  Modeling matrix-valued time series is an interesting and important research
topic. In this paper, we extend the method of Chang et al. (2017) to
matrix-valued time series. For any given $p\times q$ matrix-valued time series,
we look for linear transformations to segment the matrix into many small
sub-matrices for which each of them are uncorrelated with the others both
contemporaneously and serially, thus they can be analyzed separately, which
will greatly reduce the number of parameters to be estimated in terms of
modeling. To overcome the identification issue, we propose a two-step and more
structured procedure to segment the rows and columns separately. When
$\max(p,q)$ is large in relation to the sample size $n$, we assume the
transformation matrices are sparse and use threshold estimators for the
(auto)covariance matrices. We also propose a block-wisely thresholding method
to separate the columns (or rows) of the transformed matrix-valued data. The
asymptotic properties are established for both fixed and diverging $\max(p,q)$.
Unlike principal component analysis (PCA) for independent data, we cannot
guarantee that the required linear transformation exists. When it does not, the
proposed method provides an approximate segmentation, which may be useful for
forecasting. The proposed method is illustrated with both simulated and real
data examples. We also propose a sequential transformation algorithm to segment
higher-order tensor-valued time series.
