Adaptive Bayesian Nonstationary Modeling for Large Spatial Datasets Using Covariance Approximations

Gaussian process models have been widely used in spatial statistics but face tremendous modeling and computational challenges for very large nonstationary spatial datasets. To address these challenges, we develop a Bayesian modeling approach using a nonstationary covariance function constructed based on adaptively selected partitions. The partitioned nonstationary class allows one to knit together local covariance parameters into a valid global nonstationary covariance for prediction, where the local covariance parameters are allowed to be estimated within each partition to reduce computational cost. To further facilitate the computations in local covariance estimation and global prediction, we use the full-scale covariance approximation (FSA) approach for the Bayesian inference of our model. One of our contributions is to model the partitions stochastically by embedding a modified treed partitioning process into the hierarchical models that leads to automated partitioning and substantial computational benefits. We illustrate the utility of our method with simulation studies and the global Total Ozone Matrix Spectrometer (TOMS) data. Supplementary materials for this article are available online.高斯过程模型（Gaussian process models）已在空间统计学领域得到广泛应用，但针对超大规模非平稳空间数据集时，仍面临极为严峻的建模与计算挑战。为解决上述难题，我们提出一种贝叶斯建模方法，该方法采用基于自适应选择分区构建的非平稳协方差函数。该分区非平稳类别可将局部协方差参数整合为一个有效的全局非平稳协方差以用于预测，且允许在每个分区内估计局部协方差参数，从而降低计算成本。为进一步简化局部协方差估计与全局预测中的计算流程，我们针对所提模型的贝叶斯推断采用了全尺度协方差近似（full-scale covariance approximation, FSA）方法。本研究的一项核心贡献在于，通过将改进的树状分区过程嵌入分层模型，以随机方式对分区进行建模，从而实现自动化分区并带来显著的计算优势。我们通过仿真实验与全球总臭氧矩阵光谱仪（Total Ozone Matrix Spectrometer, TOMS）数据集验证了所提方法的实用性。本文的补充材料可在线获取。