Annals of Applied Statistics

Nonparametric spectral analysis with applications to seizure characterization using EEG time series

Li Qin and Yuedong Wang

Full-text: Open access


Understanding the seizure initiation process and its propagation pattern(s) is a critical task in epilepsy research. Characteristics of the pre-seizure electroencephalograms (EEGs) such as oscillating powers and high-frequency activities are believed to be indicative of the seizure onset and spread patterns. In this article, we analyze epileptic EEG time series using nonparametric spectral estimation methods to extract information on seizure-specific power and characteristic frequency [or frequency band(s)]. Because the EEGs may become nonstationary before seizure events, we develop methods for both stationary and local stationary processes. Based on penalized Whittle likelihood, we propose a direct generalized maximum likelihood (GML) and generalized approximate cross-validation (GACV) methods to estimate smoothing parameters in both smoothing spline spectrum estimation of a stationary process and smoothing spline ANOVA time-varying spectrum estimation of a locally stationary process. We also propose permutation methods to test if a locally stationary process is stationary. Extensive simulations indicate that the proposed direct methods, especially the direct GML, are stable and perform better than other existing methods. We apply the proposed methods to the intracranial electroencephalograms (IEEGs) of an epileptic patient to gain insights into the seizure generation process.

Article information

Ann. Appl. Stat., Volume 2, Number 4 (2008), 1432-1451.

First available in Project Euclid: 8 January 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

EEG epilepsy GACV GML locally stationary process permutation test smoothing parameter smoothing spline SS ANOVA


Qin, Li; Wang, Yuedong. Nonparametric spectral analysis with applications to seizure characterization using EEG time series. Ann. Appl. Stat. 2 (2008), no. 4, 1432--1451. doi:10.1214/08-AOAS185.

Export citation


  • Aksenova, T., Volkovych, V. and Villa, A. (2007). Detection of spectral instability in EEG recordings during the preictal period. J. Neural Engineering 4 173–178.
  • Brillinger, D. (1981). Time Series: Data Analysis and Theory, 2nd ed. Holden Day, San Francisco.
  • Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
  • D’Alessandro, M., Vachtsevanos, G., Esteller, R., Echauz, J. and Litt, B. (2001). A generic approach to selecting the optimal feature for epileptic seizure prediction. IEEE International Meeting of the Engineering in Medicine and Biology Society.
  • Fan, J. and Kreutzberger, E. (1998). Automatic local smoothing for spectral density estimation. Scand. J. Statist. 25 359–369.
  • Fan, J. and Zhang, W. (2004). Generalised likelihood ratio tests for spectral density. Biometrika 91 195–209.
  • Gao, H.-Y. (1997). Choice of thresholds for wavelet shrinkage estimate of the spectrum. J. Time Ser. Anal. 18 231–251.
  • Gu, C. (1992). Penalized likelihood regression: A Bayesian analysis. Statist. Sinica 2 255–264.
  • Gu, C. (2002). Smoothing Spline ANOVA Models. Springer, New York.
  • Guo, W. and Dai, M. (2006). Multivariate time-dependent spectral analysis using cholesky decomposition. Statist. Sinica 16 825–845.
  • Guo, W., Dai, M., Ombao, H. C. and von Sachs, R. (2003). Smoothing spline ANOVA for time-dependent spectral analysis. J. Amer. Statist. Assoc. 98 643–652.
  • Hannig, J. and Lee, T. (2004). Kernel smoothing of periodograms under Kullback–Leibler discrepancy. Signal Process. 84 1255–1266.
  • Kooperberg, C., Stone, C. J. and Truong, Y. K. (1995). Logspline estimation of a possibly mixed spectral distribution. J. Time Ser. Anal. 16 359–388.
  • Lee, T. (1997). A simple span selector for periodogram smoothing. Biometrika 84 965–969.
  • Lin, X., Wahba, G., Xiang, D., Gao, F., Klein, R. and Klein, B. (2000). Smoothing spline ANOVA models for large data sets with Bernoulli observations and the randomized GACV. Ann. Statist. 28 1570–1600.
  • Liu, A., Tong, T. and Wang, Y. (2006). Smoothing spline estimation of variance functions. J. Comput. Graph. Statist. 16 312–329.
  • Lopes da Silva, F. (1978). Analysis of EEG nonstationarities. In Contemporary Clinical Neurophysiology (W. A. Cobb and H. van Dujn, eds.) 165–179. Oxford Univ. Press.
  • McCullagh, P. and Nelder, J. (1989). Generalized Linear Models. Chapman and Hall, London.
  • Mormann, F., Andrzejak, R., Kreutz, T., Rieke, C., David, P., Elger, C. and Lehnertz, K. (2003). Automated detection of a preseizure state based on a decrease in synchronization in intracranial electroencephalogram recordings from epilepsy patients. Phys. Rev. E 67 1–10.
  • Mormann, F., Lehnertz, K., David, P. and Elger, C. (2000). Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients. Phys. D 144 358–369.
  • Ombao, H. C., Raz, J. A., Strawderman, R. L. and von Sachs, R. (2001). A simple generalised crossvalidation method of span selection for periodogram smoothing. Biometrika 88 1186–1192.
  • Ombao, H., Raz, J., von Sachs, R. and Guo, W. (2002). The SLEX model of a nonstationary time series. Ann. Inst. Statist. Math. 54 171–200.
  • Pawitan, Y. and O’Sullivan, F. (1994). Nonparametric spectral density estimation using penalized Whittle likelihood. J. Amer. Statist. Assoc. 89 600–610.
  • Qin, L. and Wang, Y. (2008). Supplement to “Nonparametric spectral analysis with applications to seizure characterization using EEG time series.” DOI: 10.1214/08-AOAS185SUPP.
  • Schiller, Y., Cascino, G. D., Busacker, N. E. and Sharbrough, F. W. (1998). Characterization and comparison of local onset and remote propagated electrographic seizures recorded with intracranial electrodes. Epilepsia 39 380–388.
  • Schiller, Y., Cascino, G. D. and Sharbrough, F. W. (1998). Chronic intracranial EEG monitoring for localizing the epileptogenic zone: An electroclinical correlation. Epilepsia 39 1302–1308.
  • Shumway, R. H. and Stoffer, D. S. (2000). Time Series Analysis and Its Applications. Springer, New York.
  • Wahba, G. (1980). Automatic smoothing of the log periodogram. J. Amer. Statist. Assoc. 75 122–132.
  • Wahba, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.
  • Wahba, G. and Wang, Y. (1993). Behavior near zero of the distribution of GCV smoothing parameter estimates for splines. Statist. Probab. Letters 25 105–111.
  • Wahba, G., Wang, Y., Gu, C., Klein, R. and Klein, B. (1995). Smoothing spline ANOVA for exponential families, with application to the Wisconsin Epidemiological Study of Diabetic Retinopathy. Ann. Statist. 23 1865–1895.
  • Winterhalder, M., Taiwald, T., Voss, H. U., Aschenbrenner-Scheibe, R., Timmer, J. and Schulze-Bonhage, A. (2003). The seizure prediction characteristic: A general framework to assess and compare seizure prediction methods. Epilepsy and Behavior 4 318–325.
  • Worrell, G. A., Parish, L., Cranstoun, S. D., Jonas, R., Baltuch, G. and Litt, B. (2004). High-frequency oscillations and seizure generation in neocortical epilepsy. Brain 127 1496–1506.

Supplemental materials