Tests for separability in nonparametric covariance operators of random surfaces

Abstract

Abstract. 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 non-separable 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.

Testing Separability of Functional Time Series

“…Kokoszka et al (2016), Gromenko et al (2016Gromenko et al ( , 2017, French et al (2016), Tupper et al (2017), Liu et al (2017), and Shang and Hyndman (2017). Testing separability of spatiotemporal functional data of the above form is investigated in Constantinou et al (2017), Aston et al (2017), and Bagchi and Dette (2017) under the assumption that the fields X n (⋅, ⋅), 1 ≤ n ≤ N, are independent. No tests are currently available for testing separability in the presence of temporal dependence across n. In a broader setting, the data that motivate this research have the form of functional panels: X n (t) = [X n1 (t), X n2 (t), ..., X nS (t)] T , 1 ≤ n ≤ N.…”

Testing Normality of Functional Time Series

“…In many settings, formulae become much simpler if normality is assumed (e.g. Panaretos et al, 2010;Kraus and Panaretos, 2012;Fremdt et al, 2013;Aston et al, 2017). A test that verifies that the assumption of normality is reasonable for a sample of curves will bolster confidence in the conclusions of these and many other functional data analysis (FDA) procedures.A well-known challenge of working with functional data is that, to perform computations, data must be reduced to finite-dimensional objects.…”

The Statistical Analysis of Acoustic Phonetic Data: Exploring Differences Between Spoken Romance Languages

“…As Aston et al . () have already found that the separability assumption does not hold for the acoustic phonetic data, it would be interesting to see how the results of this paper change without assuming a separable covariance structure (e.g. Kang et al .…”

“…The results of the paper and the definition of a path and a transformation operator from one language to another in Section 6 appear to depend considerably on this separability assumption; thus it seems necessary to conduct a test for separability (see Aston et al . () and Constantinou et al . ()).…”