Distributed Heterogeneity Learning for Generalized Partially Linear Models with Spatially Varying Coefficients

Abstract

Spatial heterogeneity is of great importance in social, economic, and environmental science studies. The spatially varying coefficient model is a popular and effective spatial regression technique to address spatial heterogeneity. However, accounting for heterogeneity comes at the cost of reducing model parsimony. To balance flexibility and parsimony, this article develops a class of generalized partially linear spatially varying coefficient models which allow the inclusion of both constant and spatially varying effects of covariates. Another significant challenge in many applications comes from the enormous size of the spatial datasets collected from modern technologies. To tackle this challenge, we design a novel distributed heterogeneity learning (DHL) method based on bivariate spline smoothing over a triangulation of the domain. The proposed DHL algorithm has a simple, scalable, and communication-efficient implementation scheme that can almost achieve linear speedup. In addition, this article provides rigorous theoretical support for the DHL framework. We prove that the DHL constant coefficient estimators are asymptotic normal and the DHL spline estimators reach the same convergence rate as the global spline estimators obtained using the entire dataset. The proposed DHL method is evaluated through extensive simulation studies and analyses of U.S. loan application data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

KEYWORDS:

• Bivariate penalized splines
• Domain decomposition
• Distributed learning inference
• Semiparametric spatial regression
• Triangulation

Supplementary Materials

In the supplemental materials, we provide the scripts and code for simulations, the technical proofs for the main theorems, an additional simulation study, and detailed descriptions of explanatory variables used in our application studies.

Acknowledgements

The authors are immensely grateful to Dr. Ming-Jun Lai from the University of Georgia for providing insight and expertise. His generous discussion and plentiful knowledge of bivariate splines have improved this study in innumerable ways. The authors also want to thank the helpful comments from the editor, associate editor, and two anonymous reviewers in improving the manuscript.

Disclosure Statement

The authors report that there are no competing interests to declare.

Additional information

Funding

This research is partially supported by National Science Foundation award DMS-2203207 (Li Wang), Simons Foundation Mathematics and Physical Sciences-Collaboration Grant for Mathematicians #963447 (Guannan Wang), and the National Institutes of Health grant NIH 1R01AG085616 (Li Wang and Guannan Wang).