Nonstationary spatial-temporal covariance models for multivariate processes on a globe

该奖项是根据2009年美国复苏和再投资法案（公法111 - 5）资助的。该项目研究地球仪上非平稳多变量过程的时空协方差函数。 地球物理和环境问题中的许多过程都以地球的大部分区域为空间域，表现出很强的空间非平稳性（特别是纬度）。此外，通常有多个感兴趣的变量，例如降水和温度之间的关系。然而，能够处理地球仪上非平稳过程的时空协方差函数并不多，而且迄今为止几乎没有针对多变量问题的时空协方差函数。在这方面，调查员开发了一个灵活的类的时空协方差函数的单变量以及多变量过程的地球仪。探讨了将关于纬度、经度和时间的微分算子应用于地球仪上各向同性过程的思想。算子的系数随纬度变化，允许灵活的单变量过程的非平稳协方差模型。它还有助于创建适合于真实的物理过程的丰富的互协方差模型类。该项目的最终目标是为多个气候模式输出建立一个联合统计模型。已经证明，这些数值气候模式具有相关的误差，这是至关重要的，有灵活的互协方差模型，以准确地模拟不同的气候模式误差之间的依赖结构。有几个有趣的计算问题，从这个应用程序中出现。特别地，研究者研究了具有特殊结构的协方差矩阵的逆的快速计算算法、协方差渐减、似然近似和缺失数据插补方法。 本研究的动机是多个气候模式输出的评估和整合的科学问题。在政府间气候变化专门委员会（IPCC）的协调下，世界各地的各种组织正在开发数值气候模式，开发和运行这些模式的成本是巨大的。然而，通常只是取这些模型的平均值，并假设它们是独立的。除了对统计领域的贡献外，拟议的研究将为气候科学家提供一个有用的工具，用于多个气候模型的相互比较。此外，它将帮助气候科学家通过准确整合多个气候模型而不仅仅是简单的平均值来提高他们对过去，当前和未来气候的理解，并使他们能够在预测中实现更精确的不确定性。拟议研究的结果将作为统计学和大气科学家学生的多学科课程。研究人员预计，这种类型的努力将促进统计学家和大气科学家之间的未来合作。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project studies classes of spatial-temporal covariance functions for nonstationary, multivariate processes on a globe. Many processes in geophysical and environmental problems these days take large portion of the Earth as their spatial domain and exhibit strong spatial nonstationarity (particularly with respect to latitude). Moreover, it is common to have multiple variables of interest, such as relationship between precipitation and temperature. However, there are not many spatial-temporal covariance functions that can deal with nonstationary processes on a globe and there are almost none developed so far for multivariate problems. In this regard, the investigator develops a flexible class of spatial-temporal covariance functions for univariate as well as multivariate processes on a globe. The idea of applying differential operators with respect to latitude, longitude, and time to an isotropic process on a globe is explored. The coefficients of the operators, varying over latitude, allow flexible nonstationary covariance models for univariate process. It also helps to create a rich class of cross covariance models suitable for real physical processes. The ultimate goal of this project is to build a joint statistical model for multiple climate model outputs. It has been demonstrated that these numerical climate models have correlated errors and it is critical to have flexible cross covariance models to accurately model the dependence structure among different climate model errors. There are several interesting computational issues that arise from this application. In particular, the investigator studies algorithms for fast computation of inverse of covariance matrix with special structures, covariance tapering, likelihood approximation, and missing data imputation method. This study is motivated by the scientific problem of evaluation and integration of multiple climate model outputs. Under the coordination of the Intergovernmental Panel on Climate Change (IPCC), various organizations over the world are developing numerical climate models and the cost to develop and run these models are enormous. However, it is common to simply take averages of these models and assume they are independent. In addition to the contribution to the field of statistics, the proposed study will provide a useful tool for climate scientists for inter-comparison of multiple climate models. Moreover, it will help climate scientists to improve their understanding of past, current, and future climate by accurately integrating multiple climate models beyond simple averages and allow them to achieve more precise uncertainty in their predictions. The results of the proposed research will be formed as multidisciplinary courses for students of both statistics and atmospheric scientists. The investigator anticipates that this type of effort will boost future collaboration between statisticians and atmospheric scientists.