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上观察到的多个变量。 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变量的结构，这迄今为止一直受到缺乏足够的模型，大规模的数据集。 该项目解决了统计科学中的这一差距，为能够模拟复杂空间依赖关系的非平稳和多变量空间模型开发了一个统一的框架。 此外，使用非平稳模型的理由一般被降级为经验结果与数据和模拟实验，本研究将开发一个同伴理论，探索这些更复杂的空间模型的相对优势。利用该项目中介绍的工具，最终的主要目标是根据大型、不规则间隔的观测网络，开发美国历史气候的网格化数据产品，采用透明的统计方法，并对这种分析中的不确定性进行正式量化。 像这样的历史数据产品在大气和气候科学领域至关重要。现代空间统计学越来越重视为涉及具有复杂依赖结构的多变量的海量空间数据集开发方法。 本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空间结构，这些结构不会在大样本量面前崩溃。 多分辨率过程有助于快速估计和计算，也有助于空间估计量渐近行为的相关理论问题。 例如，缺乏严格的非平稳方法的理论处理，目前的理解仅限于实验结果。 这项研究的同伴大样本理论的目的是确定的情况下，非平稳模型提供了有形的好处比简单的固定表兄弟。 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环境，允许对大量多变量空间数据集进行可行且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.