Distances and inference for covariance operators

Pigoli, Davide and Aston, John A.D. and Dryden, Ian L. and Secchi, Piercesare (2014) Distances and inference for covariance operators. Biometrika, 101 (2). pp. 409-422. ISSN 1464-3510

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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 statistical analysis. Distances for comparing positive-definite covariance matrices are either extended or shown to be inapplicable to functional data. In particular, an infinite-dimensional analogue of the Procrustes size-and-shape distance is developed. Convergence of finite-dimensional approximations to the infinite-dimensional distance metrics is also shown. For inference, a Fréchet estimator of both the covariance operator itself and the average covariance operator is introduced. A permutation procedure to test the equality of the covariance operators between two groups is also considered. Additionally, the use of such distances for extrapolation to make predictions is explored. As an example of the proposed methodology, the use of covariance operators has been suggested in a philological study of cross-linguistic dependence as a way to incorporate quantitative phonetic information. It is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the Romance languages.

Item Type: Article
Keywords: Distance metric; Functional data analysis; Procrustes analysis; Shape analysis
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1093/biomet/asu008
Depositing User: Dryden, Professor Ian
Date Deposited: 06 Mar 2017 10:41
Last Modified: 07 Apr 2017 04:26
URI: http://eprints.nottingham.ac.uk/id/eprint/41017

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