Estimation of Spatial Deformation for Nonstationary Processes via Variogram Alignment

Abstract

In modeling spatial processes, a second-order stationarity assumption is often made. However, for spatial data observed on a vast domain, the covariance function often varies over space, leading to a heterogeneous spatial dependence structure, therefore requiring nonstationary modeling. Spatial deformation is one of the main methods for modeling nonstationary processes, assuming the nonstationary process has a stationary counterpart in the deformed space. The estimation of the deformation function poses severe challenges. Here, we introduce a novel approach for nonstationary geostatistical modeling, using space deformation, when a single realization of the spatial process is observed. Our method is based on aligning regional variograms, where warping variability of the distance from each subregion explains the spatial nonstationarity. We propose to use multi-dimensional scaling to map the warped distances to spatial locations. We assess the performance of our new method using multiple simulation studies. Additionally, we illustrate our methodology on precipitation data to estimate the heterogeneous spatial dependence and to perform spatial predictions.

Keywords:

• Distance warping
• Functional data registration
• Nonstationarity
• Regional variograms

Supplementary Materials

Section S1 provides proofs of Properties 1 and 2. Section S2 provides extended results from the simulation study presented in Section 3. Section S3 presents an additional simulation study. Section S4 provides a discussion on robustness of our method to different subdivisions of the spatial domain. Section S5 gives a quantitative assessment of CMDS. Section S6 provides the computational time for the estimation of the deformed space in the considered simulation studies and the data application. Section S7 provides some additional results from the data application. The code and data for reproducing the results are available at https://github.com/ghulamabdul/Nonstat_Cov.

Acknowledgments

The authors would like to thank the two reviewers, an associate editor, and the editor for constructive comments and helpful suggestions.

Additional information

Funding

The work is partially supported by King Abdullah University of Science and Technology (KAUST), Office of Sponsored Research (OSR) under Award No: OSR-2019-CRG7-3800, the National Science Foundation (NSF) grants: DMS-1613054, CCF-1740761, CCF-1839252, DMS-2015226, and the National Institutes of Health (NIH) grant: R37-CA214955.