Electronic Journal of Statistics

Functional mixed effects wavelet estimation for spectra of replicated time series

Joris Chau and Rainer von Sachs

Full-text: Open access

Abstract

Motivated by spectral analysis of replicated brain signal time series, we propose a functional mixed effects approach to model replicate-specific spectral densities as random curves varying about a deterministic population-mean spectrum. In contrast to existing work, we do not assume the replicate-specific spectral curves to be independent, i.e. there may exist explicit correlation between different replicates in the population. By projecting the replicate-specific curves onto an orthonormal wavelet basis, estimation and prediction is carried out under an equivalent linear mixed effects model in the wavelet coefficient domain. To cope with potentially very localized features of the spectral curves, we develop estimators and predictors based on a combination of generalized least squares estimation and nonlinear wavelet thresholding, including asymptotic confidence sets for the population-mean curve. We derive $L_{2}$-risk bounds for the nonlinear wavelet estimator of the population-mean curve–a result that reflects the influence of correlation between different curves in the replicate population– and consistency of the estimators of the inter- and intra-curve correlation structure in an appropriate sparseness class of functions. To illustrate the proposed functional mixed effects model and our estimation and prediction procedures, we present several simulated time series data examples and we analyze a motivating brain signal dataset recorded during an associative learning experiment.

Article information

Source
Electron. J. Statist., Volume 10, Number 2 (2016), 2461-2510.

Dates
Received: July 2016
First available in Project Euclid: 7 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1473276077

Digital Object Identifier
doi:10.1214/16-EJS1181

Mathematical Reviews number (MathSciNet)
MR3545466

Zentralblatt MATH identifier
06628768

Keywords
Spectral analysis replicated time series functional mixed effects model wavelet thresholding between-curve correlation nonparametric confidence sets

Citation

Chau, Joris; von Sachs, Rainer. Functional mixed effects wavelet estimation for spectra of replicated time series. Electron. J. Statist. 10 (2016), no. 2, 2461--2510. doi:10.1214/16-EJS1181. https://projecteuclid.org/euclid.ejs/1473276077


Export citation

References

  • [1] Abramovich, F., Benjamini, Y., Donoho, D. L. and Johnstone, I. M. (2006). Adapting to unknown sparsity by controlling the false discovery rate., The Annals of Statistics 34 584–653.
  • [2] Antoniadis, A. and Sapatinas, T. (2007). Estimation and inference in functional mixed-effects models., Computational Statistics & Data Analysis 51 4793–4813.
  • [3] Aston, J., Chiou, J. M. and Evans, J. P. (2010). Linguistic pitch analysis using functional principal component mixed effect models., Journal of the Royal Statistical Society: Series C 59 297–317.
  • [4] Brillinger, D. R. (1981)., Time Series: Data Analysis and Theory. Holden-Day, San Francisco.
  • [5] Bruce, A. G. and Gao, H. Y. (1996). Understanding Waveshrink: Variance and bias estimation., Biometrika 83 727–745.
  • [6] Diggle, P. J. and Al Wasel, I. (1997). Spectral analysis of replicated biomedical time series., Journal of the Royal Statistical Society: Series C 46 31–71.
  • [7] Fiecas, M. and Ombao, H. (2016). Modeling the evolution of dynamic brain processes during an associative learning experiment., Journal of the American Statistical Association. (Accepted).
  • [8] Freyermuth, J. M., Ombao, H. andvon Sachs, R. (2010). Tree-structured wavelet estimation in a mixed effects model for spectra of replicated time series., Journal of the American Statistical Association 105 634–646.
  • [9] Gao, H. Y. (1997). Choice of thresholds for wavelet shrinkage estimate of the spectrum., Journal of Time Series Analysis 18 231–251.
  • [10] Genovese, C. R. and Wasserman, L. (2005). Confidence sets for nonparametric wavelet regression., The Annals of Statistics 698–729.
  • [11] Giacofci, M., Lambert-Lacroix, S., Marot, G. and Picard, F. (2013). Wavelet-based clustering for mixed-effects functional models in high dimension., Biometrics 69 31–40.
  • [12] Gorrostieta, C., Ombao, H., Prado, R., Patel, S. and Eskandar, E. (2012). Exploring dependence between brain signals in a monkey during learning., Journal of Time Series Analysis 33 771–778.
  • [13] Guo, W. (2002). Functional mixed effects models., Biometrics 58 121–128.
  • [14] Hernandez-Flores, C., Artiles-Romero, J. and Saavedra-Santana, P. (1999). Estimation of the population spectrum with replicated time series., Computational Statistics & Data Analysis 30 271–280.
  • [15] Higham, N. J. (2002). Computing the nearest correlation matrix; a problem from finance., IMA Journal of Numerical Analysis 22 329–343.
  • [16] Iannaccone, R. and Coles, S. (2001). Semiparametric models and inference for biomedical time series with extra-variation., Biostatistics 2 261–276.
  • [17] Jiang, J. (2007)., Linear and Generalized Linear Mixed Models and Their Applications. Springer, New York.
  • [18] Jiang, J., Luan, Y., Wang, Y. et al. (2007). Iterative estimating equations: Linear convergence and asymptotic properties., The Annals of Statistics 35 2233–2260.
  • [19] Johnstone, I. M. (2015). Gaussian Estimation: Sequence and Multiresolution Models. (Unpublished, manuscript).
  • [20] Krafty, R. T. (2016). Discriminant analysis of time series in the presence of within-group spectral variability., Journal of Time Series Analysis 37 435–450.
  • [21] Krafty, R. T., Hall, M. and Guo, W. (2011). Functional mixed effects spectral analysis., Biometrika 98 583–598.
  • [22] Krafty, R. T., Rosen, O., Stoffer, D. S., Buysse, D. J. and Hall, M. H. (2016). Conditional spectral analysis of replicated multiple time series with application to nocturnal physiology., arXiv preprint arXiv:1502.03153.
  • [23] Martinez, J. G., Bohn, K. M., Carroll, R. J. and Morris, J. S. (2013). A study of Mexican free-tailed bat chirp syllables: Bayesian functional mixed models for nonstationary acoustic time series., Journal of the American Statistical Association 108 514–526.
  • [24] Morris, J. S. (2014). Functional regression., arXiv preprint arXiv:1406.4068.
  • [25] Morris, J. S. and Carroll, R. J. (2006). Wavelet-based functional mixed models., Journal of the Royal Statistical Society: Series B 68 179–199.
  • [26] Morris, J. S., Brown, P. J., Herrick, R. C., Baggerly, K. A. and Coombes, K. R. (2008). Bayesian Analysis of mass spectrometry proteomic data using wavelet-based functional mixed models., Biometrics 64 479–489.
  • [27] Moulin, P. (1994). Wavelet thresholding techniques for power spectrum estimation., IEEE Transactions on Signal Processing 42 3126–3136.
  • [28] Nason, G. (2010)., Wavelet Methods in Statistics with R. Springer, New York.
  • [29] Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series., Journal of Time Series Analysis 17 601–633.
  • [30] Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra., The Annals of Statistics 25 38–76.
  • [31] Ombao, H., von Sachs, R. and Guo, W. (2005). SLEX analysis of multivariate nonstationary time series., Journal of the American Statistical Association 100 519–531.
  • [32] Pav, S. E. (2015). Moments of the log non-central chi-square distribution., arXiv preprint arXiv:1503.06266.
  • [33] Qin, L. and Guo, W. (2006). Functional mixed-effects model for periodic data., Biostatistics 7 225–234.
  • [34] Qin, L., Guo, W. and Litt, B. (2009). A time-frequency functional model for locally stationary time series data., Journal of Computational and Graphical Statistics 18 675–693.
  • [35] Robins, J. andvan der Vaart, A. W. (2006). Adaptive nonparametric confidence sets., The Annals of Statistics 34 229–253.
  • [36] Rudzkis, R., Saulis, L. and Statulevičius, V. (1978). A general lemma on probabilities of large deviations., Lithuanian Mathematical Journal 18 226–238.
  • [37] Searle, S. R., Casella, G. and McCulloch, C. E. (1992)., Variance Components. John Wiley & Sons, New Jersey.
  • [38] Taniguchi, M. (1979). On estimation of parameters of Gaussian stationary processes., Journal of Applied Probability 16 575–591.
  • [39] van der Vaart, A. W. (2000)., Asymptotic Statistics. Cambridge University Press, Cambridge U.K.
  • [40] Vidakovic, B. (1999)., Statistical Modeling by Wavelets. John Wiley & Sons, New York.
  • [41] Wahba, G. (1980). Automatic smoothing of the log periodogram., Journal of the American Statistical Association 75 122–132.