Oriane Dermy

Azim Roussanaly

Anne Boyer

Using Behavioral Primitives to Model Students' Digital Behavior

Keywords: Learning Analytics, Probabilistic Movement Primitives, Students' Behavioral Primitives

The rapid and intense development of distance learning in recent years has led to increasingly comprehensive solutions, recording students' activity in the form of learning traces. Adapted learning systems can exploit this data to analyze students' behavior and help them be more successful in their learning process. In this paper, we propose an approach that is new in data mining, which consists of representing the behavioral dynamics of a set of students through Behavioral Primitives. A Behavioral Primitive represents the temporal evolution of learning indicators to model the behavior of a group of students. The primitive then corresponds to a dynamic distribution, representing both the mean values and the variance of specific indicators over time. We experiment this method on a known, widely used, and open dataset (OULAD). Our results confirm the relevance of the proposed methodology to characterize students' behaviors intelligibly and visually while respecting each student's anonymity. Our work provides a powerful and explainable tool for educational actors. It allows them to analyze learners' behaviors and perform pedagogical actions.

Introduction

This paper reports a contribution to student modeling, as it describes a new approach to visualize learners' digital behavior, thanks to a dynamic representation of its evolution over time. It relies on what we call behavioral primitives to draw a realistic picture of the students' digital behavior.

The COVID-19 pandemic crisis has resulted in the intensive use of Virtual Learning Environments (VLE) in most educational institutions worldwide, leading to an increasing volume of digital traces of learners' activities. Learning Analytics (LA) provides tools to analyze and visualize this unprecedented collection of educational data, supporting a better understanding of the students' behaviors, abilities, and difficulties. A significant issue in LA deals with the release of easy-tounderstand and explainable decision support tools to teachers. Even if many approaches exist to represent dynamically and globally a set of educational data, they mainly remain hard to understand and interpret for a teacher. The challenge is there to answer the following research question: how to globally and dynamically represent a set of educational data to provide teachers with an easy-to-interpret synthetic vision of their learners?

We propose to address this global issue by designing a new approach. Using [START_REF] Treuillier | Identification of class-representative learner personas[END_REF], we first preprocess the data by 1) choosing three learning indicators that characterize learners' behavior on the VLE. These indicators are computed from the digital traces collected on the VLE. 2) taking homogeneous groups of students sharing similar behaviors according to these indicators. After this pre-processing, we define Behavioral Primitives, which could be viewed as a distribution of the evolution over time of all the selected indicators for a given group. The tool allows a global picture of digital behaviors: teachers can choose a global synthetic vision of the students population's behavior or a snapshot of a given student enabling the detection of abnormalities. To summarize, this paper proposes to contribute to the global research question by addressing a more specific research question: Do behavioral primitives allow a relevant representation of the dynamic behavior of a students group? Experiments based on the OULAD public dataset will illustrate the contributions of this paper.

The paper is organized as follows: the next section will give a global overview of our proposed approach to model learners' behavior. Section 3 starts with a description of the case study and synthesizes the main results obtained with the OULAD dataset, answering the two specific research questions. Before concluding and some perspectives on this work, we will show the different types of analysis that can be made from a teacher's perspective.

A new approach to model learners' behavior

We address our research question with a new approach, called Behavioral Primitives, which models the evolution over time of the digital behavior of a group of learners (defined by similar behavior). This model is an innovative representation of learning behavioral changes based on Probabilistic Movement Primitives [START_REF] Paraschos | Probabilistic movement primitives[END_REF]. They provide the dynamic of learners' behaviors over time. This model is a way to answer the question: "How does your learners' behavior change over time?"

We first detail this approach in the following subsection. Its potential contribution is then applied to a specific case study based on an open, well-known, and quality dataset, described in the next section.

We first give a quick overview of what exists in the literature to explain what motivated the choice of a dynamic representation of behaviors.

Related works

Existing Dynamical representation of learning behaviors. Since data mining (DM) is usually applied to big data, the results of mining techniques are sometimes complex to understand and interpret. This is why LA researchers' common preoccupation is finding ways to make the results of DM intelligible so that non-DM experts, such as teachers, can use them. Thus, the search for a visual representation of DM results has become a widely studied topic. In [START_REF] Daniel | Visual data mining techniques[END_REF], Keim et al. present different techniques to visualize the results of well-known DM algorithms.

Visualizations in LA usually allow improving the dashboards of students and teachers. Let us note that some of the visualization techniques are specific to e-education, as in [START_REF] Yoo | Student progress monitoring tool using treeview[END_REF], where Yoo et al. propose displaying a competence tree for each student. This tree represents the percentage of skills acquisition defined beforehand, and it is updated according to the student's skills acquisition over time.

Most of these visualizations are, however, static and the state of the art concerning dynamic visualizations, representing the evolution of the students' behavior over time, is more limited, even if some of them are widely used (such as the "Scatter and Column graphs," presented in [START_REF] Jin | Application of visual data mining in higher-education evaluation system[END_REF]). The most famous of these visualizations might be the "learnogramm", which represents the evolution of different data types from a student's logs over time. For example, in [START_REF] Ben-Zadok | Examining online learning processes based on log files analysis: A case study[END_REF], "learnograms" map the mined events in types of activities (simulation, game, test...) over time. In [START_REF] Hershkovitz | Developing a log-based motivation measuring tool[END_REF], "learnograms" represent the cumulative number of pages explored in different contexts to compare them. We can also note various types of time series [START_REF] Jarke | Cluster and calendar based visualization of time series data[END_REF], allowing to describe the online activity over time, where we precise the dates of exams to interpret the behavioral evolution of students. Usually, in student or teacher dashboards, the average of the students' marks (or other values) in a course is often presented on one graph over time. Another well-known type of dynamic visualization is histograms. In [START_REF] Shi | Vismooc: Visualizing video clickstream data from massive open online courses[END_REF], the authors give the number of students watching different types of videos over the past weeks using histograms. In [START_REF] Wang | A novel approach to compute pattern history for trend analysis[END_REF], an approach named "pattern histories", allows analyzing the temporal evolution of sequential pattern support (the support is the sequence number in which a pattern appears). Dermy et al. [START_REF] Oriane | A dynamic indicator to model students' digital behavior[END_REF] find the same history graphs for Time-Interval patterns.

However, in all these works, there is no representation of the dynamic of a students group over time. This is what we propose in this article, based on a new approach called behavioral primitives. To our knowledge, there is no existing research work in the field of education based on such an approach. The closest existing visualization to our approach is presented in [START_REF] Shi | Vismooc: Visualizing video clickstream data from massive open online courses[END_REF]: this visualization is called "Event graphs". The Event Graphs correspond to the maximum number of clicks the students make over time. This click number is computed for different types of videos (reading, assignment, experiment...). In the description of their visualization, the authors talk about "distribution", but in reality, it only represents the maximum student's click number at each timestamp. So it isn't a distribution modeling of the global behavior of the students. This is precisely why we propose to use behavior primitives, allowing to represent the behavioral dynamics of students group over time.

Behavioral Primitives We assume in this paper that, even if students' behavior evolves over the course, there are what we call here "Behavioral Primitives". A Behavioral Primitive corresponds to an elementary representation of the behavior of a group of students over time, meaning that the primitive encapsulates the trends (mean and variance) of the group of students' behavior over time without representing individual, specific, or isolated behaviors. Therefore, we are looking for a method to a) describe students' behavior over time and b) extract behavioral primitives that make students' behavioral changes understandable. In response to a), thanks to online student traces, we can represent the values of different types of data (number of clicks, grades, delay...) over time. We can then model this data as a trajectory, a movement. We hypothesize that discovering "behavioral primitives" is similar to finding "movement primitives", which is a vast research topic initially inspired by nature [START_REF] Simon F Giszter | Convergent force fields organized in the frog's spinal cord[END_REF]. Artificial movement primitives generate the sequence of points of a trajectory from a parametric temporal model or by a parametric dynamic system. Many methods have been proposed in the last decades, such as Gaussian Mixture Models [START_REF] Sylvain Calinon | A task-parameterized probabilistic model with minimal intervention control[END_REF][START_REF] Mohammad | Learning stable nonlinear dynamical systems with gaussian mixture models[END_REF], Dynamical Movement Primitives [START_REF] Auke | Dynamical movement primitives: Learning attractor models for motor behaviors[END_REF], Probabilistic Representation for Dynamic Movement Primitives [START_REF] Meier | A probabilistic representation for dynamic movement primitives[END_REF], or Probabilistic Movement Primitives [START_REF] Alexandros | Probabilistic movement primitives[END_REF]. A summary of these models can be found in [START_REF] Oriane | Movement Prediction: From Simple Gesture to Whole-Body Movements[END_REF]. Movement primitives are mainly used in human and robotic movement modeling. The most used of these methods is certainly the Dynamic Movement Primitives (DMPs) [START_REF] Auke | Dynamical movement primitives: Learning attractor models for motor behaviors[END_REF][START_REF] Schaal | Dynamic movement primitives-a framework for motor control in humans and humanoid robotics[END_REF][START_REF] Meier | A probabilistic representation for dynamic movement primitives[END_REF] method. However, this method requires a priori knowledge of the trajectory's final position (a position included in the trajectory model). This does not match our case, where the final student's behavior is unknown. We, therefore, use ProMPs, which are further detailed in the following subsection. We selected this method because it has been shown that it allows the recognition of a movement's intention [START_REF] Oriane | Movement Prediction: From Simple Gesture to Whole-Body Movements[END_REF], which we can translate in our context as the recognition of a student's behavior. By capturing the average of students' behavioral dynamics (which is usually done) and all the variability of these behaviors over time, we provide teachers with a much more complete view of students' learning. This full view allows them to adapt their teaching, given the average students' behavior and the whole class. Now, we will show how these behavioral primitives (ProMPs) can be implemented. What are ProMPs? Consider charts of newborn weights. They represent the average and percentiles of the classical child weights over the years to have a model of the characteristic weight of newborns. ProMPs do the same to represent the students' behavior (determined by indicators, c.f., Sec.3.2) over time. An excellent tutorial [START_REF] Ajmera | Probabilistic movement primitives[END_REF] allows readers to be more comfortable with ProMPs. Mathematically, it is a Bayesian parametric model constructed from recorded trajectories corresponding to the same movement. In our case, a "movement" corresponds to a group of students' behavioral (indicators value) evolution. This model allows learning a trajectory distribution over trajectories representing this movement. A trajectory corresponds to any evolving value(s) of one or more parameters over time (i.e., the trajectory can be multi-dimensional). For example, trajectories in Figure 2a represent the evolution of the number of clicks over time for 54 students (all students of a specific group). Note that this figure illustrates the model for scientists but is not intended for non-experts. Future results will be easier to interpret. A trajectory is modeled as:

The Probabilistic Movement Primitives Model

Ξ = [ξ(1), . . . , ξ(T )] = Φ [1:T ] ω + Ξ , (1)
where:

• Ξ is the discretized trajectory to model (here the click number over exam dates).

• ξ(t) is a vector containing the multidimensional values (here the clicks number) of Ξ at each timestamp t.

• T is the last timestamp (here the 240 th day, which is the last exam date).

• Φ t is a matrix of M Radial Basis Functions (RBFs), evaluated at time t : Φ t = [φ 1 (t), φ 2 (t), . . . , φ M (t)]. These RBFs are scattered across time.

(In Figure 2b, M=11 and the RBFs are Gaussians:

G(i) = e (t-c i ) 2 h , with c i = i-1 M , h = 1 M 2 , ∀i ∈ [1 : M ]).
• ω ∈ R X is a time-independent parameter vector weighting the Φ matrix; X = |ξ(t)| × M the dimension of the trajectory (In Figure 2c, X = 1 × 11: 11 weights model each trajectory, presented by 54 blue dotted lines).

• Ξ ∼ N (0, β) is the imprecision of the trajectory model. For each observed trajectory Ξ i , we compute the ω i parameter vector that minimizes the error between ξ i (t) and Φ t ω i + ξ , by using the Regularized Least Mean Square algorithm (Figure 2c). With n observed trajectories (here, n = 54) of a same movement, we then have a set of [ω 1 , . . . , ω n ] parameters upon which a normal distribution is computed:

p(ω) ∼ N (µ ω , σ 2 ω ), ( 2
) with µ ω = 1 n n i=1 ω i , and σ 2 ω = 1 n-1 n i=1 (ω i -µ ω ) (ω i -µ ω ).
We thus have a ProMP represented by: Ξ = Φω + Ξ . (Figure 2d presents the result of our example, where the bold line represents the average of the movement distribution, and the shaded area, the 97.5 percentile, referred as variance throughout the paper). This method gives a visual, global (with average and variance), and easy-to-interpret representation of a movement from a set of trajectories.

3 Illustration with a case study

The Open University Learning Analytics Dataset (OULAD)

In this paper, we use pre-processed data and 4 sub-datasets processed in [START_REF] Treuillier | Identification of class-representative learner personas[END_REF]. The open-access OULAD dataset [START_REF] Kuzilek | Open University Learning Analytics dataset[END_REF]

Selection of learning indicators and students' groups

Student interactions on the learning platform are associated with 20 different types of activities. Among all of these, the authors noticed that forumng, oucontent, homepage, and subpage activities are the most important predictors of engagement [START_REF] Hussain | Student Engagement Predictions in an e-Learning System and Their Impact on Student Course Assessment Scores[END_REF]. Thus, the study focused on these 4 types of educational material. Then, the authors defined different learning indicators, which correspond to relevant parameters, providing useful information about the learning quality. In this paper, we focus on 3 of their learning indicators: engagement [START_REF] Hussain | Student Engagement Predictions in an e-Learning System and Their Impact on Student Course Assessment Scores[END_REF], performance [START_REF] Arnold | Course signals at Purdue: using learning analytics to increase student success[END_REF], and reactivity [START_REF] Shirvani Boroujeni | How to Quantify Student's Regularity?[END_REF],

but teachers could choose other ones, according to their needs. In particular, we selected the most variable indicator (engagement) proposed in [START_REF] Treuillier | Identification of class-representative learner personas[END_REF], to show the ProMPs method's ability to represent the most complex data. Responsiveness to Delay between submission course-related events date and deadline Tab. 1: Definition of each learning indicator and associated features selected in OULAD.

were calculated over the total duration of the module during each exam. Knowing that there were 14 exams during this module, each indicator is calculated 14 times for each student. These 14 time-stamped values (date of the 14 exams) can be seen as a one-dimensional movement of student behavior. The "engagement" indicator is calculated over each period between two exams. The values of each indicator correspond to a dimension of the student's "behavioral movement". We thus have a multidimensional trajectory, representing a student's behavior according to all the indicators.

For each sub-dataset (see Section 3.1), we took the homogeneous subsets of students presented in [START_REF] Treuillier | Identification of class-representative learner personas[END_REF], to differentiate one group representing the larger proportion of the students (i.e., we delete outliers). Thus we have: distinction (53 students), pass (451 students), fail (357 students), and withdrawn (427 students).

The following subsection then describes how these subsets are used to model behavioral primitives.

Behavioral Primitives Application to model Student's Behavioral Changes

In the previous section, we retrieved, for each students group, a set of multidimensional trajectories. Each trajectory corresponds to the evolution of a student's behavior over time, where each dimension represents the evolution of a learning indicator over time. We now use ProMPs to compute a probabilistic distribution on each set of trajectories. Let us recall that the ProMP model contains different parameters to be tuned (c.f., Section 2.2). First of all, the number of Gaussians used to model the motion (c.f., Fig. 2), Gaussians that are spread over time (14 timestamped exams): the more Gaussians there are, the more accurate the motion is, the fewer Gaussians there are, the smoother the movement is. In our case, we use 5 Gaussians to represent the dimensions of the movement representing the performance and reactivity indicators, and 11 Gaussians to represent the dimension representing the engagement indicator.

The following sub-sections will give pieces of evidence to the second scientific question: "Do behavioral primitives allow a relevant representation of the dynamic behavior of a students group?".

Multidimensional ProMPs per Students' Group

First, we want to verify if the ProMPs allow representing in an interpretive way the behavioral tendencies of each prime students group over time. We represent the different Note: the reactivity indicator is calculated from the exams submission delay. Thus, the higher the value is, the lower student's reactivity is.

students' groups dynamically using temporal distributions (ProMPs), then we analyze the information obtained through these distributions. Figure 3 shows these results. Each sub-figure represents a students group distribution.

Explanation. Let's focus on the 1 st sub-figure (Fig. 3a). Each row corresponds to an indicator (Engagement, Performance, and Reactivity). The 1 st column corresponds to the observed data, and the 2 nd to the ProMP generated from these data. In this 2 nd column, the bold line represents the average of what students do (average value of the indicator) and, in shaded, the variance. On the x-axis, each stick corresponds to an exam number.

Distributions succeed in representing the global tendency of students' dynamical behavior. We can notice that in each sub-figure, the engagement indicator (1 st line), real data (1 st column) vary a lot from one exam (e.g., one x-axis stick represents one exam) to another. It is then difficult for teachers to interpret the students' behavioral evolution. The model (2 nd column) shows students' behavioral trends by smoothing data. This smoothing is caused by having fewer Gaussian distributions than there are notes. By putting 14 Gaussians, we obtain a more precise model, no longer representing the general trend (c.f., Picture in [START_REF] Dermy | Picture of a precise ProMP for the distinction students' group[END_REF]).

Analysis of the engagement indicator: the same tendency at different scales. By looking at the graphs representing the engagement indicator, we can notice that the ordinate axes do not all have the same scale. This implies that student groups are not equally engaged. However, we can see the same global shape over time, where at the beginning, students are the more involved, maybe because of their curiosity at the first learning session. Then, their engagement decreases still the 4 th exam, followed by a more stable period of engagement until the end of the course. A visible peak of activity appears during the period before the 9th exam, even if this peak is very small for the withdrawn students' group (c) and can be detected visually only thanks to the variance of the behavioral primitive.

Analysis of performance indicator. Since this 9th exam, the "performance" indicator decreases a little, even for the distinction (a) and pass (b) groups. This might be correlated to the "peak of activity" of the engagement indicator just before this exam. By analyzing these results, teachers could understand why. It could be because the subject of this exam requires more engagement from students because it deals with a complex learning skill.

The variance of the "performance" ProMP is less significant for students in the distinction group than those who succeed without distinction. In all cases, the averages remain high. For those who fail and withdraw, the average tends towards a bad average (around 0, and rapidly for those who withdraw).

Identification of dynamical behaviors. From these distributions, we identify two types of dynamical behavior. The first one contains the distinction (a) and pass (b) students' groups, where the performance evolution is stable and where students react quickly (with a slight delay) during the whole time. The second one contains the withdrawn (c) and fail (d) students' groups, where the performance declines to 0 more or less quickly and where the engagement is low.

Correlation between indicators. The distributions highlight that performance and reactivity are two correlated indicators: we can see that the distribution of these two indicators is symmetrical for each students group. Indeed, when means tend to 100, corresponding delays tend to 0 or even less. When the average tends to 0, related delays tend to 250, the max number of days to return the work.

About the Distinction group variance. Since the number of students in the distinction group (53) is 8.5 times smaller than the number of students in the pass group (451), one could hypothesize that it is the number of data that causes the difference in variance on the Performance ProMP's dimension (Figures 3a and3b). However, the variances of the Engagement dimension are similar, which invalidates the hypothesis. In fact, with very few students (few learning trajectories), we can model the "classical variance" of a homogeneous students' group behavior.

Next step. In these analyses, we highlight that ProMPs smooth the data variation. For some case studies, this smoothing can limit the model. Thus, a question arises. Can we adapt the model to modify the smoothness of the primitive according to the given application or the user's choice? The following sub-section answers this question by tuning the Gaussian number of this model.

Impact of the number of Gaussians on the representation of the behavioral dynamics

Second, we want to explore how the model can be tuned to represent more or less accurately the trajectory distribution. To do so, we analyze how the number of Gaussians used to represent the data impacts the final ProMP model. To clarify the presentation of the results, we focus here on the performance and reactivity indicators because the variations of these indicators' trajectories are similar and require the same number of Gaussian distributions.

We test four different models, using 3, 5, 7, and 8 Gaussian distributions, spread over time (14 timestamps for each mark). Each column illustrates such a model. The first line represents the The results show that the more Gaussian distributions there are, the more accurate the model is.

Average analysis. On each graph, the average, represented by the bold line in the middle of the distributions, becomes more and more precise when Gaussian distributions are added to the model. This is especially visible in the case of the "reactivity" indicator, where we can see that in the first column (model composed of 3 Gaussians), the model poorly represents the final variation (where students tend to submit their last assignment early). From 5 Gaussian distributions, we see that the average can describe this phenomenon.

Variance analysis. Concerning the variance, we notice that the more Gaussians there are, the more the model represents the whole variance of trajectories. However, it should be noted that the more Gaussians we add to the model, the more the computational complexity increases (expensive matrix calculation).

Limitation of the model. We can notice that the variance is symmetrical around the average, which is due to the Gaussian distribution choice. Thus, these behavioral primitives better represent groups of students whose behavioral variation among students follows a normal distribution. This is not totally the case with the Reactivity dimension of this behavioral primitive. Indeed, distinction students submit their exams on the expected days. Hence, the average curve evolves around "0" days, and some of them submit their exams in advance, so there is a variance of around 10 days below the average. Since we use Gaussian distribution, the variance also appears above the average in the behavioral primitive.

Next step. The next issue is: "can we easily compare the dynamic of the different groups?". Section 3.3.3 answer this question by overlaying distributions.

Comparison between Students' Groups

Third, we check if ProMPs can capture the behavioral differences over time between students groups. To do so, we compare the behavioral dynamics of the different students' groups by overlaying the corresponding ProMPs.

Highlight the identification of dynamical behaviors. The two tendencies presented in sub-section 3.3.1 are highlighted. For the first tendency (pass with or without distinction), the score averages remain steady and high (between 60 and 80 for those who pass without distinction and between 80 and 90 for the others), while delay averages remain close to 0. The most significant difference between these two kinds of distributions concerns the variance. The scores and delays of students who pass without distinction are more variable than those who have distinction. For the second tendency (fail with or without withdraw), the score averages fall to 0. The average of those who fail is initially higher than those who withdraw. Symmetrically, we see that their reactivity averages increase up to 240 days, which shows that they take longer and longer to submit their work until they do not submit it at all. Analysis of the engagement indicator. The behavioral primitives of those who fail and those who succeed are less distinct. Still, if we look closely, we can see that the averages of those who fail tend rapidly towards "0", as early as the 4 th exam. In contrast, the students who succeed have a similar average and variance, which remains globally the same despite the fluctuations (between 0 and 500 clicks). It should be noted that the variance of those who fail is significant and exceeds the average of those who succeed, showing that there are "fail" students that are as involved in the courses as those who succeed. This result can be seen again by looking closely at Figure 2d. In particular, in the period before the 9 th exam, although the average of those who fail is close to 0, there are students who fail who, like those who pass, have made an extra effort of engagement during this period. Finally, we want to detect the evolution of a particular student's behavior. We thus provide a tool for teachers to detect the behavioral specificity of a given student so that they can act positively to help her/him in his/her learning. Figure 6 presents in dotted lines the student's indicators evolution over time. We can verify visually that each dotted line is included in the behavioral primitive with the same color (e.g., (a) the pink dotted line is included in the pink "Fail" distribution; (b) the green dotted line is included in the green "Pass" distribution). Note that there are big dots at each exam date (each x-axis tick label). Their color corresponds to the more similar behavioral primitive at that timestamp. For example, the first big dots in (a) are green because the student has initially a behavior similar to students who pass.

Consistent behaviors. The indicators of the student who succeeds with distinction and the student who withdraws follow really accurately the corresponding behavioral primitives.

Inconsistent behaviors. Regarding the student who passed (Figure 6b), (s)he comes close at given exam periods to the distinction behavioral primitive. The light blue areas represent these periods: during the period before the 9 th exam, the student has an engagement similar to distinction students. As a result, we can see that his/her performance is similar to distinction-student regarding exams 9, 10, and 11. Perhaps, if his/her teacher had sent him/her encouraging messages during this period, the student would have been motivated to continue his/her efforts and could have succeeded with distinction.

Regarding the student who failed, her/his indicators values are also well within the corresponding students' group distribution. However, at the beginning of the year, her/his behavior could also match someone who succeeded. In the areas shown in light blue (upstream of the 8 th and 10 th exams), we see a distribution shifting from the "pass" students' group to the "fail" one. If teachers detected these periods early enough, perhaps they could have carried out a particular action to give her/him the possibility and the desire to succeed. Just after these light blue areas, the student seems to have dropped out: (s)he no longer submits her/his exams.

Conclusions and perspectives

In [START_REF] Treuillier | Identification of class-representative learner personas[END_REF], learning indicators and homogeneous students' groups have been defined, to create narrative description of these groups. These learning indicators model students' digital behaviors statically. The goal of this paper is to extend this work, by providing teachers with an overview of students' elementary behaviors over time. To do so, we looked for a visual and easy-to-understand representation of the behavioral dynamics of a group of students. This behavioral dynamic must be representative of the group, without modeling individual behaviors. To our knowledge, there is no such approach. Thus, we created one, named "Behavioral Primitive", which model the dynamic distribution of students' behavior, thanks to the ProMPs method (c.f., Sec.2.2). This method gives the average and variance of all learning indicators over time, based on Gaussian distributions.

The experiments show that Behavioral Primitives succeed in representing the dynamics of students' behavior (Sec. 3.3.1); with more or less precision depending on the needs (Sec. 3.3.2); but also to compare students' groups (Sec. 3.3.3); or even to compare a specific student with the students' groups (Sec. 3.3.4). These results show that teachers can analysis their students' behavior, enabling them to customize their course.

This new approach opens up a vast field of research, thanks to the probabilistic aspect of the model. Notably, we will attempt to statistically estimate the future student's behavior (including their grades), given the students of the previous year behaviors. Besides, ProMPs will also allows the generation of trustworthy and artificial data according to the dynamics of different students' groups. Finally, even if we have collected informally positive feedback about the visualization from teachers, we intend to conduct a study in real-life situation.

Fig. 1 :

 1 Fig. 1: From Data to a dynamic representation of students' groups' behaviors.

Fig. 2 :

 2 Fig.2: Explanation graphs of the ProMP model. In this toy sample, we represent the dynamic evolution of the click number over time (here on 20 timestamps, where the last one is at the 240 day) for the group of 54 students that succeed in their exams.

  Fail students' group.

Fig. 3 :

 3 Fig.3: Data and ProMP for each students group. Note: the reactivity indicator is calculated from the exams submission delay. Thus, the higher the value is, the lower student's reactivity is.

Fig. 4 :

 4 Fig. 4: How the number of Gaussians impacts the model of the movement.

Fig. 5 :

 5 Fig. 5: Overlay of ProMPs to compare students' Groups.

3. 3 . 4

 34 Representation of specific student behavior for each students group.

Fig. 6 :

 6 Fig. 6: Overlaying dynamics of two students' behavior (dot lines) on the students' groups distributions.

  Table 1 gives details about these 3 indicators. These three indicators

	Learning	Definition	Computed features
	indicator
	Engagement Students' activity	Number of clicks
		on the VLE
	Performance Students' learning	Scores obtained in the
		outcomes	assessments [0-100]
	Reactivity


Dirk Erhard

Julien Poisat

STRONG LARGE DEVIATION PRINCIPLES FOR PAIR EMPIRICAL MEASURES OF RANDOM WALKS IN THE MUKHERJEE-VARADHAN TOPOLOGY

. We also adapt our result to certain rescaled random walks.

Let X = (X n ) n∈N 0 be a Markov chain on a Polish space Σ with a Feller transition probability π which we assume to have a density p(x, y) with respect to a reference measure λ(dy). In the seventies, Donsker and Varadhan [START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time[END_REF](II)] showed that the empirical δ X k satisfies a large deviation principle in the space of probability measures equipped with the weak topology, when Σ is compact and p(x, y) is uniformly bounded above and below.

If the state space Σ is not compact (e.g. Σ = R d equipped with the Euclidian distance), the upper bound holds for compact sets rather than closed sets, while the lower bound still holds for any open set under the following assumption: for all x ∈ Σ, for all A ⊂ Σ such that λ(A) > 0, there exists k ∈ N such that π k (x, A) > 0, see [START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time[END_REF]Corollary 3.4 and Equation (4. [START_REF] Bolthausen | Localization of a two-dimensional random walk with an attractive path interaction[END_REF]]. In some cases, this weak large deviation principle may be upgraded to a standard (or strong) one, recovering the large deviation upper bound for all closed sets. However such cases (see e.g. the uniformity assumption in [4, Corollary 6.5.10]) fail to include many natural examples. In applications, the lack of compactness may be dealt with by adding a confining drift to the Markov chain (or diffusion) [START_REF] Donsker | Asymptotics for the polaron[END_REF] or folding on a large torus [START_REF] Bolthausen | Localization of a two-dimensional random walk with an attractive path interaction[END_REF][START_REF] Bolthausen | On self-attracting d-dimensional random walks[END_REF][START_REF] Donsker | On the number of distinct sites visited by a random walk[END_REF]. Quite recently, Mukherjee and Varadhan [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] proposed a new approach in which they embed the space of probability measures on R d into a larger space equipped with a certain topology that makes it a compact metric space. Under this new topology, they were able to prove a strong large deviation principle for the empirical measure of Brownian motion [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Theorem 4.1], which was then successfully applied to the so-called polaron problem [START_REF] Bolthausen | Mean-field interaction of Brownian occupation measures II: A rigorous construction of the Pekar process[END_REF][START_REF] König | Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional[END_REF][START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF].

In this paper we adapt and extend the work of Mukherjee and Varadhan in order to prove a strong large deviation principle for the pair empirical measure of the Markov chain (X n ) n∈N 0 , defined by (1.2)

L (2) n = 1 n n k=1 δ (X k ,X k+1 ) ,
in the case Σ = R d . Our work is motivated by the application of this LDP to the so-called Swiss cheese problem, that is the (downward) large deviation for the volume of a Wiener sausage in R d (d ≥ 2), by van den Berg, Bolthausen and den Hollander [START_REF] Van Den Berg | Moderate deviations for the volume of the Wiener sausage[END_REF]. In [START_REF] Van Den Berg | Moderate deviations for the volume of the Wiener sausage[END_REF] (as well in the discrete random walk counterpart studied a bit later by Phetpradap [START_REF]Intersections of random walks[END_REF] in his Ph.D thesis) the authors used the aformentionned folding procedure on a large torus to deal with the lack of compactness of the state space. In a future paper, we plan to apply the strong LDP for the pair empirical measure to the Swiss cheese problem in order to obtain the so-called tube property. The latter basically means that the empirical measure introduced in [START_REF] Van Den Berg | Moderate deviations for the volume of the Wiener sausage[END_REF] is close to the minimizer of the variational problem appearing in [START_REF] Van Den Berg | Moderate deviations for the volume of the Wiener sausage[END_REF]. The present paper is organized as follows. In Section 2 we introduce the relevant notations and topology. Although there is a lot in common with [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF], let us stress that this is not the Mukherjee-Varadhan topology applied to the product space R d × R d instead of R d , see also Remark 2.1 below. Our main result is stated in Section 3, see Theorem 3.1. Lower semi-continuity of the rate function is proven in Section 4. The lower and upper bounds of the LDP are proven in Sections 5 and 6, respectively, in which we use the well-known fact that (X n , X n+1 ) is itself a Markov chain. Finally we adapt our result to the case of certain rescaled random walks in Section 7

We refer the reader to [START_REF] Dembo | of Stochastic Modelling and Applied Probability[END_REF][START_REF] Frank Den Hollander | of Fields Institute Monographs[END_REF][START_REF] Deuschel | Large deviations[END_REF][START_REF] Varadhan | Large deviations[END_REF] for an account on Large Deviation Theory and [START_REF] Varadhan | The role of topology in large deviations[END_REF] for the role of topology in this theory.

Topology on the space of probability measures modulo shifts

Let M

(2) 1

:= M 1 (R d × R d ) be the space of probability measures on R d × R d and M (2) ≤1 := M ≤1 (R d × R d )
be the space of sub-probability measures on R d × R d . We consider the action of the shifts θ x,x for x ∈ R d , defined by: (2.1)

R d ×R d f (u, v)(θ x,x ν)(du, dv) = R d ×R d f (u + x, v + x)ν(du, dv) for all continuous bounded functions f : R d × R d → R and ν ∈ M (2)
≤1 . We shall denote by M

(2)

1 (resp. M (2) ≤1 ) the space of equivalence classes of M (2) 1 (resp. M (2)
≤1 ) under the collection of shifts θ x,x . For k ≥ 2, we define F k as the space of continuous functions

f : (R d ) k → R that are translation invariant, i.e. (2.2) f (x 1 + x, . . . , x k + x) = f (x 1 , . . . , x k ), ∀x, x 1 , . . . , x k ∈ R d ,
and vanishing at infinity, in the sense that

(2.3) lim max i̸ =j |x i -x j |→∞ f (x 1 , . . . , x k ) = 0. For k ≥ 1, f ∈ F 2k and α ∈ M (2)
≤1 , we write

(2.4) Λ(f, α) := f (u 1 , v 1 , . . . , u k , v k ) 1≤i≤k α(du i , dv i ),
which only depends on the orbit α.

Remark 2.1. Note that the space M

(2)

1 defined here is different from M 1 (R 2d ) defined in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]. Indeed, in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] the shifts are with respect to all directions in R 2d whereas here they are only with respect to all directions of the form (x, x) ∈ R 2d . This is very natural in view of our application to the pair empirical measure.

2.1.

Vague and weak convergence. In this section we give a slight generalisation of [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Lemma 2.2] which will be useful for later purposes. Lemma 2.2. Consider a sequence (µ n ) n∈N 0 of sub-probability measures in R d that converges vaguely to some sub-probability measure α. Then we can write µ n = α n + β n for every n ∈ N 0 , where (α n ) n∈N 0 converges weakly to α and (β n ) n∈N 0 converges vaguely to zero. Moreover, α n and β n can be chosen such that they have disjoint supports.

Proof of Lemma 2.2. The only part that was not explicitly mentioned in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] is that α n and β n can be chosen to have disjoint supports. However, this is a direct consequence of the proof in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]. Indeed, adopting their notation they defined α n as the restriction of µ n to B(0, R n ) and β n as the restriction of µ n to B(0, R n ) c , where ∥ • ∥ 2 denotes the Euclidean norm in R d and (R n ) n∈N 0 is an increasing sequence converging to infinity satisfying

(2.5) µ n (B(0, R n )) ≤ α(R d ) + 1 R n .
The claim therefore follows. ■ 2.2. Widely separated sequences. We say that two sequences (α n ) and (β n ) in M

(2) ≤1

are widely separated if, for some positive function V in F 4 ,

(2.6) lim

n→∞ (R d ) 4 V (u 1 , v 1 , u 2 , v 2 )α n (du 1 , dv 1 )β n (du 2 , dv 2 ) = 0.
The following lemma, which mimicks [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Lemma 2.4], lists the most important properties of widely separated sequences of sub-probability measures.

Lemma 2.3. Let (α n ) and (β n ) be two widely separated sequences in M

≤1 . Then, (1) For every W ∈ F 4 ,

(2.7) lim n→∞ W (u 1 , v 1 , u 2 , v 2 )α n (du 1 , dv 1 ) β n (du 2 , dv 2 ) = 0 .
(2) For every k ≥ 2 and every f ∈ F 2k ,

(2.8) lim n→∞ |Λ(f, α n + β n ) -Λ(f, α n ) -Λ(f, β n )| = 0.
Proof of Lemma 2.3. Let W ∈ F 4 be arbitrary and let V ∈ F 4 be the positive function from (2.6). Then, since V (0, v 1 , u 2 , v 2 ) is bounded from below by a positive constant on compact sets, for every ε > 0 there exists a constant C ε such that for any

v 1 , u 2 , v 2 ∈ R d , (2.9) |W (0, v 1 , u 2 , v 2 )| ≤ C ε V (0, v 1 , u 2 , v 2 ) + ε . Thus (2.10) lim sup n→∞ |W (u 1 , v 1 , u 2 , v 2 )|α n (du 1 , dv 1 )β n (du 2 , dv 2 ) ≤ C ε lim sup n→∞ V (0, v 1 -u 1 , u 2 -u 1 , v 2 -u 1 )α n (du 1 , dv 1 )β n (du 2 , dv 2 ) + ε = ε ,
and (1) follows. To see that (2) holds as well, we first note that the case k = 2 is a direct consequence of the first part of the lemma. The case k ≥ 3 follows easily: w.l.o.g, any cross-term in the expansion of Λ(f, α n + β n ) may be written as

(2.11) f (u 1 , v 1 , u 2 , v 2 , . . . , u k , v k )α n (u 1 , v 1 )β n (u 2 , v 2 ) 3≤i≤k γ n,i (u i , v i ),
where γ n,i is either α n or β n . Using translation invariance and repeating the argument used at the beginning of the proof, we see that for every ε > 0, there exists

C ε > 0 such that for all v 1 , u 2 , v 2 , . . . , u k , v k ∈ R d , (2.12) |f (0, v 1 , u 2 , v 2 , . . . , u k , v k )| ≤ C ε V (0, v 1 , u 2 , v 2 ) + ε,
which allows to conclude. ■

Here is a sufficient condition for two sequences of measures to be widely separated.

Lemma 2.4. Let (α n ) and (β n ) be two sequences in M

≤1 . If (α n ) is tight and (β n ) converges vaguely to zero, then they are widely separated.

Proof of Lemma 2.4. Let V ∈ F 4 and ε > 0. By tightness of (α n ) and boundedness of V , there exists M > 0 such that for all n ≥ 1, (2.13)

V (u 1 , v 1 , u 2 , v 2 )α n (du 1 , dv 1 )β n (du 2 , dv 2 ) ≤ |u 1 |,|v 1 |≤M (. . .) + ε∥V ∥ ∞ .
Here, (. . .) stands of course for V (u 1 , v 1 , u 2 , v 2 )α n (du 1 , dv 1 )β n (du 2 , dv 2 ). We further split the integral on the right-hand side as (2.14)

|u 1 |,|v 1 |≤M |u 2 |,|v 2 |≤2M (. . .) + |u 1 |,|v 1 |≤M |u 2 |,|v 2 |>2M (. . .) .
We claim that we can make the second term smaller than ε by choosing M even larger if necessary. Indeed, since |u 2 -u 1 | ≥ M on the domain of integration and V is vanishing at infinity the claim follows. As for the first term, it goes to zero as n goes to infinity since V is bounded and (β n ) converges vaguely to zero. ■ 2.3. Totally disintegrating sequences. We say that a sequence (µ n ) in M

≤1 is totally disintegrating if, for any r > 0, (2.15) lim

n→∞ sup x∈R d µ n (B((x, x), r)) = 0.
From now on and unless stated otherwise the balls are defined with respect to the ℓ ∞norm. Following [12, Lemma 2.3], we obtain the following result.

Lemma 2.5. The sequence

(µ n ) in M (2)
≤1 is totally disintegrating iff one of the following equivalent statements holds:

(1) There exists a positive V ∈ F 4 such that lim n→∞ Λ(V, µ n ) = 0.

(2) For any V ∈ F 4 , lim n→∞ sup x∈R d V (0, x, u, v)µ n (du, dv).

(3) For any

V ∈ F 4 , lim n→∞ Λ(V, µ n ) = 0.
Proof of Lemma 2.5. We prove that (1) ⇒ (2.15) ⇒ (2) ⇒ (3). Clearly, (3) implies [START_REF] Bolthausen | Localization of a two-dimensional random walk with an attractive path interaction[END_REF].

(i) Let us prove that (1) implies (2.15). Letting

(2.16) δ := min a,b,c∈B(0,2r) V (0, a, b, c) > 0,
we get (2.17)

V (u 1 , v 1 , u 2 , v 2 )µ n (du 1 , dv 1 )µ n (du 2 , dv 2 ) = V (0, v 1 -u 1 , u 2 -u 1 , v 2 -u 1 )µ n (du 1 , dv 1 )µ n (du 2 , dv 2 ) ≥ δ v 1 ,u 2 ,v 2 ∈B(u 1 ,2r) µ n (du 1 , dv 1 )µ n (du 2 , dv 2 ) ≥ δ sup x∈R d µ n (B((x, x), r)) 2 .
(ii) From (2.15) to (2). Let x ∈ R d . Fix M > 0 and write (2.18)

|V (0, x, u, v)|µ n (du, dv) = B((x,x),M ) |V (0, x, u, v)|µ n (du, dv) + B((x,x),M ) c |V (0, x, u, v)|µ n (du, dv).
The first term on the right hand side goes to zero uniformly in x as n tends to infinity by the boundedness of V and by (2.15), while the second term goes to zero uniformly in x as M tends to infinity by the fact that V vanishes at infinity. (iii) To go from ( 2) to (3), write

(2.19) Λ(V, µ n ) ≤ sup x,y∈R d V (0, y -x, u -x, v -x)µ n (du, dv) ≤ sup y∈R d V (0, y, u, v)µ n (du, dv) ,
which goes to zero as n → ∞, by assumption. Here, we used that µ n is a sub-probability measure to obtain the first inequality. ■

As an immediate corollary we obtain the following:

Corollary 2.6. If the sequence (µ n ) is totally disintegrating then, for any k ≥ 2 and any V ∈ F 2k , (2.20) lim n→∞ Λ(V, µ n ) = 0 .
Proof of Corollary 2.6. Apply Item (3) in Lemma 2.5 to

(2.21) W (u 1 , v 1 , u 2 , v 2 ) := sup u 3 ,v 3 ,...,u k ,v k ∈R d |V (u 1 , v 1 , . . . , u k , v k )|. ■ 2.4. Compactification. Let us define (2.22) F (2) := k≥2 F 2k ,
for which there exists a countable dense set (under the uniform metric) denoted by

(2.23) {f r (u 1 , v 1 , . . . , u kr , v kr ) : r ∈ N},
(same arguments as in [12, Section 2.2]). We define (2.24)

X (2) := ξ = { α i } i∈I : α i ∈ M (2) ≤1 , i∈I α i (R d × R d ) ≤ 1 ,
where I may be empty, finite or countable. Recall (2.4) and for any ξ 1 , ξ 2 ∈ X (2) , define

(2.25) D 2 (ξ 1 , ξ 2 ) := r≥1 1 2 r 1 1 + ∥f r ∥ ∞ α∈ξ 1 Λ(f r , α) - α∈ξ 2 Λ(f r , α) .
Proposition 2.7. D 2 is a metric on X (2) .

Proof of Proposition 2.7. This follows the same line of proof as [12, Theorem 3.1].

Step 1. If D 2 (ξ 1 , ξ 2 ) = 0 then for all k ≥ 2 and f ∈ F 2k , (2.26)

α∈ξ 1 Λ(f, α) = α∈ξ 2 Λ(f, α).
We deduce therefore that for every integer r ≥ 1,

(2.27)

α∈ξ 1 Λ(f, α) r = α∈ξ 2 Λ(f, α) r .
Indeed, define the function (for r = 2)

(2.28)

g N (u 1 , v 1 , . . . , u k , v k , u k+1 , v k+1 , . . . , u 2k , v 2k ) := f (u 1 , v 1 , . . . , u k , v k )f (u k+1 , v k+1 , . . . , u 2k , v 2k )φ u k+1 -u 1 N ,
where 0 ≤ φ ≤ 1 is equal to 1 inside a ball of radius 1 and is truncated smoothly to 0 outside a ball of radius 2. Then, g N ∈ F 4k and converges pointwise to f ⊗ f as N → ∞.

Hence, using the fact that f is bounded and dominated convergence (2.29) lim

N →∞ Λ(g N , α) = Λ(f, α) 2 .
The general case follows the same idea.

Step 2. Same as in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF].

Step 3. This step is also an adaptation of [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF], so we only sketch the arguments. We want to recover the orbit of α ∈ M

(2) ≤1 from the value of Λ(f, α) for f ∈ F 2k . Adapting [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF], from these values we get those of (2.30) k j=1 ϕ(s j , t j ), where ϕ(s, t) := e i⟨(s,t),(u,v)⟩ α(du, dv), s, t ∈ R d , provided (s j + t j ) = 0. Suppose now that

(2.31) ∀k ≥ 1, k j=1 ϕ(s j , t j ) = k j=1
ψ(s j , t j ), whenever (s j + t j ) = 0.

Following [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF], we obtain that |ϕ(s, t)| = |ψ(s, t)| and write ϕ(s, t) = ψ(s, t)χ(s, t) whenever |ϕ(s, t)| = |ψ(s, t)| ̸ = 0. As soon as (s j +t j ) = σ+τ ∈ R d , we have k j=1 χ(s j , t j ) = χ(σ, τ ), provided that the s j 's and t j 's are such that |ψ(s j , t j )| = |ϕ(s j , t j )| ̸ = 0 for all j. In particular, (2.32)

χ(s 1 + s 2 , t 1 + t 2 ) = χ(s 1 , t 1 )χ(s 2 , t 2 ),
hence as in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] we can show that χ(s, t) = e i(⟨a 1 ,s⟩+⟨a 2 ,t⟩) for some a 1 , a 2 ∈ R d . The fact that actually χ(s

1 + s 2 + C, t 1 + t 2 -C) = χ(s 1 , t 1 )χ(s 2 , t 2 ) for all C ∈ R d entails that a 1 = a 2 .
This means that α is determined up to shifts by some (a, a) ∈ R 2d , which ends the proof. ■ Proposition 2.8. The space X (2) equipped with D 2 is a compact metric space.

Proof of Proposition 2.8. This proof is an adaptation of the arguments used in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF], so we only sketch the arguments.

Step 1. Let us first show that M

(2)

1 is dense in X (2) . Let ξ = { α i , i ∈ I} ∈ X (2) and ε > 0. Pick a finite collection { α i , 1 ≤ i ≤ k} such that i>k p i < ε, where p i denotes the total mass of α i . Let α i be an arbitrary member of the orbit α i and for M > 0, let λ M be the Gaussian law on R 2d with zero mean and covariance matrix M × Id. Pick any sequence (a

i,M ) 1≤i≤k in (R d ) k such that inf i̸ =j |a i,M -a j,M | → ∞ as M → ∞. Finally, set (2.33) µ M := 1≤i≤k α i * δ (a i,M ,a i,M ) + 1 - 1≤i≤k p i λ M ∈ M (2)
1 .

Since the mutual distances of the centers of masses of the measures in the first sum increase to infinity, i.e., they are widely separated, see (2.6) and all the mass of λ M vanishes in the limit as M tends to infinity, i.e, λ M is totally disintegrating, see (2.15), we get by Lemmas 2.3 and 2.5 that for all k ≥ 1 and f ∈ F 2k , (2.34) lim

M →∞ Λ(f, µ M ) = 1≤i≤k Λ(f, α i ).
Since ε > 0 may be chosen arbitrarily small this completes the first step.

Step 2. Let us show that for any sequence (µ n ) in M

(2)

1 , there exists a subsequence along which ( µ n ) converges to some element of M

1 . Together with the first step this implies the result. We start with some preliminary considerations. We use the following concentration function:

(2.35) q µ (r) := sup x∈R d µ(B((x, x), r)), r ≥ 0, µ ∈ M (2) 1 .
Let now (µ n ) n∈N be a sequence in M

≤1 . Going over to a subsequence if necessary, we may define, by Helly's selection theorem, (2.36) q := lim r→∞ lim n→∞ q µn (r) and p := lim n→∞ µ n (R 2d ) .

If q = 0, then it follows by Corollary 2.6 that µ n → 0 in X (2) . If on the other hand q > 0, then taking a suitable translation vector (a n , a n ) ∈ R d × R d we have that for some r > 0 and all sufficiently large n the shifted measure λ n = µ n * δ (an,an) satisfies (2.37) λ n (B((0, 0), r)) ≥ q/2 .

Choosing a subsequence if needed, we may assume that (λ n ) converges vaguely to some α ∈ M

≤1 and by Lemma 2.2 we may further write λ n = α n + β n with α n converging weakly to α, β n converging vaguely to zero and α n and β n having disjoint supports. By Lemma 2.4 * , for every

V ∈ F 4 , (2.38) lim n→∞ V (u 1 , v 1 , u 2 , v 2 )α n (du 1 , dv 1 )β n (du 2 , dv 2 ) = 0 .
If additionally q = p, then no mass escapes to infinity and one can choose β n to be zero. In that case it follows that µ n → α in X (2) , similarly to [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Theorem 3.2]. To * The use of Lemma 2.4 corrects a small gap in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]. Indeed, the analogous step in the proof of [12, Theorem 3.2, Step 2] does not seem to follow from Lemma 2.4, as it is claimed therein.

conclude the result we can now proceed in very much the same way as in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]. Fix a sequence (µ n ) n in M

(2)

1 . If q ∈ {0, 1} then the above considerations imply the result. Otherwise, for some sequence (a n , a n ) ∈ R d × R d , at least along a subsequence, we have the decomposition µ n = α n + β n , where

(1) α n * δ (an,an) → α weakly as n → ∞;

(2) For every V ∈ F 4 the integral

V (u 1 , v 1 , u 2 , v 2 )α n (du 1 , dv 1 )β n (du 2 , dv 2 )
converges to zero; (3) lim n→∞ q βn (r) ≤ min{1 -q/2, q}; (4) α n and β n have disjoint support.

In the last item we used that mass q/2 has been removed from µ n , see (2.37). If the above limit is not zero one can repeat the same procedure with β n , i.e., for an appropriate shift of β n one decomposes it as a sum of two measures were one converges weakly and the other converges vaguely to zero and such that additionally the above three items are satisfied. If this process terminates after some finite number of stages k ∈ N we obtain the decomposition (2.39)

µ n = k j=1 α n,j * δ (a n,j ,a n,j ) + β n ,
such that α n,j * δ (a n,j ,a n,j ) converges weakly, q βn (r) → 0 for every r and such that for every V ∈ F 4 and 1 ≤ i < j ≤ k,

(2.40)

V (u 1 + a n,i , v 1 + a n,i , u 2 + a n,j , v 2 + a n,j )α n,i (du 1 , dv 1 )α n,j (du 2 , dv 2 ) and V (u 1 + a n,i , v 1 + a n,i , u 2 , v 2 )α n,i (du 1 , dv 1 )β n (du 2 , dv 2 )
tend to zero as n → ∞. If the process does not terminate after a finite number of stages, then one has a similar decomposition that goes by induction. We refer to the proof of [12, Theorem 3.2] for details. ■

Large Deviation Principles

Let X = (X i ) i∈N 0 be a Markov chain in R d starting from the origin and with a transition kernel π satisfying the following assumptions:

(1) (Random walk) There exists a function p : R d → R + and a reference measure λ such that π(x, A) = A p(y -x)λ(dy) for all x ∈ R d and Borel set A ⊆ R d .

(

) (Irreducibility) For all Borel set A ⊆ R d such that λ(A) > 0, there exists k ∈ N such that π k (x, A) = A p * k (y -x)λ(dy) > 0. 2
(

L n := 1 n n i=1 δ (X i-1 ,X i ) ∈ M (2) 1
the pair empirical measure associated to X and L n be its orbit in M

(2)

1 . We obtain the following: Theorem 3.1. As n → ∞, ( L n ) n∈N satisfies a strong Large Deviation Principle on the compact metric space X (2) equipped with D 2 , with speed n and rate function J (2) , where

(3.3) J (2) (ξ) := i∈I h(α i |α i,1 ⊗ π), (ξ = { α i } i∈I ),
α i being any representative of α i and α i,1 its projection onto the first d coordinates. Here, h(•|•) is the relative entropy between two measures, i.e.,

) h(µ|ν) = log dµ dν dµ , (3.4
with the understanding that the integral is infinite if the Radon Nikodym derivative above does not exist.

Remark 3.2. We note that the rate function in (3.3) is well defined due to the fact that the transition kernel π only depends on the difference of its two arguments. Indeed, this implies that inside the sum in (3.3) the choice of the element α i in the orbit α i does not matter.

Let us recall the well-known fact that the process

(X i-1 , X i ) i∈N is itself a Markov chain on R d × R d with transition kernel (3.5) π (2) (x 1 , x 2 , dy 1 , dy 2 ) := δ x 2 (y 1 )π(x 2 , dy 2 ), x 1 , x 2 , y 1 , y 2 ∈ R d .
The proof of Theorem 3.1 is split into three parts: we prove the lower semi-continuity of the rate function (Proposition 4.1) then the lower bound (Proposition 5.1) and finally the upper bound (Proposition 6.1).

We end this section with a corollary. To that end define the empirical measure L

(1)

n of (X i ) i∈N 0 via (3.6) L (1) n = 1 n n i=1 δ X i . Note that L (1)
n is simply the second marginal of L n . We further denote by L

(1)

n the orbit of L (1)
n in M 1 (R d ), the space defined in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Section 2]. See Remark 2.1 for the difference between M

(2) 1 and M 1 (R 2d ). We moreover denote by X the compactification of M 1 (R d ), equipped with the metric D, see [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Section 3] for details. The following corollary is then a direct consequence of Theorem 3.1 and the contraction principle.

Corollary 3.3. As n → ∞, ( L (1)
n ) n∈N satisfies a strong Large Deviation Principle on the compact metric space X equipped with D, with speed n and rate function J (1) defined by

(3.7) J (1) (θ) = inf{ J (2) (ξ) : ξ 1 = θ} ,
where for ξ = { α i } i∈I ∈ X (2) we denote by ξ 1 the collection {α i,2 } i∈I and α i,2 denotes the second marginal of α i (projection onto the d last coordinates).

Proof of Corollary 3.3. In order to apply the contraction principle [4, Theorem 4.2.1], we show that the second projection (onto the last d coordinates) from ( X (2) , D 2 ) to ( X , D) is continuous. The reader may check that it is actually enough to prove continuity of the projection from ( M

≤1 , D 2 ) to ( M ≤1 , D). Suppose that ( ν n (du, dv)) is a sequence of orbits of sub-probability measures on R d × R d that converges to ν(du, dv) for the D 2 metric. Let k ≥ 2 and f ∈ F k . We may write f (v 1 , . . . , v k ) = f (u 1 , v 1 , . . . , u k , v k ) where f ∈ F 2k is constant along the u-variables. Then (the choice of elements in the orbits below is not relevant),

(3.8) f (v 1 , . . . , v k ) 1≤i≤k ν(R d , dv i ) = f (u 1 , v 1 . . . , u k , v k ) 1≤i≤k ν(du i , dv i ) = lim n→∞ f (u 1 , v 1 . . . , u k , v k ) 1≤i≤k ν n (du i , dv i ) = lim n→∞ f (v 1 , . . . , v k ) 1≤i≤k ν n (R d , dv i ).
This concludes the proof. ■

Lower semi-continuity of the rate function

Proposition 4.1. The function

(4.1) ξ = { α i } i∈I ∈ X (2) → J (2) (ξ) = i∈I h(α i |α i,1 ⊗ π)
is lower semi-continuous.

Proof of Proposition 4.1. We use the notation

J (2) (α) := h(α|α 1 ⊗ π), where α ∈ M (2) ≤1
and α 1 stands for its projection on the d first coordinates along the proof. Recall (3.5).

We will also use that

(4.2) J (2) (α) = sup v log v π (2) v dα,
where the supremum runs over all bounded and Borel-measurable (or continuous and compactly supported) functions u : R d × R d → [1, +∞), see [4, Theorem 6.5.12 and Corollary 6.5.10]. Let (µ n ) be a sequence in X (2) converging to ξ = { α i } i∈I . Suppose that there exists ℓ ∈ (0, ∞) such that J (2) (µ n ) ≤ ℓ for all n large enough and let us show that J (2) (ξ) ≤ ℓ. We for now restrict to the case where for each n, µ n is made of a single orbit, so that J (2)

(µ n ) = J (2) (µ n ). Let ε > 0.
By the same arguments used in Proposition 2.8 and possibly restricting to a subsequence, we may write for some k ≥ 1,

(4.3) µ n = k i=1 α (i) n + β n ,
where α

(i) n (1 ≤ i ≤ k) and β n are sequences of sub-probability measures in R d × R d ,

and a

(i) n (1 ≤ i ≤ k) are sequences in R d .
Moreover those sequences can be chosen such that (4.4)

α (i) n * δ (a (i) n ,a (i) n ) ⇒ α i ∈ α i , n → ∞, lim n→∞ min i̸ =j |a (i) n -a (j) n | = +∞,
and (β n ) is widely separated from each (α

(i) n ), with (recall (2.35)) (4.5) lim sup n→∞ q βn (r) ≤ ε, ∀r > 0.
Moreover, by Lemma 2.2 and the construction in Step 2 of Proposition 2.8 the supports of α

n , . . . , β n are all disjoint and for each i there exists a sequence (R

n ) n∈N 0 tending to infinity such that Supp(α

(i) n ) ⊆ B((-a (i) n , -a (i) n ), R (i) n ) and the support of β n is contained in the complement of i B((-a (i) n , -a (i) n ), R (i) n ). We now define (4.6) u (n) i (x, y) = u i (x + a (i) n , y + a (i) n ), x, y ∈ R d .
Note that by the compactness of the support of u i the support of each u

(n) i is contained in a compact ball in R 2d centered around (-a (i) n , -a (i) n ).
In particular, recalling (3.5) there is R > 0 such that for all 1 ≤ i ≤ k

(4.7) Supp u (n) i ⊆ B((-a (i) n , -a (i) n ), R) , Supp π (2) u (n) i ⊆ R d × B(-a (i) n , R) , hence (4.8) 1 + u (n) i 1 + π (2) u (n) i (x, y) ̸ = 1 ⇒ y ∈ B(-a (i) n , R). Therefore (4.9) lim n→∞ log 1 + u (n) i 1 + π (2) u (n) i dα (j) n = log 1+u i 1+π (2) u i dα i (i = j) 0 (i ̸ = j).
Finally, letting (4.10)

u (n) = 1≤i≤k u (n) i ,
we obtain:

(4.11) ℓ ≥ lim inf n→∞ J (2) (µ n ) ≥ lim inf n→∞ log 1 + u (n) 1 + π (2) u (n) dµ n ≥ lim inf n→∞ 1≤i,j≤k log 1 + u (n) i 1 + π (2) u (n) i dα (j) n ≥ 1≤i≤k J (2) (α i ) -ε.
Here, the contribution coming from β n is zero because for sufficiently large n we have that β n (dx, dy) = 0 for y ∈ B(-a

(i)
n , R), which together with (4.8) shows that the contribution is indeed zero. It now remains to send ε to zero and k to infinity. Finally, to treat the general case, that is when µ n has possibly more than one orbit, the idea is the same as in the last paragraph of [12, proof of

n→∞ 1 n log P( L n ∈ G) ≥ -inf ξ∈G J (5.1) lim inf
Proof of Proposition 5.1. Let ξ = {α i , i ∈ I} be an element of X (2) such that

J (2) (ξ) < +∞, hence h(α i |α i,1 ⊗ π) < +∞ for all i ∈ I. Let U be any open neighborhood of ξ. It is enough to prove that (5.2) lim inf n→∞ 1 n log P( L n ∈ U ) ≥ -J (2) (ξ).
We proceed as in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]Lemma 4.3] and use the density of M

1 in X (2) , already proven in the proof of Proposition 2.8 (Step 1). Let k ≥ 1 and consider the sequence (µ M ) defined as in (2.33), except that we replace the totally disintegrating sequence (λ M ) by (λ M,1 ⊗ π), that is totally disintegrating as well and is such that µ M still converges in X (2) to ξ. Using the sub-additive property of ν ∈ M

(2) ≤1 → h(ν|ν 1 ⊗ π) (straightforward from (4.2)), we obtain:

(5.3) J (2) ( µ M ) = h(µ M |µ M,1 ⊗ π) ≤ 1≤i≤k h(α i |α i,1 ⊗ π) ≤ J (2) (ξ).
Thus, we have shown that there exists a sequence (µ

M ) M ∈N 0 in M (2)
1 which converges in X (2) to ξ and is such that (5.4) lim sup

M →∞ J (2) ( µ M ) ≤ J (2) (ξ) .
The lower bound now follows from the standard Large Deviation lower bound of the pair empirical measure on M

(2) 1 (see [START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time[END_REF] and the discussion in Section 1). ■

Upper bound

In this section we prove the following Proposition 6.1. For any closed set F in X (2) , (6.1) lim sup

n→∞ 1 n log P( L n ∈ F ) ≤ -inf ξ∈F J (2) (ξ).
Let U denote the space of non-negative, continuous and compactly supported functions defined on R d × R d . For any k ≥ 1, c > 0, u 1 , . . . , u k ∈ U and a 1 , . . . , a k ∈ R d , let g = g(u, c, R) : R d × R d → (0, ∞) be defined by

(6.2) g(x, y) = c + k i=1 u i (x + a i , y + a i )φ x + a i R , y + a i R , x, y ∈ R d ,
where φ is a smooth non-negative function such that 0 ≤ φ ≤ 1, φ = 1 inside the unit ball and φ = 0 outside the ball of radius two. For all µ ∈ M 1 (R d × R d ), define

(6.3) F (u, c, R, µ) = sup a 1 ,...,a k min i̸ =j |a i -a j |≥4R R d ×R d
-log π (2) g(x, y) g(x, y) µ(dx, dy).

Since F (u, c, R, •) is invariant under shifts of the form µ → µ * δ (x,x) , we may lift it up to a function F defined on M

1 . In the sequel, we write (6.4)

u i,R (x, y) := u i (x, y)φ(x/R, y/R), x, y ∈ R d .
The proof of the upper bound follows from the following three lemmas: Lemma 6.2 (Sub-exponential growth). For any choice of (u, c, R), we have

(6.5) lim sup n→∞ 1 n log E exp(n F (u, c, R, L n )) ≤ 0.
Lemma 6.3 (Lower-semicontinuous extension). If the sequence ( µ n ) converges to ξ = ( α i ) i∈I in ( X (2) , D 2 ), then

(6.6) lim inf n→∞ F (u, c, R, µ n ) ≥ Λ(u, c, R, ξ), where (6.7)
Λ(u, c, R, ξ) := sup

{ α 1 ,..., α k }⊆ξ k i=1 sup b∈R d -log π (2) (c + u i,R )(x, y) (c + u i,R )(x, y) α i (dx + b, dy + b).
Remark 6.4. Lemma 6.3 is analogous to Lemma 4.6 in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF]. However, the two suprema in (6.7) are not in the original paper. First, we add the supremum over b ∈ R d so that the quantity inside is a function of the orbit α i only rather than a function of a particular member of its orbit. This has however no consequence on the sequel of the argument in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF], since they later consider a supremum over functions (Lemma 4.7) allowing for arbitrary shifts. The other supremum is here to stress that an element of ξ is a collection of sub-probability orbits rather than a sequence. Lemma 6.5. We have (6.8) J (2) (ξ) = sup c,R>0, k≥1 u 1 ,...,u k ∈U Λ(u, c, R, ξ).

Proof of Lemma 6.2. Use that (6.9) -n

R d ×R d log π (2) g(x, y) g(x, y) L n (dx, dy) = log n i=1 g(X i-1 , X i ) π (2) g(X i-1 , X i ) , so that (6.10) E exp n R d ×R d -log π (2) g(x, y) g(x, y) L n (dx, dy) = E n i=1 g(X i-1 , X i ) π (2) g(X i-1 , X i )
.

We write the product as (6.11)

n i=1 g(X i-1 , X i ) π (2) g(X i-1 , X i ) = g(X 0 , X 1 ) π (2) g(X n-1 , X n ) n-1 i=1 g(X i , X i+1 ) π (2) g(X i-1 , X i ) ,
and since g is bounded from below by c we see that

(6.12) E n i=1 g(X i-1 , X i ) π (2) g(X i-1 , X i ) ≤ sup g c E n-1 i=1 g(X i , X i+1 ) π (2) g(X i-1 , X i ) ,
and, by the fact that Y i = (X i-1 , X i ) is a Markov chain and an induction argument the last expectation is one. This shows that the exponential rate of the right-hand side of (6.10) is zero. It therefore only remains to deal with the case in which in (6.10) an additional supremum is taken over a 1 , . . . , a k as in the statement of the result. This follows via a coarse graining argument. The idea is that by (3.1) it is exponentially unlikely that X travels in the time interval [0, n] to a distance ρ n , which allows one to restrict the supremum over a 1 , . . . , a k to balls of radius ρ n . In a very similar way as in [12, proof of Lemma 4.5] one may then conclude, so we omit the details. ■

Proof of Lemma 6.3. As it can be seen from the second step of the proof of Proposition 2.8, convergence in D 2 implies the existence of a decomposition (6.13)

µ n = k j=1 α n,j * δ (a n,j ,a n,j ) + β n ,
along subsequences as in (2.39), where, for all 1 ≤ j ≤ k

• (a n,j ) n≥1 is a sequence in R d satisfying (6.14) |a n,i -a n,j | ≥ 4R
if n large enough and i ̸ = j; • α n,j * δ (a n,j ,a n,j ) converges weakly to α j as n → ∞, where α j is some element in the orbit of α j ; • (α n,j ) and (β n ) are widely separated.

Recall (6.2). Choosing a i = -a n,i in the definition of g, we obtain (6.15) g(x, y) = c + k i=1 u i,R (x -a n,i , y -a n,i ).

By (6.14) and our assumption on φ, at most one term in the sum above can be nonzero. Also, (6.16)

π (2) u i,R (x -a n,i , y -a n,i ) = u i,R (y -a n,i , z -a n,i )p(z -y)λ(dz)
is nonzero for at most one value of 1 ≤ i ≤ k which is the same as in the above. We finally obtain: (6.17) log π (2) g(x, y) g(x, y) The supremum in the sum actually coincides with J (2) (α i ), see [4, Theorem 6.5.12 and Corollary 6.5.10]. ■ Proof of Proposition 6.1. The proof follows from Lemmas 6.2-6.5 in the exact same way as Proposition 4.4 in [START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] follows from Lemmas 4.5-4.7 therein. ■

= k i=1 log c + π (2) u i,R (x -a n,i , y -a n,i ) c + u i,R (x -a n,i ,

Adaptation to rescaled random walks

A small adaptation of the proof of Proposition 6.1 yields the same result also for a rescaled random walk. To precisely formulate the result we need to introduce more notation. Let (X i ) i∈N 0 be a random walk in Z d . Assume that its step distribution is centered and square-integrable, with (1/d)Id as covariance matrix. Let (a n ) be a sequence of positive real numbers converging to +∞ and such that a Denote by L ε n the corresponding pair empirical measure, that is (7.2) L ε n = 1 ⌊n/ℓ⌋ ⌊n/ℓ⌋ i=1 δ (X ε,n i-1 ,X ε,n i ) .

Remark 7.1. The relevant scale for potential applications to the Swiss cheese model [START_REF]Intersections of random walks[END_REF][START_REF] Van Den Berg | Moderate deviations for the volume of the Wiener sausage[END_REF] corresponds to the choice a n := n 1/d . We prove the following: Theorem 7.2. For any closed set F in X (2) ,

(7.3) lim sup n→∞ a 2 n n log P( L ε n ∈ F ) ≤ - 1 ε inf ξ∈F J (2) ε (ξ).
Here, (7.4)

J (2) ε (ξ) := i∈I h(α i |α i,1 ⊗ π ε ), (ξ = { α i } i∈I ),
where π ε the denotes the Brownian semi-group at time ε.

Proof of Theorem 7.2. It turns out that only the proof and the statement of Lemma 6.2 need to be adapted. The remaining statements are about the rate function rather than the Markov chain at hand. To that end we define (7.5) π (2) n,ε g(x, y) = E g(y, y + X ε,n 1 ) , and π (2) ε g(x, y) = E g(y, y + B ε ) .

With these notations at hand we define F as in Section 6 but with π (2) replaced by π Note that the result would be immediate from the proof if π

ε would be replaced by the transition kernel π (2) n,ε of X ε,n . Following the proof of Lemma 6.2, we write (7.7)

M i=1 g(X ε,n i-1 , X ε,n i ) π (2) ε g(X ε,n i-1 , X ε,n i ) = M i=1 g(X ε,n i-1 , X ε,n i ) π (2) n,ε g(X ε,n i-1 , X ε,n i ) M i=1 π (2)
n,ε g(X ε,n i-1 , X ε,n i ) π

(2) ε g(X ε,n i-1 , X ε,n i )

.

Since (i) g is positive, continuous and constant outside of a compact set and (ii) X ε,n n,ε g(x, y) π

(2) ε g(x, y)

-1 = 0 .

The variable x in the supremum plays no actual role and uniformity in y can be deduced from the uniform continuity of g and a coupling for which X ε,n 1 converges to B ε almost surely as n → ∞. The convergence in (7.8) allows to control the rightmost factor in (7.7). The first factor on the right hand side can be dealt with as in the proof of Lemma 6.2. ■

Date:

  March 31, 2023. (D. Erhard) Instituto de Matemática -Universidade Federal da Bahia (J. Poisat) Université Paris-Dauphine, CNRS, UMR [7534], CEREMADE, PSL Research University, 75016 Paris, France .

  Lemma 4.2]. ■ 5. Lower bound Proposition 5.1. For any open set G in X (2) ,

( 2 )

 2 (c + u)(x, y) (c + u)(x, y) α i (dx, dy).

2 n 7

 27 = o(n). Define ℓ := ℓ(ε, n) = ⌊εa 2n ⌋, M := ⌊n/ℓ⌋ and (

1 M

 1 log E exp(M F (u, c, R, L ε n )) ≤ 0 .

1

  converges in law to B ε as n → ∞,

  ) (Tightness) There exists a positive sequence (ρ n ) n∈N with lim sup n→∞ Theses assumptions include many natural examples such as the simple random walk on Z d (with the counting measure as reference measure) or the discretized Brownian motion (B iε ) i∈N 0 , where ε > 0 and B is Brownian motion (with Lebesgue measure as reference measure). Assumption (1) implies shift-invariance of the process, see Remark 3.2 below.

	Assumption (2) is used at the end of the proof of Proposition 5.1 when applying the
	standard large deviation lower bound in the usual weak topology [8, Corollary 3.4 and
	Equation (4.1)]. Assumption (3) is used during the proof of Lemma 6.2 .
	Let us denote by
	(3.2)
				1 n log ρ n =
	0 such that
	(3.1)	lim n→∞	1 n	log P sup

1≤i≤n |X i | ≥ ρ n = -∞ .

  y -a n,i ) and we can conclude almost as in[START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] Lemma 4.6]. As we already pointed out in Remark 6.4, there is more flexibility in choosing a i (see (6.2)), which explains the additional supremum (over b) in our statement, compared to[START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] Lemma 4.6]. Indeed, let b j ∈ R d for all 1 ≤ j ≤ k. Then we may choose a i = -a n,i + b i instead of a i = -a n,i . If we require that |a n,i -a n,j | ≥ 4R + max 1≤j≤k |b

j | instead of simply |a n,i -a n,j | ≥ 4R, then we finally get our claim.To see why the second supremum appears, note that if the number of elements in ξ is unbounded then in (6.13) we can actually choose the k in the sum independent of the k in (6.2). Therefore adjusting the choice of a 1 , . . . , a k we can obtain any collection { α 1 , . . . , α k } in the limit. We therefore can conclude. ■ Proof of Lemma 6.5. As in

[START_REF] Mukherjee | Brownian occupation measures, compactness and large deviations[END_REF] Lemma 4.7]

, we get

Acknowledgements 18 References 18 Acknowledgements D.E. was supported by the National Council for Scientific and Technological Development -CNPq via a Bolsa de Produtividade 303520/2019-1 and 303348/2022-4 and via a Universal Grant (Grant Number 406001/2021-9). D.E. moreover acknowledges support by the Serrapilheira Institute (Grant Number Serra-R-2011-37582). J.P. is supported by the ANR-22-CE40-0012 LOCAL.


C Bouder

G Burel

S Azou

Détermination automatique des paramètres d'une transmission numérique

Introduction

Le travail résumé ci-dessous a été réalisé dans le cadre du stage de DEA d'électronique de C. Bouder [START_REF] Bouder | Détermination automatique des paramètres d'une modulation numérique[END_REF]. Il s'agit de déterminer automatiquement les paramètres d'une transmission numérique. L'objectif étant de trouver automatiquement le type de modulation (ASK2, ASK4, PSK2, PSK4, FSK2, FSK4), la fréquence porteuse et la fréquence symbole d'un signal reçu ou intercepté.

Cette étude entre dans un cadre militaire pour la surveillance des spectres, comme dans un cadre civil pour la création d'un récepteur capable de s'adapter automatiquement à tous types de signaux.

Contexte de l'expérience

Les expériences ont été réalisées dans le cadre d'un filtrage à la réception et éventuellement à l'émission en présence de bruit pour que les résultats soient exploitables en pratique.

Présentation de l'émetteur et du récepteur

Modulations étudiées

Les modulations étudiées sont un peu différentes de celles plus couramment utilisées dites 'standards' : elles sont présentées dans l'ouvrage [START_REF] Azzouz | Automatic modulation recognition of communication sugnals[END_REF]. A partir de ces modulations on calcule des paramètres caractéristiques de chaque modulation.

Calcul des paramètres

Cinq paramètres discriminants ont été calculés à partir d'histogrammes des amplitudes, phases et fréquences instantanées du signal ramené en bande de base :

-L'écart type de la valeur absolue de l'amplitude instantanée.

-L'écart type de la valeur absolue de la phase instantanée.

-L'écart type de la phase instantanée.

-L'écart type de la fréquence instantanée.

-Le maximum de la valeur absolue de la densité spectrale de puissance de l'amplitude instantanée. Chaque modulation est alors représentée par l'ensemble des cinq paramètres qui la caractérisent.

Reconnaissance du type de modulation.

La reconnaissance du type de modulation a été orchestrée par une analyse en composantes principales des cinq paramètres et par un réseau de neurones. Chacune de ces deux méthodes prend en compte simultanément tous les paramètres caractéristiques du signal reçu, ce qui permet une analyse rapide de la transmissions numérique.

Analyse en composantes principales

On crée une base de données composée des six modulations à reconnaître, représentées par les cinq paramètres. Chaque exemple sera un point dans un espace à cinq dimensions et l'ensemble des exemples représentatifs d'une modulation formera un nuage de points. Le résultats de l'analyse de données donnera six nuages de points.

L'expérience démontre que les six nuages de points sont séparés les uns des autres, sauf pour deux modulations : PSK4 et FSK4, quelque soit le type de suréchantillonnage utilisé à l'émission. Les cinq paramètres étudiés sont donc suffisamment discriminants pour reconnaître les modulations.

Résultats du réseau de neurones

Le réseau fonctionne en deux phases : l'une d'apprentissage, l'autre de test. Pour la phase d'apprentissage il est nécessaire de fournir une base de données pour l'entrée du réseau et la base de données correspondantes de sortie. Les bases de données sont construites sur le même principe que celle de l'analyse en composantes principales. Les résultats obtenus sont de l'ordre de 80% de reconnaissance sur la base de test, pour un rapport signal à bruit de 10 dB.

Une étude plus approfondie des cinq paramètres permet de constater qu'un d'entre eux a un ordre de grandeur beaucoup plus grand que les autres, ce qui perturbe le réseau, d'où l'idée de normaliser les paramètres. Les résultats obtenus sont alors de 88%.

Pour généraliser les résultats l'expérience a été réalisée avec les constellations les plus couramment employées, les taux de reconnaissance avoisinent 97%.

Mais avant de pouvoir reconnaître les modulations, il faut déterminer les fréquences porteuse et symbole.

Estimation des fréquences porteuse et symbole.

Fréquence porteuse

La fréquence porteuse est déterminée une première fois de manière approximative par la formule de la moyenne appliquée à la densité spectrale de puissance. Une fois cette fréquence déterminée, on peut ramener le signal approximativement en bande de base et affiner l'estimation. Le signal en bande de base a pour expression :

S t a h t kT i ft k k ( ) ( ) exp( ) = -  2π où f

Conclusion

Dans l'ensemble, les résultats obtenus pour la reconnaissance du type de modulation et l'estimation des fréquences porteuse et symbole sont fiables jusqu'à un rapport signal à bruit de 10dB, avec des taux de reconnaissance supérieurs à 80%. Le calcul des fréquences permet de démoduler le signal reçu pour le ramener en bande de base et, connaissant la fréquence symbole, de retrouver la séquence émise, une fois le type de modulation déterminé.

De manière à généraliser les résultats, il faudrait prendre en compte un plus grand nombre de modulation, comme les modulations d'amplitude en quadrature, un plus grand nombre d'états, et mieux matérialiser le canal en considérant les échos. Les méthodes mises en oeuvre devraient également pouvoir se généraliser à d'autre type de transmissions comme l'étalement de spectre ou les transmissions OFDM. Ces perspectives seront étudiées dans le cadre de la thèse de C. Bouder.