Tests for separability in nonparametric covariance operators of random surfaces

TY - JOUR

T1 - Tests for separability in nonparametric covariance operators of random surfaces

AU - Aston, John A.D.

AU - Pigoli, Davide

AU - Tavakoli, Shahin

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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.

AB - 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.

KW - Acoustic phonetic data

KW - Bootstrap

KW - Dimensional reduction

KW - Functional data

KW - Partial trace

KW - Sparsity

U2 - 10.1214/16-AOS1495

DO - 10.1214/16-AOS1495

M3 - Article

AN - SCOPUS:85021405457

SN - 0090-5364

VL - 45

SP - 1431

EP - 1461

JO - ANNALS OF STATISTICS

JF - ANNALS OF STATISTICS

IS - 4

M1 - 06773279

ER -