Thierry Feuillet
email: thierry.feuillet@univ-paris8.fr.

Jean-François F Valette

Hélène Charreire

Emmanuelle Kesse-Guyot

Chantal Julia

A Vernez-Moudon

Serge Hercberg

Mathilde Touvier

Jean-Michel M Oppert

Cédric Agaesse

PhD Fabien Szabo De Edelenyi

Julien Allègre

Nathalie Arnault

Laurent Bourhis

MD Fatoumata Diallo

MD Roland Andrianasolo

MD (physicians Sandrine Kamdem

PhD Nathalie Druesne-Pecollo

Influence of the urban context on the relationship between neighbourhood deprivation and obesity

Keywords: HGAM, multilevel model, obesity, local analyses, deprivation index, urban contexts

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In this study, we have examined the nonlinear and geographically varying relationship between neighbourhood deprivation and the likelihood of being a person with overweight among participants of the French NutriNet-Santé adult cohort study (n=68,698), adjusted for age, gender and educational level. Ten urban residential contexts (e.g., suburbs, peri-urban or rural areas) were defined. We used a multilevel generalised additive modelling framework for analyses.

Results

We found that the relationship between neighbourhood deprivation and overweight differed according to urban context, in terms of both linearity and intensity. Overall, the deprivation-overweight relationship was strongly positive (with a higher prevalence of overweight in deprived neighbourhoods) in suburban areas of Paris and of other large French cities, while weak or null in small towns and rural areas, and intermediate in inner cities. In addition, we observed in suburbs of Paris and in peri-urban belts of large cities that beyond a certain level of neighbourhood deprivation, the relationship with overweight plateaued.

Discussion

In a French population from a high-income country, suburbs, as well as moderately deprived neighbourhoods of peri-urban areas of large cities, are potential targets for public health and urban planning policies aiming at preventing obesity. Our results emphasize the value of local analyses to better capture the complexity and contextual variations of socioeconomic determinants of non-communicable diseases such as obesity.

Introduction

Overweight and obesity have now reached pandemic levels. In 2016, 39% of adults aged 18 years and over were overweight (BMI ≥ 25 kg/m²) with 13% categorized as obese (BMI ≥ 30 kg/m²), representing 1.9 billion and 650 million persons, respectively [START_REF] Who | Obesity: preventing and managing the global epidemic[END_REF]. Notably, the prevalence of overweight and obesity have risen very rapidly, and globally, over the last 40 years [START_REF] Swinburn | The global syndemic of obesity, undernutrition, and climate change: The Lancet Commission Report[END_REF]. This trend is shaped by global drivers, including global markets and food systems, reflecting broad socioeconomic influences [START_REF] Jaacks | The obesity transition: stages of the global epidemic[END_REF][START_REF] Swinburn | The global obesity pandemic: shaped by global drivers and local environments[END_REF]. In addition, there is spatial variation in BMI and obesity prevalence at different geographical scales, which remains poorly understood [START_REF] Sarkar | Association between adiposity outcomes and residential density: a full-data, cross-sectional analysis of 419 562 UK Biobank adult participants[END_REF].

Such spatial patterning suggests that local moderating factors interact with global drivers to structure the actual complexity of obesity geography. Specifically, neighbourhood socioeconomic disadvantage or deprivation [START_REF] Townsend | Deprivation*[END_REF] is a major local determinant of health [START_REF] Kawachi | Neighborhoods and health[END_REF]. In middle-and high-income countries, neighbourhood socioeconomic deprivation is positively associated with body mass index (BMI) and the prevalence of obesity [START_REF] Harrington | Weighing the importance of neighbourhood: A multilevel exploration of the determinants of overweight and obesity[END_REF][START_REF] Lenthe | Neighbourhood deprivation and overweight: the GLOBE study[END_REF][START_REF] Matheson | The weight of place: A multilevel analysis of gender, neighborhood material deprivation, and body mass index among Canadian adults[END_REF]. In addition, some studies highlighted variations in the relation between deprivation and obesity according to urbanicity levels [START_REF] Ford | Neighborhood Deprivation, Supermarket Availability, and BMI in Low-Income Women: A Multilevel Analysis[END_REF][START_REF] Wang | Population-adjusted street connectivity, urbanicity and risk of obesity in the U[END_REF][START_REF] Xu | Built environment and obesity by urbanicity in the U[END_REF]. Urbanicity is typically measured using an urbanisation ratio of population living in urban areas over the total population of a region or an administrative unit (e.g. county, department, etc.), with levels being established following the classic gradation of urban to rural, as found in the US and other censuses, and as adapted by different sectors, including health [START_REF] Wang | Population-adjusted street connectivity, urbanicity and risk of obesity in the U[END_REF]. While commonly used to characterise environmental contexts, urban-rural gradients fail to capture "how local areas or neighbourhoods affect health [and] how the broader spatial context within which neighbourhoods are situated may add to or modify the effects of local areas" (Diez Roux and Mair, 2010). For example, participants may be residing in different areas that are all classified as "urban" but have substantially different population densities, which could affect the deprivation-obesity relationship. In addition, participants living in areas classified as "large metropolitan areas" may be located in the area of the primate city (e.g. Paris, London, New York) or in other large metropolitan areas (e.g. Marseilles, Birmingham, Boston), which suggests that a residential context at different levels of the urban hierarchy may also affect the deprivation-obesity relationship.

In the present study, we used a sample of 68,698 French adults to examine how the relationship between overweight and neighbourhood deprivation varies by residential context. Accordingly, a typology of residential contexts was established based on urbanicity, measured using multidimensional criteria and taking into account the hierarchy of cities in terms of socioeconomic and demographic importance.

Individual and contextual data

Study design and participants

Individual data regarding weight status and other individual covariates were collected from participants of the NutriNet-Santé study, an ongoing web-based cohort launched in France in May 2009, which focuses on relationships between nutrition and health [START_REF] Hercberg | The Nutrinet-Santé Study: a web-based prospective study on the relationship between nutrition and health and determinants of dietary patterns and nutritional status[END_REF]. Volunteers aged 18 years or older living in France and having access to the Internet fill in self-administered web-based questionnaires at baseline, as well as at regular follow-ups using a dedicated, secure website. Moreover, the participants also completed a set of questionnaires assessing demographic and socioeconomic characteristics. In addition, residential addresses were obtained from all participants, geocoded to the parcel or street level and implemented as a shapefile in a geographical information system (GIS). Of the 83,128 participants with complete individual data for this study who have completed the questionnaires between 2009 and 2014, 14,075 participants were excluded due to missing data regarding their home addresses (n=12,991), residential related variables (n=355) or BMI (n=1,084). Therefore, the sample used in the present study included 68,698 participants.

Our study was conducted according to the guidelines laid down in the Declaration of Helsinki, and all procedures were approved by the Institutional Review Board of the French Institute for Health and Medical Research (IRB Inserm #0000388FWA00005831) and the Commission Nationale Informatique et Libertés (CNIL #908450 and #909216). The study is registered at clinicaltrials.gov as NCT03335644. Electronic informed consent was obtained from each participant.

Weight status

At enrolment, participants were asked to report self-measurement of their weight and height using standardised procedures (while standing on a flat surface, lightly dressed and without shoes). In a validation study, we showed high level of agreement between anthropometric data obtained by self-report and standardised clinical measurements [START_REF] Lassale | Validity of Web-Based Self-Reported Weight and Height: Results of the Nutrinet-Santé Study[END_REF].

Body mass index (BMI) was calculated as weight [kilograms] / height² [meters]. Obesity was defined according to World Health Organization standards as BMI ≥ 30kg/m² and overweight (including obesity) as BMI ≥ 25 kg/m² [START_REF] Who | Obesity: preventing and managing the global epidemic[END_REF].

Individual covariates

Three sociodemographic variables were included in the models as covariates: age, sex and education (below high school diploma, high school diploma, and post-secondary graduate). These data were collected at baseline through self-administered questionnaires [START_REF] Hercberg | The Nutrinet-Santé Study: a web-based prospective study on the relationship between nutrition and health and determinants of dietary patterns and nutritional status[END_REF].

Neighbourhood deprivation index

We used the French deprivation index (FDep) as a measure of neighbourhood deprivation. This index was calculated following the protocol described by Rey and colleagues [START_REF] Rey | Ecological association between a deprivation index and mortality in France over the period 1997 -2001: variations with spatial scale, degree of urbanicity, age, gender and cause of death[END_REF] and was already used and compared with other indices in a similar cohort (Temam et el., 2017). Deprivation has been initially defined by Townsend as a "state of observable and demonstrable disadvantage relative to the local community or the wider society to which an individual, family or group belongs" [START_REF] Townsend | Deprivation*[END_REF]. Empirically, in our study, deprivation was quantified as the first component of principal component analysis, which included four aggregated socio-economic variables: median household income, percentage of high school graduates in the population aged 15 years and older, percentage of blue-collar workers in the active population, and unemployment rate. We used data provided by the National Institute of Statistics and Economic Studies (INSEE -https://www.insee.fr/en/accueil), the official authority collecting and disseminating information and statistics on French economics. The index was calculated at the smallest administrative unit for dissemination of French infra-municipal data, called IRIS. Typically, IRIS have a population of around 2,000 people and are thus similar to American census block groups. Each participant was assigned the value of his/her IRIS of residence.

Urban contexts

In this study, participants' neighbourhoods of residence were classified into 10 residential (iv) Inner municipalities of large urban areas, including the most important municipality (inner city) of a large urban area. A large urban area is defined as a group of contiguous municipalities, including an urban central area providing at least 10,000 jobs;

(v) Suburban municipalities of large urban areas, i.e. municipalities composing a large single build-up urban area, but excluding the inner city;

(vi) Peri-urban municipalities of large urban areas, including municipalities among which at least 40% of employed resident population work in the urban centre or in the municipalities attracted by this centre;

(vii) Municipalities of intermediate urban areas (defined as a group of contiguous municipalities including an urban central area providing from 5,000 to 10,000 jobs), whether inner cities, suburbs or peri-urban areas;

(viii) Municipalities of small urban areas (defined as a group of contiguous municipalities, including an urban central area providing from 1,500 to 5,000 jobs), whether inner cities, suburbs or peri-urban areas;

(ix) Multipolarized municipalities (MP);

(x) Rural municipalities.

More details on the procedure leading to this classification, based on a local example illustrated by maps, are provided in supplementary materials.

Statistical modelling

First, a basic logistic model was computed to explore the overall relationship (i.e. without geographical variation) between overweight, deprivation and urban context. The logistic model was run with the dichotomized dependent variable (being a person with overweight or not), while adjusting for individual covariates (gender, age and education).

Then, to explore both nonlinearity and context-specific statistical relationships between deprivation and obesity, we used a hierarchical generalised additive modelling (HGAM) framework. This method combines two common methods into a single framework: hierarchical (or multilevel) models and generalised additive models (GAM) [START_REF] Pedersen | Hierarchical generalized additive models in ecology: an introduction with mgcv[END_REF]. Specifically, GAM is a method allowing statistical relationship between independent variables and the dependent one to be described by smooth curves, usually splines [START_REF] Hastie | Generalized additive models[END_REF]. This model is expressed as follows:

∑

Where E(Y) is the expected value of the response Y (with an appropriate distribution and link function g), f j is a smooth function of the covariate x j , β 0 is an intercept term, and g -1 is the inverse link function [START_REF] Pedersen | Hierarchical generalized additive models in ecology: an introduction with mgcv[END_REF]. The smooth function f j is represented by a sum of K simpler, fixed basis functions (b j,k ) multiplied by corresponding coefficients (β j,k ), which need to be estimated: ∑ These basic functions capture the complexity of each subsequent fitted curve. Too much complexity can lead to overfitting, i.e. to very erratic curves. To avoid such an overfitting and smooth the curves, a smoothing penalty is applied to basis function coefficients [START_REF] Pedersen | Hierarchical generalized additive models in ecology: an introduction with mgcv[END_REF]. This penalty affects the shape of the curves via a smoothing parameter that is selected using restricted maximum likelihood (REML). Typically, a smoothing parameter set to 0 leads to overfitting, while a very large value forces the curve to become a straight line.

it is therefore a question of compromise between complexity and simplification of the studied statistical relationships. In this study, we used thin plate regression splines as a smooth function, as recommended for non-cyclic data [START_REF] Wood | Thin plate regression splines[END_REF]. Besides the GAM, hierarchical or multilevel models are an extension of linear models accounting for the hierarchical -nested -structure of the data [START_REF] Goldstein | Multilevel models in education and social research, Multilevel models in education and social research[END_REF]. Multilevel models specify models at a micro (e.g. individual) and at a macro (e.g. neighbourhood) level in a single framework. Accordingly, these models allow the consideration of context effects together with individual ones, avoiding both ecological and atomistic fallacies [START_REF] Owen | Under examination: Multilevel models, geography and health research[END_REF].

Moreover, in multilevel models, statistical relationships can vary by levels of the grouping variable (i.e. geographical contexts if this variable corresponds to spatial units) through random effects. The core idea of HGAMs is to combine GAM and multilevel models in order to allow varying nonlinear relationships for each level of grouping variable(s). In the present study, the grouping variable is the typology of urban contexts. More details about HGAM and the selection procedure of the final model are given in supplementary materials.

All statistical analyses were performed with R (R Development Core Team, 2018). The mgcv package [START_REF] Wood | Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models[END_REF] was used for computing HGAMs, following the codes created by Pederson and colleagues [START_REF] Pedersen | Hierarchical generalized additive models in ecology: an introduction with mgcv[END_REF].

Results

Descriptive statistics

The 68,698 French adults study participants had a mean (sd) age of 44.2 (14.5) years, a high level of education (64% of participants having completed post-secondary school) and were mostly women (77%) (Table 2). In addition, they had a mean BMI of 24 kg/m² and 31.8% of them were overweight or obese. A total of 46% of the participants lived either in inner cities of large urban areas or in their suburbs. About 22% of the participants lived in the Paris area, including 5.7% in Paris intra-muros. The remaining individuals resided either in intermediate/small urban areas, or in multipolarized/rural units (Table 1).

*** Insert Table 1 about here ***

Table 2 shows mean BMI, overweight prevalence and the neighbourhood deprivation index summary statistics grouped by urban context. It can be observed that Paris and its suburbs, as well as large urban area inner cities and their suburbs, i.e. the most urbanised contexts, are those exhibiting the lowest BMI and overweight rates, in addition to the lowest deprivation indices on average. Moreover, the variability (standard deviation) of the deprivation index is higher in cities and their suburbs compared to less densely populated areas.

*** Insert Table 2 about here ***

Results of global models

First, we computed the global generalised linear model to have an overview of average relationships over the entire sample. Specific types of urban context were included as a categorical explanatory variable, the reference category being suburbs of large urban areas.

The individual covariates were all associated with overweight, positively for age and negatively for gender (being a man is negatively associated with overweight) and education (< high school as reference category, see Figure 1). The deprivation index was also positively associated with the likelihood of being a person with overweight, meaning that the more deprived the neighbourhood, the higher the likelihood of being a person with overweight.

Regarding urban contexts, residing in

Discussion

In this study, we examined whether the relationship between neighbourhood deprivation and being a person with overweight varied in form and/or in intensity according to urban context among 68,698 French adults from the NutriNet-Santé cohort, while controlling for individual characteristics. Our main finding is that the deprivation-obesity relationship varied according to urban context. In general, our findings showed that this relationship was the strongest in suburbs of large cities (especially in the Paris area), the weakest in less urbanised contexts (small towns and rural areas), and intermediate in inner cities. This nonlinear scheme goes beyond similar studies in extending findings that were restricted only to low-income women [START_REF] Ford | Neighborhood Deprivation, Supermarket Availability, and BMI in Low-Income Women: A Multilevel Analysis[END_REF] or to geographical contexts which did not take into account the hierarchy between cities in a given urban system [START_REF] Wang | Population-adjusted street connectivity, urbanicity and risk of obesity in the U[END_REF]. Our results demonstrate the significance of the broader urban residential context, including its position in the urban system hierarchy. This key point supports the idea that within a general framework, public health policies should be adapted to local contexts for concrete application [START_REF] Swinburn | The global syndemic of obesity, undernutrition, and climate change: The Lancet Commission Report[END_REF][START_REF] Swinburn | The global obesity pandemic: shaped by global drivers and local environments[END_REF].

Some hypotheses can be proposed to explain the varying relationships between deprivation and overweight according to local residential contexts: (i) There are fewer contrasts in the least urbanised areas in terms of contextual deprivation, as indicated by the values of the deprivation index standard deviations in Table 2, while variability is as its maximum in suburban areas (especially in the suburbs of Paris). In general, social patterns in French cities are different from those observed in U.S. cities, as deprived populations in France typically live in the suburbs rather than in inner cities. The renovation process of French suburbs is most often characterised by renewed attractiveness of working-class neighbourhoods, with incomers belonging to the upper middle class [START_REF] Clerval | Gentrification, a frontier reshaping social division of urban space in the inner Paris[END_REF], even if more complex sociospatial processes are at play in some suburbs. Moreover, these neighbourhoods also increasingly gain population due to the pressure exerted by housing prices. Therefore, suburban areas -especially in Paris -are characterised by higher social heterogeneity compared to the inner-city, with a mixture of working-class, upper-middle class residents [START_REF] Clerval | Gentrification, a frontier reshaping social division of urban space in the inner Paris[END_REF]. Such differentiated deprivation variability could also partly determine contrasts on relations in other local residential contexts. (ii) Furthermore, sizes of administrative units differed between local residential contexts, with increased size observed in the least urbanised areas, which could affect the deprivation-obesity relationship. As previously suggested by Ford and Dzewaltowski [START_REF] Ford | Neighborhood Deprivation, Supermarket Availability, and BMI in Low-Income Women: A Multilevel Analysis[END_REF], the influence of neighbourhood characteristics on individual behaviours diminishes with increased neighbourhood size.

We also found non-linear deprivation-obesity relationships within specific urban contexts.

These relationships were linear in most residential contexts, but not in specific contexts such as the peri-urban belt of large urban areas and Paris suburbs. In these contexts, the relationship between deprivation and overweight was stronger for the lower half of the deprivation index values and plateaued beyond a certain threshold, towards most deprived neighbourhoods. These results would suggest that prevention policies should not only target the most deprived neighbourhoods, but also consider the entire social deprivation gradient.

To address the needs of deprived populations at a local scale and promote equity in access to healthy lifestyle options in the local environment, public health practitioners should be aware of the influence of such geographic variation. This should help design programs that would identify at-risk populations within specific geographic contexts to better support policy development, provide increased resources and improve sustainability of interventions. It could be seen as a geographical approach of the "think global, act local" framework considering the food, transportation, land-use and urban-design systems that represent the major drivers of body weight gain and obesity [START_REF] Swinburn | The global syndemic of obesity, undernutrition, and climate change: The Lancet Commission Report[END_REF].

In addition to the geographically varying relationships, our results showed an expected significant positive relationship between neighbourhood deprivation and risk of overweight, while adjusting for individual variables. Such neighbourhood inequality in overweight prevalence is in agreement with a number of previous studies in Europe [START_REF] Maier | Area level deprivation is an independent determinant of prevalent type 2 15 diabetes and obesity at the national level in Germany. Results from the national telephone health interview surveys 'German Health Update' G DA 2009 and[END_REF], as well as in the US [START_REF] Lovasi | Effect of individual or neighborhood disadvantage on the association between neighborhood walkability and body mass index[END_REF]. Differences in access to neighbourhood facilities (e.g. food outlets with healthy options, sport facilities) have been hypothesised to be related to neighbourhood inequalities in dietary intake and physical activity, favouring weight gain and obesity [START_REF] Lenthe | Neighbourhood deprivation and overweight: the GLOBE study[END_REF]. In addition, neighbourhood differences in stressors (e.g. high traffic density, lower aesthetics and crime-related safety) may contribute to increases in unhealthy behaviors, such as sedentary lifestyle [START_REF] Laraia | Place matters: Neighborhood deprivation and cardiometabolic risk factors in the Diabetes Study of Northern California (DISTANCE)[END_REF]. Further research should investigate the relationship of neighbourhood deprivation with dietary intake, physical activity and sedentary behavior according to urban context, using a similar approach as the one followed in the present paper.

One of the main strengths of our study is the use of hierarchical generalised additive modelling framework to unravel the spatial complexity of deprivation-overweight relationships. A number of previous studies have already applied the widespread GWR modelling approach for providing maps of local relationships between deprivation and obesity (see [START_REF] Oshan | A multiscale spatial analysis of obesity determinants in Phoenix, Arizona[END_REF] for a review). However, this method fails to consider the hierarchy of cities in a given geographic context. Another potential approach is the use of the random slope multilevel model [START_REF] Corsi | Weight of communities: A multilevel analysis of body mass index in 32,814 neighborhoods in 57 low-to middle-income countries (LMICs)[END_REF]; however, it does not allow exploring geographically varying nonlinear relations. Therefore, to overcome these limitations, the hierarchical generalised additive model used in the present study appears particularly suited to capture, simultaneously, spatial heterogeneity and nonlinearity of relationships.

It should be noted, however, that given the cross-sectional nature of this study, no causal pathway could be established between contextual deprivation and individual weight status.

In addition, such pathways are mediated by both residential context and obesity-related behaviors, including food choices, availability and accessibility, which were not addressed here. Also, in terms of perspective, further works could explore the possibility of declining the deprivation index according to territorial context, in particular to better depict deprivation in rural settings. Even if the use of a single, standardized index makes comparison of statistical results between spatial contexts and between studies easier, an ongoing debate still exists on the usability of "one-size-fits-all" deprivation indices comparing urban and rural area as they are often skewed (e.g. [START_REF] Burke | The development of an index of rural deprivation: a case study of Norfolk, England[END_REF]. Finally, our sample included more women than in the general population, as well as more highly educated individuals, limiting the generalisability of our findings.

Conclusion

In conclusion, this work supports the notion that using a mean, smoothed view of statistical relationships may hide local mechanisms directly derived from specific spatial contexts, and, in turn, lead to partial or biased inferences [START_REF] Feuillet | Built environment in local relation with walking: Why here and not there[END_REF][START_REF] Feuillet | Spatial heterogeneity of the relationships between environmental characteristics and active commuting: towards a locally varying social ecological model[END_REF].

  contexts, extending the two main classifications previously proposed by INSEE. The first one based on morphological and inhabitants' criteria (urban unit nomenclature: https://www.insee.fr/en/metadonnees/definition/c1501), and the other on functional criteria (urban area nomenclature: https://www.insee.fr/en/metadonnees/definition/c2070). Taken separately, these previous INSEE classifications have several limitations, as they do not distinguish the inner city from the suburbs of an urban centre. Moreover, they do not consider the Paris metropolitan area separately. Yet, in France, the Paris area represents an urban primacy, as it includes almost 20% of the French population and has a considerable economic influence (~30% of the national gross domestic product). We therefore designed a more comprehensive classification, combining demographic (number of inhabitants), morphological (built-up zone continuity) and functional (number of employed persons, commuting fluxes) thresholds, allowing our analysis to consider the hierarchy in the spectrum spanning from city centre, suburbs, and peri-urban to rural areas. The 10 resulting contexts are the following: (i) The inner city of Paris (intra-muros); (ii) Suburban municipalities of Paris, including municipalities composing the continuously built up Paris agglomeration, but excluding Paris; (iii) Peri-urban municipalities of Paris, including municipalities located in the Paris urban area and among which at least 40% of employed resident population work in Paris or in the municipalities attracted by Paris;

  Paris intra-muros or in inner cities of large urban areas were negatively associated with the likelihood of being a person with overweight compared to residing in suburbs of large urban areas (reference category). In contrast, residing in the Paris suburbs, as well as in peri-urban or in multipolarized/rural areas was positively associated with the likelihood of being overweight. In other words, living in a highly urbanised context was associated with a lower likelihood of being a person with overweight, even after adjusting for individual covariates and neighbourhood deprivation. Estimates for other types of urban context (intermediate and small urban areas) were non-significant. *** Insert Figure1about here ***4.3. Results of hierarchical generalised additive models (HGAM)GAM results are illustrated in Figure2, in which relationships are plotted for each urban context. Regarding the overweight model, smooth coefficients are significant for five urban contexts only: Paris and its suburbs, as well as all large urban areas (inner cities, suburbs and their peri-urban belt). There was no significant relation in less urbanised areas. In terms of wiggliness, most relationships had a linear shape, except in the peri-urban belt of large urban areas, where the statistical relationship was strong for the lowest values of deprivation only and tended to decrease in most deprived areas (Figure2). Also, in Paris suburbs, the deprivation-obesity relationship was slightly stronger for the lowest values of the deprivation index and weaker for the highest values. *** Insert Figure2about here ***

Figure 1 .

 1 Figure 1. Associations between individual characteristics, urban contexts and deprivation index, and being a person with overweight (BMI ≥ 25 kg/m²), estimated through a logistic model (n = 68,698). Coefficient estimates have been exponentiated (odds ratios).

Figure 2 .

 2 Figure 2. Plots of the smooth terms derived from the hierarchical generalized additive model, showing the shape of the relationships between BMI and neighbourhood deprivation index as a function of urban contexts. Estimates are adjusted for age, gender and education.

Table 2 .

 2 Our findings emphasise the importance of the local level for public health policies for preventing weight gain and obesity, taking into account the strong but varying deprivation-obesity relationship. Descriptive statistics of weight status (BMI and percent of overweight including obesity) grouped by urban contexts.

	Age (mean)	44.2 (14.5)
	Gender (% women)	77.0
	Education
	< high school (%)	2.8
	High school (%)	33.2
	Post-secondary school (%)	64.0
	Weight status
	BMI (kg/m²) (mean)	24.0
	Overweight or obese (BMI > 25) (%)	31.8
	Urban contexts
	Paris (%)	5.7
	Paris suburbs (%)	14.2
	Paris peri-urban areas (%)	2.7
	Inner cities of large urban area (%)	24.1
	Inner city suburbs of large urban area (%)	22.0
	Peri-urban areas of large urban area (%)	15.6
	Intermediate urban areas (%)	2.7
	Small urban areas (%)	2.9
	Multipolarized municipalities (%)	7.2
	Rural municipalities (%)	2.8

Moreover, our results indicate that moderately deprived neighbourhoods, such as those of Paris suburbs and peri-urban parts of large cities, should not be neglected by public policies for tackling overweight and obesity. In exhibiting geographical heterogeneity of relationships, this study reinforces the contention that health-related ecological studies should consider local nuances -i.e. spatial heterogeneity -of statistical relationships to fully capture the variation of socioeconomic determinants of non-communicable diseases such as obesity. * z-scores: high scores indicate high deprivation

Acknowledgments

The authors warmly thank all the volunteers of the NutriNet-Santé cohort. We also thank Younes Esseddik, Thi Hong Van Duong, Régis Gatibelza, and Jagatjit Mohinder (computer scientists),

coordinator) for their technical contribution to the NutriNet-Santé study.

Funding

The NutriNet-Santé study was supported by the following public institutions: Ministère de la Santé, Santé Publique France, Institut National de la Santé et de la Recherche Médicale (INSERM), Institut National de la Recherche Agronomique (INRAE), Conservatoire National des Arts et

Highlights

• Spatial patterning in the deprivation-obesity relationship remains poorly understood

• Neighbourhood deprivation-overweight relations were studied in a large adult cohort

• The relationship was strongly positive in Paris suburbs and other large cities

• The relationship was weak or null in small/rural towns and medium in inner cities

• Moderately deprived neighbourhoods are potential targets for obesity prevention CRediT author statement TF conceptualized the hypotheses and did the statistical analyses. JFV contributed to acquisition and interpretation of contextual data. JMO, HC and TF drafted the manuscript, and all authors were involved in subsequent drafts; MT and SH are guarantors for the NutriNet-santé study. All authors read and approved the final manuscript.


Anna Ouskova Leonteva

Michel Risser

Radia Hamane

Anne Jeannin-Girardon

Pierre Parrend

Pierre Collet

Anne Jeannin Girardon

A hybrid optimization tool for active magnetic regenerator

Keywords: Computing methodologies → Artificial intelligence; Modeling and simulation real-world application, hybridization, unified optimization

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Nowadays, the demand for cooling is increasing, due to the climate change and universal problems. In this context, for reducing the worldwide electricity consumption and direct greenhouse gas emissions, Active Magnetic Regenerator (AMR) refrigeration is a promising technology, which applies the phases of magnetization/demagnetization of solid refrigerants -Magneto Caloric Materials (MCMs) [START_REF] Balli | Engineering Of The Magnetic Cooling Systems: A Promising Research Axis For Environment And Energy Saving[END_REF].

Despite the promising researches, the AMR development has two principle problems that prevent its commercial production: (i) to simulate physical properties of different MCMs in good qualitative agreement with the available experimental data; (ii) to comprehensively tune many control and design parameters of the AMR model, depending on their effect on its performance. For overcoming these difficulties, related researches apply optimization algorithms to the numerical simulation models of MCMs [START_REF] Ouskova Leonteva | New Evolutionary Method for Studying Physical Properties of Magneto Caloric Materials[END_REF] and AMR [START_REF] Ganjehsarabi | Analysis and optimisation of a cascade active magnetic regenerative refrigeration system[END_REF], [START_REF] Roy | Sensitivity analysis and multiobjective optimization of a parallel-plate active magnetic regenerator using a genetic algorithm[END_REF]. The common challenge of the works aiming at optimizing the AMR performance is to set up many optimization experiments, where different number of objectives and decision variables are required.

For improving optimization process of the AMR design, we develop further the user-friendly tool presented in [START_REF] Ouskova Leonteva | New Evolutionary Method for Studying Physical Properties of Magneto Caloric Materials[END_REF] and integrate an "unified" algorithm in it. The term "unified" is borrowed from [START_REF] Seada | U-NSGA-III: A unified evolutionary algorithm for single, multiple, and many-objective optimization[END_REF] for describing the algorithms, which ensure the scalability w.r.t. the number of objectives.We scale the well-known many-objective Non-dominated Sorting Genetic Algorithm III (NSGA-III) [START_REF] Deb | An evolutionary manyobjective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints[END_REF] down to solve singleobjective problems by the hybridization of the solutions of NSGA-III with the solutions of Quantum Particle Swarm Optimization (QPSO) algorithm [START_REF] Sun | Using quantum-behaved particle swarm optimization algorithm to solve non-linear programming problems[END_REF], instead of modifying the NSGA-III structure [START_REF] Seada | U-NSGA-III: A unified evolutionary algorithm for single, multiple, and many-objective optimization[END_REF]. It allows NSGA-III to handle very small population size in single-objective cases, which is useful for optimizing time-consuming problems, like the problems of the AMR design. To our best knowledge, this work is the first attempt to apply hybridization to unified optimization.

We contribute by making a step forward the further development of AMR refrigeration for its commercial application and a new insight about hybrid method, which is capable to be employed for constituting scalable algorithms.

MATERIALS AND METHODS 2.1 The Simulation Models

The AMR model, provided by Ubiblue Company, operates in two application modes: the Magnetic Refrigeration System (MRS) and the Thermo-Magnetic energy Generator (TMG). In this case, the optimization problems of the AMR model are formulated as the maximization of efficiency and/or power density for the MRS and/or the TMG operating modes for finding optimal control and design parameters of the AMR model in both modes. From an optimization point of view, the AMR model is considered as a continuous, black-box, computationally intensive problem, where the numbers of decision variables and objectives depend on current case study.

Inside the AMR model, for reproducing physical properties of MCMs, we use a method based on the generalize magnetic simulation model provided by the Crismat laboratory [START_REF] Ouskova Leonteva | New Evolutionary Method for Studying Physical Properties of Magneto Caloric Materials[END_REF]. The optimization problems of reproducing physical properties of MCMs are formulated as the minimization of one/several differences between simulated and reference physical properties of MCMs for finding optimal parameters of the magnetic simulation model. From an optimization point of view, these problems can be considered as continuous, computationally intensive, separable or partially separable optimization problems, which can have from one to three objectives.

Hybrid Unified Optimization Algorithm

Aiming at ensuring the scalability w.r.t. the number of objectives for time-consuming problems, we develop further the idea of unified algorithm [START_REF] Seada | U-NSGA-III: A unified evolutionary algorithm for single, multiple, and many-objective optimization[END_REF], by using a hybridization method for scaling down a many-objective algorithm NSGA-III [START_REF] Deb | An evolutionary manyobjective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints[END_REF] to single-objective optimization. The hybridization is used to overcome difficulties of NSGA-III to solve single-objective problems, which are explained by too small recommended population size of NSGA-III for single objective optimization and an absence of selection pressure [START_REF] Seada | U-NSGA-III: A unified evolutionary algorithm for single, multiple, and many-objective optimization[END_REF].

The structure of the proposed algorithm, called QIU-NSA, consists of three main modules: NSGA-III, the modified QPSO and the fusion module (see Figure 1). At generation = 0, the algorithm starts in the fusion module with the definition of the following parameters: (i) the dimension of search space ( ); (ii) the dimension of objective space ( ); (iii) the vectors of boundaries (lb, ub) for each decision variable; (iv) the threshold of diversity (

) and (v) the total number of generations (

). The diversity coefficient ( ) is set to 0. The termination criterion is defined as the total number of generations. In the fusion module, the initial population P =0 are created randomly according to the defined boundaries (lb, ub) and evaluated. Then, the following steps are iterated until the termination criterion is satisfied:

(1) Make Child Population: The parent population P is sent to NSGA-III and QPSO modules in order to produce the new child populations. Each module, NSGA-III and QPSO, creates new set of solutions Q +1 -and Q +1 according its original rules and returns them into the fusion module. To summarise, the hybrid method allows to scale NSGA-III down to solve single-objective problems without any modifications in the NSGA-III structure, which excludes a risk to loss its effectiveness on multi-/many-objective problems.

Optimization Software Tool

In order to simplify a code-coupling process between each research problem and the proposed unified algorithm, we extend the optimization tool presented in [START_REF] Ouskova Leonteva | New Evolutionary Method for Studying Physical Properties of Magneto Caloric Materials[END_REF], based on the open-source EASEA (EAsy Specification of Evolutionary Algorithms) platform, by implementing QIU-NSA as a template of the EASEA platform. The EASEA provides a compiler for automatically merging a problem description file * . (the users' guide is provided in the online documentation1 ), which specifies a scientific research task, with an optimization algorithm template into a CPU parallel C++ code [START_REF] Collet | Take it EASEA[END_REF]. 3 EXPERIMENTS AND RESULTS

Validation of the Hybrid Unified Algorithm

In order to confirm an ability of QIU-NSA to solve singleobjective problems, we benchmark QIU-NSA on the Black-Box Optimization Benchmarking (BBOB) single-objective large-scale test suite [START_REF] Hansen | Real-parameter black-box optimization benchmarking: Experimental setup[END_REF] provided by the COmparing Continuous Optimizers (COCO) platform 2 . The reported results are based on 15 independent run of 15 instances of each function.

Taking into account that the AMR design problems can be large scale, we compare the performance of QIU-NSA on 640-dimensional search space with the reference results of separable CMA-ES (sepSMA) [START_REF] Ros | A simple modification in CMA-ES achieving linear time and space complexity[END_REF] and Limited Memory CMA-ES (LMCMA-ES) [START_REF] Loshchilov | A computationally efficient limited memory CMA-ES for large scale optimization[END_REF] provided by COCO platform. Looking at Figure 2, we observe that QIU-NSA demonstrates the best performance on the separable and multi-modal problems with adequate global structure. It solves 100% of separable functions with the highest accuracy (see Figure 2 (a)) that confirms its applicability for separable problems of magnetic model of MCMs. We can conclude that QIU-NSA efficiently scales NSGA-III down on the objective space and shows a good scalability w.r.t. the dimensions of the search space.

Reproducing Physical Properties of MCMs

The objective of this experiment is to validate an efficiency of QIU-NSA to find a set of the parameters of the magnetic model, which corresponds to the physical properties of the given MCM -( ( )).

Input Parameters and the objective function.

The input parameters and their boundary values are presented in Table 1.

More details about the method for studying physical properties of MCMs can be found in [START_REF] Ouskova Leonteva | New Evolutionary Method for Studying Physical Properties of Magneto Caloric Materials[END_REF]. Value 0.1 -2.5 0.1 -2.5 0.1 -2.5 = 1 -0.0-2.5

2 https://github.com/numbbo/coco

The objective function is defined as follows:

= + (1) =| (Δ ( 0 ) -Δ ( 0 )) | + | (Δ ( 1 ) -Δ ( 1 )) | (2) =| (Δ ( 0 ) -Δ ( 0 )) | + | (Δ ( 1 ) -Δ ( 1 )) | (3)
where

Δ ( 0 ), Δ ( 0 ), Δ ( 1 )
and Δ ( 1 ) is the the temperature interval width of heat capacity curve peak under different magnetic fields upon cooling and warming process.

Experimental

Result. The values of the parameters obtained by the optimization for are as follows: = 1 [T], = 0.80, 1 = 0.85, =1.00. In Figure 3 we report the temperature dependence of magnetic entropy change (Δ ), which is calculated by using the Maxwells equation from the simulated (theoretical) data with obtained parameters and from the measured data of Crismat. The agreement between the theoretical and measured data is excellent: the Mean Absolute Percentage Error (MAPE) of Δ is 0.1%. Figure 3: :The simulated and experimental temperature dependence of magnetic entropy change.

Optimization of the AMR Design

We apply the proposed tool for developing an innovative architecture of the AMR, which will be efficient for the Magnetic Refrigeration System (MRS) and the Thermo-Magnetic energy Generator (TMG) modes.

Input Parameters.

For this experiment, the most important input parameters of the AMR model, which have an impact in the both modes are selected and presented in Table 2 with their boundaries. and are internal operating conditions that can control the thermodynamic cycles of the AMR, where are the design parameters. as for the the energy efficiency of the MRS. We can conclude that for all criteria, has less impact than the others. According to Figure 4, there are several solutions, which ensure the balance between efficiency and power density for the both modes. The Pareto fronts has slightly discontinuous shape that be explained by a large number of the rejected solutions, because of their nonexistence in the both modes simultaneously. Thus, an evaluation of the energy conversion system is required by reconsidering some default parameters of the AMR model, e.g., the fluid channel thickness, which was set to optimally match a refrigeration system in this study.

CONCLUSION

Optimization of the Active Magnetic Regenerator (AMR) design and reproducing physical properties of Magneto Caloric Materials (MCMs) are the major challenges for the magnetic cooling industry. The proposed tool, thanks to the hybrid method and the EASEA platform, is a robust user-friendly instrument, which allows to easily and efficiently set up different experiments. The functionality of this tool is universal and can be easily adapted to different simulated models. It is validated that thanks to the proposed tool, it is now possible to accelerate the elaboration of a commercially available device, which will correspond to modern ecological and energy-saving requirements.

( 2 )

 2 Uniform-based Random Selection: In the fusion module, the populations Q +1 -and Q +1 are used to select 4 solutions to the next population Q +1 by the following rule: if uniformly distributed random number U (0, 1) > 0.5, the solution of Q +1 is accepted, otherwise -Q +1 -. (3) Evaluation of Q +1 : The obtained population Q +1 is evaluated and is sent back with their evaluated values to the QPSO and NSGA-III modules. (4) Selection to the Next Generation: The solutions for the next parent population P +1 are selected according to the procedure of NSGA-III. QPSO updates the local and global best solutions, according to [11]. At the end of the generation , the QPSO module returns current global solution g +1 and the NSGA-III module returns the new parent population P +1 to the fusion module.

Figure 1 :

 1 Figure 1: Simplified scheme of the unified algorithm.

Figure 2 :

 2 Figure 2: The ECDF summarized by separable and multimodal functions on 640-dimensional search space

3. 3 . 2

 32 Experimental Result.

Figure 4

 4 depicts the Pareto front of non-dominated solutions, showing in highlighted color values the MRS and TMG operating modes respectively. The Pareto front clearly reveals the conflict between the power densities and efficiency in both modes.

Table 2 :Figure 4 :

 24 Figure 4: Obtained Pareto Fronts

Figure 5 :

 5 Figure 5: The distribution of Pareto optimal points solutions

Figure 5 b

 5 ) shows that the maximum value of the recoverable mechanical power density of the TMG is 0.37 [ / 3 ] , achieved with = 1.45 [ ], = 0.15 [-], and = 17.2 [ ]. Thus, an increase of leads to increase the mechanical power density and decrease the efficiency. The maximum value of the energy efficiency of the MRS is 0.56 [-] obtained with = 13.6 [ ], = 0.11 [-] and = 1.18 [ ] (see Figure 5 c)).

  Figure 5 d) shows that the maximum value of the thermal power density of the MRS is 2, 05 [-], which is achieved with the larger value of = 17.8 [ ] and almost the same values of = 1.17 [ ] and = 0.11 [-]

Table 1 :

 1 Input parameters and their boundary values for alloy

	Parameter	1	2

http://easea.unistra.fr

ACKNOWLEDGMENTS

This work is funded through ANR project CoolMagEvo (ANR-17-CE05-0036). The ANR is gratefully acknowledged for its financial support, as well as Ubiblue Company and Crismat laboratory for their collaborations.