Representation and reconstruction of covariance operators in linear inverse problems

Abstract

We introduce a framework for the statistical analysis of functional data in a setting where these objects cannot be fully observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional data or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel statistical approach to the investigation of functional connectivity.

Analysis of task-related MEG functional brain networks using dynamic mode decomposition

“…Another possible direction that could be taken is to immediately apply the DMD framework to the estimated brain source activity time series before determining the connectivity values. Methods that carry out the two-step process (beamforming and then dimensionality reduction) simultaneously in the context of PCA have been found to produce better performance [46].…”