Monte Carlo Methods for Analysis of Large Spatial Data

空间数据集在许多科学学科中进行分析，例如生态学、地质学和环境科学。然而，经典的方法，如克立格法和贝叶斯分层高斯模型，往往由于昂贵的矩阵求逆运算而在大数据集上失效，其计算复杂度随着空间位置的增加而以三次数量级增加。为了缓解这一困难，人们在用计算方便的模型逼近原始空间模型的总体思想下，提出了各种逼近方法，如协方差锥化、低维空间过程逼近、似然逼近和马尔可夫随机场逼近。这些方法的一个普遍问题是近似值的充分性。在该方案中，研究人员提出了三种新的方法，贝叶斯辅助格法、贝叶斯选址方法和边际推理方法。贝叶斯辅助格子方法将辅助格子引入观测空间，并在辅助格子上定义了一个隐含的高斯马尔可夫随机场。通过利用高斯马尔可夫随机场的一些分析结果，贝叶斯辅助格子方法完全避免了似然估计中的矩阵求逆问题。贝叶斯选址方法将空间模型估计问题重新表述为贝叶斯变量选择问题。它在每次迭代中只处理一小部分数据，因此显著降低了数据的维度。基于Bootstrap重采样的思想，提出了边际推理方法。像贝叶斯选址方法一样，它在每次迭代中只使用一小部分数据，从而显著降低了数据的维度。值得注意的是，贝叶斯选址和边际推理方法在概念上与文献中存在的近似方法有很大的不同。现有的近似方法是使用计算方便的模型来逼近原始模型。相反，贝叶斯选址和边际推理方法寻求降低数据的维度，同时不牺牲原始模型的复杂性。在这项提案中，研究人员还将拟议的方法扩展到时空模型，并将其应用于卫星气候数据。对时空模型中缺失数据的处理进行了研究，该项目的智能价值在于为大型空间数据的统计分析提供了一些计算高效或数据降维的方法。这些新方法解决了空间数据分析中的一些核心问题，如大矩阵求逆和缺失数据填充。预计新方法将在地质统计数据、卫星气候数据和其他大型空间数据的统计分析中发挥重要作用。该项目将在空间统计和计算大气科学界产生更广泛的影响。研究成果将通过与其他学科的研究人员直接合作、会议报告、书籍和将在学术期刊上发表的论文向社区传播。该项目还将通过研究生直接参与该项目并将成果纳入本科生和研究生课程，对教育产生重大影响。

Spatial data sets are analyzed in many scientific disciplines, such as ecology, geology, and environmental sciences. However, the classical approaches, such as Kriging and Bayesian hierarchical Gaussian modeling, often break down for large data sets due to expensive matrix inverse operations, whose computational complexity increases in cubic order with the number of spatial locations. To alleviate this difficulty, various approximation approaches, such as covariance tapering, lower-dimensional space spatial process approximation, likelihood approximation and Markov random field approximations, have been proposed under the general idea of approximating the original spatial model with a computationally convenient model. A general concern on these approaches is the adequacy of approximation. In this proposal, the investigators propose three new approaches, Bayesian auxiliary lattice approach, Bayesian site selection approach and marginal inference approach. The Bayesian auxiliary lattice approach introduces an auxiliary lattice to the space of observations and defines a hidden Gaussian Markov random field on the auxiliary lattice. By using some analytical results of Gaussian Markov random fields, the Bayesian auxiliary lattice approach completely avoids the problem of matrix inversion in likelihood evaluation. The Bayesian site selection approach reformulates the problem of spatial model estimation as a problem of Bayesian variable selection. It works with only a small proportion of the data at each iteration and thus significantly reduces the dimension of the data. The marginal inference approach is proposed based on the idea of bootstrap resampling. Like the Bayesian site selection approach, it works with only a small proportion of the data at each iteration and thus significantly reduces the dimension of the data. It is worth noting that the Bayesian site selection and marginal inference approaches are conceptually very different from the approximation approaches existing in the literature. The existing approximation approaches are to approximate the original model using a computationally convenient model. Instead, the Bayesian site selection and marginal inference approaches seek to reduce the dimension of the data, while not sacrificing the complexity of the original model. In this proposal, the investigators also extend the proposed approaches to spatio-temporal models with applications to satellite climate data. How to deal with missing data for spatio-temporal models are addressed.The intellectual merit of this project is to provide some computationally efficient or data dimension reduction approaches for statistical analysis of large spatial data. The new approaches address some core problems in spatial data analysis, such as large matrix inversion and missing data imputation. The new approaches are expected to play a major role in statistical analysis of geostatistical data, satellite climate data and other large spatial data. This project will have broader impacts in both communities of spatial statistics and computational atmospheric sciences. The research results will be disseminated to the communities via direct collaboration with researchers in other disciplines, conference presentations, books, and papers to be published in academic journals. The project will have also significant impacts on education through direct involvement of graduate students in the project and incorporation of results into undergraduate and graduate courses.