Predicting Covariate-Driven Spatial Deformation for Nonstationary Gaussian Processes

TL;DR

This paper introduces a covariate-driven spatial deformation model for nonstationary Gaussian processes, enabling better predictions under changing covariate conditions by modeling deformations as functions of covariates.

Contribution

It proposes a novel covariate-dependent deformation approach using Lie algebra, with theoretical truncation and an efficient algorithm for out-of-sample prediction.

Findings

Method outperforms static deformation models in simulations.

Effective in manufacturing and geostatistics case studies.

Provides a scalable estimation-inference algorithm.

Abstract

Nonstationary Gaussian processes (GPs) are essential for modeling complex, locally heterogeneous spatial data. A common modeling approach is the spatial deformation method that warps the domain to recover isotropy. However, this static method does not account for changes in spatial correlation induced by covariates, limiting its ability to predict nonstationary GPs under new covariate conditions. To enable predictive modeling of the deformation method, we propose to model the spatial deformation as a function of covariates. The spaces of diffeomorphic deformations and Euclidean covariate vectors are connected by characterizing deformations as generated by velocity fields living in a Lie algebra. To overcome the estimation instability caused by high-order interactions between multiple covariates in a general Lie algebra, we prove that those interactions can be truncated with a moderate physical assumption. Based on the theoretical results, a concise functional form of deformations driven by multiple covariates can be established, and an efficient estimation-inference algorithm is developed for out-of-sample nonstationary GP prediction with limited covariate-deformation sample pairs. The effectiveness and generalizability of the method are demonstrated on a simulation study and two case studies, in the fields of manufacturing and geostatistics, respectively.