss, quartz arenite and amphibolite) highlight a higher difference probably related to the structure heterogeneity and segregation. As a matter of fact, rhyolite and amphibolite are respectively (i) an igneous rock that exhibits a heterogeneous nodular structure composed of quartz and feldspath crystals, and (ii) a metamorphic rock with amphibole and plagioclase feldspars weakly foliated. The distribution of these heterogeneities differs from sample to sample and could explain the measured difference. Gneiss and quartz arenite are respectively (i) a metamorphic rock with foliation and (ii) a sedimentary rock with micro-craking filled with quartz. These horizontal structures are randomly oriented into the samples and according to the Hashin-Shtrikman bounds [START_REF] Brovelli | A combination of the hashin-shtrikman bounds aimed at modelling electrical conductivity and permittivity of variably saturated porous media[END_REF], affect the measured permittivity. Relations between rock permittivity and bulk density are plotted in the Fig. 2. The three following empirical laws are studied:

= a ρ 1 (1)
The equation 1 is given by [START_REF] Ulaby | Microwave dielectric properties of dry rocks[END_REF] for terrestrial rocks and by [START_REF] Olhoeft | Dielctric properties of the first 100 meters of the moon[END_REF] for lunar materials, where a 1 = 1.96 and a 1 = 1.93, respectively;

= (a 2 + a 3 ρ) 2 (2)
The equation 2 is given by [START_REF] Shutko | Microwave radiometry of lands under natural and artificial moistening[END_REF] for dry soil, where a 2 = 1 and a 3 = 0.5;

= (a 4 + a 5 ρ) 2 -a 6 (3)
The equation 3 is given by [START_REF] Dobson | Microwave dielectric behavior of wet soil-part ii: Dielectric mixing models[END_REF] for soil having extremely low moisture content, where a 4 = 1.01, a 5 = 0.44 and a 6 = 0.062; As it can be seen on the Fig. 2, none of the previous laws is adapted to the studied rocks. Thus, the parameters are adjusted to this specific application and give the following values :

a 1 = 1.88, a 2 = 3.1, a 3 = -2, a 4 = 3.1,
a 5 = 0.89 and a 6 = 24.5. The Olhoeft's law does not fit correctly the data (the correlation coefficient is R=0.64) while the best-adjusted laws are given by both Stuko and Dobson (R=0.83). However, the same conclusion established by [START_REF] Ulaby | Microwave dielectric properties of dry rocks[END_REF] can be drawn, meaning that the variation in the density accounts for about 50% of the observed variance in the dielectric constant value. The remaining data scatter can be attributed to the mineral composition of the rocks. Some of these observations were confirmed by [START_REF] Sengwa | Dielectric properties of some minerals of western Rajasthan[END_REF] on dry minerals: at 10.1 GHz, they observed that both the density (of clay, fuller's earth, siliceous earth, tourmaline and magnesium rocks) and the chemical composition -and particularly the S iO 2 , Al 2 O 3 and FeO 3 oxides content -equally affect the measured . As the previous authors concluded, no obvious relation between the density and the imagininary part of the permittivity can be established.

These assumptions would deserve a complementary study (i) by characterizing the chemical composition and the crystalline structure and (ii) by extending the frequency range of measurements.

3. Theoritical approach and application of models

C = 1 -C air = ρ HMAS ρ HMAM (4)
Where ρ HMAM is the theoritical density of the HMA without air (namely maximum specific gravity).

It is the controlled parameter in France and other European countries.

All material contents are defined with respect to the total dry mass m s = m a + m b + m f . The weight content is simply T i = m i m s . The volumic fraction of the i th component is defined as below:

C i = C a T i a ρ a ρ i a (5)
And:

C a = C Z (6)
Where:

Z = 1 + ρ a T b ρ b T a + ρ a T f ρ f T a = 1 + C rb + C r f (7)

Model assumptions

In this section, the basic concept of electromagnetic models are taken from [START_REF] Sihvola | Electromagnetic mixing formulas and applications[END_REF], [START_REF] Sihvola | Effective permittivity of dielectric mixtures[END_REF] and [START_REF] Parkhomenko | Electrical Properties of Rocks[END_REF]. HMA is assumed following two assumptions:

First, it is considered as a multiphase media composed of binder, filler, aggregate and air, where the permittivity and volumic fraction of each component average the effective permittivity of the media. This first model family is called the Power Law (PL) model which does not differenciate host component from the guests.

Second, the HMA (Fig. 3(b)) is considered as several spherical inclusions immersed in a host matrix. In this work, the filler content is very low (about 1%) and the air is considered randomly distributed into the media. Hence, the host matrix can be either aggregate that presents the greater proportion in the mix or binder that encloses the remaining components. Under this assumption, the Unified Mixing Rules (UMR) can described the dielectric behaviour of the mix. Models acronym are detailed in table 2.

Both dielectric approaches are presented in the following section and the HMA compactness/density are calculated.

Power Law model

The PL model gives the HMA dielectric constant as a sum of each permittivity component weighted by their volumic concentration. It is defined according to the following formula:

α HMA = i C i α i (8)
The parameter α is known in the literature to have several values: when α = 1/2, (8) gives the Complex Refractive Index Model, also called CRIM model or Birchak formula [START_REF] Birchak | High dielectric constant microwave probes for sensing soil moisture[END_REF]. As explained by [START_REF] Sen | A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads[END_REF],

this assumption can be seen as a 3D-generalisation of the 1D-multilayered case of an EM field in normal incidence

where the square root of the effective permittivity is the sum of the lineic concentration multiplied by the square root permittivity of each layer. If α = 1/3, ( 8) is the Looyenga formula [START_REF] Looyenga | Dielectric constants of heterogeneous mixtures[END_REF]. Both cases α = 1 and α = -1 can be explained by considering a circuit capacitors, respectively (i) parrallel and (ii) series connected [START_REF] Sihvola | Electromagnetic mixing formulas and applications[END_REF].

Finally, for the limit α → 0, ( 8) is simply the [START_REF] Lichtenecker | Die herleitung des logarithmischen mischungsgesetzes aus allgemeinen prinzipien der stationiren strmung[END_REF] approximation given by:

ln( HMA ) = i C i ln( i ) (9)
(9) is obtained by taking into account the first order term of the Taylor series of the exponential function e αln( ) .

By combining the equations (4-8), the HMA compactness and density are calculated by the following relations:

C = ( α HMA -1)Z α a + C r f α f + C rb α b -Z (10)
ρ HMAS = ( α HMA -1)Z α a + C r f α f + C rb α b -Z ρ HMAM (11)
3.1.4. Unified mixing rule formula UMR allows writting the effective permittivity by considering a host matrix with guest inclusions [START_REF] Sihvola | Electromagnetic mixing formulas and applications[END_REF][START_REF] Sihvola | Effective permittivity of dielectric mixtures[END_REF].

HMA -m HMA + d m = i C i i -m i + d m ( 12
)
Where m is the permittivity of the host matrix m and d m = 2 m + ν( HMAm ).

After solving (12) by assuming that the matrix is the aggregate,

C = HMA -a HMA +d a -1-a 1+d a 1 Z C rb b -a b +d a + 1 Z C r f f -a f +d a -1-a 1+d a (13) ρ HMAS = HMA -a HMA +d a -1-a 1+d a 1 Z C rb b -a b +d a + 1 Z C r f f -a f +d a -1-a 1+d a ρ HMAM (14)
Where d a = 2 a + ν( HMAa ).

After solving (12) by assuming that the matrix is the binder,

C = HMA -b HMA +d b -1-b 1+d b 1 Z a -b a +d b + 1 Z C r f f -b f +d b -1-b 1+d b (15) ρ HMAS = HMA -b HMA +d b -1-b 1+d b 1 Z a -b a +d b + 1 Z C r f f -b f +d b -1-b 1+d b ρ HMAM (16)
Where

d b = 2 b + ν( HMA -b ).
The factor ν is a dimensionless parameter: ν = 0 gives the Maxwell-Garnett rule also named Rayleigh model [START_REF] Sihvola | Electromagnetic mixing formulas and applications[END_REF]. ν = 2 gives the Polder-Van Santen [START_REF] Polder | The effective permeability of mixtures of solids[END_REF] formula or Böttcher formula [START_REF] Böttcher | Theory of electric polarization[END_REF] and ν = 3 gives the Coherent potential approximation [START_REF] Sen | A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads[END_REF]. This formula is valuable for 3D inclusions meaning that they are assumed to be spherical. However, inclusions can also be in 2D such as disk shaped. According to [START_REF] Wiener | Zur theorie der refraktionskonstanten, berichte über die verhandlungen der königlich-sächsischen gesellschaft der wisseschaften zu leipzig[END_REF] and [START_REF] Yaghjian | Electric dyadic green's functions in the source region[END_REF], the difference between both shapes is the factor 2 m for 3D inclusions at the denominators which becomes m for 2D inclusions.

Models application to laboratory HMA slabs

Lab experiments were conducted on two 40*60*8 cm slabs with a SFR system composed of the PNA described in section II and mono-static ultra wideband antenna of bandpass [1.4-6 GHz]. The tested slabs are mainly composed of basalt aggregates (0/10 mm) and the weight content, permittivity and density of each component are presented in the table 3. Binder and filler permittivities have been measured by means of cylindrical cavities. In order to analyze the behaviour of the models described in section 3.1, the density and compactness are calculated for every models in a theoritical HMA permittivity range of [1-7], corresponding to air void content ranging between 100% to 0%.

This application is performed for the basalt slab described in the column 1 of table 3 (air void content of 13%) and is presented in the Fig. 4(a). and Silberstein (PL α = 1) models exhibit a huge deviation from the five other models. The highest the compactness, the closest is the HMA permittivity. For the studied slab with C = 87%, the difference of the calculated compactness can be expected to be larger than the slab with C = 95%.

Models application to laboratory HMA slabs

In this section, the HMA permittivity is measured with a SFR system. The experimental system and the theoritical background of the HMA permittivity measurement have already been well described by [START_REF] Fauchard | Non-destructive assessment of hot mix asphalt compaction/ density with a step-frequency radar: case study on a newly paved road[END_REF]. In this work, the HMA permittivity HMA is assessed thanks to the two-way travel time ∆t of the EM wave into the slab and its thickness e according to the equation: HMA = ( c∆t 2e ) 2 . The SFR permittivity measurements on the slab surfaces were performed every 5 cm along the x-axis, and every 1 cm along y-axis. In order to avoid the edge effects, a 40 cm*20 cm smaller surface is taken into account. Then, the compactness and the density of the slabs are calculated with the models PL α = 1/2 and UMR ν = [0; 2], MA and MB (with 10-11 and 13-16, respectively). UMR with 2D inclusions (disk shaped) was also tested on the slabs, but systematic offsets were observed meaning that the disk shape inclusions are not adapted for this material. This case is not presented here. The compactness and the density obtained with a gamma bench is considered as the reference value (NF EN 12697-7, 2003). The result of the measurement is represented as a color map along x and y axis. The color scale represents the assessed compactness. The whole experiment was carried out on 4 slabs of various target air void contents (87%, 90%, 92% and 95%) both on top and bottom surfaces. In this paper, only the top surface 87% and 95% slabs results are presented in the Fig. 5 and6.

In-lab results interpretation and discussion

First of all, a design flaw called rutting effect can be observed on the slabs especially for the 95% compactness slab. This is observable by a lower calculated compactness along the y-axis following two bands corresponding to the passage of the wheels during the compaction process. However this rutting effect is not observed on the gamma bench measurement, meaning that the slabs were implemented homogeneously. Hence, the explanation can only be attributed to the two-way travel time measured with the SFR that was already observed and explained by [START_REF] Fauchard | Determination of the compaction of hot mix asphalt using highfrequency electromagnetic methods[END_REF]. The table 4 summarizes the average compactness and the average density of each slabs according to the reference value (gamme bench) and the studied models. The compactness of the 87% designed slab is exactly as expected (87.2%). Withal, gamma bench measurement on the 95% designed slab exhibits a compactness of 92.6% due to a manufacturing constraint. The values obtained by gamma bench will be now considered as the true compactness.

In the present results, for both slabs and every models, the calculated compactness is close to the reference value, except for the Rayleigh model (UMR ν = 0) with binder matrix. Indeed, the UMR ν = 0 MB overestimates the compactness from 3 to 6% depending on the selected slab and exhibits a systematic shift compared to the other models. However, the UMR ν = 0 MA calculation gives excellent results for both slabs -particularly for the 87% slab (from 0.003 to the reference value) -which suggests that aggregate matrix is the appropriate choice.

The Böttcher model UMR ν = 2 MA and MB are strictly identical and as explained previously, a basic calculation of both models proves that 13-14 are equal to 15-16. In that case, the matrix nature does not affect the calculation.

UMR ν = 2 MA and MB point out excellent results as well, with a best-calculated compactness for the 95% slab (from 0.004 to the reference value).

As described earlier, the PL α = 1/2 model is a part of the group that exhibits excellent results. Consequently, CRIM model is very close to the gamma bench compactness (from 0.007 to 0.011). In that formalism, the PL does not distinguish the host matrix from guest inclusions. Previous works [START_REF] Fauchard | Non-destructive assessment of hot mix asphalt compaction/ density with a step-frequency radar: case study on a newly paved road[END_REF] shown that PL α = 1/2 is a relevant model for HMA compactness assessment. As UMR ν=[0,2] MA are relatively similar to the reference value and the PL α = 1/2, it could suggest that aggregate as host matrix is a valid choice.

For the present lab experiment, the slabs were manufactured with basalt rock extracted from a single vein and controlled by means of cylindrical cavities. This leads to the assumption that the aggregate permittivity is constant into the whole slab, tacking into account the standard deviation of 2%. Following this consideration, the PL α = 1/2 and the UMR ν=[0,2] MA models might be considered as equivalent for the lab compactness/density assessment.

Al-Qadi et al. ( 2010) have shown for their on-site application that the Rayleigh model with binder matrix gave closest results from the reference value than the CRIM and Böttcher models. In another work, [START_REF] Leng | Development and validation for in situ asphalt mixture density prediction models[END_REF] concluded that the Böttcher model was the most accurate for their on-site case study. carried out on-site compactness assessment by using PL α = 1/2 in great accordance with standard nuclear and 249 core testing. For all of these approaches, the aggregate permittivity was back-computed according to density cali- bration by core drilling. Due to the multiple parameters which could influence the accuracy of the HMA measured 251 permittivity, the same conclusion can be drawn, i.e. the three aforementionned models can be usable for the in-situ 252 compactness/density assessment.

Conclusion

The objective of this work has been to characterize the dielectric properties of rocks used in pavement assessment in order to calculate the HMA compactness from electromagnetic mixing models. The dielectric investigation has

shown that the studied rocks can be considered either as near perfect dielectric or low loss dielectric with a real permittivity ranges from 4 to 9. Application of [START_REF] Olhoeft | Dielctric properties of the first 100 meters of the moon[END_REF], [START_REF] Shutko | Microwave radiometry of lands under natural and artificial moistening[END_REF] and [START_REF] Dobson | Microwave dielectric behavior of wet soil-part ii: Dielectric mixing models[END_REF] fitting pointed out that the real permittivity is dependent upon the density and the chemical composition of the rock.

Further studies need to be performed concerning the structural and chemical composition of the rocks in order to support these assumptions. It could be of great interest to carry out the dielectric behaviour on a wider frequency range so that the behaviour of dry rocks at high frequencies could be better understood. The comparison of the two electromagnetic mixing model families recently used for HMA assessment has been performed on laboratory slabs whose the permittivity of each component was measured by means of cylindrical cavities (binder, filler and aggregate). Slabs were controlled by gamma bench measurement as standard for comparison with applied models.

This investigation highlights that the PL α = 1/2 (CRIM), UMR ν=0 MA (Rayleigh) and UMR ν=2 MA (Böttcher) models gave excellent results in correlation with the reference compactness. Thereby, it was concluded that choosing aggregate as matrix for the UMR family is more appropriate than binder. In consideration of the uncertainties of radar measurement or permittivity measurement for both on site and lab experiments, the three models i.e. PL α = 1/2 (CRIM), UMR ν=0 MA (Rayleigh) and UMR ν=2 MA (Böttcher) can be considered as equivalent. These models deserve to be applied on different slab natures in order to study their deviation and further in-situ investigations should be undertaken for validation.

Table 1: First column: the rock number (x-axis in Fig. 1(a), 1(b) and 1(c)), the name, the measurement date (2013 and 2015), the quarry and the rock feature -subsequent columns: real and imaginary part of the permittivity measured at 0.57 GHz, 1 GHz, 1.7 GHz, 1.15 GHz, 2.63 GHz and 4.13 GHz in cylindrical cavities, corresponding loss tangent tanδ and density ρ.

Rock number -nature (quarry, date, feature)

Figure 1 :

 1 Figure 1: Dielectric characterization of 24 rocks obtained by means of cylindrical cavities at 0.57 GHz, 1 GHz, 1.7 GHz, 1.15 GHz, 2.63 GHz and 4.13 GHz. Rock details are given in the table 1.

Figure 2 :

 2 Figure 2: Real part of the permittivity as a function of the density according to Olhoeft, Shutko and Dobson laws (dashed lines) and then adjusted (solid lines).

  3.1. Definition and presentation of the models 3.1.1. Definition of HMA density and compactness HMA is considered as 4-phase mixture composed of binder, filler, aggregate and air. Density and compactness HMA have already been studied by[START_REF] Al-Qadi | In-place hot-mix asphalt density estimation using ground-penetrating radar[END_REF],[START_REF] Leng | Development and validation for in situ asphalt mixture density prediction models[END_REF] for 3-phase mixture and by[START_REF] Fauchard | Determination of the compaction of hot mix asphalt using highfrequency electromagnetic methods[END_REF][START_REF] Fauchard | Non-destructive assessment of hot mix asphalt compaction/ density with a step-frequency radar: case study on a newly paved road[END_REF] for 4-phase mixture. Here is a brief remind of these basic principles. Density ρ, weight content T , volume V and real part of the permittivity of each component are depicted in the Fig.3(a). The indexes b, f, a, and air refer respectively to binder, filler, aggregate and air. All densities and weight contents are provided by the road manufacturer for on site control, except the air void content that determines the mechanical objective of the road construction. In Europe, aggregates and filler densities are measured according to the NF-EN 18545 standard.The bitumen density is measured according to the NF-EN 15326+A1 standard. The HMA formulation is controlled (I) in laboratory, every year by an independent organism, and (ii) on site, by cores sampling every time a road is manufactured. Permittivity of each component are measured in laboratory. The related critical parameter to assess on newly-paved roads are (i) the density of the mix including air (namely bulk specific gravity) ρ HMAS or (ii) the compactness C 2 of the HMA, related to each other by the following relation:

Figure 3

 3 Figure 3: a) HMA representation and distribution of its components, and b), dielectric 4-phase mixture where the matrix permittivity m can be either aggregate or binder.

  (a) HMA model behaviour from 100% to 0% air void content.(b) HMA model behaviour in the investigation area.

Figure 4 :

 4 Figure 4: Bulk specific gravity/compactness as a function of the real part of HMA permittivity (87% basalt slab): the red lines refer to the Power Law model, the blue lines to the Unified Mixing Rule with aggregate matrix and the green lines to Unified Mixing Rule with binder matrix.

Figure 5 :

 5 Figure 5: Assessed compactness on the 87% slab by means of SFR system obtained with a) the reference gamma bench system, b) the CRIM model (PL α = 1/2), c) the Rayleigh model with aggregate matrix ([UMR ν = 0 MA), d) the Rayleigh model with binder matrix ([UMR ν = 0 MB) and e) the Böttcher model with aggregate and binder matrix (UMR ν = 2 MA and MB).

Figure 6 :

 6 Figure 6: Assessed compactness on the 95% slab by means of SFR system obtained with a) the reference gamma bench system, b) the CRIM model (PL α = 1/2), c) the Rayleigh model with aggregate matrix ([UMR ν = 0 MA), d) the Rayleigh model with binder matrix ([UMR ν = 0 MB) and e) the Böttcher model with aggregate and binder matrix (UMR ν = 2 MA and MB).

Table 3 :

 3 Mix design of the studied HMA slabs

	Aggregates	Basalt (0/10)	Basalt (0/10)
	Designed compaction	0.87	0.95
	T a	0.927	0.932
	T b	0.056	0.058
	T f	0.017	0.01
	a	(7.71±0.20)-j(0.13±0.08)	(7.71±0.20)-j(0.13±0.08)
	b	(2.50±0.02)-j(0.003±0.002) (2.50±0.02)-j(0.003±0.002)
	f	(7.2±0.02)-j(0.01±0.005)	(7.2±0.02)-j(0.01±0.005)
	ρ a (g/cm 3 )	2.890	2.890
	ρ b (g/cm 3 )	1.030	1.030
	ρ f (g/cm 3 )	2.635	2.635

Table 4 :

 4 Compactness (C) and density (ρ) results on HMA slabs designed for an air void content of 13% and 5%. In red color, the reference gamma bench (γ) measurement. Subsequent columns: PL model for α = 1/2 and UMR models for ν = 0 (Rayleigh) and ν = 2 (Böttcher), both for aggregate matrix (MA) and binder matrix (MB).

	Slabs Gamma bench	PL	UMR	UMR	UMR	UMR
		γ	α = 1/2	ν = 0 MA	ν = 0 MB	ν = 2 MA	ν = 2 MB
	C 87%	0.872±0.027	0.883±0.020 0.875±0.022 0.929±0.017 0.888±0.019 0.888±0.019
	ρ 87%	2.285±0.071	2.315±0.053 2.292±0.057 2.434±0.043 2.325±0.051 2.325±0.051
	C 95%	0.926±0.014	0.919±0.026 0.913±0.028 0.958±0.021 0.922±0.025 0.922±0.025
	ρ 95%	2.421±0.036	2.398±0.041 2.386±0.073 2.504±0.054 2.410±0.066 2.410±0.066

For rocks, the relative density ρ/ρ water is named density with ρ water = 1g/cm 3 at 4 • C

Compactness can be expressed from 0 to 1 (unitless) or from 0 to 100 %.

ACKNOWLEDGEMENTS

A sincere thanks goes to the reviewers for their relevant comments to improve the manuscript. The authors would like to thank Cyril Ledun (Cerema) and Henri Mauduit (Université de Rouen) for their help to the permittivity measurements. They also express their gratitude to Aminata Diouf (INSA de Rouen) for her precious help. This study would not have been possible without the financial support of USIRF (Union des syndicats des industries routières franc ¸aises and Région Haute-Normandie). We are very grateful to the COST Action TU1208 "Civil Engineering Applications of Ground Penetrating Radar".