A crucial assumption to reduce computational complexity in spatial-temporal data analysis is separability, which factors the covariance structure into a purely spatial and a purely temporal component. In this paper, we develop statistical inference tools for validating this assumption for a second-order stationary process under both domain-expanding-infill asymptotics and domain-expanding asymptotics. In contrast to previous work on this subject, the methodology neither requires the assumption of normally distributed data, nor uses spectral methods. Our approach is based on nonparametric estimates of measures for the deviation between the covariance matrix and separable approximations, which vanish if and only if the assumption of separability is satisfied. We derive the asymptotic distributions of appropriate estimators for these measures with non-standard limiting distributions and use these results to develop inference tools for validating the assumption of separability. More specifically, we derive confidence intervals for the deviation measures, tests for the hypothesis of exact separability, and for the hypothesis that the deviation from separability is smaller than a prespecified threshold.

翻译：降低时空数据分析计算复杂度的关键假设是可分离性，该假设将协方差结构分解为纯空间分量与纯时间分量。本文针对二阶平稳过程，在区域扩展-填充渐近与区域扩展渐近两种框架下，构建了验证该假设的统计推断工具。与以往研究相比，本方法既不要求数据服从正态分布假设，也不采用谱分析方法。我们的研究基于协方差矩阵与其可分离近似之间偏离程度的非参数估计量——该偏离量在且仅当可分离性假设成立时为零。我们推导了这些估计量在非标准极限分布下的渐近分布，并利用这些结果构建了验证可分离性假设的推断工具。具体而言，我们推导了偏离程度的置信区间，建立了精确可分离性假设的检验方法，以及偏离程度小于预设阈值的假设检验方法。