Abstract

We address the problem of analysing the variation within a sample of stochastic processes, particularly focusing on their second-order covariance structure. Covariance operators are primarily known in functional data analysis for the crucial role they play in the Karhunen-Loeve expansion. However, these operators themselves may exhibit variability and the analysis of such variability is often necessary to complement the inferential picture based only on first-order statistical analysis. Because of their non-linearity, covariances necessitate tailored statistical techniques to assess their fluctuations. Such techniques, if suitably chosen, grant access to powerful statistical procedures. Detecting structural breaks in higher-order moments is relevant in many applications, and it is not obvious how to determine them by pure visual inspection. In this talk, we leverage the geometric properties of the space of functional covariances as well as results from optimal transport theory, in order to develop a novel approach for the detection of break-points in the covariance of a sample of functional data. Exploiting the interplay between geometry, statistics and functional analysis, we aim to provide a framework for inference for covariance operators. All algorithms presented are illustrated using real data and are implemented in the R package fdWasserstein.