Modeling Multivariate and Space-Time Processes: Foundations and Innovations

温度和压力的地球物理过程通常是高度相关的，并且在空间上随时间演变，具有复杂的结构。例如，许多大气过程，如湍流过程，可以表现出长期的相关性，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往往呈指数级衰减，因此是不充分的。许多地球物理过程产生的数据通常是连续索引的，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元和时空协方差函数及其理论性质，以表征复杂行为，如远程依赖和不对称，并开发用于估计平滑行为和远程依赖的稳健估计程序。该项目还将开发和分发用户友好的开源软件，促进其广泛应用于复杂的数据分析问题，并为下一代统计学家和数据科学家提供培训机会。该项目由统计计划和促进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远程依赖性的多元和时空过程。该框架集成并扩展了在比例混合建模和客观贝叶斯文献中出现的强大技术。一种尺度混合技术用于构造新的多元和时空协方差函数，并提供了灵活的特性，包括任意平滑、远程依赖和不对称。将为从起源/尾部行为和筛选效应等方面有原则地、统一地研究所得协方差的实际用途提供理论基础，并为内插和外推设置下的预测精度提供理论见解。目的利用贝叶斯推理，在允许估计平滑参数和尾衰参数的参考先验条件下，实现高斯过程在超几何协方差函数合流下的鲁棒参数估计。发展的统计理论和推理工具将为空间统计和相关领域使用协方差模型建模多元和时空过程提供新的基础。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。

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.