  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.
