Spatial Homogeneity Pursuit of Regression Coefficients for Large Datasets

Abstract

Spatial regression models have been widely used to describe the relationship between a response variable and some explanatory variables over a region of interest, taking into account the spatial dependence of the observations. In many applications, relationships between response variables and covariates are expected to exhibit complex spatial patterns. We propose a new approach, referred to as spatially clustered coefficient (SCC) regression, to detect spatially clustered patterns in the regression coefficients. It incorporates spatial neighborhood information through a carefully constructed regularization to automatically detect change points in space and to achieve computational scalability. Our numerical studies suggest that SCC works very effectively, capturing not only clustered coefficients, but also smoothly varying coefficients because of its strong local adaptivity. This flexibility allows researchers to explore various spatial structures in regression coefficients. We also establish theoretical properties of SCC. We use SCC to explore the relationship between the temperature and salinity of sea water in the Atlantic basin; this can provide important insights about the evolution of individual water masses and the pathway and strength of meridional overturning circulation in oceanography. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Keywords:

• Fused lasso
• Minimum spanning tree
• Penalized least squares
• Spatially clustered coefficient
• Varying coefficient regression

Additional information

Funding

This work was supported by NSF [grant nos. DMS-1343155, DMS-1622433, and DMS-1737885] and NIH [grant no. R01-CA19776101].