CAREER: New Directions in Spatial Statistics

德维伊斯过程（英语：de Wijs process）（也被称为高斯自由场）是一个基本的空间过程，它是基于格点的高斯马尔可夫随机场的标度极限，并在二维中推广了布朗运动。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，以及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，有很大的需要弥合这一差距，制定一个原则性的框架，统计和空间模型的推断，并追求新的计算，使这种推断可行。 该项目将考虑通过共轭梯度和其他方法制定de Wijs过程的适当泛函，以构建有用的随机场和新的无矩阵计算，并将专注于开发新的科学应用领域。拟议的研究还将为过去几十年来许多空间统计研究人员讨论的理论和计算问题提供新的思路和更深入的理解。 新的无矩阵计算将为研究参数自举方法和多尺度建模提供进一步的动力，并构建一类新的非高斯随机场。 该项目将有助于提高对环境生物测定、地下水砷污染和星系分布等研究的科学认识。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过程是将布朗运动从时间推广到空间的一个基本空间过程。 使用de Wijs过程作为基本构建块，该项目将开发新的数学，并导出快速，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测定、地下水砷污染和星系分布的研究中获得更好的科学理解。 在这个项目中开发的统计数据和计算也将与环境或全球变化以及健康研究中出现的各种研究问题特别相关。最后，该项目将通过开发新的研究生和本科生课程，整合研究和教育活动，并为研究生和本科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.