  We extend the principal component analysis (PCA) to second-order stationary
vector time series in the sense that we seek for a contemporaneous linear
transformation for a $p$-variate time series such that the transformed series
is segmented into several lower-dimensional subseries, and those subseries are
uncorrelated with each other both contemporaneously and serially. Therefore
those lower-dimensional series can be analysed separately as far as the linear
dynamic structure is concerned. Technically it boils down to an eigenanalysis
for a positive definite matrix. When $p$ is large, an additional step is
required to perform a permutation in terms of either maximum cross-correlations
or FDR based on multiple tests. The asymptotic theory is established for both
fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity.
Numerical experiments with both simulated and real data sets indicate that the
proposed method is an effective initial step in analysing multiple time series
data, which leads to substantial dimension reduction in modelling and
forecasting high-dimensional linear dynamical structures. Unlike PCA for
independent data, there is no guarantee that the required linear transformation
exists. When it does not, the proposed method provides an approximate
segmentation which leads to the advantages in, for example, forecasting for
future values. The method can also be adapted to segment multiple volatility
processes.
