Hierarchical models for Large Geostatistical Datasets with Applications to Forestry and Ecology

该提案为对可从大量地点获得的点参照空间数据进行统计推断提供了一个综合框架。统计理论用于开发数学上正式但计算上可行的方法，可以有广泛的应用。通过马尔可夫链蒙特卡罗（MCMC）方法实现的分层模型已成为特别受欢迎的空间建模，因为它们的灵活性和能力，以适应模型，这将是不可行的与经典的方法，以及他们的避免可能不适当的渐近。然而，拟合层次空间模型往往涉及昂贵的矩阵分解，其计算复杂度随着空间位置的数量以立方阶增加，使得这种模型对于大型空间数据集不可行。这种计算负担加剧了多变量设置与几个空间相关的响应变量，也当数据收集在频繁的时间点和时空过程模型使用。研究人员提出了一类基于随机过程的模型，该随机过程是将原始过程投影到低维子空间上的结果。研究人员将这些模型称为预测过程模型，并建议探索其理论特性。PI的长期目标是开发一套完整的统计方法，在林业和生态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区别在于，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。这解决了统计上的讽刺，即大型数据集正是允许对丰富关联结构进行统计估计的地方。 重点是即使使用功能中等的计算工具也可以执行的模型，因此可以供大量研究人员使用，随着地理信息系统和全球定位系统等空间参照技术的日益普及和可用性，这些技术可以用简单的手持设备确定地理坐标，如今，各种学科的科学家和研究人员都可以访问大量的地理编码数据。更广泛的影响，建议的方法是最好的评估连接本研究的结果与GIS对人类社会的广泛认可的影响。从确定健康标准的空间差异到更精确的天气预测，GIS技术如今几乎应用于社会的各个领域。通过挽救调查人员使用临时和定性的方法，往往带来虚假的故事，所提出的方法可以在环境研究中产生深远的有益影响，可能触及社会的意想不到的角落。考虑这样一种情况：由于模型不足，生态学家无法识别多个物种之间的关键共生关系。数学形式主义，其所有的复杂性，最大限度地减少这种错误所产生的定性技术目前流行的林业和生态分析。这样的问题和其他一些科学问题需要正式的空间分析，利用大型数据集所携带的信息的全部力量。它们包括但不限于公共和环境卫生、气象学、工程学、地球科学等，其基本目标是相同的：使用有助于改善人类社会的新发现。

This proposal lays down a comprehensive framework for carrying out statistical inference on point-referenced spatial data that are available from a large number of locations. Statistical theory is used to develop mathematically formal but computationally feasible methods that can have a broad range of applications. Hierarchical models implemented through Markov chain Monte Carlo (MCMC) methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well their avoidance of possibly inappropriate asymptotics. However, fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is aggravated in multivariate settings with several spatially dependent response variables and also when data is collected at frequent time points and spatiotemporal process models are used. The investigators propose a class of models based upon a stochastic process that results from projecting the original process onto a lower-dimensional subspace. The investigators term these models as predictive process models and propose to explore their theoretical properties. 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 and ecology. A recurrent underlying theme of the proposed methods that distinguishes it 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 statistical estimates of rich association structures are permissible. The emphasis is on models that can be executed even with moderately powerful computing tools and so would be accessible to a large number of researchers.With the increasing popularity and availability of spatial referencing technologies such as Geographical Information Systems (GIS) and Global Positioning Systems (GPS) that can identify geographical coordinates with a simple hand-held device, scientists and researchers in a variety of disciplines today have access to large amounts of geocoded data. 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. By redeeming the investigators from using ad-hoc and qualitative methods that often bring out spurious stories, the proposed methods can have far reaching beneficial effects in environmental research that potentially touch unexpected corners of society. Consider a situation where an ecologist is unable to recognize critical symbiotic relationships between multiples species, due to inadequate models. Mathematical formalism, for all its complexities, minimizes such errors arising from qualitative techniques currently prevalent in forestry and ecological analysis. Such and several other scientific problems require formal spatial analysis, harnessing the full power of the information that large datasets carry. They include, but are not limited to, public and environmental health, meteorology, engineering, geosciences and so on, where the fundamental goal is the same: use new findings that will help improve human society.