Fast simulation of Gaussian random fields with flexible correlation models in Euclidean spaces

2025-12-21

AI Summary

Efficient simulation of Gaussian random fields with flexible correlation structures remains a key challenge in spatial statistics and uncertainty quantification. This paper overcomes the traditional limitation of the spectral turning bands (STB) method—which has been restricted to Matérn covariances—by introducing a unified framework supporting three important classes: Matérn, Kummer–Tricomi (with polynomial decay or long-range dependence, represented via Beta-prime mixtures), and Gauss–Hypergeometric (with compact support, including the generalized Wendland family, represented via Beta/Gasper mixtures). The proposed algorithm ensures numerical stability and achieves optimal O(N) linear computational complexity. Integrated with weighted pairwise composite likelihood estimation and parametric bootstrap, it enables standard error estimation and model selection on large-scale climate datasets. This advancement substantially enhances the scalability and practical applicability of Gaussian random field simulation in real-world settings.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The efficient simulation of Gaussian random fields with flexible correlation structures is fundamental in spatial statistics, machine learning, and uncertainty quantification. In this work, we revisit the emph{spectral turning-bands} (STB) method as a versatile and scalable framework for simulating isotropic Gaussian random fields with a broad range of covariance models. Beyond the classical Matérn family, we show that the STB approach can be extended to two recent and flexible correlation classes that generalize the Matérn model: the Bummer-Tricomi model, which allows for polynomially decaying correlations and long-range dependence, and the Gauss-Hypergeometric model, which admits compactly supported correlations, including the Generalized Wendland family as a special case. We derive exact stochastic representations for both families: a Beta-prime mixture formulation for the Kummer-Tricomi model and complementary Beta- and Gasper-mixture representations for the Gauss-Hypergeometric model. These formulations enable exact, numerically stable, and computationally efficient simulation with linear complexity in the number of spectral components. Numerical experiments confirm the accuracy and computational stability of the proposed algorithms across a wide range of parameter configurations, demonstrating their practical viability for large-scale spatial modeling. As an application, we use the proposed STB simulators to perform parametric bootstrap for standard error estimation and model selection under weighted pairwise composite likelihood in the analysis of a large climate dataset.

Problem

• Simulate Gaussian random fields with flexible correlation models efficiently
• Extend spectral turning-bands method to new covariance families like Bummer-Tricomi
• Enable exact, stable simulation for large-scale spatial modeling applications

Innovation

• Spectral turning-bands method for isotropic Gaussian random fields
• Exact stochastic representations for flexible correlation models
• Linear complexity simulation with numerically stable algorithms