Generalized state-space models for modeling nonstationary EEG time-series
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.
Keywordstime-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.
- 2.Akaike, H., Nakagawa, T.: Statistical Analysis and Control of Dynamic Systems. Kluwer, Dordrecht (1988)Google Scholar
- 5.Box, G.E.P., Jenkins, G.M.: Time Series Analysis, Forecasting and Control, 2. edn. Holden-Day, San Francisco (1976)Google Scholar
- 6.Durbin, J., Koopman, S.J.: Time Series Analysis by State Space Methods. Oxford University Press, Oxford, New York (2001)Google Scholar
- 10.Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton, New Jersey (1994)Google Scholar
- 12.Kailath, T.: Linear Systems. Information and System Sciences Series. Prentice-Hall, Englewood Cliffs (1980)Google Scholar
- 13.Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Engin. 82, 35–45 (1960)Google Scholar
- 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
- 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
- 18.Protter, P.: Stochastic Integration and Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York (1990)Google Scholar
- 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
- 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.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