Geometric methods for the study of covariance matrices
Studies of fMRI functional connectivity typically involve the analysis of high-dimensional covariance matrices, encoding the BOLD signal covariance between brain regions or voxels. An increasingly popular approach this problem, which I’ve used extensively in my imaging work, uses methods which respect the natural Riemannian-manifold structure of the space of covariance matrices. Although an enormous body of work has already translated common statistical models (e.g. linear models, PCA) over to this setting, there are still relatively few convenient software tools for non-specialists. More importantly, there are multiple nagging issues that have received almost no almost no rigorous study – such as the development of error metrics for the distortions caused by tangent space methods, or methods for quantifying “region blurring” after centering by parallel transport.
My own R toolbox spdm implements most of the basic operations needed by these methods.