  We propose a model for time series taking values on a Riemannian manifold and
fit it to time series of covariance matrices derived from EEG data for patients
suffering from epilepsy. The aim of the study is two-fold: to develop a model
with interpretable parameters for different possible modes of EEG dynamics, and
to explore the extent to which modelling results are affected by the choice of
manifold and its associated geometry. The model specifies a distribution for
the tangent direction vector at any time point, combining an autoregressive
term, a mean reverting term and a form of Gaussian noise. Parameter inference
is carried out by maximum likelihood estimation, and we compare modelling
results obtained using the standard Euclidean geometry on covariance matrices
and the affine invariant geometry. Results distinguish between epileptic
seizures and interictal periods between seizures in patients: between seizures
the dynamics have a strong mean reverting component and the autoregressive
component is missing, while for the majority of seizures there is a significant
autoregressive component and the mean reverting effect is weak. The fitted
models are also used to compare seizures within and between patients. The
affine invariant geometry is advantageous and it provides a better fit to the
data.
