Hierarchical models for Large Geostatistical Datasets with Application

该建议为从大量地点获得的点参考高维空间数据进行统计推断奠定了一个全面的框架。建议的重点是方法论，而不是纯理论或纯应用。因此，统计理论被用来发展数学上形式化但计算上可行的方法，可以有广泛的应用。将探索理论推导和新结果，这些结果将增强当前的方法（包括PI在先前nsf资助的研究中的发现），但始终牢记实践空间分析。该方法的基本框架是利用将原过程投影到低维子空间得到的低秩空间过程。PI打算探索低秩空间过程关于不同度量的近似性质。项目的长期目标是开发一套完整的统计方法，在林业、生态学和更广泛的环境科学的各种实验中估计空间模型。所提出的方法与现有方法的不同之处在于，建模者不需要牺牲建模的丰富性作为对大型数据集的妥协。这解决了统计学上的讽刺，即大型数据集恰恰是可以有效检测复杂关系的地方。现代空间技术，如地理信息系统（GIS）和全球定位系统（GPS）通常用一个简单的手持设备识别地理坐标。因此，今天不同学科的科学家和研究人员可以前所未有地访问地理编码数据。随着数据在观测地点的数量和每个地点的观测数量方面变得越来越高维，科学家们正在寻求对复杂关系的假设。反过来，这些产生了相当复杂的层次模型，即使对于中等规模的数据集，计算成本也很高。该团队认识到需要对大型多元空间数据进行统计建模，并提出了一种基于模型的设置来处理各种各样的大型地质统计数据集。尽管一些更严肃的统计建模将需要多处理器功能，但本项目的重点是使用功能适中的计算工具实现的方法。因此，所提议的方法将可供大量研究人员使用。通过将本研究的结果与GIS对人类社会的广泛认可的影响联系起来，可以最好地评估所提出方法的更广泛影响。从确定卫生标准的空间差异到更精确的天气预报，地理信息系统技术如今几乎用于社会的每个领域，所提出的方法可以在环境研究中产生深远的有益影响，可能触及社会意想不到的角落。

This proposal lays down a comprehensive framework for carrying out statistical inference on point-referenced high-dimensional spatial data available from a large number of locations. The focus of the proposal is methodological rather than purely theoretical or purely applied. Thus, statistical theory is used to develop mathematically formal but computationally feasible methods that can have a broad range of applications. Theoretical derivations and new results that will enhance current methods (including findings by the PI in prior NSF-funded research) will be explored, but always keeping in mind the practicing spatial analyst. The basic framework is to use a low-rank spatial process obtained by projecting the original process onto a lower-dimensional subspace. The PI intends to explore approximation properties of the low rank spatial process with regard to different metrics. The long-term goal of the PI is to develop a full suite of statistical methods that estimate spatial models in a wide variety of experiments in forestry, ecology and the broader environmental sciences. A recurrent underlying theme of the proposed methods that makes it different from existing methods is that the modeler does not need to sacrifice richness in modeling as a compromise for the large datasets. This resolves the statistical irony that large datasets are precisely where complex relationships can be detected effectively.Modern spatial technologies such as Geographical Information Systems (GIS) and Global Positioning Systems (GPS) routinely identify geographical coordinates with a simple hand-held device. Consequently, scientists and researchers in a variety of disciplines today have access to geocoded data as never before. With data becoming increasingly high-dimensional both in terms of number of observed locations and the number of observations per location, scientists are seeking to hypothesize complex relationships. These, in turn, yield rather complex hierarchical models that are computationally expensive even for moderately sized datasets. This team recognises a need for statistical modeling of large multivariate spatial data and proposes a model-based setup to tackle a wide variety of large geostatistical datasets. Although some of the more serious statistical modeling will require multi-processor capabilities, the emphasis on this project is on methodology implementable with moderately powerful computing tools. The proposed methodologies would, therefore, be accessible to a large number of researchers. The broader impact of the proposed methods is best assessed by connecting the outcome of this research with the widely recognized impact of GIS on human society. From identifying spatial disparities in health standards to more precise weather predictions, GIS technology is used today in almost every sphere of society and the proposed methods can have far reaching beneficial effects in environmental research that potentially touch unexpected corners of society.