  In this paper, we explore dimension reduction for time series of functional
data within both stationary and non-stationary frameworks. We introduce a
functional framework of generalized dynamic principal component analysis
(GDPCA). The concept of GDPCA aims for better adaptation to possible
nonstationary features of the series. We define the functional generalized
dynamic principal component (GDPC) as static factor time series in a functional
dynamic factor model and obtain the multivariate GDPC from a truncation of the
functional dynamic factor model. GDFPCA uses a minimum squared error criterion
to evaluate the reconstruction of the original functional time series. The
computation of GDPC involves a two-step estimation of the coefficient vector of
the loading curves in a basis expansion. We provide a proof of the consistency
of the reconstruction of the original functional time series with GDPC
converging in mean square to the original functional time series. Monte Carlo
simulation studies indicate that the proposed GDFPCA is comparable to dynamic
functional principal component analysis (DFPCA) when the data generating
process is stationary, and outperforms DFPCA and FPCA when the data generating
process is non-stationary. The results of applications to real data reaffirm
the findings in simulation studies.
