Analysing Spatial Heterogeneity using Optimal Transport

The article presents a general method based on optimal transport theory for quantifying spatial heterogeneity in group distributions. Three indices are proposed allowing for a multi-scale analysis. The global index extends the one introduced by various authors in the study of spatial or temporal concentration. From it, we derive two local indices: an non-negative and a signed indices, which both arise from the optimal coupling of the observed distributions with a null distribution which we defined. We discuss the choice of the optimal coupling among the many possible ones, and its impact on the sensitivity of our local indices to different situations of heterogeneity. All these indices take strongly into account the entire spatial structure of the group distributions. Moreover, all these indices have the advantage to be expressed in meaningful units (meters, travel time), making them easier to interpret. To illustrate the method, numerical applications are made on the first round of the 2022 French presidential election results in Paris, where the city is divided into units corresponding to polling stations while groups represent running candidates.