Observations of groundwater pollutants, such as arsenic or Perfluorooctane sulfonate (PFOS), are riddled with left censoring. These measurements have impact on the health and lifestyle of the populace. Left censoring of these spatially correlated observations are usually addressed by applying Gaussian processes (GPs), which have theoretical advantages. However, this comes with a challenging computational complexity of $\mathcal{O}(n^3)$, which is impractical for large datasets. Additionally, a sizable proportion of the data being left-censored creates further bottlenecks, since the likelihood computation now involves an intractable high-dimensional integral of the multivariate Gaussian density. In this article, we tackle these two problems simultaneously by approximating the GP with a Gaussian Markov random field (GMRF) approach that exploits an explicit link between a GP with Mat\'ern correlation function and a GMRF using stochastic partial differential equations (SPDEs). We introduce a GMRF-based measurement error into the model, which alleviates the likelihood computation for the censored data, drastically improving the speed of the model while maintaining admirable accuracy. Our approach demonstrates robustness and substantial computational scalability, compared to state-of-the-art methods for censored spatial responses across various simulation settings. Finally, the fit of this fully Bayesian model to the concentration of PFOS in groundwater available at 24,959 sites across California, where 46.62\% responses are censored, produces prediction surface and uncertainty quantification in real time, thereby substantiating the applicability and scalability of the proposed method. Code for implementation is made available via GitHub.

翻译：地下水污染物（如砷或全氟辛烷磺酸PFOS）的观测数据普遍存在左删失现象，此类测量结果对公众健康与生活方式具有重要影响。现有研究通常采用高斯过程（Gaussian processes, GPs）处理具有空间相关性的左删失观测数据，该方法虽具有理论优势，但面临$\mathcal{O}(n^3)$的高计算复杂度难题，难以适用于大规模数据集。此外，当数据中左删失观测值占比过高时，似然计算需涉及多元高斯密度的高维不可解积分，进一步加剧了计算瓶颈。本文通过引入高斯马尔可夫随机场（GMRF）逼近高斯过程，利用Matérn相关函数与随机偏微分方程（SPDE）框架下的GMRF显式关联性，同时解决上述两大难题。我们创新性地在模型中引入基于GMRF的测量误差项，不仅简化了删失数据的似然计算，更显著提升模型运算速度且保持卓越精度。与当前处理删失空间响应的最先进方法相比，本方法在多种仿真场景下展现出鲁棒性与显著的计算可扩展性。最终，将该全贝叶斯模型应用于加利福尼亚州24,959个采样点（其中46.62%响应值存在删失）的地下水中PFOS浓度预测，在实时生成预测表面与不确定性量化的同时，充分验证了所提方法的适用性与可扩展性。相关实现代码已在GitHub开源。