Predicting Covariate-Driven Spatial Deformation for Nonstationary Gaussian Processes

2026-04-29

AI Summary

Traditional spatial deformation methods struggle to model covariate-driven nonstationary spatial dependencies and exhibit limited generalization. This work proposes a covariate-driven diffeomorphic spatial deformation framework that represents the deformation as a function of covariates, generating smooth and invertible mappings via velocity fields in a Lie algebra. To enhance stability and generalizability, the method incorporates a physics-informed truncation strategy for high-order interaction terms. It is the first approach to enable nonstationary Gaussian process extrapolation under covariate conditioning, demonstrating superior small-sample predictive performance on both synthetic data and real-world applications in manufacturing and geostatistics.

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.

Problem

• nonstationary Gaussian processes
• spatial deformation
• covariate-driven
• predictive modeling
• spatial correlation

Innovation

• nonstationary Gaussian processes
• spatial deformation
• covariate-driven
• Lie algebra
• diffeomorphic mapping