Raphaël Didier
email: raphael.didier@univ-lorraine.fr

A community's resilience to the covid-19 crisis The Florain monetary community

Keywords: local currency, crisis, resilience, covid-19, SSE, governance, sociocracy, territory, values, monetary community. JEL codes: A14, E42, R11

In this paper, we study the resilience of the community of individual users of a French local currency in the face of an abrupt halt in exchanges during the first containment related to the covid-19 pandemic (March 17, 2020 to May 11, 2020). Our study is based on the local currency of the Nancy Basin, the Florain, for which we have a field survey conducted before the pandemic, three interviews conducted between the first and second containment in France (May 12 and October 30, 2020), observations and figures obtained during participation in the association's general assembly and publications found on the structure's blog. This allowed us to highlight sociological factors (feeling of being a consum'actor, social representations of members and existence of an identity niche among active volunteers) and organizational factors (ethos of active volunteers, sociocratic mode of governance of the association and inscription of the Florain community in a life basin) that contribute to community resilience. However, in the particular case of a local currency, we show that there is also a predominant institutional dimension, linked to its social and political nature, which makes the community of individual users a local monetary community.

In its April 2021 World Economic Outlook, the IMF recalled that, in addition to the worrying human toll of covid-19, major uncertainties remain regarding the global economic and social outlook. This crisis is the result of governments' desire to halt production in the name of the best interests of human life, which limits economic analyses based on the usual conceptual tools of economists [START_REF] Boyer | Les Capitalismes à l'épreuve de la pandémie[END_REF]. While the social, health, and economic impacts associated with covid-19 are felt around the world, but to varying degrees, communities certainly have a critical role to play in the process of emerging from the crisis [START_REF] South | Sustaining and strengthening community resilience throughout the COVID-19 pandemic and beyond[END_REF].

The ability of communities to cope with the stresses resulting from these social and economic changes invites the mobilization of the psychological concept of individual resilience (Cyrulnik and Jorland, 2012), to be extended to communities. The literature on community resilience has long been split between the resilience of ecological systems [START_REF] Holling | Resilience and stability of ecological systems[END_REF], defined as their ability to absorb change while maintaining the same relationships among populations, and social resilience (Johnston and Paton, 2001), which is the community's ability to draw on its internal resources to cope with change. In light of modern challenges (climate change, constraints on financial and natural resources, wars...), [START_REF] Adger | Social and ecological resilience: Are they related?[END_REF] showed that these two forms of resilience must be thought of together and focused on the links between them. [START_REF] Berkes | Community Resilience: Toward an Integrated Approach[END_REF] continued this line of thought and proposed an integrated approach to resilience. In this article, we will focus solely on community (social) resilience, which is as much a theoretical framework as a social process [START_REF] Edge | Understanding community resiliency in rural communities through multimethod research[END_REF]. Various academic works on community (social) resilience by Ross et al. (2010), [START_REF] Norris | Community resilience as a metaphor, theory, set of capabilities, and strategy for disaster readiness[END_REF], or [START_REF] Edge | Understanding community resiliency in rural communities through multimethod research[END_REF] have sought to list the main factors of community resilience. But to our knowledge, no study has yet been conducted on the resilience of a community of individual users of a local currency. However, the first confinement (March 17, 2020to May 11, 2020) in France was a shock for many of them, in particular Florain, a local currency in the Nancy Basin created in 2017. Indeed, as the Florain only exists in paper form for the time being, the confinement has de facto blocked its circulation, insofar as the closure of shops except for essential goods has not only deprived users of places to buy, but also of exchange counters allowing the replenishment of bills.

Our article therefore proposes to analyze the main dimensions of the resilience of a local currency user community. To do so, we have at our disposal a field survey conducted before the pandemic, three interviews conducted between the first and second confinements in France (May 12 and October 30, 2020), observations and figures obtained during participation in the association's general assembly and publications found on the structure's blog. Our article is divided into three parts. In the first part, we will define the monetary community, the concept of community resilience and show that the key factors of this resilience can be classified in two main categories, sociological and organizational. In the second part, we will present our field of study, the Florain, our survey and the interviews conducted. Finally, in the third part, we will show that beyond the sociological factors (the feeling of being a consum'actor, the social representations of the members and the existence of an identity niche among the active volunteers) and organizational factors (the ethos of the active volunteers, the sociocratic mode of governance of the association and the inclusion of the Le Florain community in a catchment area), the resilience of the monetary community of Le Florain is above all the result of an institutional dimension, linked to the social and political nature of the currency.

From the community to the monetary community

The community cannot be reduced to the sum of the individuals that compose it, in that it has an existence of its own that derives from the more or less strong ties between people. This is why Tönnies (1887/2010) proposed a typical ideal classification between community and society, the former being characterized by inter-individual relations based on affective, geographical, social or cultural proximity, where individual autonomy is necessarily limited by the unity sought, while the latter leaves more room for free association and individual choice. We find again the opposition between tradition and modernity, which Durkheim (1889[START_REF] Durkheim | De la division du travail social[END_REF] summarizes with the following formula: "Gesellschaft implies a circle of men who, as in Gemeinschaft, live and dwell in peace next to each other but, instead of being essentially united, are on the contrary essentially separated, and whereas in Gemeinschaft they remain united despite all distinctions, here they remain distinct despite all ties.".

As it stands, such a classification, static in Tönnies, is not able to describe a community of users of a currency, insofar as it can be observed that the latter borrows characteristics from both ideal types. The classification proposed by Weber (1922Weber ( /1971) ) between "communalization" (Vergemeinschaftung) and "sociation" (Vergesellschaftung) appears much more relevant for our purpose, since it characterizes social relations based on a subjective feeling2 of belonging to a community in the first case and on a deliberate choice of participation in a common interest in the second. It should be noted that for Weber, communalization and sociation can coexist3 and that the latter can be linked to rationality in value, i.e., to the adherence of members to considerations of a higher order, which we can describe as ethical. In the case of legal tender, however, ethical trust is a matter of a higher order -based, for example, on national symbolswhich ensures that the currency retains its status as a common good (Aglietta and Orléan, 2002). In this way, the community of users of legal tender is similar to a society whose specificity is that it is built on a social object, money, which is the "materialization of the whole" (Aglietta and Orléan, 2002, p. 100). Thus defined, the community of users of legal tender becomes a monetary community. And as the work of the French institutionalist monetary school [START_REF] Alary | Les divers courants de l'institutionnalisme monétaire : un état des lieux[END_REF] shows, the specificity of a monetary community is that the strong sense of belonging of its members results from the very nature of money.

Community Resilience

Resilience is a term that comes from the Latin resilientia (to bounce back), which in the 19th century referred to the ability of a material to withstand shocks in the physical sciences. This physical phenomenon was then used as a "metaphor to illustrate the idea that a human being can resist a trauma, hold on and start again" (Cyrulnik and Jorland, 2012, p.8). According to these authors, the seminal work on resilience is that of psychologist Emmy Werner (Werner and Smith, 1983), who conducted a study of 700 children on the island of Hawaii in the 1950s. These children presented multiple risks (abandonment, health deficiencies, emotional deficiencies, etc.), but this did not prevent a significant proportion of them from finding family and professional stability once they became adults. She therefore deduced that there is a phenomenon capable of opposing a certain form of psychological and social determinism. This concept, originally linked to the individual, was then extended to ecosystems [START_REF] Holling | Resilience and stability of ecological systems[END_REF] and to social groups, when the latter have to face natural disasters in particular (Johnston and Paton, 2001). The resulting definitions of resilience are therefore numerous and may only partially overlap. For example, [START_REF] Holling | Resilience and stability of ecological systems[END_REF] defines the resilience of an ecological system as "a measure of the persistence of systems and their ability to absorb changes and disturbances while maintaining the same relationships among populations or state variables" which, it should be noted, does not imply a return to the original state. Johnston and Paton (2001) define it as "the ability of the community to draw on its internal resources and skills to manage the demands, challenges and changes it encounters. The Community Resilience Manual published by the Canadian Centre for Community Renewal (cited in [START_REF] Berkes | Community Resilience: Toward an Integrated Approach[END_REF] defines a community as resilient if "it takes intentional steps to improve the personal and collective capacity of its citizens and institutions to respond to and influence social and economic change". [START_REF] Adger | Social and ecological resilience: Are they related?[END_REF] describes social resilience as "the ability of groups or communities to cope with external stresses and disruptions resulting from social, political, and environmental change". [START_REF] Edge | Understanding community resiliency in rural communities through multimethod research[END_REF] further note that community resilience is as much "a theoretical framework as it is a social process, explaining community responses to external forces such as economic downturns, natural disasters, or other threats to sustainability".

We believe it is relevant to draw on the definitions of [START_REF] Adger | Social and ecological resilience: Are they related?[END_REF] and Johnston and Paton (2001), the former highlighting the nature of the external shocks that a community may experience, the latter affirming the importance of drawing first on internal resources to ensure resilience. This will allow us to propose a categorization of the key factors of community resilience.

Key factors in community resilience

Defining community resilience as "the ability of a community not only to cope with adversity, but also to draw strength from it" Edge, Joyce and Kulig (2008) used three different methods (interviews, survey and database analysis) to study the resilience of three communities in Alberta between 2003 and 2005. The first two were rural communities -small towns far from major commuting areas -and the third was an urban neighbourhood. In all three cases, local governments supported the study. Each study community was selected based on the shocks it had experienced: fighting the development of an intensive livestock operation, closing a mine when the community was heavily dependent on mining and logging, and merging with a neighbouring town. The main factors of community resilience, as listed by the study, can be classified into two main categories: sociological factors (sense of belonging to the community, community solidarity, shared vision, good understanding among members) and organizational factors (leadership, transparent problem-solving process, ability to cope with change, ability to deal with internal divisions).

With government and non-government officials, Ross et al. (2010) studied social resilience, defined as "how individuals, communities and societies adapt, transform and potentially become stronger when faced with environmental, social, economic or political challenges". Their field of study was northern Queensland, Australia. On the basis of interviews and data studies, they highlight the existence of several key factors in community resilience, which we can again classify into sociological factors (a diverse and innovative economy that guarantees opportunities) and organizational factors (governance that facilitates change in the community, a strong link between people and the land4 , the community's own infrastructure, the ability to maintain and develop community networks, and skills/knowledge that are shared and passed on). [START_REF] Norris | Community resilience as a metaphor, theory, set of capabilities, and strategy for disaster readiness[END_REF] conducted a 3.5 year study to identify factors that contribute to people's positive adaptation to life in a rural community, in this case Stanthorpe, a small town in South East Queensland, Australia. Based on interviews 5 and case studies, they created a resilience toolkit -defined as "the ability of an individual or community to cope with stress, overcome adversity, or adapt positively to change" -with 11 of the most commonly cited resilience factors to improve the resilience of communities and individuals: Sociological factors (development of social networks and support for individuals, positive vision of the future, definition of a goal to pursue, recognition of diversity within the community, shared belief) and Organizational factors (implementation of activities that promote new experiences and learning, learning together from past experiences and crises, environment and lifestyle, infrastructure and services that are useful to members, support of local businesses and organization of experience exchanges, existence of a leader within the community).

Thus, the factors contributing to the resilience of a community can be classified into two broad categories, sociological factors and organizational factors. In the rest of this article, we will use this categorization to study the key factors of the resilience of the Florain community, a local and complementary citizen currency in the Nancy area.

The Florain user community

The July 31, 2014 law on Social and Solidarity Economy (SSE) gave a legal framework to local currencies in France, defining them as payment instruments that are complementary to the official currency, which can only circulate in a restricted geographical area and concern only a small number of goods and services.

An associative local currency

After a two-year period of maturation of the project, during which participatory meetings were multiplied to decide, among other things, on the governance of the association, the name of the local currency and the charter of values, the Florain was officially put into circulation in October 2017. In accordance with the law, an association 6 (status under French law 1901), whose sole purpose is the management of the Florain, which circulates according to the statutes revised in 2020 on a now enlarged area: "the living area of Nancy and Bar-le-Duc". The association is made up of associate members (individuals, professional service providers who accept the Florain as payment for their products or services) and non-adherent members (ex-officio members) who are local authorities, financial backers contributing to the project, partner organizations or any other organization designated by the steering committee. The association considers as active members the employee, the possible interns and civic services, as well as the active volunteers. In total, there are about twenty-five very involved people who contribute to the smooth running of the association. Until 2019, Florain has seen a continuous increase in the number of members (individuals and professionals) and in the amount of money in circulation. At the end of 2019, just before the covid-19 pandemic, the association managing the local currency announced 102,000 Florains in circulation, only in paper form and 12 exchange counters. 195 professional providers accepted the Florain in payment and the community of individual users, legally constituted of only members 7 , 531 individuals.

5 An exercise was conducted with a group of students from Stanthorpe High School. The students were given the concepts of resilience and had to describe and photograph what these concepts meant to them. 6 Article 2 of the association's statutes states: "the purpose of this association is to experiment with and then extend the use of a local and complementary citizen currency in the Nancy and Bar-le-Duc areas, called the "Florain", so as to encourage production and consumption approaches that respect social, environmental and economic values, as described in the association's charter". 7 You have to be a member to be able to exchange your euros for guilders at an exchange counter.

From the first thoughts in 2015 about creating a local currency, sociocratic governance emerged as a way to manage what would become the Florain, insofar as it is, to paraphrase [START_REF] Parker | The question of organization: A manifesto for alternatives[END_REF] 8 , an "organizational response to political questions". Sociocracy is conceived as "a mode of governance and decision-making that allows an organization to behave like a living organism, that is, to self-organize" [START_REF] Buck | La sociocratie : les forces créatives de l'autoorganisation[END_REF]. Ultimately, sociocracy is a form of participatory management, which does not eliminate hierarchy but rather superimposes it [START_REF] Romme | The sociocratic model of organizing[END_REF], and is based on four fundamental characteristics [START_REF] Endenburg | Construction Principles and Design Rules in the Case of Circular Design[END_REF]: organization in circles, consent, election without candidates, and the double link.

The circle is a functional, self-organizing work group whose members make strategic decisions about their work unit. The circle therefore has a semi-autonomy, which makes it the time-space of strategic decision-making based on the equivalence of the participants (unus inter pares). When it comes to choosing a person from the circle to take on a particular role (meeting secretary, representative to another circle, etc.), there is an election without candidates9 [START_REF] Buck | La sociocratie : les forces créatives de l'autoorganisation[END_REF]. Decisions in Circles are made by consent10 , which means that a proposal is adopted if and only if there are no further reasoned objections within the Circle. An argumentative objection is thus seen as constructive, since it justifies the circle's consideration of it and then modifying or even deleting the initial proposal [START_REF] Buck | La sociocratie : les forces créatives de l'autoorganisation[END_REF].

Finally, in order to articulate the circles between them and thus obtain governance, sociocracy foresees the representation of the lower level circle in the higher level circle by two persons, a procedure called double linkage. In the case of Florain, the colleges 11 designate the members who will represent them in the strategic committee and the accreditation commission (figure 1):

Survey and interviews

At the beginning of 2019, the Florain's management association agreed that we conduct a survey among the Florain's private users. We thus prepared a questionnaire in three parts (see appendix), with the objective of recovering usable data on many fields: the uses of the Florain (9 questions), the perception of the Florain (5 questions maximum) and the users of the Florain (11 questions maximum). While many of the questions classically asked respondents to choose from among several proposed answers, others offered them the possibility of expressing themselves freely through a text or of classifying words that come to mind when the concept of money is mentioned. When several questions related to the same subject, we tried to group them together as much as possible in a table, for which it was proposed to choose a nuanced answer ranked between 1 and 5. We improved the questionnaire by submitting it in advance to the association's steering committee, in order to obtain their comments, criticisms and proposals for improvement. The complete questionnaire was then programmed on the LimeSurvey software, which provides the necessary tools to anonymously process the answers to the different types of questions and to generate a link in the communication plan.

The survey took place between July 16 and October 1st, 2019, with a follow-up via newsletter and Facebook twice in September 2019. Due to the absence of an updated email address file, the steering committee suggested that we send the link of the questionnaire to all supporters, first through a mention in the newsletter concomitantly with a publication on their Facebook page, and finally through an article on their blog. 87 people responded in part or in full to the questionnaire, but only 65 declared themselves to be members of Florain for the year 2019 and 5 did not even answer this question. Of the 65 people who declared themselves members of the Florain in 2019, 57 answered all of the questions submitted to them. The association told us that as of December 31, 2019, the Florain has 531 individual members. In doing so, since we do not know their number at the end of the survey on October 1, 2019, we calculate the lower bound of the response rate (Figure 2), which in any case seems close to that obtained by Fare (2012): Our results are thus based on the 57 complete responses.

In addition to this survey, we conducted 15 interviews with individual Florain members. For this article, we have retained those conducted between the first and second confinement in France -i.e. between May 12 and October 30, 2020 -namely three semi-directive interviews: two with different founding members (and still active volunteers), designated by MF1 and MF2, and one with the employee. And on February 14, 2021, we attended the general assembly of the association, which made a first assessment of the shock undergone and of the projects to come, including the digitalization of the local currency.

Our analysis is therefore based on the following observations: 57 responses to questionnaires before the pandemic, three interviews between the first and second confinement in France, our participation in the general assembly on February 14, 2021 and publications on the association's blog.

The resilience of the Florain community

The community of private users of the Florain suffered a shock during the first confinement in France (March 17, 2020to May 11, 2020). Indeed, as the Florain only existed in paper form at that time, the lockdown effectively blocked its circulation, as the closure of shops except for basic goods not only deprived users of places to buy, but also of exchange counters allowing the replenishment of bills. This was particularly detrimental at this time of year, as spring was very conducive to shopping after the long winter in Lorraine, which was marked by a continental climate. But as soon as the first confinement ended, the community quickly sought to resume its exchanges, which testifies to a capacity for resilience, the dimensions of which we will specify.

The sociological dimension of community resilience in Florain

According to [START_REF] Dubuisson-Quellier | La consommation engagée[END_REF], consumption can be seen as a "space of contestation or social mobilization" within which consumers refuse certain modes of consumption linked to, for example, big box stores while seeking to create exchange alternatives at the local level.

Local currencies, because of their legal link to the euro, are not in a position to propose a monetary system that competes with the legal tender, but rather seek to modify consumption patterns, by "reinforcing the social link through our exchanges" and by "prioritizing local consumption through short circuits of proximity and quality 12 ". In so doing, users of a local currency, who before joining were merely consumers separated from each other by market logic, become consum'actor re-affiliated with a collective. This means that they are morally committed to a certain mode of consumption that is in line with the civic values contained in the local currency's charter of values. This can be seen very clearly in the case of the Florain (Figure 3):

Figure 3 -The feeling of being a good citizen by using a local currency

Question 10 of the survey: on a scale of 1 to 5 (1 strongly disagree, 5 strongly agree), would you say that using the Florain is a civic act? Rather than a boycott, which consists of refusing to buy a product to protest against the policy of a company or a state, the user of a local currency practices a sort of buycott 13 , a neologism that designates the tendency to buy products from companies that behave ethically, in this case those that are members of the local currency network. This is the type of "better consumption" that emerges from our survey of Florain users (Figure 4):

1
12 Preamble of the Manifesto for Citizen Complementary Local Currencies (MLCC) written by the MLCC network and available at this address: http://monnaie-locale-complementaire-citoyenne.net/adhesion-mlcc. 13 Dubuisson-Quellier (2018) reminds us that the buycott was already implemented in the 19th century with the buyers' leagues, which demanded the respect of certain social norms concerning, for example, decent working conditions. The boycott/buycott opposition is reminiscent of the opposition made by Hischman (1970) between exit and voice.

Figure 4 -The feeling of consuming better by using a local currency

Question 13 of the survey: on a scale of 1 to 5 (1 strongly disagree, 5 strongly agree), would you say that the Florain allows for better consumption? This form of consumer engagement through the use of a local currency appears to us to be an important factor in community resilience, in that it provides a strong motivation to continue the local currency experience even after a fairly long shock such as that associated with confinement.

1
In addition, as part of our survey, we asked respondents to write down three words that spontaneously come to mind when they think of Florain (Figure 5). Out of 74 different words, there were 59 hapax14 . We then asked them to rank the three words they had written down according to the importance given to them. The words local, ecology and ethics appear most often in the first rank. Finally, by crossing the frequencies of occurrence and the rank of importance of the words, we attempt to approach the central core of the social representations15 of individual users of the local currency, despite the small number of respondents (57). Such social representations shared by a large number of local currency users are a powerful way of building a common reality, or even a common identity, or of reinforcing it if it already exists [START_REF] Moscovici | Social representation and social explanation[END_REF], which helps to ensure the resilience of the community, as shown by the work of [START_REF] Edge | Understanding community resiliency in rural communities through multimethod research[END_REF]. We find the following words: local/locale, ecology/ecological, solidarity/solidarity, all of which appear explicitly in the Florain charter of values and are found in all local currency charters. They constitute a sufficiently strong cement in view of the current challenges (ecological transition, territorial development, the need to invent new forms of solidarity, etc.) to retain users -and perhaps even find new ones -despite the economic and health crisis.

Moreover, according to Abric's (1994) structural approach, any social representation is composed of a stable central core and a peripheral system, which serves to match an individual's social representations with the situations he or she faces. From this definition, we believe that another reason for the resilience of a community of local currency users is that the latter is able to act on the social representations of its users. More specifically, building on the work of Moliner ( 2015), who showed that by acting on the peripheral system it is possible to modify the representations of the central core, [START_REF] Fare | La portée des initiatives de monnaies sociales complémentaires en termes de transformation des représentations sociales, Communication au colloque francophone international cultures, territoires et développement durable[END_REF] highlight the two channels through which membership in a local currency can modify the social representation that users have of the currency: an argumentative process and, above all, the experimentation of new practices on a daily basis within a community. In a second stage, this evolution should allow users to question and modify their relationship to market16 exchange [START_REF] Tichit | Représentations sociales de la monnaie : contraste entre les citoyens et les porteurs de monnaies locales[END_REF], by "re-embedding" the economic in the social (Polanyi, 1944(Polanyi, /1985) ) or even by bringing out alternative paradigms to the market order.

Finally, beyond the spatial proximity between all the users, there is also an emotional17 and social18 proximity between the active volunteers. Thus, although this local currency functions according to a societal principle with free membership and participatory operation of the management association, we note differentiated forms of commitment: many users participate little or not at all in the operation of the Florain, while others have an almost political commitment within the structure [START_REF] Didier | Le profil socioéconomique des utilisateurs de monnaies locales en France. Le cas particulier du Florain à Nancy[END_REF]. To account for the particularly militant commitment of Florain's active volunteers in a world characterized by the weakening of lasting commitments, Ion (1992) uses the concepts of "distanced commitment" and "identity niches". The first one allows to show that from now on "the sacrifice of the private life on the altar of the Cause is not any more of setting" (Ion, 1992, p.82) and that the commitment is seen as reversible, which characterizes rather well the simple users of a local currency, who do not seek a priori a binding commitment. On the other hand, Ion (1992) shows that even within modern groups, "identity niches" can appear, i.e. "places where collective identity is reinforced when individual identities are difficult to live with, and where some of the members try to perpetuate an old-fashioned way of functioning, preserving in particular a strong internal sociability (a "private we") that allows for identity reassurance mechanisms" (Ion, 1992, p.91). And it seems to us that it is precisely this "private we" that contributes to the overall resilience of the community of Florain users, insofar as it allows for a form of mechanical solidarity within this group [START_REF] Durkheim | De la division du travail social[END_REF][START_REF] Durkheim | De la division du travail social[END_REF], linked to the values defended. This leads to a collective consciousness that facilitates the collective work that is essential to find the means to overcome a crisis.

Thus, the fact of being a member of a community of consum'actors, the social representations of the members and the existence of an identity niche are three key factors in the resilience of the community of individual users of Florain. They constitute the sociological dimension.

The organizational dimension of Florain's community resilience

The governance of a community plays a critical role in its resilience, as shown by [START_REF] Edge | Understanding community resiliency in rural communities through multimethod research[END_REF] and [START_REF] Norris | Community resilience as a metaphor, theory, set of capabilities, and strategy for disaster readiness[END_REF], among others. In the case of the Florain, we observed that the sociocratic governance of the management association allowed each member of the operational team, during and after the first containment, to keep in mind the association's operating structure, to cooperate, to communicate and to make sense of events. Therefore, important decisions concerning the daily management of the association could be taken quickly, since each person of the operational team knew perfectly the role he had to play. In particular, the decision to put the employee on short-time work was made in less than two weeks and the employee agreed to continue working a few hours a week to coordinate the work of the volunteers and update the association's administrative documents. Thanks to the association's governance, the entire user community was able to resume its exchanges after the shock.

It is clear that the individual behavior of the members (ethos), especially that of the active volunteers as they ensure the daily functioning of the structure, is a key factor in community resilience. It should also be noted that, as in many associations, the line between active volunteer and employee is tenuous [START_REF] Hély | À travail égal, salaire inégal : ce que travailler dans le secteur associatif veut dire[END_REF], especially since in the case of Florain the employee is a former volunteer. This allows the employee to carry out what the members of the Florain call "the work of binding the members together", which is decisive during a crisis such as that of covid-19, which weakens physical contacts. It is true that during the first confinement, all the members of the community found themselves isolated at home, even though the Florain has always wanted to be a convivial association, both in the common sense of pleasant relations among the members of a group, and in the sense given to it by [START_REF] Illich | La convivialité[END_REF], i.e., the capacity of a structure to be at the service of human beings and even to be on their scale.

To remedy this, the active volunteers made an effort to contact the providers by telephone, in order to maintain the link and see to what extent it was possible to help them at least in terms of communication. But it is the use of digital communication tools (Zoom, Jitsi...), once mastered by the majority, that allowed regular contacts. This use continued after the first confinement, leading to a mixture of remote and face-to-face communications/meetings, which enabled the members of the community to reconnect.

Moreover, the active volunteers of the Florain feel that the association has allowed them to express their skills effectively up to now, which makes them more inclined to accept possible changes resulting from the crisis that could contribute to the resilience of the Florain community. These changes are all the more accepted because they are discussed in the different governance circles where everyone has the opportunity to express themselves according to the principle of equivalence of individuals in sociocracy [START_REF] Romme | The sociocratic model of organizing[END_REF]. In the end, each Florain volunteer has a proven autonomy within his or her circle, so that, as a whole, they declare that they do within the association what they wanted to do when they joined it. This true satisfaction in terms of real individual freedom within the association is reminiscent of the notion of "capability" developed by [START_REF] Sen | Capability and Wellbeing[END_REF].

In addition, there is a strong link between the Florain community and its territory, if only because of the law, which obliges a local currency to circulate in a restricted geographical area. This implies a spatial proximity between the members. But it is the search for a social marking of this space, not unlike the social marking of legal tender described by [START_REF] Zelizer | La signification sociale de l'argent[END_REF], that makes a local currency specific. Some thinkers of local currency, like [START_REF] Lepesant | Éléments d'analyse pour se repérer dans un projet de MLC[END_REF], wonder whether we should not rather speak of "territorial currency", reminding us that "a catchment area 19 is a social and political construction rather than a geographically defined 'location'". This concept of catchment area -sometimes referred to as "living area", particularly in the case of the Florain -seems to us to be relevant to describe the way in which a local currency socially appropriates its territory, transforms it and accepts in return that it must adapt to it.

As far as the Florain is concerned, although the majority of service providers are located within a 40 km radius of the city of Nancy (figure 6), our survey shows that the "living area" of the Florain is immediately perceived by users as being included in the former administrative region of Lorraine, to the point that they feel they are contributing to the development of Lorraine by using the Florain in this small living area. In other words, the community of individual users of the Florain has been built on a strong geographical, historical and social identity, which is also reflected in the charter of values ("Relocate the economy by supporting local businesses in order to boost employment"), the publications of the association in the media (blog, Facebook, interviews in the regional press...) and our interviews conducted before the pandemic ("Let's think local, let's think Florain!" a volunteer told us). Such a territorial identity seems to us to be another key factor in the resilience of a community, a result that is found for communities of various natures [START_REF] Berkes | Community Resilience: Toward an Integrated Approach[END_REF]. Finally, the community of users of a local currency may also decide to forge more or less strong links with other actors in its catchment area, members of other communities. In their national survey, [START_REF] Blanc | Les monnaies locales en France : un bilan de l'enquête nationale 2019-20[END_REF] show that 90% of the responding local currencies have at least one relationship with another actor in the territory 20 , essentially with structures that are part of the transition or with local associations (culture, organic farming, short circuits, participatory housing, third-party sites, etc.). Some have links with more institutional partners (consular chambers, CRESS, traders' unions, etc.) and a minority are developing relationships with other social money systems. These are all opportunities to build a network of inter-community links. These links, even if they are weak in terms of frequency of contact, emotional intensity, intimacy and reciprocity of services rendered, can become strong [START_REF] Granovetter | The Strength of Weak Ties. Mark S. Granovetter[END_REF], in that they allow the ideas and values promoted by the local currency to reach people who are socially distant from the members. This increases the number of supporters and eventually users, which can contribute to the community resilience of the local currency, especially since some of them are also likely to provide material or human assistance to the organization [START_REF] Norris | Community resilience as a metaphor, theory, set of capabilities, and strategy for disaster readiness[END_REF]. Le Florain was able to count on a large donation from a member company during the first lockdown and on state aid (partial unemployment and solidarity funds) as well as specific aid from France Active Lorraine to partially compensate for the drop in revenue linked to the first lockdown.

19 "Bassin de vie" in French. 20 Even if they point out a lack of diversity in the partners of local currencies.

Thus, the ethos of the active volunteers, the sociocratic governance of the association and the inclusion of the community of individual users of Florain in a catchment area are three key factors of community resilience. They constitute the organizational dimension.

The institutional dimension of Florain's community resilience

Of course, the resilience of the community of individual Florain users is the result of sociological and organizational factors. But, unlike other communities, its resilience cannot be understood without adding an institutional dimension that is consubstantial with money. Generally speaking, economic institutions are a set of rules, norms, conventions and practices that structure relations between economic actors. This definition thus makes it possible to affirm that money is an economic institution and even an "element of permanence in a world in perpetual transformation" (Chavance, 2018, p.101). In particular, local currencies, because of their rules of issue and circulation, can therefore be seen as local economic institutions, which structure relations between members on the scale of a living area. Thus, the community of individual users of a local currency is from the outset part of a project to change society with a strong political dimension, even if it is at the local level [START_REF] Blanc | La communauté comme construction monétaire[END_REF]. This is reminiscent of the genesis of 22 social currencies, of which local currencies are a part, which, according to [START_REF] Laacher | Les systèmes d'échange local (SEL) : entre utopie politique et réalisme économique[END_REF], find a "prehistory in a long tradition of revolutionary utopia (Marx, Fourier, Proudhon, Owen, Gesell to mention only the main theorists), for whom social change passed through a work of domestication of the insolent power of money, through a desire to invert the links of subordination between the economy and politics. [START_REF] Guyomart | De l'État-souverain à la souveraineté subsidiaire des monnaies locales complémentaires[END_REF] sees "a political government of money by a group sharing common values". This multiplication of local monetary initiatives driven by the desire to "reappropriate the use of money by the citizen23 " is reminiscent, to some extent, of the concept of empowerment24 developed in the United States in the 1970s, to designate the search for greater participation of the population (with real power) in order to achieve local economic and social objectives.

According to [START_REF] Rocher | Introduction à la sociologie générale[END_REF], the notion of value corresponds to "the way of being or acting that a person or a community recognizes as ideal and that makes desirable or estimable the beings and behaviors to which it is attributed. We deduce that the system of values supported by a local currency, in that it is the result of a bottom-up25 approach, is capable of creating ethical trust (Aglietta and Orléan, 1998), which complements methodical trust -resulting from the fact that exchanges are repeated in good conditions -and hierarchical trust -linked to the guarantee provided by a superior authority26 recognized by users as inspiring trust. In this way, the community of individual users of the local currency takes on a particular dimension. Given that the means of payment are organized and structured by the managing association, it can be said that it a community of payment. If we add ethical trust, we obtain a local monetary community [START_REF] Blanc | La communauté comme construction monétaire[END_REF], which is not only associated with market relationships, but also with relationships of reciprocity, redistribution and solidarity. Our survey has shown that individual users of Florain recognize themselves quite well in this concept of local monetary community: Such a shared feeling of belonging to a local monetary community, which results from the very nature of the currency, constitutes in our opinion a strong cement contributing to community resilience. It is, in effect, a matter of its members pursuing a sort of ideal based on strong shared values, even when it has become impossible to use the local currency. We conclude that the resilience of the community of users of a local currency has as its main dimension the existence of this local monetary community, a product of the very nature of the local currency.

Conclusion

In this article, we sought to analyze the resilience of the community of individual users of a French local currency in the face of the sudden halt in exchanges during the first containment related to the covid-19 pandemic (March 17, 2020to May 11, 2020). Our study is based on the local currency of the Nancy Basin, the Florain, for which we have a field survey conducted before the pandemic, three interviews conducted between the first and second containment in France (May 12 and October 30, 2020), observations and figures obtained during participation in the association's general assembly and publications found on the structure's blog. This allowed us to highlight that community resilience was the result of sociological factors such as the feeling of being a consum'actor, shared social representations and the existence of an identity niche among active volunteers, but also of organizational factors resulting from the ethos of active volunteers, the sociocratic mode of governance of the association and the inclusion of the Florain community in a living area. But in the particular case of a local currency, we have shown that there is an institutional dimension, linked to its social and political nature, which makes the community of individual users a local monetary community. The latter is based on an ideal (linked to strong shared values), which members are keen to defend in the event of a shock, making it a key dimension of the community resilience of a local currency.

Although our study has only focused on one local currency, it has allowed us to analyze this case in depth. It would therefore be relevant to test this institutional dimension in the case of other French local currencies, and then internationally, in order to have an exhaustive multidimensional analysis of the resilience of a community of individual users of a local currency.

Figure 1 -

 1 Figure 1 -Florain's governance

Figure 2 -

 2 Figure 2 -Lower bound of the response rate

Figure 5 -

 5 Figure 5 -Frequency of occurrence of words spontaneously cited by respondents when thinking about the Florain

Figure 6 -

 6 Figure 6 -Florain users' feeling of contributing to the development of LorraineQuestion 13 of the survey: on a scale of 1 to 5 (1 strongly disagree, 5 strongly agree), would you say that by using the Florain you participate in the development of Lorraine?1 2 3 4 5 3,51 % 3,51 % 1,75 % 26,32 % 64,91 % [ Source : Survey on Florain by the author ]

Figure 8 -

 8 Figure 8 -Florain users' sense of belonging to a monetary community Survey question 10: On a scale of 1 to 5 (1 strongly disagree, 5 strongly agree), would you say: by using Florain, I feel part of a community 1 2 3 4 5 5,26 % 5,26 % 19,30 % 31,58 % 38,60 % [ Source : Survey on Florain by the author ]

  The particularity of local currencies is that they are built on strong values (preservation of the environment, development of the territory, social justice, reappropriation of the currency, etc.), which can be found, among other things, in a charter of values specific to each of them 21 , the drafting of which has mobilized many volunteers. We can see in this a confirmation of what[START_REF] Inglehart | The Silent revolution[END_REF] calls a "silent revolution", i.e. the preference of citizens for non-materialistic values, such as freedom of expression and quality of life, to the detriment of the materialistic values of the post-war period. Our survey has shown in particular the importance given by Florain users to these values (here the example of social justice):

	Figure 7 -Florain users' emphasis on social justice
	Survey question 10: On a scale of 1 to 5 (1 strongly disagree, 5 strongly agree), would you say
	that Florain works for social justice?
	1	2	3	4	5
	5,26 %	3,51 %	12,28 %	38,60 %	40,35 %
	[ Source : Survey on Florain by the author ]

This subjective feeling is based on a traditional or affective dimension.

Family relations are a good example, since they have a strong communal dimension, but they are also governed by rules (family law, civil law, etc.) with a social dimension.

"healthy people, healthy country".

[START_REF] Parker | The question of organization: A manifesto for alternatives[END_REF] are actually interested in the types of organization related to anarchism.

In practice, there are several ways to proceed. One way is to ask the facilitator to introduce the position or role to be filled, and then each member of the circle proposes a name and justifies his choice. At the end of the round, everyone can change their vote, provided they give their reasons.

In the case of Florain, the statutes (article

11) provide that in the event of "the impossibility of obtaining the consent of the assembly, and only on items that cannot be postponed (due, for example, to scheduling imperatives), decisions may be taken, exceptionally, by a majority of those present". 11 Florain's governance is made up of 5 colleges, with one person belonging to only one college: college of the operational team, college of service providers, college of users, college of charter guarantors, college of partners.

A hapax, short for hapax legomenon, is a word that has only one occurrence.

Following Jodelet (2003), we can define social representation as "a form of knowledge, socially elaborated and shared, having a practical aim and contributing to the construction of a reality common to a social group".

[START_REF] Tichit | Représentations sociales de la monnaie : contraste entre les citoyens et les porteurs de monnaies locales[END_REF] has shown in the particular case of the local currency la doume, that the members of the association managing the project do not have the same perceptions of the currency as the general public.

The affective closeness between active volunteers is evident from our interviews and is characterized, among other things, by the organization of convivial moments.

This social proximity is notably characterized by an overrepresentation of highly educated people (nearly 50% of the respondents to our survey have a degree higher than 5 years).

Preamble of the Manifesto for Citizen Complementary Local Currencies (CCLC) written by the MLCC network.

Bacque and Biewener (2013) remind us that the concept of empowerment is flourishing in the United States in all fields, from local government to social action and education. Historically, it was used by feminist activists, the popular education movement, battered women's groups, and black movement activists to claim political representation for their communities.

In contrast to the traditional top-down approach, where management makes a decision and the lower levels of the hierarchy implement it, the bottom-up approach starts with the citizens closest to the ground. The latter then build together answers to the various problems, which will be implemented by the higher levels.

In the case of a local currency, the hierarchical trust results from the close link between the local currency unit and the euro, in that the managing association is obliged to deposit the euro counterparty in a reserve fund, usually with an ethical financial institution.

Appendix -The Survey Questionnaire

Le Florain and you

This questionnaire, entirely anonymous, is carried out by Raphaël DIDIER, doctoral student at the BETA laboratory of the University of Lorraine (Nancy). It is part of a university research on local currencies and should help to better understand the socio-economic profile of Florain users.

The data collected will be subject to statistical processing for academic purposes in accordance with the regulations in force, while guaranteeing anonymity and non-disclosure to the general public. The results will be presented to the Florain association.

For the success of this work, it is important to have many participants and that they answer all the questions (estimated time of only 15 minutes). Because of the construction of this survey, it is not possible to move on to the next question without having answered the previous ones. Rest assured, you can save your answers at any time and resume later.

We thank you in advance for your participation.

For any question: raphael.didier@univ-lorraine.fr

The uses of Florain

Your perception of Florain

On a scale of 1 to 5 (1 strongly disagree, 5 strongly agree), you would say:

Choose the appropriate answer for each item: Choose the appropriate answer for each item: We thank you very much for answering this questionnaire! Very soon, you will be invited to discover the results of this survey during a restitution within the Florain association.


Wencan Zhu

C Éline

AND L Évy-Leduc

Nils Tern Ès

A VARIABLE SELECTION APPROACH FOR HIGHLY CORRELATED PREDICTORS IN HIGH-DIMENSIONAL GENOMIC DATA

Keywords: variable selection, highly correlated predictors, genomic data

In genomic studies, identifying biomarkers associated with a variable of interest is a major concern in biomedical research. Regularized approaches are classically used to perform variable selection in high-dimensional linear models. However, these methods can fail in highly correlated settings. We propose a novel variable selection approach called WLasso, taking these correlations into account. It consists in rewriting the initial highdimensional linear model to remove the correlation between the biomarkers (predictors) and in applying the generalized Lasso criterion. The performance of WLasso is assessed using synthetic data in several scenarios and compared with recent alternative approaches. The results show that when the biomarkers are highly correlated, WLasso outperforms the other approaches in sparse high-dimensional frameworks. The method is also successfully illustrated on publicly available gene expression data in breast cancer. Our method is implemented in the WLasso R package which is available from the Comprehensive R Archive Network.

Introduction

The identification of prognostic genomic biomarkers (i.e. biomarkers associated with a variable of interest, for example a clinical endpoint in clinical trials) has become a major concern for the biomedical research field. Indeed, prognostic biomarkers may help to anticipate the prognosis of individual patients and may also be useful to understand a disease at a molecular level and possibly guide for the development of new treatment strategies [START_REF] Kalia | Biomarkers for personalized oncology: recent advances and future challenges[END_REF]).

To this end, statistical variable selection approaches are widely used to identify a subset of biomarkers in high-dimensional settings where the number of biomarkers p is much larger than the sample size n. Several reviews focused on this topic [START_REF] Saeys | A review of feature selection techniques in bioinformatics[END_REF] and [START_REF] Heinze | Variable selection -a review and recommendations for the practicing statistician[END_REF] for example). Commonly used techniques include hypothesisbased test: t-test [START_REF] Mcdonald | Handbook of Biological Statistics 2nd Edition[END_REF]), wrapper approaches [START_REF] Saeys | A review of feature selection techniques in bioinformatics[END_REF]): forward, backward selection, and penalized approaches: Lasso [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]), Elastic-net [START_REF] Zou | Regularization and variable selection via the elastic nets[END_REF]), SCAD [START_REF] Fan | Variable selection via nonconcave penalized likelihood and its oracle properties[END_REF]) among others. Hypothesis tests are limited to independently consider associations for each biomarker thus neglecting potential relationships between them. Wrapper approaches often show high risk of overfitting and are computationally expensive for high-dimensional data [START_REF] Smith | Step away from stepwise[END_REF]). More efforts have been devoted to penalized methods, given the attractive feature of automatically performing variable selection and coefficient estimation simultaneously [START_REF] Fan | Statistical challenges with high dimensionality: Feature selection in knowledge discovery[END_REF]). We shall thus focus on this type of approaches in the following.

Let us consider the following linear regression model:

(1) y = Xβ + .

where y = (y 1 , . . . , y n ) T is the variable to explain (clinical endpoint), X = (X 1 , . . . , X p ) is the design matrix containing the expression of biomarkers such that the correlation matrix of its columns is Σ, β = (β 1 , . . . , β p ) T is a sparse vector to estimate, namely with a majority of null coefficients, and is the error term. The Lasso penalty is a well-known approach to estimate β with a sparsity enforcing constraint. It consists in minimizing the following penalized least-squares criterion [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]):

(2) L λ (β) = y -Xβ 2 2 + λ β 1 , where • 2 is the Euclidean norm and β 1 = p k=1 |β k |. However, the Lasso has several drawbacks in highly correlated settings [START_REF] Zou | Regularization and variable selection via the elastic nets[END_REF]) such as the violation of the Irrepresentable Condition (IC) defined in [START_REF] Zhao | On model selection consistency of lasso[END_REF]. The authors of this article prove that this condition is necessary and sufficient to recover the support of β, namely to retrieve the null and non null components in the vector β and thus to provide a sign consistent estimator. This condition is defined as follows. Let S = {j, β j = 0} be the set of active variables, S c the set of non-active variables and X A the submatrix of X containing only the indices of columns which are in the set A. Then, the design matrix X satisfies the IC if, for some constant η ∈ (0, 1],

(3) X T S c X S (X T S X S ) -1 sign(β S ) j ≤ 1 -η, for all j, where sign(x) = 1, if x > 0, -1 if x < 0 and 0 if x = 0. Intuitively, this condition means that the correlation between the active and non active explanatory variables is smaller that the correlation between the active explanatory variables. Hence, this condition is most likely to be violated when the correlations between non active and active variables are large. In high-dimensional genomic data, this condition is difficult to guarantee as the correlation between biomarkers is usually high [START_REF] Michalopoulos | Human gene correlation analysis (hgca): A tool for the identification of transcriptionally co-expressed genes[END_REF]). [START_REF] Wang | Precision lasso: Accounting for correlations and linear dependencies in high-dimensional genomic data[END_REF] tested the irrepresentable condition on several publicly available genomic data and highlighted that the condition is violated in almost all the datasets investigated.

Methods have been proposed to deal with the issue of high correlations between the biomarkers. Preconditioning the Lasso is one of them. It consists in transforming the given data X and y before applying the Lasso criterion. For example, [START_REF] Jia | Preconditioning the lasso for sign consistency[END_REF] and [START_REF] Wang | High dimensional ordinary least squares projection for screening variables[END_REF] proposed to left multiply X, y and thus in Model (1) by specific matrices to remove the correlations between the columns of X. A major drawback of the latter, called HOLP (High dimensional Ordinary Least squares Projection), is that the preconditioning step may increase the variance of the error term and thus may alter the variable selection performance. Another recently published method named Precision Lasso [START_REF] Wang | Precision lasso: Accounting for correlations and linear dependencies in high-dimensional genomic data[END_REF]) proposes to handle the correlation issue by assigning similar weights to correlated variables. This approach revealed better performance than the other methods when the biomarkers were highly correlated. However, it failed in more favorable cases when the biomarkers are not correlated.

In this paper, we propose an alternative and novel approach, called Whitening Lasso (WLasso), to take into account the correlations that may exist between the predictors (biomarkers). Our method proposes to transform Model (1) in order to remove the correlations existing between the columns of X and thus to "whiten" them and make the IC valid but without changing the error term . This prevents us from noise inflation, see (4). Then, the variable (biomarker) selection is performed thanks to the generalized Lasso criterion devised by [START_REF] Tibshirani | The solution path of the generalized lasso[END_REF]. The full details of our method are provided in Section 2. An extensive simulation study is presented in Section 3 to assess the selection performance of our approach and to compare it to other methods in different settings. WLasso is also applied to a publicly available dataset in breast cancer in Section 4. Finally, we discuss our findings and give concluding remarks in Section 5.

Methods

In this section, we propose a novel variable selection approach called WLasso (Whitening Lasso) which consists in removing the correlations existing between the biomarkers (columns of X) and in applying the generalized Lasso criterion proposed by [START_REF] Tibshirani | The solution path of the generalized lasso[END_REF] for variable selection purpose.

2.1. Model Transformation. Inspired by the literature on preconditioning, we propose to rewrite Model (1) in order to remove the correlation existing between the columns of X. More precisely, let Σ -1/2 := U D -1/2 U T where U and D are the matrices involved in the spectral decomposition of the symmetric matrix Σ given by: Σ = U DU T . We then denote X = XΣ -1/2 . Therefore, (1) can be rewritten as follows:

(4) y = X β + ,

where β = Σ 1/2 β := U D 1/2 U T β.
With such a transformation, since the n rows x 1 , . . . , x n of X are assumed to be independent Gaussian random vectors with a covariance matrix equal to Σ, the covariance matrix of the rows of X is equal to identity and the columns of X are thus uncorrelated. The advantage of such a transformation with respect to the preconditioning approach proposed by [START_REF] Wang | High dimensional ordinary least squares projection for screening variables[END_REF] is that the error term is not modified thus avoiding an increase of the noise which can overwhelm the benefits of a well conditioned design matrix.

To illustrate the benefits of our methodology, observations y were generated according to Model (1) with p = 500, n = 50, β having 10 non null components which are equal to 2 and with Σ defined by (5) Σ = Σ 11 Σ 12 Σ T 12 Σ 22 where Σ 11 is the correlation matrix of active variables with off-diagonal entries equal to α 1 , Σ 22 is the one of non active variables with off-diagonal entries equal to α 3 and Σ 12 is the correlation matrix between active and non active variables with entries equal to α 2 . In the case where (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), Figure 1 displays the percentage of components j for which the Irrepresentable Condition (3) is not satisfied from 100 replications. We can see from this figure that our approach (WLasso) dramatically improves the number of indices j for which the IC condition is satisfied. The results are even better than those obtained by the transformation proposed by HOLP [START_REF] Wang | High dimensional ordinary least squares projection for screening variables[END_REF]).

The following illustrations of Section 2 are obtained from observations y generated according to the previous scenario.

2.2. Estimation of β. In order to estimate β with a sparsity enforcing constraint on β, we use the generalized Lasso criterion proposed by [START_REF] Tibshirani | The solution path of the generalized lasso[END_REF] which consists in minimizing the following criterion with respect to β:

y -Xβ 2 2 + λ Dβ 1 ,
where D is a specific matrix. Note that this criterion boils down to the classical Lasso criterion if D is the identity matrix. In Model (4), we thus propose to minimize the following criterion with respect to β:

(6) L gen λ ( β) = y -X β 2 2 + λ Σ -1/2 β 1 ,
which guarantees a sparsity enforcing constraint on β thanks to the 1 penalty. We thus obtain

β 0 (λ) = Argmin β L gen λ ( β).
To estimate β, we will not directly use β 0 (λ) but the following modified estimator which can be seen as a thresholding of the components of β 0 (λ). For K in {1, . . . , p}, let Top K is defined by: ( 7)

β (K) j (λ) =    β 0j (λ), j ∈ Top K Kth largest value of | β 0j |, j ∈ Top K .
Figure 1. Proportion of components j such that (3) is violated. These results were obtained from 100 replications.

The choice of K is explained in Section 2.4.

Figure 2 displays the average of (| β

( K) j (λ) -β j (λ)|) 1≤j≤p
for all the values of λ that are considered. We can see from this figure that the thresholding improves the estimation of β.

2.3. Estimation of β. As previously, to estimate β, we will first consider β 0 = Σ -1/2 β and then apply a thresholding strategy. Thus, we propose to estimate β by β = ( β

( M ) j ) 1≤j≤p
where β (M ) j is defined by: ( 8)

β (M ) j (λ) = β 0j (λ), j ∈ Top M 0, j ∈ Top M .
The choice of M is explained in Section 2.4.

As we can see from Figure 3, more true non null (active) components of β (true positive) and more true null (non active) components of β (true negative) can be retrieved with β than with β 0 . 2.4. Choice of the parameters. To choose the parameters K and M in ( 7) and ( 8) for each λ, we use a strategy based on the Mean Squared Error (MSE). We shall first explain the strategy that we used for choosing K.

Let MSE K (λ) = y -X β (K) (λ) 2 2 ,
where y, X and β (K)

(λ) are defined in (1), ( 4) and ( 7), respectively and

K(λ) = Argmin K ≥ 1 s.t. MSE K+1 (λ) MSE K (λ)
≥ γ , where γ ∈ (0, 1).

Large values of γ will lead to large values of K(λ) and thus to a weak thresholding of the estimator of β. In practice, as it is shown in Section 3, taking γ in (0.9,0.99) provides satisfactory and almost similar results.

For the choice of M (λ), we use the same procedure except that MSE K (λ) is replaced by

(9) MSE M (λ) = y -X β (M ) (λ) 2 2 ,
where y, X and β

(M )
(λ) are defined in ( 1) and ( 8), respectively. Both criteria are displayed in Figure 4 for a value of λ which is chosen according to the strategy explained in Section 3.2.

Estimation of Σ.

Since the matrix Σ is unknown in practice, it has to be estimated. In the particular situation where Σ has the block structure described in ( 5), we propose the following strategy. Firstly, we compute the empirical correlation matrix as follows. Let S be the sample p × p covariance matrix defined by where x i denotes the ith row of X defined in (1). The corresponding p × p sample correlation matrix R = (R i,j ) is defined by:

S = 1 n -1 n i=1 (x i -x) (x i -x) , with x = 1 n n i=1 x i ,
(10) R i,j = S i,j σ i σ j , ∀1 ≤ i, j ≤ p,
where

σ 2 i = 1 n -1 n =1 (X ,i -X i ) 2 , with X i = 1 n n =1 X ,i , ∀1 ≤ i ≤ p.
Secondly, the two groups (or clusters) of active and non active biomarkers are obtained by using a hierarchical clustering with the complete agglomeration method. Thirdly, the entries of Σ are computed by averaging the values of R within the groups. More precisely, let ρ i,j denote the value of the entries in the block having its rows corresponding to Cluster i and its columns to Cluster j. Then, for a given clustering C:

(11) ρ i,j =            1 #C(i)#C(j) k∈C(i), ∈C(j) R k, , if C(i) = C(j) 1 #C(i)(#C(i)-1) k∈C(i), ∈C(i),k = R k, , if C(i) = C(j)
,

where C(i) denotes the cluster i, #C(i) denotes the number of elements in the cluster C(i) and R k, is the (k, ) entry of the matrix R defined in (10).

We illustrate the performance of our method in Figure 5 in the case where Σ has the structure ( 5) with (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7). We can see from this figure that the proposed methodology for estimating the correlation coefficients within the blocks of Σ is very efficient.

2.6. Summary of the WLasso method. The WLasso method can be summarized as follows:

• First step: Estimation of the matrix Σ by Σ, see Section 2.5. • Second step: Transformation of Model (1) into Model (4) to remove the correlation existing between the columns of X, see Section 2.1 where Σ is replaced by Σ. • Third step: Estimation of β defined in (4), see Section 2.2. • Fourth step: Estimation of β defined in (1), see Section 2.3 and identification of its null and non null components.

Numerical experiments

We performed numerical experiments to assess the performance of the WLasso and to compare it with other recent approaches.

All simulated datasets were generated from Model (1) in which the number of predictors (biomarkers) p is equal to 100, 200, 500, 1000 or 2000 and the sample size n is equal to 50 or 100. We randomly chose 10 non null coefficients among the p coefficients of β which correspond to the active biomarkers, thus considering different sparsity levels. The value b of the non null coefficients is equal to either 0.5 or 1 to consider different signal-to-noise ratios.

Regarding the correlation matrix Σ which contains the correlation values between the biomarkers, namely the correlations between the columns of the design matrix X, several structures were considered:

• Block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7) and (0.5, 0.7, 0.9); • Independent setting where Σ is the identity matrix. The results that are presented hereafter are obtained from 100 replications.

3.1. Estimation of Σ. To evaluate the impact of the estimation of Σ, simulations were performed to compare the performance of WLasso when Σ is known and when it is estimated. The results are displayed in Figure 6 for several values of γ (0.9, 0.95, 0.97) which is a parameter appearing in Section 2.4. In the top left part of this figure the largest difference between the True Positive Rate (TPR) and False Positive Rate (FPR) is displayed for several values of p and for n = 50. In the top right and bottom parts of the figure, the corresponding TPR and FPR are displayed, respectively. We can see from this figure that for the value of λ maximizing the difference between TPR and FPR and for all the values of γ, all the active variables are properly retrieved without selecting non active variables when Σ is known. In the case where Σ is estimated by using the approach described in Section 2.5, 75% of the active variables are recovered and less than 1% of non active variables are wrongly estimated as active variables. Note that the results displayed in Figure 6 are obtained when (b, n) = (0.5, 50) but we obtained similar conclusions for (b, n) = (1, 50), (b, n) = (0.5, 100) and (1, 100).

3.2. Choice of λ. For tuning the parameter λ involved in our methodology, we propose choosing the value which minimizes MSE M (λ) (λ) defined in (9). In Figure 13 of the Supplementary material, we compare the performance of our approach with this choice of λ (solid line) to the optimal one obtained when λ is chosen to yield the largest difference between the TPR and the FPR (dotted line). We can observe from this figure that, for the different values of γ, the TPR is a little bit smaller when λ is estimated but that the FPR is quite similar. Similar results were obtained for (b, n) = (1, 50), (b, n) = (0.5, 100) and (1, 100). However, for larger b or n, the difference between the performance obtained by the optimal choice of λ and by our choice of λ is smaller.

3.3.

Comparison with other methods. In this section, we compare our methodology with other approaches: the classical Lasso described in [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] and two recently proposed methods aiming at handling the correlations between the columns of the design matrix X: HOLP and Precision Lasso proposed by [START_REF] Wang | High dimensional ordinary least squares projection for screening variables[END_REF] and [START_REF] Wang | Precision lasso: Accounting for correlations and linear dependencies in high-dimensional genomic data[END_REF], respectively. This comparison is performed by computing the TPR and FPR of these approaches for different values of the parameters involved in each of them.

The grid of λ for the classical Lasso and for our approach is provided by the glmnet and genlasso R packages, respectively. Concerning the Precision Lasso, we found for each value of n and p the λ min and λ max leading to p non null estimated coefficients and p null estimated coefficients, respectively. Then, we chose 100 values of λ uniformly distributed in the interval [λ min , λ max ] and we used the light implementation of the Precision Lasso. As for HOLP, β is estimated by βHOLP = X T (XX T ) -1 y. Then, for each s in {1, . . . , p}, the components of β which are estimated as non null are the s largest among the | βHOLP,j |, where βHOLP,j denotes the jth components of βHOLP . In this case, the parameter controlling the sparsity level of the estimator of β is s. It has a similar role as λ in the previous approaches.

Figure 7. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), b = 0.5 and n = 50.

The corresponding results are displayed in Figures 7 and8 in the case where n = 50 and b = 0.5 and Σ has the block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7) and (0.5, 0.7, 0.9), respectively. The top left part of these figures displays the largest difference between TPR and FPR for different values of p, which corresponds to an optimal choice of the parameters. For WLasso, we also display the results obtained when the parameter λ is chosen by using the strategy proposed in Section 3.2, γ = 0.95 and Σ is estimated using the procedure explained in Section 2.5. The corresponding TPR and FPR for each method are displayed in the top right part and bottom part of the figures, respectively. Note that we also conducted experiments in the case where b = 1. Since the conclusions are very similar, the corresponding figures are given in the Supplementary material.

We can see from Figures 7 and8 that WLasso outperforms the other methods: the TPR is one of the largest while the FPR is the smallest. HOLP has a larger TPR than WLasso. However, the associated FPR is much larger. It has moreover to be noticed that Lasso, HOLP and Precision Lasso are favored with respect to WLasso since their parameters were Figure 8. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.5, 0.7, 0.9), b = 0.5 and n = 50. chosen to optimize their performance in terms of TPR and FPR whereas, in WLasso, the parameter λ was chosen by using the strategy of Section 3.2 and Σ was estimated.

Figure 9 displays the results when the sample size n is increased and equal to 100. We observe from this figure that the overall performance has been improved and that our approach outperforms the others especially in the case where p is large. Similar results are obtained in the case where b = 1 and (α 1 , α 2 , α 3 ) = (0.5, 0.7, 0.9). We refer the reader to the Supplementary material for further details.

Figure 10 displays the performance of the different methodologies in the case where Σ = Id, n = 50 and b = 0.5, that is in the case where there is no correlation between the biomarkers (columns of X). We can see from this figure that even in this case, our method, which is designed for handling the correlation between the biomarkers, obtains similar results as the Lasso except for small values of p. However, when λ is chosen to obtain optimal results in terms of the difference between TPR and FPR, our approach achieves the best performance with HOLP. In the case where n = 100, our approach obtains the best results, see the Supplementary material. "maxsteps" (maximum number of steps/λs considered in the algorithm) involved in the genlasso R package. The timings were obtained on a workstation with 8GB of RAM and Intel Core i5 (2.4GHz) CPU. We can see from this figure that it takes only 6 minutes for processing data with our approach when n = 50 and p = 2000.

Moreover, we can observe from Figure 12 that the most time consuming step of WLasso is the one where the generalized Lasso criterion is used (blue part in Figure 12). However, the computational time of this step was divided by two when the parameter "maxsteps" was changed from 2000 (default value) to 500 without changing the variable selection results.

Application to gene expression data in breast cancer

We applied the previously detailed methods to publicly available data at Gene Expression Omnibus database (www.ncbi.nlm.nih.gov/geo), with accession code GSE2990, see [START_REF] Sotiriou | Gene Expression Profiling in Breast Cancer: Understanding the Molecular Basis of Histologic Grade To Improve Prognosis[END_REF]. A total of n = 189 tumor samples from patients with breast cancer were available and their microarray data have been collected on 2,283 probes. Expression data has been preprocessed and normalized as in the original publication. A filtering step based on the interquartile range (IQR) was considered to remove some probes as in [START_REF] Gentleman | Bioinformatics and Computational Biology Solutions Using R and Bioconductor (Statistics for Biology and Health)[END_REF]. We removed probes with IQR < 1.5 and those which lack of annotation. The remaining p = 1, 112 probes were then standardized. The goal of the application is to identify genes associated with the development of breast cancer. To this end, the expression of the gene BRCA1 (BReast CAncer gene 1) was considered as the variable y to explain (response variable). BRCA1 is a well known human tumor suppressor gene that helps to repair damaged DNA, thus the mutation of this gene can notably increase the risk Figure 12. Time allocation for each part of our method WLasso for p = 2000 and two values of the parameter "maxsteps". The time spent in the first three steps described in Section 2.6 is displayed in the first three bars in red, green, blue, respectively except the correction stage of Steps 3 and 4 (Equations ( 7) and ( 8)) which is given in the fourth bar in purple.

of breast cancer, see [START_REF] Rosen | BRCA1 gene in breast cancer[END_REF]. The standardized 1,112 probes were considered as explanatory variables and their correlation is displayed in the Supplementary material.

We compared the performance of the approaches investigated in Section 3 in terms of genes selection. For our approach WLasso, we used the methodology described in Section 2.5 for estimating the correlation matrix Σ. Assuming that it has the block-wise correlation structure (5), we estimated the coefficients α 1 , α 2 and α 3 by α1 = 0.17, α2 = 0.21 and α3 = 0.52, respectively. As for the Lasso method, the parameter λ was chosen by crossvalidation and the number of variables to be selected was fixed to 50 for Precision Lasso and HOLP in order to select approximately the same number of variables as with the Lasso.

Table 1 given in the Supplementary material provides the list of genes corresponding to the selected probes for each method. Unfortunately, HOLP could not provide any results since it requires the computation of the inverse of the matrix XX T which is not invertible in this case. The matrix X T is indeed not full rank in this dataset. The database for annotation, visualization and integrated discovery (DAVID) version 6.8 proposed by [START_REF] Dennis | David: Database for annotation, visualization, and integrated discovery[END_REF] was used to highlight the genes potentially related to the development of breast cancer, which correspond to true positives. Based on this tool, 8 genes were considered as true positives for WLasso and Lasso which have only one common gene: GSTT1. However, Lasso selected more false positives than WLasso (44 vs. 26). Among the 50 variables selected by Precision Lasso, 6 are identified as true positives by DAVID including the wellknown ESR1 gene in breast cancer. Based on this annotation, Precision Lasso detected less true positives than Lasso. Moreover, Wlasso identified more true positives with less false positives than Precision Lasso. Nevertheless, this application is for illustration purpose only and not for a fair evaluation or comparison of the methods.

Conclusion

In this paper, we proposed an innovative, efficient and fully data-driven method to deal with the variable selection issue in high-dimensional frameworks where the active variables are highly correlated with the non-active ones which is implemented in the WLasso R package available from the CRAN. The proposed WLasso method has been assessed and compared with other methods in a simulation study with several scenarios. In the highly correlated setting, WLasso successfully identifies more true positives with limited false positives as compared with the classical Lasso. Contrary to HOLP, WLasso still works when several columns are linearly dependent and does not suffer from the inflation of noise introduced by the preconditioning. Compared with the recent Precision Lasso approach, which aims to deal with the same issue, WLasso obtained better results in terms of selection accuracy in the different settings considered. WLasso is also very computationally efficient and demonstrated its abilities to properly identify genes related to breast cancer from a publicly available gene expression dataset. However, the following directions could be considered to improve its performance.

Firstly, the method that we used for estimating Σ could be improved by using more sophisticated approaches such as [START_REF] Perrot-Dockès | Estimation of large block structured covariance matrices: Application to "multi-omic" approaches to study seed quality[END_REF]. Secondly, our way of choosing the parameter λ for the final model selection could also be improved by considering crossvalidation or stability selection. Until now, a simple approach has been considered to avoid computation time and performed quite well especially for moderate to high simple size. Thirdly, most of the computational time of our approach is spent in the application of the generalized Lasso criterion. Hence, for an application to genomic datasets having more than twenty thousands of variables, it could be worth speeding it up. This will be the subject of future work.

Supplementary material

This supplementary material provides additionnal figures and a table for the paper: "A variable selection approach for highly correlated predictors in high-dimensional genomic data".

Figure 13 illustrates Section 3.2. Figures 14,15,16,17,18,19,20 and 21 provide similar results as those displayed in Figure 7 of the paper in the following cases:

• Block-wise correlation structure for Σ with (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0. 1 give additionnal information for the Application Section. Figure 16. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.5, 0.7, 0.9), b = 1 and n = 50.

Figure 17. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 of the paper (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) of the paper with parameters (α 1 , α 2 , α 3 ) = (0.5, 0.7, 0.9), b = 0.5 and n = 100.

Figure 18. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 of the paper (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) of the paper with parameters (α 1 , α 2 , α 3 ) = (0.5, 0.7, 0.9), b = 1 and n = 100. Table 1. Selected genes from each method, genes related to breast cancer are in bold.

  be the set of indices corresponding to the K largest values of the components of | β 0 |, then the estimator of β is β = ( β

Figure 2 .

 2 Figure 2. Boxplots of the average of (| β 0j (λ) -β j (λ)|) 1≤j≤p (left) and (| β

Figure 3 .

 3 Figure 3. Number of True Positive and True Negative for β in red and β 0 in blue for a given vector of observations y.

Figure 4 .

 4 Figure 4. MSE K (λ) (left) and MSE M (λ) (right) for λ chosen thanks to the strategy explained in Section 3.2 for a given vector of observations y. The vertical dotted lines correspond to K(λ) and M (λ), respectively.

Figure 5 .

 5 Figure 5. Estimation of the parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7). The horizontal dotted lines correspond to the true values of the parameters. These results are obtained from 100 replications for each value of p.

Figure 6 .

 6 Figure 6. max(TPR-FPR) and the corresponding True Positive Rate (TPR) and False Positive Rate (FPR) for (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), b = 0.5 and n = 50. Dotted line: Σ, solid line: Σ.

3. 4 .

 4 Numerical performance. Figure11displays the computational times of our approach implemented in the R package WLasso for different values of p and of the parameter

Figure 9 .

 9 Figure9. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), b = 0.5 and n = 100.

Figure 10 .

 10 Figure 10. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ = Id, b = 0.5 and n = 50.

Figure 11 .

 11 Figure 11. Computational time of our approach WLasso when the parameter "maxsteps" has the default value, namely 2000 (dotted line) and when maxsteps=500 (solid line).

  7) with b = 1 and n = 50 b = 1 and n = 100 • Block-wise correlation structure for Σ with (α 1 , α 2 , α 3 ) = (0.5, 0.7, 0.9) with b = 1 and n = 50 b = 0.5 and n = 100 b = 1 and n = 100 • Σ is equal to identity with b = 1 and n = 50 b = 0.5 and n = 100 b = 1 and n = 100 Figures 22 and Table

Figure 13 .

 13 Figure 13. (TPR-FPR) and the corresponding True Positive Rate and False Positive Rate for (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), b = 0.5 and n = 50. Dotted line: optimal choice of λ, solid line: choice of λ explained in Section 3.2 of the paper.

Figure 14 .

 14 Figure14. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 of the paper (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) of the paper with parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), b = 1 and n = 50.

Figure 15 .

 15 Figure15. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and (TPR-FPR) for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ has the block-wise correlation structure defined in (5) with parameters (α 1 , α 2 , α 3 ) = (0.3, 0.5, 0.7), b = 1 and n = 100.

Figure 19 .

 19 Figure 19. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and TPR-FPR for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 of the paper (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ = Id, b = 1 and n = 50.

Figure 20 .

 20 Figure 20. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and TPR-FPR for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 of the paper (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ = Id, b = 0.5 and n = 100.

Figure 21 .

 21 Figure 21. Top left: max(TPR-FPR) for Lasso, HOLP, Precision Lasso (PL) and TPR-FPR for WLasso obtained for the λ chosen by the strategy proposed in Section 3.2 of the paper (solid line). Results obtained for the optimal choice of λ for WLasso (dotted line). Corresponding TPR (top right) and FPR (bottom) when Σ = Id, b = 1 and n = 100.

Figure 22 .

 22 Figure 22. Heatmap of correlations between the probes.

  , RIOK3, RGS2, USP1, TRIM14, LPL, GGH, GSTT1, SMN1, MMP11, PSMB9, DST, RAD54L, TFAP2A, DSC2, GABRP, INPP4B, GHR, CD36, PTP4A3, ASS1, H2BFS, ANXA3, CXCL12, TPX2, CACNA1D, GOLGA8A, ATG5, SC5D, PTN, C1R, WWP1, DPT, MUC1, LRRC15, SMA4, CCZ1B, ASNSD1, COPS4, PSD3, VAV3, MS4A4A, KLF13, QPCTL, PLA2G12A, MUM1 PL H2AFZ, BTG2, SDC1,IGF2R, INSIG1, ALCAM, SDHD, ACADM, FBLN1, SNRPE, N4BP2L2 , SLC25A37, IGF1R, PPFIBP1, SYT1, SORBS2, AGA, IFI44L, TFF3, DSC2, MYB,ESR1, SETBP1, FGFR2, GOLGA8A, DNAJB6 , CD24,HLA-DQB1, JUP, SULF1, RDX, COL14A1, NFIB, COL6A1, KRT6B, NBPF10, N4BP2L2, DGKH, RGS1, ZSCAN18, LZTFL1, GREM1, C1orf115, DUSP12, KLHL2, ARMC9, DENND1B, TMPRSS3, WIZ, NTRK2 HOLP

Funding

This work was supported by the Association Nationale Recherche Technologie (ANRT).