Generalized Spatially Varying Coefficient Models

Abstract

In this article, we introduce a new class of nonparametric regression models, called generalized spatially varying coefficient models (GSVCMs), for data distributed over complex domains. For model estimation, we propose a nonparametric quasi-likelihood approach using the bivariate penalized spline approximation technique. We show that our estimation procedure is able to handle irregularly-shaped spatial domains with complex boundaries. Under some regularity conditions, the estimator for the coefficient function is proved to be consistent in the L₂ sense and its convergence rate is established. We develop a numerically stable algorithm using penalized iteratively reweighted least squares method to estimate the coefficient functions in GSVCMs. To gain efficiency in the computation for large-scale data, we further propose a QR decomposition-based algorithm, which requires only sub-blocks of the design matrix to be computed at a time, so that it allows efficient estimation of GSVCMs for large datasets with modest computer hardware. The finite sample performance of the GSVCM and its estimation method is examined by simulations studies. The proposed method is also illustrated by an analysis of the crash data in Florida. Supplementary materials for this article are available online.

Keywords:

• Bivariate penalized spline
• Generalized cross-validation
• QR decomposition
• Quasi-likelihood
• Triangulation

Supplementary Materials

Technical material: The file GSVCM_sub_final.pdf (PDF file) provides the detailed notations and detailed proof of Theorem 1 in the article.

Codes and data: The file contains the codes and data for Sections 4 and 5. A README file describes the contents. (Code.zip)

R package BPST: R package “BPST” used to perform the proposed method in the article is available in the following website: https://github.com/funstatpackages/BPST.

Acknowledgments

The authors would like to thank the editor, Dr. Tyler McCormick, an associate editor, and three referees for their constructive comments that significantly improved an earlier version of this article.

Additional information

Funding

Li Wang’s research was supported in part by National Science Foundation awards DMS-1542332 and DMS-1916204.