Valérie Deldrève
email: valerie.deldreve@irstea.fr

Cécilia Claeys

Are National Parks Inherently Unequal? The 2006 French Park Reform and its Initial Implementation in Mainland France São os Parques Nacionais inerentemente desiguais? A reforma de 2006 dos Parques Franceses e sua implementação inicial na França continental

Keywords: sociology, French national park, environmental inequality, environmental effort, environmental justice

In France, like in other parts of the world, environmental-protection policies appear to have taken note of local resistance driven by the eviction of populations and restrictions on their use of certain areas. The amendment to the law on French national parks (2006 reform) as such recognized that local populations can help to protect natural environments, as well as their right to participate in defining the charter (or spatial plan) of new parks. This article analyses the creation and local acceptance of the Calanques National Park, a peri-urban park and the first of a new generation of parks in mainland France. Our approach is based on the analytical framework of environmental effort. Using a qualitative corpus of semi-structured interviews and direct observation, we explore three analytical axes. We begin by analyzing the consultation process conducted during the creation of the park. This points up inequalities in access to the public sphere and underscores that only a few, wellendowed user-organizations truly helped define the "good use" of the Calanques that became a reference in the park's charter. Then we examine how such unequal user participation impacted the wording used in the charter and the distribution of environmental effort demanded of different users. Lastly, the first two observations lead us to question the acceptability of the effort required and its division among users, particularly since the consultation process has been strongly criticized. We question how fairly the effort required by the park's creation was distributed, particularly given the lack of information and limited legitimacy afforded to the consultation process. Finally, we look at the blaming that has taken place amongst different users of the park -a process compounded for many of those interviewed by their fear of being deprived (unjustifiably, they felt) of their freedom to access nature.

Introduction

In France, the creation of national parks under the 1960 Parks Law has been the source of much conflict. Some of the country's parks remain mired in conflict, while the creation of other parks has been blocked altogether [START_REF] Larrère | Histoire(s) et mémoires des parcs nationaux[END_REF]. In an attempt to overcome the local resistance that inevitably accompanies having a natural site classified as a national park, the French government passed a new law in 2006 which grants greater power to local elected representatives, explicitly recognizes the rights and knowledge of local users, and invites them to participate in actually defining the park project. Our research into the creation of the new Calanques National Park, 1 however, indicates that the application of these new principles actually tends to reinforce the environmental inequality that they were meant to reduce. How should we interpret this apparent paradox?

The goal of this paper is to outline the interactive processes that worked to strengthen local environmental inequalities during the creation of the Calanques National Park. To do so, we will use the analytical framework of environmental effort. Environmental effort may be defined as the socially differentiated and potentially unfair contribution of social actors to public environmental protection policies [START_REF] Deldrève | Produire des inégalités environnementales justes… ou injustes[END_REF]. We will begin by looking at both the weight of procedures and the more structural factors underpinning the inequality of access to the public sphere. We will 1 CDE-Calanques: A new national park in the Calanques? French government CDE research programme, 2008-2011, V. Deldrève (Irstea) and P. Deboudt (Université de Lille 1) ed., FHUVEL: Human activities and vulnerability of coastal areas French government Liteau research programme, 2010-2013 (T. Tatoni ed., with C. Claeys). The analytical framework used to examine environmental inequalities and effort is that of the Effijie project (ANR Socenv, Candau andDeldrève, eds., 2014-2018): Environmental effort as inequality: Justice and iniquity in the name of the environment. For a comparative analysis of biodiversity and water policies in mainland France and its overseas departments.

then show how the procedures adopted for dialogue tend to reproduce the asymmetry of the public sphere (in the Habermasian sense) that developed throughout the 20 th century around the protection of the Calanques.2 Next, we will examine who was actually involved in defining and distributing such effort; how this dialogue helped legitimize a "good use" of the Calanques and grant certain privileges; and how it reinforced certain types of inequality in access to the Calanques based on tradition and merit. National parks are one of the most restrictive tools for protecting nature and the effort they require is in large part translated into regulatory terms. In the last section, we will present a typology of points of view based on the users interviewed, regardless whether or not they were represented in the consultation phase. This typology will allow us to examine the criteria and principles that individuals use to deem fair or unfair the feared and/or expected bans and restrictions on use -i.e., the effort potentially required of users -inherent to national parks.

Theoretical framework

Why talk about environmental inequality or environmental effort as part of protecting the environment? Environmental inequalities have become of increasing interest to the field of public policy and research. This has been true since the 1980s and 1990s in the United States, under the impetus of the environmental justice movement, and in Europe since the 2000s through the register of sustainable development. Semantics (ecological vs. environmental) aside, these inequalities are defined in the broadest sense as inequalities in the relationship that populations and social groups have with their environment. They relate to the impact that such groups have on their environment via their production and consumption patterns. They are also connected to their exposure to risk, access to natural resources and amenities, ability to take environmental action and to benefit from policies in this field. Such inequalities are compoundable and bound up with other types of inequality (e.g., socio-economic and cultural); they are a facet of environmental justice that public policy cannot ignore (including for reasons of efficiency). So, concretely, who has to support the effort that such policies require of current generations to protect the environment? Economists differ in their findings on this topic. Some argue that it is the wealthiest who offset their higher impact by contributing more to protection measures via their resources [START_REF] Lipietz | Économie politique des écotaxes[END_REF]; others claim that it is the poorest who contribute more despite the fact that their impact is lesser, as is the benefit they draw from it [START_REF] Pye | Addressing the social dimensions of environmental policy: A study on the linkages between environmental and social sustainability in Europe[END_REF]Martinez-Alier, 2008). Local protests over French national park policy [START_REF] Larrère | Histoire(s) et mémoires des parcs nationaux[END_REF] and its 2006 reform encourage us to look more closely at this controversial debate. The Calanques National Park -which is a product of the 2006 reform and is located on the doorstep of an agglomeration of two million inhabitants -is a particularly salient example since it touches directly on the effort required of locals and users of the Calanques.

We will argue that "environmental effort" is the source of much inequality. In practical terms, this concept refers to all positive and passive actions (e.g. non-consumption of a given resource) carried out to preserve natural resources and amenities. In terms of protected areas, environmental effort may come in the form of rules and regulations via ac-cess rights or the stipulation of which activities are authorized within a given zone. It can also involve restrictions to access. Restrictions are not always physical, however, since they may also be based on the value attached to relevant local amenities (natural beauty or the possibility for recreational activities, for example). Moreover, such value may differ depending on social category and type of use -and it is also dependent on cultural norms, which may place greater value on one type of use over another [START_REF] Deldrève | Produire des inégalités environnementales justes… ou injustes[END_REF].

It is therefore necessary to ask who defines what are "good uses" of nature and a fortiori of protected areas. We posit that inequalities in effort are directly connected to inequalities in participation as a peer in social life, in the sense defined by Nancy [START_REF] Fraser | Rethinking the Public Sphere: A Contribution to the Critique of Actually Existing Democracy, Social Text[END_REF]. The misrepresentation (Fraser, 2005)3 generated by consultation procedures therefore helps reinforce environmental inequalities since certain groups are not able to defend their vision and uses of the environment in the public policy sphere. More broadly, inequalities in participation are connected to structural inequalities -both socio-economic and cultural -which also result in unequal access to the "public sphere" in the Habermasian sense4 or, to paraphrase Iris Marion [START_REF] Young | Justice and the Politics of Difference[END_REF] and Nancy [START_REF] Fraser | Rethinking the Public Sphere: A Contribution to the Critique of Actually Existing Democracy, Social Text[END_REF], in a nonrecognition of the diversity of publics.

Methodology

We have been conducting sociological research into the Calanques National Park since 2008.

This began in the context of two research programmes, one focused on the consultation process surrounding the creation of the park (Deldrève and Deboudt, 2012) and the other interested in land-and sea-based uses of the territory of the future park (Tatoni et al., 2013) (Figure 1). This paper is based on a corpus that combines data collected during direct observation and from a qualitative survey (using semi-structured interviews). Between 2008 and 2012, we observed thirty-two dialogue meetings and conducted two hundred and thirty interviews with managers, experts, actors from the non-profit sector, and participating and non-participating users of the Calanques and surrounding islands. Among the users interviewed who did not participate in the consultation process, 125 were met directly onsite, primarily in one land-based area and three sea-based locations. The chosen land area -the Marseilleveyre massif between the Mont-Rose and Marseilleveyre Calanques -is one of the entry points into the Calanques closest to Marseille. The first sea-based location is a popular diving spot called Les Farillons, within the Riou archipelago. The second is Sormiou Bay, the largest natural anchorage area in the Calanques, famous among yachters; the third location is a natural anchorage zone in the Frioul archipelago particularly near the main town harbours. We recorded the interviews and meetings, and all interviews were transcribed.

We then thematically analysed this data-set manually and using nVivo© software.

Unequal access to the public sphere: who defines environmental effort?

Since the turn of the last century, the Calanques have been the focus of a great deal of protest by recreational groups (e.g. hikers, climbers, followed later by local hut owners and renters). Like in other parts of the world, such upper-and middle-class groups have banded together to oppose urbanization and industrial activity that pose a threat to the natural environment. Such groups were key to having the Calanques classified as a natural heritage site in 1973. They were also instrumental in the eventual creation of a national park, and some of them were board members of the Public Interest Group ("GIP")5 formed in 1999 to manage the Calanques' transition to National Park status.

Part of the process of creating a national park required a dialogue with local stakeholders, the goal of which was to collectively establish a charter -i.e., a territorial roadmap for the park.6 There were two guiding principles to how the consultation stage was constructed. The first, a reference to the Grenelle de l'Environnement (French political meetings in 2007 that resulted in environmental commitments), required having the five types of civil society actors represented (elected officials; government representatives; professionals; inhabitant-, user-and environmental protection associations; and researchers and technical experts). The GIP managerial team and its director (a senior government official) further identified an overarching category -comprised of owners, and private and especially public managers of the Calanques 7 -which was defined as having "priority" in the consultation process and with whom it communicated regularly during meetings prior to the consultation workshops. The second principle required the representation of a majority of users and activities from the sites. In keeping with these procedural norms, members of the GIP managerial team specifically identified groups of people who appeared most organized and legitimate.

As a result, the aforementioned recreational users of the Calanques -who are both active and resourceful -were essentially first in line. When the National Park was created, there were no recent data on the types of people that visit the Calanques. Thus, the wide variety of social backgrounds of such visitors and the diverse activities they take part in were not identified as part of the dialogue process. Further, no effort was made to redress the imbalance caused by the overwhelming presence of certain users in this process. This same group, whose leaders belonged to both user and resident organizations, did not shy away from exerting their influence. They did not hesitate to go "over the heads" of the GIP and contact local and national elected officials (and even the media) when they felt that their views were not given sufficient consideration by scientific and technical representatives.

Eventually, an agreement was struck between these users and the GIP team, whose job was to apply IUCN and French government philosophy and ecological specifications pertaining to national parks. Despite a series of controversies (e.g. over the impact of different practices), an "apparent consensus" [START_REF] Urfalino | La décision par consensus apparent. Nature et propriétés[END_REF] was reached and the practices of these recreational stakeholders were recognized as environmentally-friendly, "traditional" and "worthy" (Deldrève and Deboudt, 2012). Such practices were reframed through the lens of an ecological imperative: "We were the trailblazers of sustainable development", "our practices are ecocompatible" (hut owners' representative); and their impact -questioned by researchers -was described as minimal and responsible: "We fish responsibly, we know what we're doing, we do not stress the ecosystem so as not to disrupt it" (fisher, small leisure boat). The term "traditional" was never precisely defined and was taken to mean "longstanding, and passed down through generations": "Going down to the hut is a tradition -my tradition, and it comes from my grandparents. I spent all my holidays here as a child. Everyone knew each other. Now I come down with my grandchildren and the tradition lives on" (hut owner in Sormiou). However, research has shown that local power meetings (e.g., the hunting group, climbing group, etc.). In all, this represented roughly 150 meetings and 500 hours of debate, in addition to the drop-in office hours which were not particularly well attended. 7 This included the Office National des Forêts (National Forest Office), Marseille town council, Conservatoire du Littoral (coastal protection agency), Bouches du Rhône General Council, Cassis town council, Ministry of Defence, Electricité de France and private owners who had created an association since 2000.

dynamics also work to shape who is recognized as indigenous and which practices are liable to be classified as "traditional" [START_REF] Retière | Autour de l'autochtonie. Réflexions sur la notion de capital social populaire[END_REF]. In the case of the Calanques, some long-standing but littlevalued practices were not considered "traditional" despite the fact that they had been passed down for generations. During the dialogue process, many uses of the park were criticized without taking into account the modern technological developments tied to other "traditional" activities. The activities given the most merit were those involving physical exertion, such as walking and climbing (Figure 2), those that involved "roughing it", like staying in a hut with no electricity or running water, and those that required specialized knowledge of the local area, such as hunting (Figure 3), gathering food and fishing.

The attribution of merit to some activities occurred at the expense of other, less demanding activities, like tourist boating (Figure 4), picnicking and sunbathing. The allegedly "good uses" (Massena-Gourc, 1994) of the park had a relationship with nature that was primarily ascetic and contemplative (Figure 5) [START_REF] Fabiani | L'amour de la nature[END_REF]. The fact that this category was comprised almost exclusively of activities enjoyed by individuals of middle and FIGURE 2 -Rock climbing (Claeys, 2015). upper socio-economic status worked to create social inequality and further drove environmental inequality through the attribution of merit to certain activities in public policy.

Unequal access to the Calanques: who should shoulder the environmental effort?

When the charter for the Calanques National Park was drawn up, the three-faceted recognition enjoyed by some groups (as ecological, traditional and "worthy") meant that they were able to more easily defend the privileged status they had attained during the dialogue process. For example, the charter forbids fires within the park -with the exception of barbecues at people's huts. All dogs are to be kept on a lead -except those used for hunting. Initially, "traditional" fishing competitions were allowed, but spearfishing competitions were not. 8 Moreover, the road leading to the Sormiou Calanque (closed in summer to limit the risk of fire and regulate access to the Calanque and maritime public property) remained under the authority of the mayor9 and, according to our observations, the hut renters' association and its manager, who have the final say over who is granted access.

The lack of voices defending alternative uses of the national park contributed to such uses losing credibility. The users involved in these other activities tended to be young or low income local residents, as well as tourists. Their activities were deemed the cause of "excessive amounts of visitors" or labelled "environmentally harmful and antisocial". While perception alone could not stop these groups from accessing the park, their activities were viewed as a nuisance that needed to be controlled and contained. And as a result of such perception, greater environmental effort was required of the members of these less privileged groups. Our research, as well as that conducted by researchers in the fields of environmental justice [START_REF] Taylor | The Rise of Environmental Justice Paradigm. Injustice framing and the Social Construction of Environmental Discourses[END_REF][START_REF] Evans | Nature" et la justice environnementale[END_REF] and political ecology [START_REF] Peluso | Situer les political ecologies: l'exemple du caoutchouc[END_REF] point up that such problems are never purely environmental. One example of a particular group being subject to this kind of criticism is the case of the young, low-income residents who live near the Sormiou Calanque. These residents congregate in the area to socialize and partake in leisure activities, much to the displeasure of local hut renters: "The kids go directly to the beach, that's all they've got […] they walk there or hitchhike […] like I used to, also. I did that and so I taught my kids to do the same and so that's what they do" (resident of La Cayolle); "[…] Now, if I want to hang with all my buddies, I go to Sormiou" (young inhabitant of La Cayolle, forester).

There have been complaints about arguments and public disorder, sometimes escalating into physical violence: "I called the police. They come down here, they damage everything, insult you. And if you say anything, they set things on fire" (hut owner in Sormiou). The council where these people live has long suffered from a bad reputation in the public eye and is reputed to be an unsafe place to visit: "It's Chicago […]. You can't let tourists go there, they'll get robbed" (resident from a neighbouring district). At the same time, the environmental value of the district -historically 8 The ban on competitions was later extended to include all categories in the final version of the park's charter. designed as a place of social relegation successively populated by waves of immigrants -has increased over time, first as a result of its proximity to the Calanques, and then with the pending creation of a national park. Because of this, it is becoming more and more attractive to property developers and people of wealthier social standing [START_REF] Hérat | Une politique de nature à l'épreuve du territoire: le quartier de la Cayolle et la calanque de Sormiou (Marseille)[END_REF]. This process seems to be pushing the most economically distressed populations out of the district, i.e. farther from the natural amenities that originally formed the backdrop for their living conditions and leisure activities: "My father would say, 'come on, get ready, we're going to the hill…let's go take a little walk'. […] Now they [her children] go to Sormiou because Sormiou's their spot" (resident of La Cayolle).

The 2006 French National Park Law calls for the rights and knowledge of local users to be recognized. Its application in the case of the Calanques National Park, however, was based on local power relationships and dominant norms regarding "good uses" that determined which local users were able to benefit from such recognition [START_REF] Ginelli | Bien pratiquer la na-ture… pour protéger les Calanques ?[END_REF]. These norms were apparently consensual during the dialogue process, as they were more broadly in the public sphere, including within the protest movement against the creation of the national park. This movement was driven by a group composed of professional stakeholders who felt they were not sufficiently consulted (such as aquatic sports professionals) who were quickly joined by "traditional" recreational users who dissociated themselves more or less sustainably from the GIP (e.g., climbers opposed to the initial plan for a strict nature reserve; the majority of hut owners and renters; and hunters). Such protest came therefore not from a "weak public" or a "counterpublic" in the sense coined by Nancy Fraser; rather, it was conducted under the same banner as those participating in the consultation process and users who supported the park -i.e., it advocated respect for worthy, local habits and customs.10 But, while partisans of the park's creation felt that the park and the authority of its director (appointed by the government) would ensure the protection of the Calanques and their traditions (notably against the incessant push of urbanization projects), its opponents feared that the director would overshadow the authority of local representatives and place "in the hands of a senior government official" the future of their traditional practices, already threatened by the tourist appeal of the national park label.

And yet beyond the visible public sphere itself, discord over the park's creation was based less on the image of the park and the role of the government than on the regulation that inherently comes with a national park.

Acceptance/unacceptance of environmental effort and feelings of injustice

The rules and regulations inherent to the creation of the Calanques National Park can be interpreted as a legal and potentially coercive formalization of the definition of a "good use" of nature and of its implications in terms of distributing environmental effort. The stricter regulations that would come with the Park's creation were of great concern to all users, both those who participated and non-participants in the consultation process. The different arguments advanced do not fit neatly into simple categories of "for" or "against" the regulations, however. They can be categorized into five ideal-types.

In support of more regulation: From this perspective, greater regulation surrounding access to the Calanques and its archipelagos in the context of the national park's creation was desired or at least seen as inevitable. There were two levels of support for this position, which need to be further broken down.

-A (vindictive) request for greater regulation: Such calls for greater regulation were mainly aimed at other users, thus pointing up the existence of conflicts of use [START_REF] Claeys | Protected Areas and Overuse in the Context of Socio-Natural Changes: An Interdisciplinary French Case Study[END_REF]; all types of users could be accused or the accuser. For example, professional and amateur fishermen blamed each other for destroying fish resources; divers and recreational boaters mutually accused each other of damaging the sea bed with their moorings, while together they condemned jet-ski users who seemed to embody "bad use" given the noise pollution and excessive speed of their activity: "It's dangerous because you don't need a real licence. People think they're on a circuit but there are people out there swimming, spearfishing and freediving, etc. It's totally nuts, and very dangerous-and not sufficiently regulated" (a recreational boater). "In a national park, a sport that disrupts nature and all other users should not be allowed…" (a swimmer and walker).

-(Resigned) acceptance of increased regulation: Here, the increased regulation that comes with the creation of a national park was seen as inevitable and people as such felt they would need to resignedly adapt their practices. It was as such that this recreational boater mentioned the predictable restrictions to anchoring in open roadstead: "Outer mooring damages the sea bed. So it's going to be forbidden one of these days. It's absolutely inevitable […] to accept such constraints if we want to still be able to enjoy this in twenty or thirty years". Similarly, this amateur gatherer-collector acknowledged that, "Everyone needs to make an effort. We don't have a choice if we want to keep our Calanques. Sure, the park is a pain if we can't pick and gather or go where we like… But if we want to keep all this, we can say that, despite the constraints, it's a lesser evil."

A rejection of greater regulation: Opposition to increased regulation was also based on two more or less radical positions.

-Existing regulation deemed sufficient: From this perspective, regulation per se was not the issue, but the existing regulations were deemed sufficient. This view tended to be accompanied by remarks underscoring that it would suffice to correctly enforce the existing legislation rather than create more rules. A hut owner as such noted, "It's a conservation area. It was created years ago. So why go and add another layer and more constraints now?" A walker and recreational boater also mentioned an economic argument: "Particularly since there are already laws in place and they are not enforced. So what is the point of creating others? What's the point of creating a park that's going to cost hundreds of millions of euros a year when everything already exists? Why not just enforce what already exists?" -A rejection of the principle of regulation: This opposition to regulation was directly tied to a refusal to let go of one's own use of the Calanques and its archipelagos. Users defended the environmentally-friendly nature of their activities in the Calanques. They underscored their concern for the site's conservation and deemed it more efficient than any coercive regulatory tool could be. This position included forms of protest against public authorities as well as against researchers and scientific experts. One recreational boater as such criticized the connivance between decision-makers and researchers: "When these people become hon-est… I'm talking mostly about the politicians and maybe also the researchers who don't want to upset the politicians because it could lead to less money for them." Such opposition also took the form of identity-based claims: "We are the guardians of the Calanques," said one local; "We are the sentinels of the sea" said a fisherman. As such, they felt they should remain the sole trustees of "good use" and its prescription. "We go see the tourists and we tell them not to leave their garbage lying around, or to be careful -some go off wearing sandals, without any water or a map. We also help prevent the risk of fires. And now they want to teach us lessons like children. But we didn't wait for the park" (hut owner, hunter).

For an educational and incentive-based alternative: This position was a compromise between the acceptance and refusal of regulation. It advocated for a sort of third way based on information, awareness-raising and education amongst users. This walker weighed prevention and repression: "So, yes, work to prevent rather than repress." Another reasoned in terms of education: "But you need to educate people about respecting the site…" One diver suggested, "that we not head in the direction of more restrictions and regulations. I think it is more important to inform, to communicate […] sincerely. If we explain things a bit, the reasoning behind things, everyone is capable of understanding, especially us, those involved with the coastal life of the Bay of Marseille. It's our work, so we all have an interest in protecting it." Awareness-raising should as such be the first goal of the park: "If the park bans something but no one explains why, it won't work. People will always find a way to do it…" (a walker, educator).

While the position of those interviewed towards regulation was certainly influenced by their use and profile, it can by no means be reduced to such variables alone. The diversity of points of view was visible both amongst users described as "traditional" during the consultation process and amongst other users. The acceptance of effort was rooted not only in people's relationship with the natural environment, but also in their relationship with others. Much of the discourse was tinged with distinctions between "good" and "bad" users and the refusal to partake in the shared effort was in large part based on the belief that this distinction was unfair. In conducting interviews, two types of feelings of injustice emerged. The first set was related to the conditions in which effort was defined and who had to support it -and it was directly tied to criticism about the consultation process. The second set pertained more directly to the way this effort was distributed and to the related sense of inequality therein. Feelings of injustice were at times reinforced by a sense of lost legitimacy and even of dispossession.

During the interviews, users were asked about their awareness of the on-going consultation process over the national park project. Overall, awareness of the consultation process was quite limited. Of the users met in the field that were aware of the consultation process, most were critical of it. Their criticism hinged on 3 main points: 1. The unrepresentative nature of the population invited to take part in the consultation meetings, deemed too restrictive and not representative of the whole community: "I believe that hut owners are consulted, but, personally, I've never seen anyone doing surveys in the street or…" (a walker); "No one ever consulted us about anything at all, there are CIQs (citizen committees), I think, but they're limited…" (an inhabitant of a local neighbourhood in the Calanques). 2. The fact that some stakeholders could speak louder than others and were heard at the expense of other user groups: "The hunters want to hunt, the fishers want to fish, the walkers want to walk, everybody […], because you always see the same people in these discussions, it's odd, isn't it? You always see the same faces, be it the politicians or the others there, you see their photos in the paper and they're in the discussions but, in the end, nothing ever moves forward" (a walker). 3. The "divisive alliance" between policy-makers and GIP scientific advisers, who had allegedly decided everything in advance: "I attended two things and it was a non-consultation process. It was… They convened people to say the public was consulted, but the project was already finalized and… Afterwards, there was a general uproar, so they pretended to convene all the different players, but everything was already decided, so… In my opinion, they gave a little bit, some food for thought, and they backtracked on two or three things" (a recreational boater).

Many of the users we met were particularly reticent about accepting the environmental effort required by the creation of the Calanques National Park since they felt their impact on the environment was insignificant compared to the impact of other sources of disruption and pollution. While there was disagreement between them over whether it was underwater hunters, recreational fishermen or professional traditional fisherman who depleted more fish stocks, or whether it was the climbers or walkers who caused more damage to the fauna and flora, everyone agreed that their respective impact needed to be put into perspective compared to the waste from the treatment plant for the city of Marseille that is dumped into the Cortiou Calanque or the industrial red mud that flows into the Bay of Cassis. The most vindictive comments came from sea users involved in leisure boating, scuba diving and fishing: "Based on that, I find it completely abnormal that there is waste like that and then they go and make a natural park, you see. You get the impression that they're passing the buck to others, you know, to the people" (couple of recreational boaters). Some recreational fishermen expressed their dismay: "We're just out to catch a couple of fish, we will never deplete the sea, I promise you […]. They're going to ban it on us and that's enough… You'll see, things are going to get tough then."

Users felt that it was environmentally pointless and unfair to force them to pollute-or fish less when there were factories that dumped their sludge directly into the water of the future park, and industrial fishing boats trawling off-shore. Some users as such questioned the relevance of a park given the economic interests and asymmetrical relationships: "What's the point of a park if there's no way to stand up to the big guys? It's always the little guys who get hit" (traditional fisherman). Making an effort in this context was seen as bearing responsibility for the damage done by others.

Lastly, this sense of injustice tended to be exacerbated in the case of a peri-urban national park given the spatial proximity between the city and the protected area (Figure 6), as well as due to the multiple and even ambiguous status of this nature for urban users who viewed it as either -or at once both -remarkable and ordinary (Figure 6), wild and domestic [START_REF] Claeys | Marseille et ses natures: perméabilités spatiales, segmentations sociales[END_REF].

In other words, it is particularly frustrating to have to change the way you use a place when it is nearby and familiar. The issue hinged on both the sense of injustice over who was responsible for environmental problems, and on the sense that an indigenous relationship with nature -one that was special and experienced as "free" -was being challenged for reasons that remained incomprehensible or unacceptable. "The Calanques are our garden and we have taken care of them. We have always been able to pick things and fish… Now they're going to tell us to stop or to stay on the paths. Do you find that fair?" (a local, hunter's wife); "They're going to treat us like Indians and make a park for the tourists" (hut owner). Without addressing the issue of who owns the land, the views collected during the interviews point up several types of appropriation: "It's our Calanque [it belongs to us, hut owners]"; "It's the garden of Marseille's inhabitants" (Figure 6); "(…) The people need their Calanques, because it's their Calanques… It's freedom… You see, it's their oxygen… Freedom of movement…"; "The Calanques belong to everyone."

Equality and freedom of access for all or only for the inhabitants of Marseille were mentioned but not retained as principles of justice during the consultation process, in which the of good use and the distribution of effort were mainly conceived, as we saw above, through the lens of tradition and merit.

Conclusion

The 2006 reform to park policy embraced a more procedural approach [START_REF] Lascoumes | Le bien commun comme construit territorial, identités d'action et procédures[END_REF] and as such strengthened the power of actors from the public sphere -and, in this sense, it fostered democratic change. But it also reinforced a vision of democracy that was limited to "strong publics" [START_REF] Fraser | Rethinking the Public Sphere: A Contribution to the Critique of Actually Existing Democracy, Social Text[END_REF], users who were defending a natural site while simultaneously protecting their use of it. Whether this result is common to all "new generation" national parks or those affected by the 2006 reform remains to be proven. Our findings nonetheless echo research conducted into the history of nature protection around the world, shown to be strongly influenced by recreationist elites and their vision of the world (e.g., [START_REF] Cronon | The trouble with wilderness: or, getting back to the wrong nature[END_REF][START_REF] Benjaminsen | Discours et pratiques de conservation en Afrique[END_REF][START_REF] Selmi | L'émergence de l'idée de parc national en France. De la protection des paysages à l'expérimentation coloniale[END_REF]. In this sense, the result of the 2006 reform is more part of a continuum than an actual break with the past.

The 2006 reform also implicitly helped bolster environmental inequalities since it institutionalized a definition of "good use" that complied with the norms of certain groups. Environmental norms are FIGURE 7 -An entrance into the national park (Deldrève, 2015).

not always legal in nature. They are invisible, most often a tacit part of behaviour and help shape access to natural sites [START_REF] Evans | Nature" et la justice environnementale[END_REF]. But in a national park, they also have a regulatory dimension since it is necessary to distribute environmental effort.

If, more than personal interest alone, it is the principles used to justify the effort required that broadly condition its acceptance, then the challenge to come for the park will be to broaden the discursive sphere in order to re-open debate over what is fair. There is of course a risk to broadened dialogue since it will re-open debate in areas where was hard to reach. But such consensus was built on the exclusion of groups [START_REF] Rancière | La mésentente[END_REF] who are also asked to bear the burden of effort. Raising awareness about the topic of protection -a "priority" for some users and on the national park's agenda -might help encourage consent from those groups least familiar with these issues. That will not suffice, however, because people's relationship with regulation cannot be reduced to a question of awareness-raising alone. Rather, it touches on equity, both in terms of the distribution of effort and in terms of helping to define it. In this sense, the park cannot afford to shy away from the question of environmental justice.

FIGURE 1 -

 1 FIGURE 1 -The Calanques National Park (www.calanques-parcnational.fr/fr/mediatheque/cartotheque).

FIGURE 3 -

 3 FIGURE 3 -Sormiou huts (Cresp, 2012).

FIGURE 4 -

 4 FIGURE 4 -Tourist boating and swimmers (Cresp, 20120).

FIGURE 5 -

 5 FIGURE 5 -An amateur painter(Claeys, 2015)

FIGURE 6 -

 6 FIGURE 6 -Marseille from the National Park (Cresp, 2012).FIGURE7-An entrance into the national park(Deldrève, 2015).

The Calanques are a group of rocky coastal inlets located near the city of Marseille, in southern France.

Nancy Fraser argues that there are two such types of injustice. The first is "ordinary" and arises when populations are denied the opportunity to participate as peers in decisions that affect them, despite being recognized as members of the "political community". The second is more obscure and covert; it is the effect of a misframing which excludes from the decision-making community populations that "should" participate and are as such made invisible.

Let us recall that, for Jürgen[START_REF] Habermas | L'espace public. Archéologie de la publicité comme dimension constitutive de la société bourgeoise[END_REF][START_REF] Habermas | L'espace public. Archéologie de la publicité comme dimension constitutive de la société bourgeoise[END_REF]), the public sphere is a freely accessible space for rational debate and permanent, public deliberation. It is independent from public authorities, guided by the principle of publicity and as such equipped with critical power, a power of communication capable of opposing the more administrative power the State.

A GIP brings together legal entities governed by both public law, such as the government and local municipalities from the area, and private law (individuals). The general assembly of the Calanques' GIP (40 members) and its board of directors (24 members) is as such comprised of three colleges: the government and state-owned enterprises; local municipalities; and "civil society", i.e. the non-profit sector and private owners.

The consultation process lasted a year and was divided into five thematic workshops: "land uses", "water uses", "organizing the management of the core area", "understanding the heritage and character", and "ecological solidarity or the ideal peripheral area". Territorial workshops were also conducted alongside the above in the form of public meetings and private fora for elected officials. Finally, there were also several bilateral

According to the 2006 law on parks and following an amendment proposed by Jean-Claude Gaudin (the current mayor of Marseille) pertaining to municipalities of over 500 000 inhabitants, the municipality retains its authority to police matters related to the management of roads within the park's core area.

As an example, the goal set by the collective that served as an umbrella group for opposition to the park's creation was to "Protect the practice of cultural, professional, sporting and leisure activities while respecting nature and our traditions within the Calanques massif, harbour of Marseille and along the coast, both on land and at sea" (source: http://www.amisradecalanques.fr/?page_id=58).

processo de consulta. Finalmente, olhamos para as acusações de culpa trocadas entre os diferentes usuários do parque, um processo agravado para muitos dos entrevistados pelo medo de serem privados (injustificadamente, eles sentiram) de sua liberdade de acesso à natureza.


Fei Zheng
email: fei.zheng@doctorant.ec-lyon.fr

Stéphane Derrode
email: stephane.derrode@ec-lyon.fr

Wojciech Pieczynski
email: wojciech.piezynski@telecom-sudparis.eu

Fast Exact Filtering in Generalized Conditionally Observed Markov Switching Models with Copulas

Keywords: Markov switching model, CGLSSM, CGOMSM, GCOMSM, Copulas, Optimal filter, Triplet Markov chain

We deal with the problem of statistical filtering in the context of Markov switching models. For X N 1 hidden continuous process, R N 1 hidden finite Markov process, and Y N 1 observed continuous one, the problem is to sequentially estimate X N 1 and R N 1 from Y N 1 . In the classical "conditional Gaussian Linear state space model" (CGLSSM), where (R N 1 , X N 1 ) is a hidden Gaussian Markov chain, fast exact filtering is not workable. Recently, "conditionally Gaussian observed Markov switching model" (CGOMSM) has been proposed, in which (R N 1 , Y N 1 ) is a hidden Gaussian Markov chain instead. This model allows fast exact filtering. In this paper, using copula, we extend CGOMSM to a more general one, in which (R N 1 , Y N 1 ) is a hidden Markov chain (HMC) with noise of any form and the regimes are no need to be all Gaussian, while the exact filtering is still workable. Experiments are conducted to show how the exact filtering results based on CGOMSM can be improved by the use of the new model.

Introduction

Consider three random processes X N 1 = (X 1 , . . . , X N ), R N 1 = (R 1 , . . . , R N ) and Y N 1 = (Y 1 , . . . , Y N ). Each X n , R n , Y n takes their values in R m , Ω = {1, . . . , K} and R q respectively. The problem that we deal with is to find the unobservable (or hidden) processes (R N 1 , X N 1 ) from the observation

Y N 1 = y N 1 .
In the model we propose, we assume, as it is usually made, that both triplet (X N 1 , R N 1 , Y N 1 ) and R N 1 are Markov chains. The first markovianity then implies that the couple (

X N 1 , Y N 1 ) is Markovian conditionally on R N 1 . The distribution of (X N 1 , R N 1 , Y N 1
) is defined by the initial distribution p (x 1 , r 1 , y 1 ) and the transitions p x n+1 , r n+1 , y n+1 |x n , r n , y n which will be taken of the form p (r n+1 |r n ) p x n+1 , y n+1 |r n+1 n , x n , y n consistently with the Markovianity of R N 1 . Here (r n , r n+1 ) is denoted by (r n+1 n ) for short. In the "conditionally Gaussian observed Markov switching model" (CGOMSM) proposed in [START_REF] Abbassi | Optimal filter approximations in conditionally Gaussian pairwise Markov switching models[END_REF] and applied to general non-linear systems in [START_REF] Gorynin | Exact fast smoothing in switching models with application to stochastic volatility[END_REF][START_REF] Gorynin | Fast filtering in switching approximations of nonlinear Markov systems with applications to stochastic volatility[END_REF], the transitions p x n+1 , y n+1 |r n+1 n , x n , y n are assumed to be Gaussian with linear regimes. The aim of the paper is to extend the CGOMSM, which allows exact filtering, to a more general one in which p y n+1 |r n+1 n , y n are no longer limited to be Gaussian and the regime G (x n+1 |x n , r n+1 n , y n+1 n ) are no longer necessarily to be linear on the observations. The new model, called "Generalized conditionally observed Markov switching model" (GCOMSM), benefits from the copulas, which has been widely used in statistical finance for dependence description [START_REF] Choroś | Copula estimation. Copula theory and its applications[END_REF][START_REF] Roger | An introduction to copulas[END_REF]. Copulas were firstly introduced into hidden Markov chain (HMC) with dependent noise by [START_REF] Brunel | Unsupervised signal restoration using hidden Markov chains with copulas[END_REF], and importance of their role in segmentation efficiency is shown in [START_REF] Derrode | Unsupervised data classification using pairwise Markov chains with automatic copulas selection[END_REF][START_REF] Derrode | Unsupervised classification using hidden Markov chain with unknown noise copulas and margins[END_REF]. However, to our best knowledge, no work considers them in switching state-space models. In the proposed GCOMSM, p y n+1 |r n+1 n , y n is much more flexible compare to the original CGOMSM making use of copula. Experiments are conducted to show the interest of the new model with comparison to the result given by traditional Gaussian linear assumptions.

The paper is organized as follows. In next Sections, we recall CGOMSM, specify the general GCOMSM and show how fast optimal filtering and smoothing runs in this new model. Experiments are displayed and analyzed in the third Section. Finally, the conclusion and perspectives are given in the last Section four.

Generalized conditionally observed Markov switching model (GCOMSM)

Let us consider a CGOMSM, in which the Markov triplet (X N 1 , R N 1 , Y N 1 ) distribution is defined by p (x 1 , r 1 , y 1 ) = p (r 1 ) p (x 1 , y 1 |r 1 ) and transitions of the form p x n+1 , r n+1 , y n+1 |x n , r n , y n = p (r n+1 |r n ) p x n+1 , y n+1 |x n , y n , r n+1 n . Both p (x 1 , y 1 |r 1 ) and p x n+1 , y n+1 |x n , y n , r n+1 n are Gaussian. In [START_REF] Abbassi | Optimal filter approximations in conditionally Gaussian pairwise Markov switching models[END_REF], the CGOMSM is described by linear regime:

X n+1 Y n+1 = F xx (R n+1 n ) F xy (R n+1 n ) 0 F yy (R n+1 n ) F (R n+1 n ) X n Y n + U n+1 V n+1 , (1)
with F (R n+1 n ) an appropriate system transition matrix, and U n+1 , V n+1 represents the independent Gaussian zero-mean noise which are independent from

T n = (X n , R n , Y n ).
We see in CGOMSM, the pair (R N 1 , Y N 1 ) is a Markov chain, and

p x n+1 , y n+1 r n+1 n , x n , y n = p y n+1 r n+1 n , y n p x n+1 x n , r n+1 n , y n+1 n , (2)
which makes p r N 1 y N 1 can be computed, thus the exact filtering is feasible.

Under CGOMSM, p y n+1 |r n+1 n , y n is Gaussian and

G (x n+1 |x n , r n+1 n , y n+1 n
) is linear on x n , y n and y n+1 . However, to maintain the feasibility of exact filtering, the Gaussian setting and linear form are not necessary conditions.

Definition of Generalized conditionally observed Markov switching model (GCOMSM)

The Generalized conditionally observed Markov switching model (GCOMSM) which extends the CGOMSM considers still the triplet (X N 1 , R N 1 , Y N 1 ) a Markov chain, defined by p (x 1 , r 1 , y 1 ) and transition of the form

p x n+1 , r n+1 , y n+1 |x n , r n , y n =p (r n+1 |r n ) p y n+1 r n+1 n , y n p x n+1 x n , r n+1 n , y n+1 n , (3)
Unlike in CGOMSM, p y n+1 |r n+1 n , y n is enriched by copula represented as: ) in CGOMSM is extend to

p y n+1 r n+1 n , y n =f r n+1 y n+1 |r n+1 n c n+1 F l n+1 y n |r n+1 n , F r n+1 y n+1 |r n+1 n |r n+1 n , (4)
x n+1 = A n+1 r n+1 n , y n+1 n x n + B n+1 r n+1 n , y n+1 n + ν n+1 (5)
in GCOMSM, in which A n+1 (•) and B n+1 (•) can be any function forms of r n , r n+1 , y n , y n+1 . ν n+1 ∼ N (0, V n+1 (r n+1 n )). Integrally, they can be also written as

x n+1 ∼ N A n+1 r n+1 n , y n+1 n x n + B n+1 r n+1 n , y n+1 n , V n+1 r n+1 n . (6)

Fast exact filtering in GCOMSM

The Markov property of (

R N 1 , Y N 1 ) in GCOMSM leads to p x n r n+1 n , y n+1 1 = p (x n |r n , y n 1 ). Besides, since p (x n+1 |x n , r n+1 n , y n+1 n
) is Gaussian defined as (6), we have

E X n+1 x n , r n+1 n , y n+1 n = A n+1 r n+1 n , y n+1 n E [X n |r n , y n 1 ] + B n+1 r n+1 n , y n+1 n . (7) Then E X n+1 r n+1 , y n+1 1 is computable from E [X n |r n , y n 1 ] with E X n+1 r n+1 , y n+1 1 = rn p r n r n+1 , y n+1 1 A n+1 r n+1 n , y n+1 n E [X n |r n , y n 1 ] + B n+1 r n+1 n , y n+1 n (8) in which p r n r n+1 , y n+1 1 is computable because of the Markovianity of R N 1 , Y N 1 . More precisely, we can write p r n r n+1 , y n+1 1 = p r n+1 n , y n+1 1 rn p r n+1 n , y n+1 1 , (9)
and p r n+1 n , y n+1 1 can be calculated recursively with

p r n+1 n , y n+1 1 = r n-1 p r n-1 , r n+1 n , y n+1 1 = r n-1 p r n n-1 , y n 1 p (r n+1 |r n ) p y n+1 r n+1 n , y n , (10)
Finally, the filtering is given by

E X n+1 y n+1 1 = r n+1 p r n+1 y n+1 1 E X n+1 r n+1 , y n+1 1 . (11)
3 Example of GCOMSM and experiment on the matched exact filtering

We present here an example to show the flexibility of the proposed GCOMSM as well as the performance of the matched exact filtering. We focus on the timeindependent case of the general GCOMSM, which means that the parameters depend only on the switches (r n+1 n ). For simplification, we assume that (R N 1 , Y N 1 ) is stationary reversible, which means that p y n+1 |r n+1 n = p y n+1 |r n+1 , therefore the "left" and "right" margins in (4) are equal. Under these assumptions the equation ( 4) and ( 6) can be written as

p y n+1 r n+1 n , y n = f r n+1 y n+1 c r n+1 n F rn (y n ) , F r n+1 y n+1 (12)
x n+1 ∼ N A r n+1 n y n+1 n x n + B r n+1 n y n+1 n , V r n+1 n . ( 13
)
In place of the time dependence in original definition, the dependence on switches are moved to subscript of all functions. For this example, we assume that R N 1 has two component values Ω = {1, 2}. And for each j, k ∈ Ω, f j (y n ) = f rn=j (y n ), c j,k F j (y n ) , F k y n+1 = c rn=j,r n+1 =k F j (y n ) , F k y n+1 with F j , C j,k the associated CDF in [START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF]. In (13), the abbreviation is taken in the same way: A j,k (y n+1 n ) = A rn=j,r n+1 =k, (y n+1 n ), so as B j,k (y n+1 n ) and V j,k . The parameters of p y n+1 |r n+1 n , y n which are set to be non-Gaussian as 1. α1 and β1 are the shape parameters, loc1 and scale1 are short for location and scale. 2. β2 represents the shape parameter, loc2 and scale2 for location and scale.

-Margins: f 1 (y n ) = Beta 1 {α 1 = 0.9, β 1 = 0.9, loc 1 = -4, scale 1 = 6}, f 2 (y n ) = Fisk 2 {β 2 = 4, loc 2 = -2.7, scale 2 = 2.4}. -Copulas: c 1,1 {•, •} = Arch14 3 {•, •|α 1,1 = 3}, c 2,2 {•, •} = FGM {•, •|α 2,2 = 0.5}, c 1,2 {•, •} = c 2,1 {•, •} = Clayton {•, •
3. Short for Archimiedean copula, order: 14.

  where we use f l n+1 (y n |r n+1 n ) and f r n+1 y n+1 |r n+1 n to denote the probability density function (PDF) of the left and right margins respectively. Similarly, F l n+1 (y n |r n+1 n ), F r n+1 y n+1 |r n+1 n are their associated cumulative distribution function (CDF), while c n+1 (•, •|r n+1 n ) represents the density of the two-dimensional copula conditionally on switches. The copula above then completes the two margins to form a joint distribution p (y n+1 n |r n+1 n ) which can theoretically embrace any distribution form. Moreover, the simple linear regime G (x n+1 |x n , r n+1 n , y n+1 n ) which corresponds to p (x n+1 |x n , r n+1 n , y n+1 n

  |α 1,2 = 4.7}. The marginal and joint distribution are displayed in Figure1a, 1b.p (x n+1 |x n , r n+1 n , y n+1 n ) is set with A j,k (y n+1 n ) = a j,kx n , simple non-linear function on y n , y n+1 that B j,k (y n+1 n ) = b j,k y n y n+1 , and in which the parameters are assigned as -a j,k :a 1,1 = 0.2, a 1,2 = 0.4, a 2,1 = 0.6, a 2,2 = 0.8, -b j,k : b 1,1 = 0.7, b 1,2 = 0.5, b 2,1 = 0.6, b 2,2 = 0.9, -V j,k : V 1,1 = V 2,2 = 1.0, V 1,2 = V 2,1 = 0.8.

  (a) Margins (f1 (y n ):Beta, f2 (y n ):Fisk). (b) joint distribution of f1,2 y n , y n+1 . (c) Histogram of x N 1 . (d) Histogram of y N 1 .

Figure 1 :

 1 Figure 1: Distributions and histograms of simulated GCOMSM data.

2000 samples are simulated according to the above setting of GCOMSM. We see from the histograms of the simulated data illustrated in Figure 1c, 1d that they are hardly to be approximated by Gaussian mixtures with small component number. Exact filtering for GCOMSM is applied on y N 1 to restore the hidden r N 1 (decided by maximum posterior mode criterion from p (r n |y n 1 )) and x N 1 . For comparison, we conducted also the filtering based on Gaussian assumptions (both margins and copulas are assumed to be Gaussian) by using Maximum likelihood (ML) and Pseudo-Likelihood Maximization (PLM) [START_REF] Kim | Comparison of semiparametric and parametric methods for estimating copulas[END_REF] for Gaussian parameter estimation of margins and copulas applied on data. Restoration results are average of 100 independent experiments, illustrated in Table 1. We can see that the exact filtering performs well on restoring the hidden switches and states for GCOMSM, while the Gaussian assumption is obviously inferior comparing to the exact filter. Their performance can also be told from the trajectories. Figure 2 illustrates an trajectory example from one instance among the 100 experiment.

Conclusion

In this work, copula is introduced in the recent "conditionally Gaussian observed Markov switching model" (CGOMSM), and fuse to a more general one called "generalized conditionally observed Markov switching model" (GCOMSM). Experiments verify the capability of GCOMSM to work on data under flexible distributions. The fast exact filtering for GCOMSM can be much less time consuming comparing to using other non-Gaussian models which do not allow exact filtering and Monte-Carlo methods are needed to be applied. The future work may contain the model identification (identifying the margins, copulas [START_REF] Kodjo | Unsupervised image segmentation with pairwise Markov chains based on nonparametric estimation of copula using orthogonal polynomials[END_REF][START_REF] Derrode | Unsupervised classification using hidden Markov chain with unknown noise copulas and margins[END_REF][START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF], and also the conditional regime functions [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF]) of GCOMSM, and application of the model on non-Gaussian non-linear data restoration. In addition, smoothing can also be a perspective of interest.


Haoyu Li

Stéphane Derrode

Lamia Benyoussef

Wojciech Pieczynski
email: wojciech.pieczynski@telecom-sudparis.eu

Unsupervised Pedestrian Trajectory Reconstruction from IMU Sensors

Keywords: Pedestrian Navigation, Stance Detection, Inertial Sensor, HMMs

This paper presents a pedestrian navigation algorithm based on a foot-mounted 9DOF Inertial Measurement Unit, which provides accelerations, angular rates and magnetics along 3-axis during the motion. Most of algorithms used worldwide are based on stance detection to reduce the tremendous integration errors, from acceleration to displacement. As the crucial part is to detect stance phase precisely, we introduced a cyclic left-to-right style Hidden Markov Model that is able to appropriately model the periodic nature of signals.

Stance detection is then made unsupervised by using a suited learning algorithm. Then, assisted by a simplified error-state Kalman filter, trajectory can be reconstructed. Experimental results show that the proposed algorithm can provide more accurate location, compared to competitive algorithms, w.r.t. ground-truth obtained from OpenStreet Map.

Introduction

In recent years, Pedestrian Navigation System (PNS) has gained a lot of attention and been investigated extensively with various kinds of sensors like Inertial Measurement Units (IMUs), camera-based systems and WIFI-based ones. Among these sensors, IMUs have great advantages as they are small and can be wore on the body. They also do not need to pre-install devices like cameras or WIFI systems, and can be used both indoor and outdoor. With the kinematics information acquired from IMUs, it is theoretically possible to transfer the signals from sensor frame to earth frame, or called global frame, based on the sensor orientation, and then to compute velocity and displacement of the motion. Therefore, the exactness of IMUs-based PNS algorithms highly depends on the accuracy of orientation estimation and displacement computation.

When a person is walking, his foot swings in the air and does not move when attached to the ground, alternately. Consequently, one step can be broken down into four phases [START_REF] Mannini | A hidden Markov model-based technique for gait segmentation using a foot-mounted gyroscope[END_REF]: stance, push-up, swing and step-down phases. Stance phase is also called zero-velocity state, as the foot is not moving. The most common way to detect stance phase is to set thresholds for both acceleration and angular rate. To avoid manually setting thresholds, unsupervised learning methods can be used. J. Taborri et al. proposed an HMM-based distributed classifier for rehabilitation application [START_REF] Taborri | A novel HMM distributed classifier for the detection of gait phases by means of a wearable inertial sensor network[END_REF]. H. Pham et al. [START_REF] Pham | A LLE-HMM-based framework for recognizing human gait movement from EMG[END_REF] introduced a LLE-HMM framework and use EMG signals for gait recognition. A foot-mounted gyroscope for stance detection is implemented in [START_REF] Mannini | A hidden Markov model-based technique for gait segmentation using a foot-mounted gyroscope[END_REF], however because of the weak initialization, parameter learning fails at some time.

In order to reconstruct the trajectory, an algorithm called Pedestrian Dead-Reckoning (PDR) has been proposed, which computes the displacement by estimating and integrating each step length and heading. PDR is very easy to be implemented and used in many applications, but its error particularly depends on the sensor employed. E. Foxlin [START_REF] Foxlin | Pedestrian tracking with shoe-mounted inertial sensors[END_REF] firstly proposed a PNS algorithm that applies Extended Kalman Filter (EKF) to estimate the error and uses ZUPT approach to reduce the large integration error from acceleration to velocity and then to displacement. He called it INS-EKF-ZUPT (IEZ). Then, different strategies were proposed according to different measurements during the stance phase. S. Rajagopal [START_REF]Personal dead reckoning system with shoe mounted inertial sensors[END_REF] supposes the angular rate during stance phase to be zero and proposes a Zero Angular Rate Update (ZARU) algorithm to compute trajectory. The orientation error, particularly the yaw error, can also be estimated and added into the EKF measurement with the help of digital compass [START_REF] Norrdine | Step detection for ZUPT-Aided inertial pedestrian navigation system using footmounted permanent magnet[END_REF][START_REF] Jiménez | Indoor pedestrian navigation using an INS/EKF framework for yaw drift reduction and a foot-mounted imu[END_REF].

In this paper, we propose an adaptive stance detection algorithm that uses unsupervised parameter learning algorithm, and emploies a simplified Error-state Kalman Filter for PNS trajectory reconstruction. The remaining of the paper is organized as follows. First, a left-to-right HMM is presented to detect stance with a specific initialization algorithm for unsupervised parameter learning. Then a simplified errorstate Kalman filter is developed to compensate for integration errors. An experiment is conducted on true pedestrian data and the proposed algorithm is compared with other algorithms.

Stance Detection with Left-to-Right HMM

Precise stance detection plays a critical role in ZUPT algorithm, since if the detection result is wrong (i.e. the swing phase is regarded as a stance phase), then the velocity will be wrongly compensated to zero while the foot is still moving. 4}, where Ω represents the stance, push-up, swing and step-down phases respectively. Assuming a discrete time independent Markov process, X can be parameterized by an initial probabilities vector π = p(x 1 ) and a transition matrix A = p(x 2 |x 1 ). In a cyclic Left-to-Right HMM (LR-HMM), this transition matrix has the following particular shape, only allowing to switch from one class to the next:

Let start assuming a hidden Markov chain model with observations

Y = {Y 1 , . . . , Y N }, each Y n ∈ R, and with unknown states X = {X 1 , . . . , X N }, each X n = k ∈ Ω = {1, . . . ,
A =     1 -∆ 2 ∆ 2 0 0 0 1 -∆ 3 ∆ 3 0 0 0 1 -∆ 4 ∆ 4 ∆ 1 0 0 1 -∆ 1     , (1)
with ∆ k = p(x n = k|x n-1 ) the transition probability from state k-1 to state x n = k.

Here, the N observations represent gyroscope measurements y n = s ω xn s ω yn s ω zn , which represent the angular rate of the sensor projected in sensor frame. The distributions of observations conditional to classes are assumed to be Gaussian

p (y n | x n = k) ∼ N (µ k , Σ k ), (2)
where µ k (3 × 1 vector) and Σ k (3 × 3 matrix) are the mean and co-variance of observations corresponding to state k. So that the LR-HMM model we deal with is parametrized by the following set of parameters

Θ = {π k , ∆ k , µ k , Σ k } k∈Ω .
All the parameters can be learned using the well-known Baum-Welch algorithm, which is based on the EM principle (Expectation-Maximization) for finding the maximum likelihood iteratively, starting from an initial guess Θ (0) of parameters and stopping after a criterion or a maximum number of iterations is reached.

Commonly, the initialization is performed using Kmeans algorithm. However, in LR-HMM the state transition has a specific transition order and structure (see eq. ( 1)) that Kmeans is not able to provide since it does not takes into account past observations. As an illustration, Fig. 1 shows the gyro observations. The observation values close to zero indicate the stance phase. It can be seen that angular rate goes across zero at transition between two states, then Kmeans wrongly classifies the state as stance phase.

To find the true transition order form Kmeans classification results, we propose the algorithm sketched in the diagram 2. Firstly, we filter the angular rate by using a low-pass Butterworth filter, and select the most significant axis, i.e. the axis whose signal has the largest magnitude, like the blue dashed line in Fig. 1. Secondly, we search for the movement durations between every stance phase that lasts longer than a threshold, e.g. 0.3s for example, and then we find the states corresponding to all the peaks and valleys in every movement duration, the peaks and valleys indicate the states of non-stance phases. Thirdly, we sort these states by time and only keep the first one if one state repeats twice or more. Thus, a list of nonstance states order in every movement duration can be acquired. At last, we count the most firstly appeared state in the order list, which means the push-up phase, the state of swing phase and step-down phase can be derived in the same way. In our example, after the disposal of Kmeans results the re-ordered state transition is 1 → 2 → 3 → 4 (Orange line in Fig. 1). This guess of states is then used to initialize EM for parameter learning.

Error-state Kalman Filter

By knowing the stance phases, velocity is assumed to be zero and the velocity integration error can be acquired easily, but it is not able to obtain displacement integration error directly. Therefore the displacement integration error should be estimated appropriately, since it derives from velocity error and the correlation between velocity and displacement is determined by the integration function (5), thus it can be estimated by an appropriate way. Error-state Kalman filter is firstly introduced in [START_REF] Foxlin | Pedestrian tracking with shoe-mounted inertial sensors[END_REF] and gives a strategy to estimate the displacement integration error. In this work, We employ a simplified error-state Kalman filter (9 dimensional), since orientation estimation is performed independently by a gradient descent algorithm based quaternion method [START_REF] Madgwick | Estimation of IMU and MARG orientation using a gradient descent algorithm[END_REF]. The error-state only takes into account the acceleration (a), velocity (v) and displacement (r):

δη n = δη n|n = δ e r n , δ e v n , δ s a n (3)
The error-state transition equation writes

δη n|n-1 = Φ n δη n-1|n-1 + w n-1 , (4)
Where the superscripts e and s represent the earth frame and sensor frame respectively. w n represents the process noise with covariance matrix Q n = E (w n w n ) and where the error-state transition matrix Φ n is a 9 × 9 matrix given by

Φ n =   I 3×3 ∆t • I 3×3 0 3×3 0 3×3 I 3×3 ∆t • e s C n 0 3×3 0 3×3 I 3×3   . ( 5
)
Matrix Φ n is time-variant and depends on the value of e s C n , which represents the rotation matrix required to convert vectors from sensor frame to earth frame at time n. It is derived from the estimated quaternion [START_REF] Wang | Quaternion-based Kalman filter for AHRS using an adaptivestep gradient descent algorithm[END_REF]. Now, the measurement equation writes

z n = Hδη n|n + υ n , (6)
where z n is the measurement from sensor, H = [0 3×3 , I 3×3 , 0 3×3 ] is the measurement matrix, and υ n is the measurement noise, assumed Gaussian with covariance R n .

Because error-state measurement is only available when xn is detected as stance phase, the error-state is only updated during this period. The error-state measurement in stance phase is z n = e v n -[0, 0, 0] T (zero represents the real velocity), thus by the prediction and estimation in Kalman Filter [START_REF] Simon | Optimal state estimation: Kalman, H infinity, and nonlinear approaches[END_REF], the velocity and displacement can be compensated by e v n -δ e v n and e r n -δ e r n respectively, δ e r n and δ e v n in error-state δη n should be reset to zero after the compensation.

Experimental Results

An experiment was conducted on a road nearby the campus of École Centrale de Lyon, Ecully, France. The ground truth is obtained from Openstreet Map , the total travel distance is 1075m, the data was stored in the IMU embedded SD card. The sampling rate was set to 100 Hz, the range of accelerometers, gyroscopes and magnetometers were set to 8g, 1000deg/s and 2.5Ga respectively 1 . Before starting to walk, a short standing time period without motion is necessary for initializing the quaternion corresponding to the earth frame (North-West-Up coordinate system), the magnetic declination at Lyon is 1.2 • . The LR-HMM method we propose is tested and compared to another threshold based stance detection method detailed in [START_REF] Qiu | Inertial/magnetic sensors based pedestrian dead-reckoning by means of multi-sensor fusion[END_REF]. Parameters of both algorithms are learned or tuned for getting the best result. The total steps number in experiment is 1458, so the steps number of one foot is 729. Compared with the threshold based stance detection method, LR-HMM obtains a more regular stance pattern, rarely makes a false negative detection or misses one step (Fig. 4). Table . 1 reports the missing number and false negative detection number of both algorithms.

Trajectory reconstruction is done by different algorithms, including the proposed algorithm in this paper and a commonly used IEZ algorithm (15 dimensional error-state Kalman Filter). In Fig. 5, the proposed algorithm (HMM+GDA) makes a travel distance of 1077.7m, the relative travel distance error is 0.25%, the End-to-End error is 23.3m. From Tab. 2, in spite of the fact that the End-to-End error of HMM+IEZ is a little smaller than the proposed algorithm, the Dynamic Time Warping (DTW) distance of the proposed algorithm is smaller than HMM+IEZ. This interesting point means that the trajectory derived from the proposed algorithm is closer to the ground truth. And furthermore, when tuning parameters for all algorithms, HMM+GDA has the most robust performance for trajectory reconstruction.

Conclusion

We present an algorithm for trajectory reconstruction from a foot-mounted IMU sensor. The basic difficulty of such an algorithm mainly relies on minimizing the double integration required to calculate the displacement (earth frame) from the observed kinematic signals (sensor frame). We propose an algorithm that is mainly unsupervised and relies on a cyclic Left-Right HMM to mimic the periodicity of the step phases during a walk.

It seems that our algorithm performs better than the state-of-the-art methods, but we still need to experiment further the algorithm, to study large-scale motion, and also to study the performance with respect to the elevation when the terrain is not plane. We also plan to investigate if using two sensors together on both two feet can improve trajectory reconstruction results. In that case we have to study possible interference between the two sensors, and their influence on the reconstructed trajectory.

Figure 1 :

 1 Figure 1: Re-ordered initial state sequence: 1→2→3→4.

Figure 2 :

 2 Figure 2: Diagram of state transition re-ordering from Kmeans classification results.

Figure 3 :

 3 Figure 3: Shimmer3 sensor and placement on the shoe.

Figure 4 :

 4 Figure 4: Stance detection, step #1 represents the stance.

Figure 5 :

 5 Figure 5: Trajectory of different algorithms compared with the ground truth.

Table 1 :

 1 Stance Detection Error.

	Algorithm Steps Number Missing Number False Negative (rate in %)
	LR-HMM	735	0	6 (0.823)
	Threshold	724	17	14 (1.920)

Table 2 :

 2 PNS Trajectory Error.

		HMM+GDA Threshold+GDA HMM+IEZ Ground Truth
	End-to-End Error(m)	23.31	24.76	16.11	0
	End-to-End Positioning Accuracy(%)	2.16	2.30	1.49	0
	Travel Distance(m)	1077.69	1131.19	1078.52	1075
	Relative Error of Travel Distance(%)	0.25	5.22	0.32	0
	DTW Distance	8784	8617	12809	-