The prevalence of multivariate space-time data collected from monitoring networks and satellites, or generated from numerical models, has brought much attention to multivariate spatio-temporal statistical models, where the covariance function plays a key role in modeling, inference, and prediction. For multivariate space-time data, understanding the spatio-temporal variability, within and across variables, is essential in employing a realistic covariance model. Meanwhile, the complexity of generic covariances often makes model fitting very challenging, and simplified covariance structures, including symmetry and separability, can reduce the model complexity and facilitate the inference procedure. However, a careful examination of these properties is needed in real applications. In the work presented here, we formally define these properties for multivariate spatio-temporal random fields and use functional data analysis techniques to visualize them, hence providing intuitive interpretations. We then propose a rigorous rank-based testing procedure to conclude whether the simplified properties of covariance are suitable for the underlying multivariate space-time data. The good performance of our method is illustrated through synthetic data, for which we know the true structure. We also investigate the covariance of bivariate wind speed, a key variable in renewable energy, over a coastal and an inland area in Saudi Arabia. The Supplementary Material is available online, including the R code for our developed methods.

翻译：从监测网络和卫星收集或由数值模型生成的多变量时空数据的普遍性，使得多变量时空统计模型备受关注，其中协方差函数在建模、推断和预测中起着关键作用。对于多变量时空数据，理解变量内和变量间的时空变异性是采用实际协方差模型的基础。同时，通用协方差的复杂性往往使得模型拟合极具挑战性，而简化的协方差结构（包括对称性和可分离性）可降低模型复杂度并简化推断过程。然而，在实际应用中需要仔细检验这些性质。在此项工作中，我们正式定义了多变量时空随机场的这些性质，并利用函数数据分析技术将其可视化，从而提供直观的解释。随后，我们提出了一种严格的基于秩的检验程序，以判断简化的协方差性质是否适用于基础的多变量时空数据。通过已知真实结构的合成数据，我们验证了该方法具有良好的性能。我们还研究了沙特阿拉伯沿海和内陆地区的二元风速（可再生能源的关键变量）的协方差。补充材料包括我们开发方法的R代码，可在线上获取。