Title: Statistics on functional data and 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.