Estimation of Spatial Deformation for Nonstationary Processes via Variogram Alignment

TL;DR

This paper introduces a novel method for modeling nonstationary spatial processes by aligning regional variograms and using multi-dimensional scaling to estimate deformation functions, enabling better spatial predictions.

Contribution

The paper presents a new approach for estimating spatial deformation in nonstationary processes using variogram alignment and multi-dimensional scaling, applicable with a single data realization.

Findings

Effective in simulation studies

Improves spatial prediction accuracy

Captures heterogeneous spatial dependence

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, at a fundamental level, 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 asses 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.