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的多变空间时间数据或从数字模型中产生的多变空间时间数据普遍存在,已引起对多变空间时空统计模型的极大注意,其中共变功能在建模、推断和预测方面发挥着关键作用。对于多变空间时间数据而言,了解变量内部和跨变量时空变异,对于采用现实的共变模式至关重要。同时,通用变量的复杂性往往使模型非常具有挑战性,简化变量结构,包括对称和分离,可以降低模型的复杂性,并便利推论程序。然而,在实际应用中,需要仔细研究这些属性。在此处介绍的工作中,我们正式界定了多变空时空随机域的这些属性,并使用功能数据分析技术来直观这些属性,从而提供直观的解释。我们随后提出了严格的按级计算的测试程序,以判断现有简化的共变异性特性是否适合基础的多变式空间数据,可以降低模型的复杂性,并便利推断程序。在实际应用中,需要仔细研究这些特性。我们的方法在可变式的海域中,通过一个可变的海中的方法,通过一个可变式数据,我们所开发的可变式的海中的一种关键的海中的方法,用来测量一个可变式数据,一个可变式的海中一个可变的轨道,一个可变式的系统,一个可变式的系统,用来测量一个可变式的轨道区域。</s>