Spatial clustering of count differences via Skellam mixtures and Potts model

We propose a novel spatial finite mixture model for the analysis and clustering of integer-valued differences between Poisson counts, motivated by applications where the scientific interest lies in net differences—such as mortality gaps or migration flows—rather than marginal event counts. Our approach models the observed differences using a mixture of Skellam distributions, providing a natural probabilistic framework for discrete, signed outcomes. Spatial dependence among regions is accommodated via a local Potts-type conditional specification on the latent cluster allocation, encouraging spatial coherence in the clustering structure. Parameter estimation is performed through maximum (pseudo) likelihood, via an EM-type algorithm through a mean-field approximation, with cluster assignment updates based on local maximization or mean-field approximations, and uncertainty quantified via bootstrap. Through simulation studies, we demonstrate the method’s ability to recover both true clustering and underlying parameters in scenarios with spatial structure and well-separated clusters. We further illustrate the approach using data on male-female neoplasms mortality gaps across Italian provinces, uncovering interpretable spatial patterns in excess mortality. The proposed methodology extends existing spatial mixture models to settings with integer-valued differences and provides a flexible tool for spatially structured analysis of comparative risks.