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.

翻译：本文提出了一种取值于黎曼流形的时间序列模型，并将其应用于癫痫患者脑电图数据衍生的协方差矩阵时间序列拟合。本研究具有双重目标：其一，针对脑电图动态的不同可能模式，开发具有可解释参数的模型；其二，探究建模结果受流形选择及其关联几何结构影响的程度。该模型为任意时间点的切向方向向量设定了分布，融合了自回归项、均值回复项及高斯噪声形式。参数推断通过最大似然估计实现，并比较了使用协方差矩阵标准欧几里得几何与仿射不变几何所得的建模结果。研究结果可区分患者癫痫发作与发作间期：在发作间期，动态呈现强均值回复成分而缺少自回归成分；而在多数发作期，存在显著自回归成分且均值回复效应较弱。拟合模型还用于比较患者内部及患者间的癫痫发作特征。仿射不变几何更具优势，能为数据提供更优拟合效果。