可扩展的贝叶斯SPDE建模用于受 censoring 的空间响应
Computationally Scalable Bayesian SPDE Modeling for Censored Spatial Responses
AI总结：
本文提出了一种基于GMRF的贝叶斯SPDE建模方法，用于处理受左删失的空间响应，解决了计算复杂性和高维积分问题，提高了模型效率和精度。
摘要：
地下水污染物观测，如砷或全氟辛基磺酸（PFOS）的观测，充斥着左删失。这些测量对公众的健康和生活方式有影响。这些空间相关观测的左删失通常通过高斯过程（GPs）来处理，这具有理论优势。然而，这带来了计算复杂度为O(n³)的挑战，对于大规模数据集来说是不切实际的。此外，大量数据被左删失会进一步造成瓶颈，因为似然计算现在涉及了一个不可处理的高维多元高斯密度积分。在本文中，我们同时解决这两个问题，通过用高斯马尔可夫随机场（GMRF）方法近似GP，该方法利用了GP与Matérn相关函数和通过随机偏微分方程（SPDEs）构建的GMRF之间的显式联系。我们引入了一个基于GMRF的测量误差到模型中，这减轻了删失数据的似然计算，大幅提高了模型的速度，同时保持了良好的精度。我们的方法在各种模拟设置中展示了稳健性和显著的计算可扩展性。最后，该完全贝叶斯模型对加利福尼亚州24,959个地点的PFOS浓度拟合，其中46.62%的响应被删失，产生了实时的预测表面和不确定性量化，从而证明了所提方法的适用性和可扩展性。实现代码通过GitHub提供。
英文摘要：
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 Matérn 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.
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