Abstract

We will present a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. The framework incorporates the intrinsically non-linear structure of covariances by means of the Bures-Wasserstein metric geometry, a.k.a. Otto's calculus. We will discuss how to define and estimate means, covariances, and spectral representations. Our motivation comes from modern problems in functional data analysis, where one observes operator-valued random processes–for instance when analysing dynamic functional connectivity and fMRI data, or when analysing multiple functional time series in the frequency domain. Nevertheless, our framework is also novel in the finite-dimensions (matrix case), and we demonstrate what simplifications can be afforded then. Based on joint work with Leonardo Santoro (EPFL).