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Distances and inference for covariance operators

Pigoli, Davide; Aston, John A.D.; Dryden, Ian L.; Secchi, Piercesare

Authors

Davide Pigoli

John A.D. Aston

IAN DRYDEN IAN.DRYDEN@NOTTINGHAM.AC.UK
Professor of Statistics

Piercesare Secchi



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.

Journal Article Type Article
Publication Date Apr 17, 2014
Journal Biometrika
Print ISSN 0006-3444
Electronic ISSN 1464-3510
Publisher Oxford University Press
Peer Reviewed Peer Reviewed
Volume 101
Issue 2
APA6 Citation Pigoli, D., Aston, J. A., Dryden, I. L., & Secchi, P. (2014). Distances and inference for covariance operators. Biometrika, 101(2), https://doi.org/10.1093/biomet/asu008
DOI https://doi.org/10.1093/biomet/asu008
Keywords Distance metric; Functional data analysis; Procrustes analysis; Shape analysis
Publisher URL https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/asu008
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0





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