In this paper, we consider multivariate functional time series with a two-way dependence structure: a serial dependence across time points and a graphical interaction among the multiple functions within each time point. We develop the notion of dynamic weak separability, a more general condition than those assumed in literature, and use it to characterize the two-way structure in multivariate functional time series. Based on the proposed weak separability, we develop a unified framework for functional graphical models and dynamic principal component analysis, and further extend it to optimally reconstruct signals from contaminated functional data using graphical-level information. We investigate asymptotic properties of the resulting estimators and illustrate the effectiveness of our proposed approach through extensive simulations. We apply our method to hourly air pollution data that were collected from a monitoring network in China.

翻译：本文研究具有双向依赖结构的多元函数时间序列：时间点间的序列依赖与各时间点内多个函数间的图形交互作用。我们提出了动态弱可分性的概念——此条件较现有文献假设更为一般，并藉此刻画多元函数时间序列的双向结构。基于所提出的弱可分性，我们构建了函数图形模型与动态主成分分析的统一框架，并进一步扩展该框架以利用图形层级信息从受污染函数数据中实现信号的最优重构。我们研究了所得估计量的渐近性质，并通过大量仿真实验验证了所提方法的有效性。最后，我们将该方法应用于中国监测网络采集的每小时大气污染数据。