  The blind source separation model for multivariate time series generally
assumes that the observed series is a linear transformation of an unobserved
series with temporally uncorrelated or independent components. Given the
observations, the objective is to find a linear transformation that recovers
the latent series. Several methods for accomplishing this exist and three
particular ones are the classic SOBI and the recently proposed generalized FOBI
(gFOBI) and generalized JADE (gJADE), each based on the use of joint lagged
moments. In this paper we generalize the methodologies behind these algorithms
for tensor-valued time series. We assume that our data consists of a tensor
observed at each time point and that the observations are linear
transformations of latent tensors we wish to estimate. The tensorial
generalizations are shown to have particularly elegant forms and we show that
each of them is Fisher consistent and orthogonal equivariant. Comparing the new
methods with the original ones in various settings shows that the tensorial
extensions are superior to both their vector-valued counterparts and to two
existing tensorial dimension reduction methods for i.i.d. data. Finally,
applications to fMRI-data and video processing show that the methods are
capable of extracting relevant information from noisy high-dimensional data.
