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

该提案为对大量地点提供的以点为基准的空间数据进行统计推断制定了一个全面的框架。统计理论被用来开发数学形式但在计算上可行的方法，这些方法可以有广泛的应用范围。通过马尔可夫链蒙特卡罗(MCMC)方法实现的分层模型已经成为空间建模的特别流行的模型，因为它们具有灵活性和能力来拟合经典方法不可行的模型，并且避免了可能不适当的渐近。然而，拟合分层空间模型通常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三次数量级增加，使得这种模型对于大型空间数据集是不可行的。在具有几个空间相关响应变量的多变量环境中，以及在频繁的时间点收集数据和使用时空过程模型时，这种计算负担会加剧。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较低维子空间的结果。研究人员将这些模型称为预测性过程模型，并建议探索它们的理论性质。PI的长期目标是开发一整套统计方法，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一个反复出现的基本主题是，建模者不需要牺牲建模的丰富性来折衷大型数据集。这解决了一个具有统计学讽刺意味的问题，即大型数据集正是允许对丰富的关联结构进行统计估计的地方。随着地理信息系统(GIS)和全球定位系统(GPS)等空间参考技术的日益普及和可用性，可以通过简单的手持设备识别地理坐标，今天不同学科的科学家和研究人员可以访问大量的地理编码数据。通过将这项研究的结果与公认的地理信息系统对人类社会的影响联系起来，可以最好地评估拟议方法的更广泛影响。从识别健康标准的空间差异到更精确的天气预报，地理信息系统技术今天几乎应用于社会的每一个领域。通过弥补调查人员使用特别和定性的方法经常带来虚假故事，拟议的方法可以在潜在触及社会意外角落的环境研究中产生深远的有益影响。考虑这样一种情况：由于模型不完善，生态学家无法认识到多种物种之间的关键共生关系。数学形式主义，尽管其所有的复杂性，最大限度地减少了目前在林业和生态分析中流行的定性技术所产生的错误。这类和其他几个科学问题需要正式的空间分析，充分利用大型数据集所承载的信息的全部力量。它们包括但不限于公共和环境健康、气象学、工程学、地球科学等，这些领域的基本目标是相同的：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.