It can be clearly seen from the map of the natural breaks for the predicted values (Fig 7.) that the distance doesn’t explain the population density and this for the whole territory. (Note that the natural breaks have been implemented manually according to the ranges obtain in Qgis). One color only has been attributed to the entire cells when trying to explain the population’s spread by the distance to nature zone. Not surprisingly, the standard deviation map of the residuals of the predicted values show higher value in the dense spots (Fig. 8). Any additional information could be extracted from mapping residuals. 
 Considering the neighbours values in the linear regression doesn’t bring significant improvement. Indeed, the correlation factor has increased by factor of 10 but still stay very low and close by zero (Fig. 4). Coefficient and variables obtained from the linear methods are shown on Fig. 4. The decision was taken to not illustrate in this study, the results of the natural breaks maps of the predicted values as well as the standard deviation of the residuals from this weighted method as the results obtained were not much different from the first approach. 

   Discussion

The main result obtained was a correlation null between the distance to the first greenbelt and the residents for both weighting and not weighting regression linear analysis between the two variables. Even with trying different weight ID or methods, results would be the same. Thus, the studied variable tend to not influence the population repartition. This result would be totally different if the repartition of the population was not concentrated on few spots but more arrange smoother among the territory. Meaning our conclusion is specific to the case of Vernier, very particular, and cannot explained a global behavior. Further analysis has to be implemented in order to deeply assess if the independent variable is not significant to explain the repartition of the population over the territory. Using a T student’s law could be good approximation for a such analysis because it focus on giving a significance at a local scale.   
By mapping the results, we now learn which areas are privileged in terms of proximity to vegetation synonym of calm and healing. Even extremely densely populated, le Lignon is at the first place for access to vegetation. Les Libellules, Balexert and les Avanchets are on contrary, discriminate.  

Conclusion

The hypothesis established was not confirmed from the case of study. The presence of green, seemed to not have influence the  spatial distribution of habitants in the territory of Vernier. However, this first manipulation of data had allow to report important information of the territory and its population: an inequality exists in term of access to forest and park. Rich in the information, an index of vulnerability could be developed in future analysis considering the persons living far away from the first green areas as being more susceptible to  stress, insomnia and asthma. This could be useful to suggest family with babies or old person to live in certain places instead of others. However,  other sources of vulnerabilities should be included (distance to the public transport, etc) in order to have an index as realistic as possible.