Global and local indicators of spatial connectivity for areal data

RESUMO

Methods for areal data commonly describe spatial structure through global autocorrelation measures or local indicators based on neighboring values. In many applications, however, it is also relevant to know whether high- or low-valued areas form connected spatial structures. Motivated by topological data analysis, we develop global and local indicators of spatial connectivity for observations defined on an areal adjacency graph. We construct thresholded spatial graphs and use the Betti-0 curve to record their number of connected components. We further derive a local decomposition in which each area's contribution to the Betti-0 curve is expressed as an activation term minus a merging term. Each component fusion contributes to the merging term through a symmetric allocation between the entering area and the neighboring areas through which the fusion occurs. Integrating the merging contributions yields area-level indicators of participation in the connectivity of high- or low-valued regions. Global and local inference is based on random relabeling, with a global test for departures from spatial exchangeability and a conditional test for the local indicators. We illustrate the framework using COVID-19 incidence in Italian provinces during two epidemic waves. Significant departures toward greater connectivity were found for both high- and low-incidence areas, with the largest discrepancy corresponding to high incidence during the first wave. The proposed indicators complement conventional measures of spatial association and are applicable whenever the spatial coherence of threshold exceedances or deficits is of interest.