Spatial modelling often uses Gaussian random fields to capture the stochastic nature of studied phenomena. However, this approach incurs significant computational burdens (O(n3)), primarily due to covariance matrix computations. In this study, we propose to use a low-rank approximation of a thin plate spline as a spatial random effect in Bayesian spatial models. We compare its statistical performance and computational efficiency with the approximated Gaussian random field (by the SPDE method). In this case, the dense matrix of the thin plate spline is approximated using a truncated spectral decomposition, resulting in computational complexity of O(kn2) operations, where k is the number of knots. Bayesian inference is conducted via the Hamiltonian Monte Carlo algorithm of the probabilistic software Stan, which allows us to evaluate performance and diagnostics for the proposed models. A simulation study reveals that both models accurately recover the parameters used to simulate data. However, models using a thin plate spline demonstrate superior execution time to achieve the convergence of chains compared to the models utilizing an approximated Gaussian random field. Furthermore, thin plate spline models exhibited better computational efficiency for simulated data coming from different spatial locations. In a real application, models using a thin plate spline as spatial random effect produced similar results in estimating a relative index of abundance for a benthic marine species when compared to models incorporating an approximated Gaussian random field. Although they were not the more computational efficient models, their simplicity in parametrization, execution time and predictive performance make them a valid alternative for spatial modelling under Bayesian inference.

翻译：空间建模常采用高斯随机场来刻画研究现象的随机特性。然而，该方法因协方差矩阵计算而面临显著的计算负担（O(n³)）。本研究提出在贝叶斯空间模型中将薄板样条的低秩近似作为空间随机效应，并将其统计性能与计算效率同近似高斯随机场（基于SPDE方法）进行比较。通过截断谱分解对薄板样条的稠密矩阵进行近似，计算复杂度降为O(kn²)次操作（k为节点数）。采用概率统计软件Stan中的哈密顿蒙特卡洛算法进行贝叶斯推断，从而评估所提模型的性能与诊断效果。模拟研究表明：两种模型均能准确恢复数据生成参数；但采用薄板样条的模型在实现链收敛方面较之近似高斯随机场模型具有更优的执行时间。此外，对于来自不同空间位置的模拟数据，薄板样条模型展现出更优的计算效率。在实际应用中，相较于采用近似高斯随机场的模型，以薄板样条作为空间随机效应的模型在估算底栖海洋物种相对丰度指数时可产生相似结果。尽管这类模型并非计算效率最优的模型，但其参数化简洁性、执行时间与预测性能使其成为贝叶斯推断框架下空间建模的有效替代方案。