This work is focused on constructing space-time covariance functions through a hierarchical mixture approach that can serve as building blocks for capturing complex dependency structures. This hierarchical mixture approach provides a unified modeling framework that not only constructs a new class of asymmetric space-time covariance functions with closed-form expressions, but also provides corresponding space-time process representations, which further unify constructions for many existing space-time covariance models. This hierarchical mixture framework decomposes the complexity of model specification at different levels of hierarchy, for which parsimonious covariance models can be specified with simple mixing measures to yield flexible properties and closed-form derivation. A characterization theorem is provided for the hierarchical mixture approach on how the mixing measures determine the statistical properties of covariance functions. Several new covariance models resulting from this hierarchical mixture approach are discussed in terms of their practical usefulness. A theorem is also provided to construct a general class of valid asymmetric space-time covariance functions with arbitrary and possibly different degrees of smoothness in space and in time and flexible long-range dependence. The proposed covariance class also bridges a theoretical gap in using the Lagrangian reference framework. The superior performance of several new parsimonious covariance models over existing models is verified with the well-known Irish wind data and the U.S. air temperature data.

翻译：本研究致力于通过层次混合方法构建时空协方差函数，该方法可作为捕捉复杂依赖结构的基础模块。该层次混合方法提供了一个统一的建模框架，不仅构造了一类具有闭式表达的新非对称时空协方差函数，还提供了相应的时空过程表示，进一步统一了多种现有时空协方差模型的构建方式。该层次混合框架将模型设定的复杂性分解至不同层次，可通过简单的混合测度设定简约的协方差模型，从而获得灵活的性质与闭式推导。本文给出了层次混合方法的特征定理，阐明混合测度如何决定协方差函数的统计性质。研究讨论了基于此方法衍生的若干新协方差模型的实际应用价值。同时，本文提出一个定理用于构建广义有效的非对称时空协方差函数类，其在空间与时间维度上可具有任意且可能不同的平滑度以及灵活的长程依赖性。所提出的协方差类还弥补了拉格朗日参考框架应用中的理论空白。通过经典爱尔兰风速数据与美国气温数据的验证，多个新简约协方差模型相较于现有模型展现出更优越的性能。