A three domain covariance framework for EEG/MEG data

In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.Comment: 25 pages, 8 figures, 1 tabl

Group Symmetry and non-Gaussian Covariance Estimation

We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments

Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics