Spatial Linear Regression with Covariate Measurement Errors: Inference and Scalable Computation in a Functional Modeling Approach

Abstract

For spatial linear models, the classical maximum-likelihood estimators of both regression coefficients and variance components can be biased when the covariates are measured with errors. This work introduces a theoretically backed-up estimation framework for the spatial linear errors-in-variables model in a functional approach. Compared with the structural models, the functional approach treats the unobserved true covariates as fixed unknown parameters without imposing additional structures, thus leading to more robust parameter inference. Our model parameters are estimated simultaneously based on a set of unbiased estimating equations. Under some regularity conditions, we prove the consistency of the proposed estimating-equation estimators and derive their asymptotic distribution. In addition, a consistent variance estimator is developed for the estimating-equation estimators. To handle large spatial datasets, we provide two approaches to obtain scalable estimations based on our proposed estimating equations, where the required computational time and storage are reduced to be linear with sample size for each estimating-function evaluation. Simulation studies under different settings show that our estimators are consistent and the scalable algorithms work well. Finally, the proposed method is applied to studying the relationship between Arctic sea ice and related geophysical variables. Supplementary materials for this article are available online.

KEYWORDS:

• Arctic sea ice
• Covariance approximation
• Errors-in-variables model
• Gaussian process
• Spatial process
• Unbiased estimating equation

Acknowledgments

The authors thank the Editor, the Associate Editor, and two anonymous reviewers for their comments that helped us significantly improve this work. This research was supported by Public Computing Cloud, Renmin University of China. The authors are listed alphabetically. The research of Shiyuan He was partially supported by National Natural Science Foundation of China (No.11801561). The research of Bohai Zhang was partially supported by National Natural Science Foundation of China (No.11901316), and the Fundamental Research Funds for the Central Universities, Nankai University (63201156), China. The authors report there are no competing interests to declare.

Supplementary Materials

Technical material: The file (Supp_MESE.pdf, PDF file) provides technical details of the theoretical results and algorithm implementation. The file also contains supplementary results of the numerical studies.

Code: The archive file (Code_MESE.zip, ZIP file) contains the R code implementing the proposed method with several illustrative examples. A README file is included to describe its contents.

Additional information

Funding

The authors thank the Editor, the Associate Editor, and two anonymous reviewers for their comments that helped us significantly improve this work. This research was supported by Public Computing Cloud, Renmin University of China. The authors are listed alphabetically. The research of Shiyuan He was partially supported by National Natural Science Foundation of China (No.11801561). The research of Bohai Zhang was partially supported by National Natural Science Foundation of China (No.11901316), and the Fundamental Research Funds for the Central Universities, Nankai University (63201156), China. The authors report there are no competing interests to declare.