regTPS-KLE: A Novel Approach To Approximate A Gaussian Random Field for Bayesian Spatial Modeling

TL;DR

This paper tackles scalable Bayesian spatial modeling by approximating Gaussian random fields (GRFs) through a novel regTPS-KLE approach. By defining the covariance as the inverse of a regularized elliptic operator $L_{\alpha} = \\mathbf{I} + \alpha \\Delta^{2}$, it derives a Mercer-like spectral representation that combines thin plate splines with a Karhunen–Loève expansion, including explicit handling of the polynomial null space. The authors develop a Ritz–Galerkin discretization using TPS basis functions, employ adaptive truncation to retain variance, and implement a non-centered Bayesian inference scheme with efficient MCMC leveraging precomputed eigenstructure. Through simulations with Matérn and Exponential covariances and a NO$_2$ real-data application, regTPS-KLE demonstrates competitive predictive accuracy relative to SPDE while offering superior computational efficiency and automatic dimension reduction, highlighting its practical appeal for large-scale spatial analysis.

Abstract

Gaussian random field is a 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 processes, its practical application depends on the explicit or implicit definition of the covariance structure. In this work, we propose a novel approach, referred to as regTPS-KLE, for approximating a Gaussian random field by explicitly constructing its covariance via a regularized thin plate spline (TPS) kernel. Because TPS kernels are conditionally positive definite and lack a direct spectral decomposition, we formulate the covariance as the inverse of a regularized elliptic operator. To evaluate its statistical performance, we compare its predictive accuracy and computational efficiency with a Gaussian random field approximation constructed using the stochastic partial differential equations (SPDE) method and implemented within an MCMC algorithm. In simulation studies, the predictive differences between the SPDE and regTPS-KLE models were minimal when the spatial field was generated using Matèrn and exponential covariance functions, while regTPS-KLE models consistently outperformed the SPDE approach in terms of computational efficiency. In a real data application, regTPS-KLE exhibits superior predictive accuracy compared with SPDE models based on leave-one-out cross-validation while also achieving improved computational efficiency.

Figures (12)

• Figure 1: Posterior (blue color) and prior distributions (red color) for the penalty parameter $\alpha$. The vertical dashed lines indicates the quantiles (0.2 and 0.8) associated with the posterior distributions.
• Figure 2: Posterior means $|\mathbb{E}[z_k \mid \mathbf{y}]|$ versus prior standard deviations $\sqrt{\lambda_{k,\alpha}}$ in a $\log_{10}$ scale for all scenarios. The red circles indicates the null space modes ($k \leq 3$) with unit prior variance and blue triangles indicates the penalized modes with $\lambda_{k,\alpha} = (1 + \alpha v_k)^{-1}$. The dashed line is the theoretical prior expectation $\mathbb{E}[|z_k|] = \sqrt{2\lambda_{k,\alpha}/\pi}$ while the gray region is $\pm 2\text{SD}$ band.
• Figure 3: Cumulative proportion of variance explained by the truncated KLE. The dashed horizontal line indicates the 99% threshold used to select the truncation level.
• Figure 4: The penalty eigenvalues $v_k$ increase rapidly after the null-space components, reflecting increasing roughness of higher-order TPS basis functions. Small values of $v_k$ correspond to smooth, low-frequency modes, whereas large values penalize highly oscillatory components
• Figure 5: Normalized estimation errors of the SPDE (left side) and regTPS-KLE model (right side) relative to the true Matérn covariance. Spatial patterns of the heatmap reveal underestimation (blue) and overestimation (red) for each model.