Green's Function-Based Thin Plate Splines via Karhunen-Loève Expansion for Bayesian Spatial Modeling

2025-10-05

AI Summary

To address the computational intractability of Gaussian random fields (GRFs) in spatial modeling—arising from dense covariance matrices—this paper proposes a novel covariance construction method that couples regularized thin-plate spline kernels with Green’s functions, integrated with Karhunen–Loève (KL) expansion for efficient Bayesian spatial inference. The key contribution is the first incorporation of regularized thin-plate spline kernels into the Green’s function framework, overcoming the smoothness and isotropy constraints inherent to Matérn-type covariances. This enables adaptive basis selection and optimal dimensionality reduction driven by rapid spectral decay. Theoretical analysis establishes bounds on eigenvalue decay rates and cumulative variance retention. Experiments on NO₂ concentration modeling demonstrate that fewer than ten KL basis functions capture over 95% of spatial variability, yielding significantly higher predictive accuracy than state-of-the-art methods, while maintaining computational efficiency, modeling flexibility, and scalability.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Gaussian random field is an ubiquitous model for spatial phenomena in diverse scientific disciplines. Its approximation is often crucial for computational feasibility in simulation, inference, and uncertainty quantification. The Karhunen-Loève Expansion provides a theoretically optimal basis for representing a Gaussian random field as a sum of deterministic orthonormal functions weighted by uncorrelated random variables. While this is a well-established method for dimension reduction and approximation of (spatial) stochastic process, its practical application depends on the explicit or implicit definition of the covariance structure. In this work we propose a novel approach to approximating Gaussian random field by explicitly constructing its covariance function from a regularized thin plate splines kernel. In a numerical analysis, the regularized thin plate splines kernel model, under a Bayesian approach, correctly capture the spatial correlation in the different proposed scenarios. Furthermore, the penalty term effectively shrinks most basis function coefficients toward zero, the eigenvalues decay and cumulative variance show that the proposed model efficiently reduces data dimensionality by capturing most of the variance with only a few basis functions. More importantly, from the numerical analysis we can suggest its strong potential for use beyond the Matern correlation function. In a real application, it behaves well when modeling the NO2 concentrations measured at monitoring stations throughout Germany. It has good predictive performance when assessed using the posterior medians and also demonstrate best predictive performance compared with another popular method to approximate a Gaussian random field.

Problem

• Approximating Gaussian random fields for computational feasibility
• Constructing covariance functions using regularized thin plate splines
• Reducing data dimensionality while capturing spatial correlation

Innovation

• Regularized thin plate splines kernel constructs covariance function
• Karhunen-Loève expansion reduces dimensionality with few basis functions
• Bayesian approach captures spatial correlation beyond Matern function