Kriging is an established methodology for predicting spatial data in geostatistics. Current kriging techniques can handle linear dependencies on spatially referenced covariates. Although splines have shown promise in capturing nonlinear dependencies of covariates, their combination with kriging, especially in handling count data, remains underexplored. This paper proposes a novel Bayesian approach to the low-rank representation of geoadditive models, which integrates splines and kriging to account for both spatial correlations and nonlinear dependencies of covariates. The proposed method accommodates Gaussian and count data inherent in many geospatial datasets. Additionally, Laplace approximations to selected posterior distributions enhances computational efficiency, resulting in faster computation times compared to Markov chain Monte Carlo techniques commonly used for Bayesian inference. Method performance is assessed through a simulation study, demonstrating the effectiveness of the proposed approach. The methodology is applied to the analysis of heavy metal concentrations in the Meuse river and vulnerability to the coronavirus disease 2019 (COVID-19) in Belgium. Through this work, we provide a new flexible and computationally efficient framework for analyzing spatial data.

翻译：克里金法是地统计学中预测空间数据的成熟方法。现有克里金技术能够处理空间参考协变量的线性依赖关系。尽管样条函数在捕捉协变量非线性依赖方面展现出潜力，但其与克里金法的结合——特别是在处理计数数据方面——仍未得到充分探索。本文提出了一种新颖的贝叶斯方法，用于实现地理可加模型的低秩表示，该方法通过整合样条函数与克里金技术，能够同时处理空间相关性和协变量的非线性依赖关系。所提出的方法适用于许多地理空间数据集中固有的高斯分布数据与计数数据。此外，通过对特定后验分布采用拉普拉斯近似，显著提升了计算效率，相比贝叶斯推断中常用的马尔可夫链蒙特卡洛方法具有更快的计算速度。通过模拟研究评估了该方法的性能，验证了所提方法的有效性。本方法被应用于默兹河重金属浓度分析以及比利时2019冠状病毒病（COVID-19）脆弱性评估研究。通过这项工作，我们为空间数据分析提供了一个新颖灵活且计算高效的框架。