Electronic Journal of Statistics

Data-driven shrinkage of the spectral density matrix of a high-dimensional time series

Mark Fiecas and Rainer von Sachs

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Time series data obtained from neurophysiological signals is often high-dimensional and the length of the time series is often short relative to the number of dimensions. Thus, it is difficult or sometimes impossible to compute statistics that are based on the spectral density matrix because estimates of these matrices are often numerically unstable. In this work, we discuss the importance of regularization for spectral analysis of high-dimensional time series and propose shrinkage estimation for estimating high-dimensional spectral density matrices. We use and develop the multivariate Time-frequency Toggle (TFT) bootstrap procedure for multivariate time series to estimate the shrinkage parameters, and show that the multivariate TFT bootstrap is theoretically valid. We show via simulations and an fMRI data set that failure to regularize the estimates of the spectral density matrix can yield unstable statistics, and that this can be alleviated by shrinkage estimation.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 2975-3003.

First available in Project Euclid: 15 January 2015

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Bootstrap high-dimensional time series shrinkage estimation spectral analysis


Fiecas, Mark; von Sachs, Rainer. Data-driven shrinkage of the spectral density matrix of a high-dimensional time series. Electron. J. Statist. 8 (2014), no. 2, 2975--3003. doi:10.1214/14-EJS977. https://projecteuclid.org/euclid.ejs/1421330627

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  • [1] Berk, K. N. (1974). Consistent autoregressive spectral estimates., The Annals of Statistics 2 489–502.
  • [2] Berkowitz, J. and Diebold, F. X. (1998). Bootstrapping multivariate spectra., Review of Economic and Statistics 80 664–666.
  • [3] Bickel, P. and Freedman, D. (1981). Some asymptotic theory for the bootstrap., The Annals of Statistics 9 1196–1217.
  • [4] Böhm, H. andvon Sachs, R. (2008). Structural shrinkage of nonparametric spectral estimators for multivariate time series., Electronic Journal of Statistics 2 696–721.
  • [5] Böhm, H. andvon Sachs, R. (2009). Shrinkage estimation in the frequency domain of multivariate time series., Journal of Multivariate Analysis 100 913–935.
  • [6] Brillinger, D. (2001)., Time Series: Data Analysis and Theory. Society for Industrial and Applied Mathematics.
  • [7] Dahlhaus, R. (2000). Graphical interaction models for multivariate time series., Metrika 51 157–172.
  • [8] Dai, M. and Guo, W. (2004). Multivariate spectral analysis using Cholesky decomposition., Biometrika 91 629–643.
  • [9] Dette, H. and Paparoditis, E. (2009). Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities., Journal of the Royal Statistical Society B 71 831–857.
  • [10] Fiecas, M. and Ombao, H. (2011). The generalized shrinkage estimator for the analysis of functional connectivity of brain signals., The Annals of Applied Statistics 5 1102–1125.
  • [11] Fiecas, M., Ombao, H., Linkletter, C., Thompson, W. and Sanes, J. (2010). Functional connectivity: Shrinkage estimation and randomization test., NeuroImage 5 1102–1125.
  • [12] Fiecas, M., Ombao, H., van Lunen, D., Baumgartner, R., Coimbra, A. and Feng, D. (2013). Quantifying temporal correlations: A test-retest evaluation of functional connectivity in resting-state fMRI., NeuroImage 65 231–241.
  • [13] Fox, M. D. and Greicius, M. (2010). Clinical applications of resting state functional connectivity., Frontiers in Systems Neuroscience 4 1–13.
  • [14] Fox, M. D. and Raichle, M. E. (2008). Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging., Nature Reviews Neuroscience 8 700–711.
  • [15] Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates., The Annals of Statistics 20 121–145.
  • [16] Friston, K., Frith, C., Liddle, P. and Frackowiak, R. (1993). Functional connectivity: The principal-component analysis of large (PET) data sets., Journal of Cerebral Blood Flow and Metabolism 13 5–14.
  • [17] Guo, W. and Dai, M. (2006). Multivariate time-dependent spectral analysis using Cholesky decomposition., Statistica Sinica 16 825–845.
  • [18] Jentsch, C. and Kreiss, J.-P. (2010). The multiple hybrid bootstrap – resampling multivariate linear processes., Journal of Multivariate Analysis 101 2320–2345.
  • [19] Kirch, C. and Politis, D. N. (2011). TFT-bootstrap: Resampling time series in the frequency domain to obtain replicates in the time domain., The Annals of Statistics 39 1427–1470.
  • [20] Krafty, R. and Collinge, W. (2013). Penalized Multivariate Whittle Likelihood for Power Spectrum Estimation., Biometrika 100 447–458.
  • [21] Kreiss, J. P. and Paparoditis, E. (2003). Autoregressive-aided periodogram bootstrap for time series., Annals of Statistics 31 1923–1955.
  • [22] Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices., Journal of Multivariate Analysis 88 365–411.
  • [23] Mallows, C. L. (1972). A note on asymptotic joint normality., Annals of Mathematical Statistics 43 508–515.
  • [24] Ombao, H., Raz, J., Strawderman, R. and von Sachs, R. (2001). A simple generalised crossvalidation method of span selection for periodogram smoothing., Biometrika 88 1186–1192.
  • [25] Pourahmadi, M. (2011). Covariance estimation: The GLM and regularization perspectives., Statistical Science 26 369–387.
  • [26] Salvador, R., Suckling, J., Schwarzbauer, C. and Bullmore, E. (2005). Undirected graphs of frequency-dependent functional connectivity in whole-brain networks., Philosophical Transactions of the Royal Society B 360 937–946.
  • [27] Salvador, R., Sarró, S., Gomar, J. J., Ortiz-Gil, J., Vila, F., Capdevila, A., Bullmore, E., McKenna, P. J. and Pomarol-Coltet, E. (2010). Overall brain connectivity maps show cortico-subcortical abnormalities in schizophrenia., Human Brain Mapping 31 2003–2014.
  • [28] Shehzad, Z., Kelly, A. M. C., Reiss, P. T., Gee, D. G., Gotimer, K., Uddin, L. Q., Lee, S. H., Marguilies, D. S., Roy, A. K., Biswal, B. B., Petkova, E., Castellanos, F. X. and Milham, M. P. (2009). The resting brain: Unconstrained yet reliable., Cerebral Cortex 19 2209–2229.
  • [29] Shrout, P. E. and Fleiss, J. L. (1979). Intraclass correlations: Uses in assessing rater reliability., Psychological Bulletin 86 420–428.
  • [30] Tzourio-Mazoyer, N., Landeau, B., Papathanassiou, D., Crivello, F., Etard, O., Delcroix, N., Mazoyer, B. and Joliot, M. (2002). Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain., NeuroImage 15 273–289.