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扩展子集，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，以深入了解潜在的现象。统计分析需要灵活的非平稳和多元结构，迄今为止，由于缺乏适合大数据集的模型，这些结构受到阻碍。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依赖性的非平稳和多元空间模型开发了一个统一的框架。此外，使用非平稳模型的理由通常归结为数据和模拟实验的经验结果；本研究将为探索这些更复杂的空间模型的相对好处发展一个配套理论。使用本项目中引入的工具，最终的主要目标是开发美国历史气候的网格数据产品，该产品基于大型、不规则间隔的观测网络，具有透明的统计方法和对这种分析中的不确定性的正式量化。诸如此类的历史数据产品在大气和气候科学领域具有至关重要的意义。现代空间统计越来越注重开发涉及具有复杂依赖结构的多个变量的大规模空间数据集的方法。本研究旨在通过多分辨率过程建立一个通用框架，用于模拟非平稳和多元空间结构，这些结构在面对大样本量时不会崩溃。多分辨率过程有助于快速估计和计算，也有助于解决空间估计量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。本研究的配套大样本理论旨在确定非平稳模型比简单的平稳表亲提供切实好处的情况。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Matern，允许在这些常见的协方差类别下对空间平滑进行理论处理。此外，该项目将把多分辨率过程的概念推广到多变量设置，允许对大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.