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.

翻译：高斯随机场是众多科学领域中空间现象的普遍模型。其近似对于模拟、推断及不确定性量化的计算可行性至关重要。Karhunen-Loève展开为将高斯随机场表示为由不相关随机变量加权的确定性正交函数之和，提供了理论上的最优基。尽管这是用于（空间）随机过程降维与近似的成熟方法，但其实际应用取决于协方差结构的显式或隐式定义。在本工作中，我们提出了一种新方法，称为regTPS-KLE，通过正则化薄板样条（TPS）核显式构建协方差来近似高斯随机场。由于TPS核是条件正定的且缺乏直接的谱分解，我们将协方差公式化为一个正则化椭圆算子的逆。为评估其统计性能，我们将其预测精度和计算效率与使用随机偏微分方程（SPDE）方法构建并在MCMC算法中实现的高斯随机场近似进行了比较。在模拟研究中，当空间场使用Matèrn和指数协方差函数生成时，SPDE与regTPS-KLE模型之间的预测差异极小，而regTPS-KLE模型在计算效率方面始终优于SPDE方法。在实际数据应用中，基于留一交叉验证，regTPS-KLE展现出相较于SPDE模型更优的预测精度，同时亦实现了更高的计算效率。