## The Annals of Statistics

### Tests for separability in nonparametric covariance operators of random surfaces

#### 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 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

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

Dates
Revised: June 2016
First available in Project Euclid: 28 June 2017

https://projecteuclid.org/euclid.aos/1498636862

Digital Object Identifier
doi:10.1214/16-AOS1495

Mathematical Reviews number (MathSciNet)
MR3670184

Zentralblatt MATH identifier
06773279

#### Citation

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. https://projecteuclid.org/euclid.aos/1498636862

#### References

• Aston, J. A. D. and Kirch, C. (2012). Evaluating stationarity via change-point alternatives with applications to fMRI data. Ann. Appl. Stat. 6 1906–1948.
• Aston, J. A. D., Pigoli, D. and Tavakoli, S. (2017). Supplement to “Tests for separability in nonparametric covariance operators of random surfaces.” DOI:10.1214/16-AOS1495SUPPA, DOI:10.1214/16-AOS1495SUPPB.
• Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• Chen, K., Delicado, P. and Müller, H.-G. (2016). Modeling function-valued stochastic processes, with applications to fertility dynamics. J. R. Stat. Soc. Ser. B. Stat. Methodol. To appear. doi:10.1111/rssb.12160.
• Constantinou, P., Kokoszka, P. and Reimherr, M. (2015). Testing separability of space–time functional processes. Available at arXiv:1509.07017.
• Cressie, N. and Huang, H.-C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. J. Amer. Statist. Assoc. 94 1330–1340.
• Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York.
• Fuentes, M. (2006). Testing for separability of spatial-temporal covariance functions. J. Statist. Plann. Inference 136 447–466.
• Garcia, D. (2010). Robust smoothing of gridded data in one and higher dimensions with missing values. Comput. Statist. Data Anal. 54 1167–1178.
• Genton, M. G. (2007). Separable approximations of space–time covariance matrices. Environmetrics 18 681–695.
• Gneiting, T. (2002). Nonseparable, stationary covariance functions for space–time data. J. Amer. Statist. Assoc. 97 590–600.
• Gneiting, T., Genton, M. G. and Guttorp, P. (2007). Geostatistical space–time models, stationarity, separability, and full symmetry. Monogr. Statist. Appl. Probab. 107 151.
• Gohberg, I., Goldberg, S. and Kaashoek, M. A. (1990). Classes of Linear Operators. Vol. I. Operator Theory: Advances and Applications 49. Birkhäuser, Basel.
• Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 109–126.
• Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer, New York.
• Kadison, R. V. and Ringrose, J. R. (1997). Fundamentals of the Theory of Operator Algebras: Elementary Theory. Vol. I. Graduate Studies in Mathematics 15. Amer. Math. Soc., Providence, RI.
• Lindquist, M. A. (2008). The statistical analysis of fMRI data. Statist. Sci. 23 439–464.
• Liu, C., Ray, S. and Hooker, G. (2014). Functional principal components analysis of spatially correlated data. Available at arXiv:1411.4681.
• Lu, N. and Zimmerman, D. L. (2005). The likelihood ratio test for a separable covariance matrix. Statist. Probab. Lett. 73 449–457.
• Mas, A. (2006). A sufficient condition for the CLT in the space of nuclear operators—Application to covariance of random functions. Statist. Probab. Lett. 76 1503–1509.
• Mitchell, M. W., Genton, M. G. and Gumpertz, M. L. (2005). Testing for separability of space–time covariances. Environmetrics 16 819–831.
• Pigoli, D., Aston, J. A. D., Dryden, I. L. and Secchi, P. (2014). Distances and inference for covariance operators. Biometrika 101 409–422.
• Rabiner, L. R. and Schafer, R. W. (1978). Digital Processing of Speech Signals 100. Prentice-hall, Englewood Cliffs.
• Ramsay, J. O., Graves, S. and Hooker, G. (2009). Functional Data Analysis with R and MATLAB. Springer, New York.
• Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer, New York.
• Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
• Ringrose, J. R. (1971). Compact Non-self-Adjoint Operators. Van Nostrand Reinhold, London.
• Secchi, P., Vantini, S. and Vitelli, V. (2015). Analysis of spatio-temporal mobile phone data: A case study in the metropolitan area of Milan. Stat. Methods Appl. 24 279–300.
• Simpson, S. L. (2010). An adjusted likelihood ratio test for separability in unbalanced multivariate repeated measures data. Stat. Methodol. 7 511–519.
• Simpson, S. L., Edwards, L. J., Styner, M. A. and Muller, K. E. (2014). Separability tests for high-dimensional, low-sample size multivariate repeated measures data. J. Appl. Stat. 41 2450–2461.
• Tang, R. and Müller, H.-G. (2008). Pairwise curve synchronization for functional data. Biometrika 95 875–889.
• Tavakoli, S. (2016). covsep: Tests for determining if the covariance structure of 2-dimensional data is separable. R package version 1.0.0. Available at https://CRAN.R-project.org/package=covsep.
• Tibshirani, R. J. (2014). Past, present, and future of statistical science. In Praise of Sparsity and Convexity (X. Lin, C. Genest, D. L. Banks, G. Molenberghs, D. W. Scott and J.-L. Wang, eds.) 497–506. Chapman & Hall, London.
• Wang, J.-L., Chiou, J.-M. and Mueller, H.-G. (2016). Functional data analysis. Annual Review of Statistics and Its Application 3 257–295.
• Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J. and Evans, A. C. (1996). A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping 4 58–73.
• Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.

#### 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.