Generalized state-space models for modeling nonstationary EEG time-series

  • A. Galka
  • K.K.F. Wong
  • T. Ozaki
Part of the Springer Series in Computational Neuroscience book series (NEUROSCI, volume 4)


In this chapter we discuss a comprehensive framework for decomposing nonstationary time-series into a set of constituent processes. Our methodology is based on autoregressive moving-average (ARMA) modeling and on state-space modeling. For the purpose of modeling nonstationary phenomena, such as sudden phase transitions in dynamical behavior, we employ “generalized autoregressive conditional heteroscedastic modeling” (GARCH modeling), a technique originally introduced in the field of financial data analysis; however, this technique first needs to be generalized to the case of state-space modeling. Models are obtained through maximum-likelihood estimation; the innovation approach to time-series prediction helps us to derive an approximative expression for the likelihood of given data. We present three application examples relevant to the analysis of nonstationary phenomena in EEG time-series; the first case is the transition from the conscious state into anesthesia in a human patient, the second is the transition into epileptic seizure in a human patient, and the third is the transition between two sleep stages in a sheep fetus. The modeling algorithm does not require any prior information on the timing of such nonstationary phenomena.


time-series analysis ARMA model state-space model Kalman filtering innovation approach nonstationarity GARCH model 



The work of A. Galka was supported by Deutsche Forschungsgemeinschaft (DFG) through SFB 654 “Plasticity and Sleep”. The anesthesia EEG data set was kindly provided by W.J. Kox and S. Wolter, Department of Anesthesiology and Intensive Care Medicine, Charité-University Medicine, Berlin, Germany, and by E.R. John, Brain Research Laboratories, New York University School of Medicine, New York, USA. The fetal sleep data set was kindly provided by A. Steyn-Ross, Department of Engineering, University of Waikato, New Zealand. The epilepsy EEG data set was kindly provided by K. Lehnertz and C. Elger, Clinic for Epileptology, University of Bonn, Germany.


  1. 1.
    Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Contr. 19, 716–723 (1974)CrossRefGoogle Scholar
  2. 2.
    Akaike, H., Nakagawa, T.: Statistical Analysis and Control of Dynamic Systems. Kluwer, Dordrecht (1988)Google Scholar
  3. 3.
    Åström, K.J.: Maximum likelihood and prediction error methods. Automatica 16, 551–574 (1980)CrossRefGoogle Scholar
  4. 4.
    Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327 (1986), doi: 10.1016/0304-4076(86)90063-1CrossRefGoogle Scholar
  5. 5.
    Box, G.E.P., Jenkins, G.M.: Time Series Analysis, Forecasting and Control, 2. edn. Holden-Day, San Francisco (1976)Google Scholar
  6. 6.
    Durbin, J., Koopman, S.J.: Time Series Analysis by State Space Methods. Oxford University Press, Oxford, New York (2001)Google Scholar
  7. 7.
    Engle, R.F.: Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008 (1982), doi: 10.2307/1912773CrossRefGoogle Scholar
  8. 8.
    Galka, A., Yamashita, O., Ozaki, T.: GARCH modelling of covariance in dynamical estimation of inverse solutions. Physics Letters A 333, 261–268 (2004), doi: 10.1016/j.physleta.2004.10.045CrossRefGoogle Scholar
  9. 9.
    Gupta, N., Mehra, R.: Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculations. IEEE Trans. Autom. Contr. 19, 774–783 (1974), doi: 10.1109/TAC.1974.1100714CrossRefGoogle Scholar
  10. 10.
    Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton, New Jersey (1994)Google Scholar
  11. 11.
    Kailath, T.: An innovations approach to least-squares estimation – Part I: Linear filtering in additive white noise. IEEE Trans. Autom. Control 13, 646–655 (1968), doi: 10.1109/TAC.1968.1099025CrossRefGoogle Scholar
  12. 12.
    Kailath, T.: Linear Systems. Information and System Sciences Series. Prentice-Hall, Englewood Cliffs (1980)Google Scholar
  13. 13.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Engin. 82, 35–45 (1960)Google Scholar
  14. 14.
    Lévy, P.: Sur une classe de courbes de l’espace de Hilbert et sur une équation intégrale non linéaire. Ann. Sci. École Norm. Sup. 73, 121–156 (1956)Google Scholar
  15. 15.
    Milton, J.G., Chkhenkeli, S.A., Towle, V.L.: Brain connectivity and the spread of epileptic seizures. In: V.K. Jirsa, A.R. McIntosh (eds.) Handbook of Brain Connectivity, pp. 477–503. Springer-Verlag, Berlin, Heidelberg, New York (2007)CrossRefGoogle Scholar
  16. 16.
    Ozaki, T., Valdes, P., Haggan-Ozaki, V.: Reconstructing the nonlinear dynamics of epilepsy data using nonlinear time-series analysis. J. Signal Proc. 3, 153–162 (1999)Google Scholar
  17. 17.
    Penny, W.D., Stephan, K.E., Mechelli, A., Friston, K.J.: Comparing dynamic causal models. NeuroImage 22, 1157–1172 (2004), doi: 10.1016/j.neuroimage.2004.03.026CrossRefPubMedGoogle Scholar
  18. 18.
    Protter, P.: Stochastic Integration and Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York (1990)Google Scholar
  19. 19.
    Rauch, H.E., Tung, G., Striebel, C.T.: Maximum likelihood estimates of linear dynamic systems. American Inst. Aeronautics Astronautics (AIAA) Journal 3, 1445–1450 (1965)Google Scholar
  20. 20.
    Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)CrossRefGoogle Scholar
  21. 21.
    Shephard, N.: Statistical aspects of ARCH and stochastic volatility. In: D.R. Cox, D.V. Hinkley, O.E. Barndorff-Nielsen (eds.) Time Series Models in Econometrics, Finance and Other Fields, pp. 1–67. Chapman & Hall, London (1996)Google Scholar
  22. 22.
    Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W., Wilcocks, L.C.: Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. I. A thermodynamics analogy. Phys. Rev. E 64, 011917 (2001), doi: 10.1103/PhysRevE.64.011917Google Scholar
  23. 23.
    Su, G., Morf, M.: Modal decomposition signal subspace algorithms. IEEE Trans. Acoust. Speech Signal Proc. 34, 585–602 (1986)CrossRefGoogle Scholar
  24. 24.
    West, M.: Time series decomposition. Biometrika 84, 489–494 (1997)CrossRefGoogle Scholar
  25. 25.
    Wong, K.F.K., Galka, A., Yamashita, O., Ozaki, T.: Modelling nonstationary variance in EEG time-series by state space GARCH model. Computers Biol. Med. 36, 1327–1335 (2006), doi: 10.1016/j.compbiomed.2005.10.001CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of NeurologyUniversity of KielKielGermany
  2. 2.Massachusetts General HospitalHarvard Medical SchoolBostonUSA
  3. 3.Tohoku UniversityAoba-kuJapan

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