D Pigoli, JAD Aston, IL Dryden and P Secchi
Distances and Inference for Covariance Functions
Abstract: A framework is developed for inference concerning the covariance operator of a functional random process, where the covariance operator itself is an object of interest for the statistical analysis. Distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable for functional data. In particular, an infinite dimensional analogue of the Procrustes size and shape distance is developed. The convergence of the finite dimensional approximations to the infinite dimensional distance metrics is also shown. To perform inference, a Fr´echet estimator for the average covariance function is introduced, and a permutation procedure to test the equality of the covariance operator between two groups is then considered. The proposed techniques are applied to two problems where inference concerning the covariance is of interest. Firstly, in data arising from a study into cerebral aneurysms, it is of interest to determine
whether two groups of data can be combined when comparing with a third group. For this to be done, it is necessary to assess whether the covariance structures of the two groups are the same or different. Secondly, in a philological study of cross-linguistic dependence, the use of covariance operators has been suggested as a way to incorporate quantitative phonetic information. It is shown that distances between languages derived from phonetic covariance functions can provide insight into relationships between the Romance languages.
Keywords: Distance Metrics; Functional Data Analysis; Procrustes Analysis; Shape Analysis