Markov Random Fields, Geostatistics and Matrix-Free Computation

在过去的几十年里，空间统计在农业、流行病学、地质学、图像分析和环境科学中发挥着越来越重要的作用。PI先前的研究为连接空间统计的两个主要分支即马尔可夫随机场和地质统计学以及推进快速统计计算提供了新的视角。目前，许多重要的科学应用都需要使用复杂的空间模型及其多变量和时空版本。然而，这些复杂空间模型的统计计算仍然是一个挑战。该项目对这些复杂的空间和时空模型产生了新的数学理解，从而为以最小存储推进各种可扩展统计计算开辟了可能性。该项目将有助于增进对诸如砷和镁污染、地下水水化学分析以及美国阿片类药物过量病例的时空变化等研究的科学理解。该项目汇集了来自不同科学领域的数学和计算知识，以开发空间统计和推理的原则框架。本研究旨在对(i)高邻域阶高斯马尔可夫随机场的构造，（ii）两个或多个空间变量的联合建模，以及（iii）复杂时空模型提供新的认识。提出了新的无矩阵计算来推进统计推断。这些计算不仅包括最好的线性无偏预测和残差最大似然估计，而且还包括可扩展的哈密顿蒙特卡罗方法。应用将包括绘图(1)地下水中的重金属污染和(2)美国各地药物过量病例的地理差异。该项目还旨在通过开发空间统计和可扩展计算的短期课程和案例研究，以及为研究生提供宝贵的培训和学习机会，将研究和教育活动结合起来。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the past few decades, spatial statistics has become increasingly important in agriculture, epidemiology, geology, image analysis and environmental science. PI's prior research provided new perspectives in connecting two major branches of spatial statistics, namely the Markov random fields and geostatistics and in advancing fast statistical computations. At present, many important scientific applications demand use of complex spatial models and their multivariate and spatial-temporal versions. However, statistical computations of these complex spatial models have remained a challenge. The project derives new mathematical understanding on these complex spatial and spatial-temporal models, which then opens up the possibility of advancing various scalable statistical computations with minimal storage. The project will contribute to obtaining enhanced scientific understanding in studies such as arsenic and magnesium contamination and hydro-chemical analysis of groundwater and spatial and spatial temporal variations in opioid overdose cases in the United States.The project brings together mathematical and computational knowledge from different scientific fields to develop principled frameworks for spatial statistics and inference. The research aims to provide new understanding on (i) constructions of higher neighborhood order Gaussian Markov random fields, (ii) joint modeling of two or more spatial variables, and (iii) complex spatial-temporal models. Novel matrix-free computations are proposed to advance statistical inference. These computations include not just best linear unbiased predictions and residual maximum likelihood estimation, but also scalable Hamiltonian Monte Carlo methods. Applications will include mapping (1) heavy metal contamination in groundwater and (2) geographic variations in drug overdose cases across the United States. The project also aims to integrate research and educational activities through developing short courses and case studies on spatial statistics and scalable computation, and through providing valuable training and learning opportunities for graduate students.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.