Spatially-Encouraged Spectral Clustering

Spatially-constrained clustering is a central concern in geographic data science. It finds applications in epidemiology, sociological neighborhood study, criminology, operations research, statistics, and econometrics, to name a few. One recently-developed method provides a powerful technique that ostensibly allows contiguity and attribute similarity to be balanced. Using a single parameter, this method suggests that geography and feature information can be balanced to generate regions that compromise on geographic regularity and feature coherence. In this paper, I investigate a free parameter omitted in their discussion. I find this free parameter has a significant impact on the solution structure; in addition to the single parameter considered therein, this parameter also affects the balance of contiguity and attribute similarity in identified clusters. This is because what the technique uses are two separate kernels --- one for spatial similarity and one for attribute similarity. Both kernels affect solution quality. Exploiting this realization, I create a generalization that leverages this behavior and apply it in two empirical examples with data of differing spatial support. This generalization, called ``spatially-encouraged spectral clustering,'' embeds the fact that spatial and attribute information can be combined in a variety of ways, and parameters must be available for both sets of information in order to use them effectively in the quintessential geographic data science problem of cluster detection.