Modeling Multivariate and Space-Time Processes: Foundations and Innovations

温度和压力的地球物理过程往往是高度相关的，并随着时间的推移在空间中演变，结构复杂。例如，许多大气过程，如湍流过程，可以表现出长程相关性，随着距离的增加，相关性衰减缓慢。虽然现有的协方差模型在描述这些过程的平滑行为方面是成功的，但这些模型中的相关性通常以指数速度衰减，因此是不够的。许多地球物理过程产生的数据经常被连续索引，并在许多学科中表现出复杂的依赖结构，包括地球物理学，生态学，环境和气候科学，工程学，公共卫生，经济学，政治学和商业科学。该项目将开发新的多变量和时空协方差函数及其理论特性，以表征长期依赖和不对称等复杂行为，并开发用于估计平滑行为和长期依赖的稳健估计程序。该项目还将开发和分发方便用户的开放源码软件，促进广泛采用该软件解决复杂的数据分析问题，并为下一代统计人员和数据科学家提供培训机会。该项目由统计计划和刺激竞争研究的既定计划（EPSCoR）共同资助。该项目将开发理论基础和统计模型，用于使用基于模型的框架推断具有长期依赖性的多变量和时空过程。该框架集成和扩展了规模混合建模和客观贝叶斯文献中出现的强大技术。尺度混合技术被用来构建新的多变量和时空协方差函数，并提供灵活的属性，包括任意平滑，长程依赖，和不对称。将提供理论基础，以原则性和统一的方式研究所得协方差的实际用途，如原点/尾部行为和筛选效应，并提供理论见解预测精度在内插和外推设置。目的利用Bayes推断，在融合超几何协方差函数下，在允许估计光滑性参数和尾衰减参数的情况下，给出高斯过程的鲁棒参数估计。开发的统计理论和推理工具将为空间统计和使用协方差模型的相关领域的多变量和时空过程建模提供新的基础。该奖项反映了NSF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。

Geophysical processes for temperature and pressure are often highly correlated and are evolving in space over time with complex structures. For instance, many atmospheric processes such as turbulent processes can exhibit long-range dependence with correlation decays slowly as distance increases. While existing covariance models are successful in describing the smoothness behavior of these processes, the correlation in these models often decays exponentially fast and hence is inadequate. The data resulting from many geophysical processes are often continuously indexed and exhibit complicated dependence structures in many disciplines, including geophysics, ecology, environmental and climate sciences, engineering, public health, economics, political sciences, and business science. This project will develop new multivariate and space-time covariance functions with their theoretical properties to characterize complex behaviors such as long-range dependence and asymmetry and develop robust estimation procedures for estimating smoothness behaviors and long-range dependence. The project will also develop and distribute user-friendly open-source software, facilitate its broad adoption for complex data analytical problems, and provide training opportunities for next-generation statisticians and data scientists. This project is jointly funded by the Statistics Program and the Established Program to Stimulate Competitive Research (EPSCoR). This project will develop theoretical foundations and statistical models for inferring multivariate and space-time processes with long-range dependence using a model-based framework. This framework integrates and extends powerful techniques arising in the literature on scale-mixture modeling and objective Bayes. A scale-mixture technique is used to construct new multivariate and space-time covariance functions and offers flexible properties including arbitrary smoothness, long-range dependence, and asymmetry. Theoretical foundation will be provided to study the practical usefulness of the resultant covariances in a principled and unified manner in terms of several properties such as origin/tail behaviors and screening effect and offer theoretical insights on prediction accuracy in both interpolative and extrapolative settings. Objective Bayes inference is used to enable robust parameter estimation for Gaussian processes under the confluent hypergeometric covariance function with the reference prior in which the smoothness and tail-decay parameters are allowed to be estimated. The developed statistical theory and inferential tools will provide new foundations for modeling multivariate and space-time processes in spatial statistics and related areas that use covariance models.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.