Distances and inference for covariance operators

By Davide Pigoli, John A. D. Aston, Ian L. Dryden, Piercesare Secchi. 2014

117 citations recorded by the source corpus.

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 Frechet 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.

Subfields: Artificial Intelligence, Food Science, Signal Processing

View Paper: https://www.semanticscholar.org/paper/9595e02d6e2fed1bbf58602dc5f4d304956d0fec

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Citations in the corpus

Fréchet regression for random objects with Euclidean predictors (2016, background): “Positive definite matrices have been studied previously from the random object perspective under different metrics (Arsigny et al. 2007; Pigoli et al. 2014).”
Bootstrap-based testing of equality of mean functions or equality of covariance operators for functional data (2016, unknown): “Finally, permutation tests for equality of covariance operators applied to different distance measures between two covariance functions have been considered by Pigoli et al. (2014).”
Tests for separability in nonparametric covariance operators of random surfaces (2015, background): “unds can be better explored in a two dimensional time-frequency domain. In particular, we consider here the dataset of 23 speakers from ﬁve different Romance languages that has been ﬁrst described in Pigoli et al. (2014). The speakers were recorded while pronouncing the words corresponding to the nu”