The Annals of Statistics

Tests for separability in nonparametric covariance operators of random surfaces

John A. D. Aston, Davide Pigoli, and Shahin Tavakoli

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The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional data analysis settings, where this assumption is most relevant. We propose here to test separability by focusing on $K$-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full nonseparable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.

Article information

Ann. Statist., Volume 45, Number 4 (2017), 1431-1461.

Received: September 2015
Revised: June 2016
First available in Project Euclid: 28 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 62F40: Bootstrap, jackknife and other resampling methods

Acoustic phonetic data bootstrap dimensional reduction functional data partial trace sparsity


Aston, John A. D.; Pigoli, Davide; Tavakoli, Shahin. Tests for separability in nonparametric covariance operators of random surfaces. Ann. Statist. 45 (2017), no. 4, 1431--1461. doi:10.1214/16-AOS1495.

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Supplemental materials

  • Supplementary material: “Tests for separability in nonparametric covariance operators of random surfaces”. Background technical results, implementation details, additional simulation studies and additional proofs.
  • Research data supporting “Tests for separability in nonparametric covariance operators of random surfaces”. R package implementing the methodology of the paper. Data and script for reproducing the numerical simulations and figures of the paper. Data generated for the paper and used in the phonetic application, and script to reproduce it.