Non-stationary Spatial Modeling Using Fractional SPDEs

2025-12-21

AI Summary

This paper addresses the challenge of jointly modeling fractional smoothness and spatially varying anisotropy in nonstationary spatial processes. We propose a novel Gaussian random field (GRF) framework based on a coefficient-varying fractional stochastic partial differential equation (SPDE). Our approach unifies the modeling of the fractional differentiation order, spatially varying anisotropy, range, and variance within a single coherent SPDE formulation. To mitigate parameter estimation bias and overfitting, we introduce spectral parameterization and a joint regularization prior. The method leverages sparse matrix computations, automatic differentiation, and gradient-based optimization, with the Continuous Ranked Probability Score (CRPS) as the primary evaluation metric. Empirical results demonstrate robust estimation of key nonstationary parameters from datasets with over 500 observations. In ocean salinity prediction, anisotropy dominates performance; in precipitation forecasting, fractional smoothness is more critical. Our method achieves significantly lower MSE and CRPS compared to state-of-the-art alternatives.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We construct a Gaussian random field (GRF) that combines fractional smoothness with spatially varying anisotropy. The GRF is defined through a stochastic partial differential equation (SPDE), where the range, marginal variance, and anisotropy vary spatially according to a spectral parametrization of the SPDE coefficients. Priors are constructed to reduce overfitting in this flexible covariance model, and parameter estimation is done with an efficient gradient-based optimization approach that combines automatic differentiation with sparse matrix operations. In a simulation study, we investigate how many observations are required to reliably estimate fractional smoothness and non-stationarity, and find that one realization containing 500 observations or more is needed in the scenario considered. We also find that the proposed penalization prevents overfitting across varying numbers of observation locations. Two case studies demonstrate that the relative importance of fractional smoothness and non-stationarity is application dependent. Non-stationarity improves predictions in an application to ocean salinity, whereas fractional smoothness improves predictions in an application to precipitation. Predictive ability is assessed using mean squared error and the continuous ranked probability score. In addition to prediction, the proposed approach can be used as a tool to explore the presence of fractional smoothness and non-stationarity.

Problem

• Modeling non-stationary spatial data with varying anisotropy and smoothness
• Estimating parameters efficiently while preventing overfitting in flexible models
• Assessing the application-dependent importance of fractional smoothness and non-stationarity

Innovation

• Spatially varying anisotropy via fractional SPDEs
• Gradient-based optimization with automatic differentiation
• Penalization to prevent overfitting in flexible models