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以复杂的结构演变。例如，许多大气过程，如湍流过程，可能表现出长期相关性，相关性随着距离的增加而缓慢衰减。虽然现有的协方差模型成功地描述了这些过程的平滑行为，但这些模型中的相关性通常以指数级快速衰减，因此是不够的。许多地球物理过程产生的数据经常被连续编入索引，并在许多学科中显示出复杂的相关性结构，包括地球物理、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。这个项目将开发新的具有理论性质的多变量和时空协方差函数来表征复杂的行为，如长期相关性和非对称性，并开发稳健的估计程序来估计平滑行为和长期相关性。该项目还将开发和分发用户友好的开放源码软件，促进其在复杂数据分析问题上的广泛采用，并为下一代统计学家和数据科学家提供培训机会。该项目由统计项目和已建立的激励竞争性研究项目(EPSCoR)共同资助。该项目将开发理论基础和统计模型，以便使用基于模型的框架推断具有长期相关性的多变量和时空过程。该框架集成和扩展了文献中关于尺度混合建模和目标贝叶斯的强大技术。尺度混合技术被用来构造新的多元和时空协方差函数，并提供了包括任意光滑性、长期相关性和非对称性在内的灵活性质。本文将提供理论基础，以原则性和统一性的方式从起源/尾部行为和筛选效应等几个方面研究合成协方差的实际有用性，并对内插和外推环境下的预测精度提供理论见解。目的利用贝叶斯推断，在具有参考先验的合流超几何协方差函数下，实现高斯过程的稳健参数估计，其中允许估计光滑性和尾部衰减参数。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中的多变量和时空过程建模提供新的基础。这一奖项反映了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。

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.