Queen rook methods
Brouillet, C. Introduction Living in a green neighborhoods is source of serenity and well-being.  A number of studies have been conducted in order to examined the benefits of natural environment on humain health. The significant demographic growth we have experienced in recent years and the rural exodus have drastically transformed our territory. Cities have been concreted and green spaces reduced to minimum.  Knowing the capacity of the nature to increase human's quality of life, people are attracted by nature and tend to live as close as possible from it. The following study aims to establish a correlation for the territory of Vernier between the distance to the natural areas with the residential poles.  Short distance, dense regions and long distance, sparsely populated regions. Which can be translated in a mathematical point of view in a negative correlation between the two variables. In paralell to that, we would like to know which regions are the most discriminate in terms of access to nature. Access to nature is a parameter underestimate in the assessment of the precariousness of a group of persons or of a whole region. The territory of Vernier was considered in 2011 as the denser commune of the Canton of Geneva with a population showing the highest signs of precariousness in the whole Canton (8 of the 22 sub-sectors showing the lowest signs of precariousness are in the commune of Vernier) \cite{ocstat2012}. Index of precarity developed by the Office fédérale de la Statistique (OCSTAT) has taken into consideration four types of insecurity: family, housing, monetary and job. Nevertheless, since few years, access to vegetation is increasingly seen as an indispensable aspect to quantify when analysis our's population well-being. In 2012, a research developed an index helping to identity at small scale critical neighborhood and environment which need green areas in priority, by taking into consideration the type of the vegetation and the proximity to green \cite{Gupta_2012}. Improvements still need to be done in order to include social aspects into the model.     Initial data  A set of data including a population file, a group of two vegetation layers of the Geneva's region and a vector file highlighting  the limits of the municipality of Vernier were used to perform the analysis. Polygon layers for the vegetation over the territory were downloaded from the open data catalogue of the Geneva Canton (Carte de la végétation au 1/25 000, direct link to the website: www.ge.ch/sitg). A ground resolution of 5 m at a scale of 1:25000 was used.  The vector file with the limits of the municipality of Vernier have been furnished by the Laboratory of Geographic Information Systems  of EPFL (LASIG). The total permanent resident population for the year 2015 was investigated. Statistic was issued from the Geostat plateform managed by the the Federal Office of Statistiques, OFS (www.geostat.admin.ch).  Data collection was achieved by geocoding on the basis of the Federal Register of Buildings and Housing (RegBL).Note that all the data were georeferenced using the same coordinate systems  CH1903+LV03.   Methods In order to evaluate the correlation between populaation repartition and the distance to the closest green area, initial data were loaded on QGIS first and secondly on the GeoDa software. At first,  on the base of the hectometric geometry of the population file, a regular grid surrounding the Vernier municipality, was created (resolution of 100*100). Combining different Qgis functions allowed to obtain for each cell the distance from the closest green area. The grid was then exported on GeoDa where all the principal statistic analysis, as well as maps, were computed. First, a linear regression was established to explore the relation between our variables: \(y_j=\beta_0+\beta_1x_{1j}+\epsilon_j\), where the variable \(y_j\), correspond to the total population and the variable \(x_j\) to the distance from the green region and \(\epsilon_j\) to the residus of each location j. Based on a stationarity hypothesis  of the observed spatial phenomenon, this statistical analysis doesn't seem realistic for the study. Poor results obtain from this hypothesis push to put in place a regression with dependent spatially weighted variable in order to obtain a regression line for each spatial unit: \(y_j=\beta_0+\beta_1x_{j1}+\rho w_iy_i+\ \epsilon_j\) , with  \(w_i\) the weight of the spatial unit i relative to the spatial unit j and \(\rho\) the spatial lag computed by taking the weighted average of the neighboring cells.