regTPS: a novel approach to approximate a Gaussian random field for Bayesian spatial modeling

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–Loéve Expansion provides an optimal low-rank representation of a (spatial) stochastic process, but its practical implementation relies heavily on how the underlying covariance structure is specified. In this work, we propose a novel approach called regTPS, which constructs a spatial covariance by formulating the inverse of a regularized thin plate spline operator. By computing the eigendecomposition of the regularized biharmonic operator, regTPS induces a smoothness prior that hierarchically allocates variance across spatial frequencies. We evaluate the statistical performance and computational efficiency of regTPS relative to the established stochastic partial differential equations approach within an MCMC algorithm in two different simulation designs. Across various simulation scenarios, the regTPS method achieved predictive accuracy comparable to the SPDE approach. Moreover, regTPS substantially reduced the effective parameter dimension, with reductions of approximately 43–52% for large sample sizes. Also, within a mesh independent simulation setting, regTPS consistently maintains its predictive accuracy under both Matérn and exponential covariance functions, keeping an optimal balance between basis-dimension reduction and sampling efficiency. Lastly, application to real NO₂ NO 2 data concentrations using 10-fold and leave-one-out cross-validation establishes that regTPS obtain a competitive predictive accuracy against SPDE and nearest-neighbor Gaussian process models. Thus, regTPS presents a rigorous, low-dimensional, and computational efficient alternative for Bayesian spatial modeling.