other disciplines such as spatial economics or anthropology also use these tools. As this book is not intended to develop sophisticated mathematical and statistical methods, it is nevertheless necessary to review a few tools in order to get the most out of the often tedious mapping work. It would be a pity to miss some conclusions because of a lack of knowledge of these tools. To do this, it is important to be familiar with spatial autocorrelation and related indices: Moran index, Geary index. Less specific to spatial analysis, however, the Gini coefficient (Gini 1921) is used in particular to show income inequalities by location as was done for the city of Athens (Panori and Psycharis 2018).
Autocorrelation expresses the correlation of a variable with itself. This correlation can be measured either over time by comparing successive values of the variable (temporal autocorrelation) or over time by measuring the variable in different locations (spatial autocorrelation) (Oliveau 2017).
Spatial data are characterized by their great heterogeneity, which is systematic, while temporal data encounter this type of difficulty less frequently (Jayet 2001).
A distinction is made between heterogeneity of size (geographical entities such as cities, regions or countries are very diverse in size), heterogeneity of shape (regions do not have the same contours), heterogeneity of position (a northern region and a southern region of the same size and shape are not comparable), heterogeneity of structure in terms of qualification, economic activity or the size of establishments (Jayet 2001). It is therefore appropriate to use certain tools such as autocorrelation measurements to measure this heterogeneity.
Moran’s I (Moran 1950) and Geary’s C (Geary 1954), also known as Geary’s contiguity ratio, are the main tools for measuring autocorrelation and the following formulations have been rewritten (Cliff and Ord 1973). Here is Moran’s I index (Oliveau 2017):
where
– zi and zj are the coordinates of geographical entities;
– zi is the value of the variable for entity i, its mean being
– i is the geographical entity;
– j is the neighbor of entity i;
– n is the total number of geographical entities in the sample;
– m is the total number of pairs of neighbors;
– w is the weighting matrix, the elements of which take, for example, the value 1 for the neighboring i, j and 0 otherwise.
Moran’s I formula compares the difference between the ratio of the value of a variable concerning an individual to the mean of these values, to the ratio of the value of the same variable for neighboring individuals to the same mean. This I index takes these values between –1 (negative spatial autocorrelation) and +1 (positive spatial autocorrelation). But the value of I can exceed 1 or be less than –1 (Oliveau 2011). A non-zero measure of this index shows a contiguous effect between close spaces:
– if I > 0, the contiguous spaces have similar measures of the variable;
– if I < 0, it means the absence of the significant variation or disparate values;
– if I is close to 0, there is no negative or positive spatial autocorrelation.
But the measurement of this index I is not without criticism and may not clearly reflect spatial structures. There are three limitations to the measurement of Moran’s I (Oliveau 2011):
– the measurement of spatial autocorrelation obtained from Moran’s I is global unlike Geary’s C which provides a local measurement of spatial autocorrelation. This global character of Moran's I can lead not to a precise spatial structure, but to two different spatial configurations:- a configuration highlighting a central pole;- the presence of two peripheral poles;
– Moran’s I considers the deviation from the mean without looking at neighboring individuals, but also with values close to the mean;
– Moran’s I is sensitive, on the one hand, to the level of observation, and on the other hand, to the mode of neighborhood chosen.
Moran’s I measured the spatial coherence of a chain of stores with a measure of the network’s territorial coverage by relative entropy (Rulence 2003). The influence of time on spatial dependence has been estimated by data on property prices (Devaux and Dubé 2016).
Once again, Moran’s I has the disadvantage of being too global when looking at spatial structures over small areas (Brunet and Dollfus 1990), which requires the use of Geary’s C index. The Geary C index (Geary 1954) is also used to measure spatial autocorrelation and is presented as follows (Oliveau 2017) following a rewrite (Cliff and Ord 1973):
where
– zi and zj are the coordinates of geographical entities;
– zi is the value of the variable for entity i, its mean being
– i is the geographical entity;
– j is the neighbor of entity i;
– n is the total number of geographical entities in the sample;
– m is the total number of pairs of neighbors;
– w is the weighting matrix, the elements of which take, for example, the value 1 for the neighboring i, j and 0 otherwise.
Unlike Moran’s I, Geary’s C value can indicate a positive autocorrelation if it is less than 1 (the minimum being 0), a negative autocorrelation with a value greater than 1 (the maximum being 2), or the absence of autocorrelation if it takes the value 1. Moran’s I’s and Geary’s C’s were used, for example, to examine the spatial distribution of employment in public services in 124 European regions (Rodriguez and Camacho 2008). Frequent use of these indices has shown that Moran’s I is more powerful than Geary’s C (Jayet 2001) and, for this reason, is the most widely used. However, the objective of the research must be considered and if it is more focused on the discovery of small spatial structures, Geary’s C will be more appropriate. These indices are primarily used as tests to assess the presence or absence of spatial autocorrelation, which have only an asymptotic value, meaning that they can only be used with a large amount of data (Jayet 2001).
The Gini coefficient (Gini 1921), or Gini concentration index, measures the distribution of income and wealth in a given population. It is widely used in research work and has undergone many changes (Giorgi and Gigliarano 2017). It is sometimes attached to work measuring autocorrelation when considering the population studied in a given space. The Gini coefficient actually measures wage inequality and varies from 0, a situation where there is perfect equality between all wages and 1 situation as unequal as possible, in other words, where all wages are zero except one. The higher the Gini coefficient, the more inequality there is in the distribution of wages3.
1.4.2.3. Simulation systems
Spatial simulations based on multi-agent systems
