Highly Multivariate Geo-Statistics Using Graphical Models

林业、生态学、气候科学、环境健康和许多其他领域的研究人员经常使用空间统计分析在数千个地点收集的地理标记数据。现代地理信息系统(GIS)能够同时测量每个位置的许多不同变量。这预示着空间统计向多变量范式的转变。对所有变量的联合分析有助于确定不同变量的共同地理模式和来源。在这个项目中，PI追求能够充分处理这种高度多变量的地理空间数据集的新出现的复杂性的统计方法。这些创新包括a)利用关于变量之间相关性的现有科学信息，b)确保算法的计算可伸缩性，以及c)改善多变量分析结果的可解释性。拟议创新的起源在于与气候建模、空气和水质量有关的实质性问题。这些研究领域研究了21世纪人类社会面临的一些最具威胁的挑战。该项目开发的统计方法将使这些领域的从业人员能够利用有限的计算资源进行高度多变量的空间分析。该项目还为研究生提供了在许多不同和基本的统计学领域以及高级统计计算方面进行培训的机会。多变量GP通常是通过混合单变量GP来创建的，这些GP混淆了每个合成曲面的单独空间特征。像多变量MATERN GP这样的直接结构更容易解释，但需要复杂的参数约束，提供的灵活性很小，无法利用关于变量间相关性的先验信息。PI提出了一种创建多变量GP的新方法，该方法赋予每个曲面一个可解释的GP度量，该度量具有曲面特定的方差、光滑度和空间衰减性，但也能够将变量之间的依赖网络整合到构造中。贯穿始终的一个反复出现的主题是对图形模型的多功能开发。定义在空间、时间和变量域上的图被用来创建在解释、计算和结构方面具有所需性质的多变量GP。另一个随之而来的主题是利用GP的标准分解将离散结构扩展到定义良好的连续随机过程，从而能够在任何新位置进行预测。将探索新的、简单但有效的参数估计策略。最后，PI分别关注非欧几里德空间域，如河口和河网。新的单变量GP将被设计出来，以尊重这些区域的复杂轮廓。随后，图形模型的和谐应用将创建多变量局部平滑GP来分析这些域上的多变量空间数据。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Researchers in forestry, ecology, climate sciences, environmental health, and many other fields routinely analyze geo-tagged data collected at thousands of locations using spatial statistics. Modern Geographical Information Systems (GIS) are empowered to simultaneously measure many different variables at each location. This heralds a shift towards a multivariate paradigm in spatial statistics. A joint analysis of all the variables help identify common geographical patterns and sources for the different variables. In this project, the PI pursues statistical methodology that can adequately address the emerging complexities of such highly-multivariate geospatial datasets. The innovations include a) utilizing available scientific information about the dependence among variables, b) ensuring computational scalability of the algorithms, and c) improving interpretability of findings from the multivariate analysis. The genesis of the proposed innovations lies in substantive questions related to climate modeling, air and water quality. These research domains study some of the most threatening challenges to the human society in the twenty-first century. The statistical methods developed in this project will enable practitioners in these fields to conduct highly-multivariate spatial analysis using modest computing resources. The project also provides the opportunity to train graduate students in many diverse and essential areas of statistics as well as in advanced statistical computing.Gaussian Processes (GPs) have long been used for modeling multivariate spatial surfaces. Multivariate GPs are often created by mixing univariate ones which obfuscate the individual spatial characteristics of each resultant surface. Direct constructions like the multivariate Matern GP are more interpretable but entail complex parameter constraints offering little flexibility to exploit prior information about inter-variable dependence. The PI proposes a novel procedure to create multivariate GPs that endows each surface with an interpretable GP measure with surface-specific variance, smoothness and spatial decay, but also enables incorporating the dependency network among the variables into the construction. A recurrent theme throughout is the versatile exploitation of graphical models. Graphs defined in space, time and variable domains are used to create multivariate GPs with desirable properties in terms of interpretation, computation and structure. Another accompanying theme is utilizing a standard decomposition of GPs to extend the discrete construction to well-defined continuous stochastic processes, thereby enabling predictions at any new location. Novel, simple, but efficient strategies will be explored for parameter estimation. Finally, the PI separately focuses on non-Euclidean spatial domains like estuaries and river networks. New univariate GPs will be devised that respect the complicated contours of these domains. Subsequently, harmonious application of graphical models will create multivariate locally smooth GPs to analyze multivariate spatial data on such domains.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.