  We propose a two-step procedure to model and predict high-dimensional
functional time series, where the number of function-valued time series $p$ is
large in relation to the length of time series $n$. Our first step performs an
eigenanalysis of a positive definite matrix, which leads to a one-to-one linear
transformation for the original high-dimensional functional time series, and
the transformed curve series can be segmented into several groups such that any
two subseries from any two different groups are uncorrelated both
contemporaneously and serially. Consequently in our second step those groups
are handled separately without the information loss on the overall linear
dynamic structure. The second step is devoted to establishing a
finite-dimensional dynamical structure for all the transformed functional time
series within each group. Furthermore the finite-dimensional structure is
represented by that of a vector time series. Modelling and forecasting for the
original high-dimensional functional time series are realized via those for the
vector time series in all the groups. We investigate the theoretical properties
of our proposed methods, and illustrate the finite-sample performance through
both extensive simulation and two real datasets.
