The concept of spatial dependency

In a very general way, the strength of the spatial dependency is a measure expressing the relationship between the variation of properties and the spatial proximity. In the case of a continuous spatial distribution, this relation can be expressed by a continuous function of the numerical difference of properties compared to the distance. Thus, one can see that the closer two places are in space, the more the difference in their property is weak (or the larger their similarity is). This continuous function also makes it possible to model the particular way in which distance acts on the importance of the difference (the two concepts of "range" and "function of distance decay" identified by the variogramme make it possible to account for spatial dependency). It will thus be shown that:

• The spatial proximity is measured by means of the Euclidean distance between pairs of places. This concept can be enriched by the directional contribution or the orientation in order to check the property of anisotropy.
• The variation of the properties is based on their numerical difference (interval of values) because the measurement level is cardinal and thus continuous.

1.2.2a Descriptors of spatial dependency for discontinuous distributions

Spatial proximity refers to point, linear or areal features. Particularly for the last two types, the proximity can be expressed simply by using only topological descriptors (contiguity, order of vicinity), because their size, form and orientation are variable. The most common descriptor is the Join Count Statistic Join Count Statistic (an index of contiguity or of adjacency).

The variation of the properties between contiguous objects is expressed in a different manner, according to the level of measurement of the considered phenomenon:

• at the nominal level, we will consider only the similarity or the difference of values for contiguous features.
• at the ordinal level, as the importance of the difference of values expresses a difference of ranks; this can be taken into account, in relation to the measuring unit of this hierarchy.
• at the cardinal level, it is possible to determine the difference of values of each contiguous features.

Thus we will retain two types of indices of spatial dependency, also called indices of spatial autocorrelation, based on the property of adjacency. The first is the Join Count Statistic (coefficient of adjacency), adapted to numerical properties measured at a nominal level. The seconds is Moran's I Coefficient, or alternatively, the Gaery Ratio.