Collaborative Research: Theory and Methods for Massive Nonstationary and Multivariate Spatial Processes

空间统计领域是统计科学的一个不断扩大的子集，在地球物理、环境、生态和经济科学等各种专业中有许多应用。这些科学中的现代数据集通常涉及在数千到数百万个不规则间隔的地理位置观察到的多个变量。相关的科学目标包括表面估计、随机模拟和统计建模，以洞察潜在现象。统计分析需要灵活的非平稳和多变量结构，到目前为止，由于缺乏适合大规模数据集的模型，这一点一直受到阻碍。该项目解决了统计科学中的这一空白，为能够模拟复杂空间相关性的非平稳和多变量空间模型开发了一个统一的框架。此外，使用非平稳模型的合理性通常归因于通过数据和模拟实验得出的经验结果；这项研究将为探索这些更复杂的空间模型的相对好处开发一个配套理论。使用该项目中介绍的工具，最终的主要目标是为美国的历史气候开发一个网格数据产品，其基础是具有透明的统计方法和对这种分析中的不确定性进行正式量化的不规则分布的大型观测网络。这样的历史数据产品在大气和气候科学领域具有至关重要的作用。现代空间统计学越来越关注开发涉及多个变量和复杂相关结构的海量空间数据集的方法。这项研究旨在通过多分辨率过程建立一个通用的框架，用于建模非平稳和多变量的空间结构，在面对大样本时不会崩溃。多分辨过程有助于快速估计和计算，也适用于空间估计器的渐近行为的相关理论问题。例如，缺乏对非平稳方法的严格理论处理，目前的理解仅限于实验结果。这项研究的伴随大样本理论旨在确定在哪些情况下，非平稳模型比更简单的平稳模型提供切实的好处。一个相关的目标是现有空间结构的近似理论；特殊的多分辨率结构可以近似现有的协方差，如MATRUN，允许在这些常见的协方差类别下进行空间平滑的理论处理。此外，该项目将把多分辨率过程的概念推广到多变量环境中，允许对海量多变量空间数据集进行可行和灵活的基于推理的建模。

The field of spatial statistics is an expanding subset of statistical science with numerous applications in a wide variety of specialties such as geophysical, environmental, ecological and economic sciences. Modern datasets in these sciences often involve multiple variables observed at thousands to millions of irregularly spaced geographical locations. Associated scientific goals include surface estimation, stochastic simulation and statistical modeling to gain insight of underlying phenomena. Statistical analyses require flexible nonstationary and multivariate constructions, which have heretofore been hampered by a lack of models adequate for datasets of large magnitude. This project addresses this gap in statistical science, developing a unifying framework for nonstationary and multivariate spatial models capable of modeling complex spatial dependencies. Additionally, the justification for the use of nonstationary models is generally relegated to empirical results with data and simulation experiments; this research will develop a companion theory for exploring the relative benefit of these more complex spatial models. Using the tools introduced in this project, the final major goal is to develop a gridded data product for the historical climate of the United States based on large, irregularly spaced observational networks with transparent statistical methodology and formal quantification of the uncertainty in such an analysis. Historical data products such as this are of crucial importance in the fields of atmospheric and climate sciences.Modern spatial statistics has increased focus on developing methods for massive spatial datasets that involve multiple variables with complex dependency structures. This research aims to foster a common framework via multiresolution processes for modeling nonstationary and multivariate spatial structures that does not break down in the face of large sample sizes. Multiresolution processes lend themselves to fast estimation and computation, and also to the linked theoretical questions of asymptotic behavior of spatial estimators. For example, there is a lack of rigorous theoretical treatment of nonstationary approaches, with current understanding limited to experimental results. The companion large sample theory of this research is aimed at identifying situations in which nonstationary models provide tangible benefits over simpler stationary cousins. A linked goal is approximation theory for existing spatial constructions; special multiresolution constructions can approximate existing covariances such as the Matern, allowing for a theoretical treatment of spatial smoothing under these common classes of covariances. Additionally, the project will generalize the notion of a multiresolution process to the multivariate setting, allowing for feasible and flexible inference-based modeling of massive multivariate spatial datasets.