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

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。这个项目研究全球非平稳、多变量过程的时空协方差函数类。当今地球物理和环境问题中的许多过程都以地球大部分地区为空间域，并表现出很强的空间非平稳性(特别是关于纬度)。此外，通常会有多个感兴趣的变量，例如降水和温度之间的关系。然而，能够处理地球上非平稳过程的时空协方差函数并不多，而且到目前为止，几乎还没有发展出用于多变量问题的协方差函数。在这一点上，研究者开发了一类灵活的时空协方差函数，适用于地球上的单变量和多变量过程。探索了将关于纬度、经度和时间的微分算符应用于地球上的各向同性过程的想法。算子的系数随纬度变化，允许单变量过程的灵活的非平稳协方差模型。它还有助于创建适用于实际物理过程的丰富的交叉协方差模型。该项目的最终目标是为多个气候模型的输出建立一个联合统计模型。结果表明，这些数值气候模式存在相关误差，建立灵活的交叉协方差模型以准确地模拟不同气候模式误差之间的相关性结构是至关重要的。这个应用程序产生了几个有趣的计算问题。特别研究了具有特殊结构的协方差矩阵求逆的快速算法、协方差锥化算法、似然逼近算法和缺失数据补偿算法。这项研究的动机是多个气候模式输出的评估和集成的科学问题。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都在开发气候数值模式，而开发和运行这些模式的成本是巨大的。然而，简单地取这些模型的平均值并假设它们是独立的是很常见的。除了对统计领域的贡献外，拟议的研究还将为气候科学家提供一个有用的工具，用于多种气候模型的相互比较。此外，它将帮助气候科学家通过准确地整合多种气候模型来提高他们对过去、现在和未来气候的理解，而不是简单的平均水平，并使他们能够在预测中实现更精确的不确定性。拟议的研究结果将形成为统计学和大气科学家学生的多学科课程。这位研究人员预计，这类努力将促进统计学家和大气科学家之间未来的合作。

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.