CAREER: New Directions in Spatial Statistics

De Wijs过程(在统计物理学中也称为高斯自由场)是一个基本的空间过程，它是基于格子的高斯马尔可夫随机场的标度极限，是二维布朗运动的推广。然而，目前统计物理学中的高斯自由场理论(包括后来的随机场理论)和现代概率论与目前基于格点的高斯马尔可夫随机场的空间统计实践之间存在着很大的差距。因此，迫切需要弥合这一差距，为空间模型的统计和推断制定一个原则性框架，并寻求使这种推断可行的新的计算方法。该项目将考虑制定适当的De Wijs过程泛函，以通过共轭梯度和其他方法构建有用的随机场和新的无矩阵计算，并将重点放在开发新的科学应用领域。拟议的研究还将为过去几十年空间统计学中许多研究人员讨论的理论和计算问题提供新的线索，并使其得到更深入的理解。新的无矩阵计算将进一步推动参数Bootstrap方法和多尺度建模的研究，并构造一类新的非高斯随机场。该项目将有助于增进对环境生物分析、地下水砷污染和星系分布的研究的科学认识。空间统计领域的进展很重要，因为新的统计方法可以应用于天文学、农业、生物医学成像、计算机视觉、气候和环境研究、流行病学和地质学等领域的广泛科学问题。De Wijs过程是从时间到空间推广布朗运动的一个基本空间过程。该项目将以De Wijs进程为基础，开发新的数学，并得出快速、高效和大规模的统计计算，以便以实用的方式回答各种科学问题。这将导致连续统空间数据和空间点模式分析的新发展，并将使我们能够在环境生物分析、地下水砷污染和星系分布的研究中获得更好的科学理解。本项目将开发的统计和计算也将特别适用于环境或全球变化以及健康研究中出现的各种研究问题。最后，该项目将通过开发新的研究生和本科课程来整合研究和教育活动，并将为研究生和本科水平的学生提供宝贵的培训和学习机会。

The de Wijs process (also known as the Gaussian free field in statistical physics) is a fundamental spatial process that arises as the scaling limit of lattice based Gaussian Markov random fields and generalizes Brownian motion in two-dimensions. However, at present, there is a wide gap between the theory of Gaussian free field (including the subsequent theory of random fields) in statistical physics and modern probability, and the current practice of spatial statistics via lattice based Gaussian Markov random fields. Thus, there is great need to bridge this gap to develop a principled framework for statistics and inference of spatial models and to pursue novel computations that make such inferences feasible. This project will consider formulating appropriate functionals of the de Wijs process to construct useful random fields and novel matrix-free computations via conjugate gradient and other methods, and will focus on developing new areas of scientific applications. The proposed research will also shed new light on and allow deeper understanding of theoretical and computational issues discussed by many researchers in spatial statistics in the past decades. Novel matrix-free computations will provide further impetus to study parametric bootstrap methods and multi-scale modeling, and to construct a new class of non-Gaussian random fields. The project will contribute to obtaining enhanced scientific understanding in studies of environmental bioassays, arsenic contamination of groundwater and distributions of galaxies. Advances in the field of spatial statistics are important because new statistical methods can be applied to a wide range of scientific questions in fields such as astronomy, agriculture, biomedical imaging, computer vision, climate and environmental studies, epidemiology and geology. The de Wijs process is one fundamental spatial process that generalizes Brownian motion from time to space. Using the de Wijs process as a fundamental building block, this project will develop novel mathematics and derive fast, efficient and large-scale statistical computations so that various scientific questions can be answered in a practical way. This will lead to new developments for the analysis of continuum spatial data and spatial point patterns, and will allow us to obtain enhanced scientific understanding in studies of environmental bioassays, arsenic contamination of groundwater and distributions of galaxies. The statistics and the computations that will be developed in this project will also be particularly relevant for various research problems that arise in environmental or global change, and in health studies. Finally, this project will integrate research and educational activities through the development of new graduate and undergraduate courses and will also provide valuable training and learning opportunities for students at graduate and undergraduate levels.