Principal Components Analysis of Periodically Correlated Functional Time Series

Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic operator-valued filters, score processes, and the inversion formulas. We show that these objects are defined via a convergent series under a simple condition requiring summability of the Hilbert-Schmidt norms of the filter coefficients and that they possess optimality properties. We explain how the Hilbert space theory reduces to an approximate finite-dimensional setting which is implemented in a custom-build |R| package. A data example and a simulation study show that the new methodology is superior to existing tools if the functional time series exhibits periodic characteristics.