In multivariate spatio-temporal analysis, we are faced with the formidable challenge of specifying a valid spatio-temporal cross-covariance function, either directly or through the construction of processes. This task is difficult as these functions should yield positive definite covariance matrices. In recent years, we have seen a flourishing of methods and theories on constructing spatio-temporal cross-covariance functions satisfying the positive definiteness requirement. A subset of those techniques produced spatio-temporal cross-covariance functions possessing the additional feature of nonstationarity. Here we provide a review of the state-of-the-art methods and technical progress regarding model construction. In addition, we introduce a rich class of multivariate spatio-temporal asymmetric nonstationary models stemming from the Lagrangian framework. We demonstrate the capabilities of the proposed models on a bivariate reanalysis climate model output dataset previously analyzed using purely spatial models. Furthermore, we carry out a cross-validation study to examine the advantages of using spatio-temporal models over purely spatial models. Finally, we outline future research directions and open problems.

在多元时空分析中，我们面临着直接或通过构建过程指定有效时空互协方差函数的巨大挑战。由于这些函数应产生正定协方差矩阵，因此此任务很困难。近年来，我们看到了构造满足正定性要求的时空互协方差函数的方法和理论的兴旺发展。这些技术的子集产生时空互协函数，具有非平稳性的附加特征。在这里，我们提供了有关模型构建的最新方法和技术进步的评论。此外，我们引入了一类来自拉格朗日框架的多元时空非对称非平稳模型。我们在以前使用纯空间模型进行分析的双变量再分析气候模型输出数据集上证明了所提出模型的功能。此外，我们进行了一项交叉验证研究，以检验使用时空模型相对于纯空间模型的优势。最后，我们概述了未来的研究方向和未解决的问题。