Manifold-adapted models for time series of covariance and correlation matrices with applications to Electroencephalography data

Abstract

EEG (electroencephalography) is a method for recording the brain’s electrical activity in real-time by placing electrodes on the subject’s scalp or, in some circumstances, surgically within the cranium, allowing for the measurement of neural signals. Analysing dynamic brain patterns with EEG is crucial for diagnosing and treating epilepsy. Statistical analysis of EEG data commonly relies on p × p covariance (or correlation) matrices derived from pre-processed signals, where p represents the number of electrodes. However, the space of covariance (or correlation) matrices is not a vector space with the usual additive structure, and so analysis of samples or time series of covariance (or correlation) matrices must make some geometrical assumptions about the underlying space. Fortunately, both the set of p × p non-singular covariance and correlation matrices form Riemannian manifolds. Riemannian geometry provides a systematic mathematical framework that allows conventional linear statistical methods to be adapted to the non-linear geometrical setting. The number of electrodes p can be large, necessitating dimensional reduction to make analysis more computationally tractable. Moreover, electromagnetic artefacts and high correlations between sets of electrodes can result in rank deficiencies in the observed matrix-valued time series. These singularities make analysis more difficult and represent redundancies in the data. To address this, we employ dimensional reduction techniques by identifying linear combinations of channels and selecting channel subsets. These approaches ensure that the reduced time series of covariance (or correlation) matrices remain strictly positive definite, and the data thereby lie on certain smooth manifolds with the natural Riemannian structure. Our focus is on modelling time series {Si : i = 1, . . . , n} of full-rank covariance (or correlation) matrices. This data can be examined within one of three spaces: C +(p) ⊂ S +(p) ⊂ Sym(p). In this context, Sym(p) comprises symmetric matrices and is equipped with the Frobenius norm for Euclidean geometry. S +(p) represents the space of symmetric positive definite matrices, equipped with an affine invariant metric that preserves invariance under affine transformations, especially for high-magnitude covariance matrices. By factoring out variances from covariance matrices, the set of full-rank correlation matrices is represented in the quotient geometry, denoted as C +(p). Thus, we intrinsically analyse EEG matrix-valued time series data within these three spaces. Although manifold-valued data have gained substantial attention and applications in various fields recently, the literature on manifold-valued time series remains limited. This research aims to address two main objectives. First, we aim to develop manifold-adapted models for time series of matrix-valued EEG data with interpretable parameters for different possible modes of EEG dynamics. 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. This model effectively captures a wide range of potential dynamics governing the evolution of EEG data, from a smooth progression along geodesics to a noisy mean-reverting random walk within the underlying manifold. Secondly, we aim to explore the extent to which modelling results are affected by the choice of the manifold and its associated geometry. Manifold-adapted models are implemented in different tangent spaces of Sym(p), S +(p), and C +(p). This enables modelling time series of covariance matrices in Sym(p) and S +(p), and time series of correlation matrices in Sym(p), S +(p), and C +(p). Note that Sym(p) can be specified as a Riemannian manifold and is convenient to present it in that way for comparison with geometries on S +(p) and C +(p). The comparison of these geometries sheds light on their relative advantages. To handle the potentially large number of parameters, we simplify the general manifoldadapted model to two simpler models with fewer parameters. These simplified coefficients reveal the relative coefficients of each dynamics mode at each time point for each pair of electrodes. Parameter inference is carried out through maximum likelihood estimation. The Mahalanobis distance serves as a metric to gauge the dissimilarity of seizures based on estimated coefficients and their asymptotic covariance matrices. The results effectively discriminate between epileptic ictal (during a seizure) and interictal (between seizures) periods in patients and quantify the dissimilarity among seizures. The affine invariant geometry and quotient geometry also provide a better fit for time series of covariance matrices and correlation matrices, respectively. In this research, we primarily construct manifold-adapted models for time series of covariance and correlation matrices derived from EEG data for epilepsy patients. We also contribute to the research on correlation matrix space by introducing a quotient metric inspired by the affine invariant metric in the covariance matrix space. The geometric concepts within the Riemannian structure of three spaces open avenues for future work related to non-Euclidean statistical models using manifold-valued data.