Abstract
Decomposition of multivariate time series data into independent source components forms an important part of preprocessing and analysis of time-resolved data in neuroscience. We briefly review the available tools for this purpose, such as Factor Analysis (FA) and Independent Component Analysis (ICA), then we show how linear state space modelling, a methodology from statistical time series analysis, can be employed for the same purpose. State space modelling, a generalization of classical ARMA modelling, is well suited for exploiting the dynamical information encoded in the temporal ordering of time series data, while this information remains inaccessible to FA and most ICA algorithms. As a result, much more detailed decompositions become possible, and both components with sharp power spectrum, such as alpha components, sinusoidal artifacts, or sleep spindles, and with broad power spectrum, such as FMRI scanner artifacts or epileptic spiking components, can be separated, even in the absence of prior information. In addition, three generalizations are discussed, the first relaxing the independence assumption, the second introducing non-stationarity of the covariance of the noise driving the dynamics, and the third allowing for non-Gaussianity of the data through a non-linear observation function. Three application examples are presented, one electrocardigram time series and two electroencephalogram (EEG) time series. The two EEG examples, both from epilepsy patients, demonstrate the separation and removal of various artifacts, including hum noise and FMRI scanner artifacts, and the identification of sleep spindles, epileptic foci, and spiking components. Decompositions obtained by two ICA algorithms are shown for comparison.
Similar content being viewed by others
References
Aït-Sahalia, Y., & Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financ. Econ., 83, 413–452.
Akaike, H. (1974a). Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes. Ann. Inst. Stat. Math., 26, 363–387.
Akaike, H. (1974b). A new look at the statistical model identification. IEEE Trans. Autom. Control, 19, 716–723.
Akaike, H., & Nakagawa, T. (1988). Statistical analysis and control of dynamic systems. Dordrecht: Kluwer Academic.
Allen, P. J., Josephs, O., & Turner, R. (2000). A method for removing imaging artifact from continuous EEG recorded during functional MRI. NeuroImage, 12, 230–239.
Åström, K. J. (1980). Maximum likelihood and prediction error methods. Automatica, 16, 551–574.
Attias, H., & Schreiner, C. E. (1998). Blind source separation and deconvolution: the dynamic component analysis algorithm. Neural Comput., 10, 1373–1424.
Baldick, R. (Ed.) (2006). Applied optimization: formulation and algorithms for engineering systems. Cambridge: Cambridge University Press.
Bar-Shalom, Y., & Fortmann, T. (1988). Tracking and data association. San Diego: Academic Press.
Barros, A. K., & Cichocki, A. (2001). Extraction of specific signals with temporal structure. Neural Comput., 13, 1995–2000.
Basilevsky, A. (1994). Statistical factor analysis and related methods: theory and applications. New York: Wiley-Interscience.
Beckmann, C., & Smith, S. (2004). Probabilistic independent component analysis for functional magnetic resonance imaging. IEEE Trans. Med. Imaging, 23, 137–152.
Beckmann, C., & Smith, S. (2005). Tensorial extensions of independent component analysis for multisubject FMRI analysis. NeuroImage, 25, 294–311.
Belouchrani, A., Abed-Meraim, K., Cardoso, J.-F., & Moulines, E. (1997). A blind source separation technique using second order statistics. IEEE Trans. Signal Process., 45, 434–444.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econom., 31, 307–327.
Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis, forecasting and control. San Francisco: Holden-Day.
Brockwell, P. J., & Davis, R. A. (1987). Time series: theory and methods. Berlin: Springer.
Cheung, Y. M., & Xu, L. (2003). Dual multivariate auto-regressive modeling in state space for temporal signal separation. IEEE Trans. Syst. Man Cybern., 33, 386–398.
Choi, S., Cichocki, A., Park, H., & Lee, S. (2005). Blind source separation and independent component analysis: a review. Neural Inf. Process. Lett. Rev., 6, 1–57.
Chui, C. K., & Chen, G. (1999). Springer series in information sciences : Vol. 17. Kalman filtering: with real-time applications (3rd ed.). Berlin: Springer.
Cichocki, A., & Amari, S. (2002). Adaptive blind signal and image processing. Chichester: Wiley.
Comon, P. (1994). Independent component analysis, a new concept? Signal Process., 36, 287–314.
Delorme, A., Sejnowski, T., & Makeig, S. (2007). Enhanced detection of artifacts in EEG data using higher-order statistics and independent component analysis. NeuroImage, 34, 1443–1449.
Durbin, J., & Koopman, S. J. (2001). Time series analysis by state space methods. Oxford: Oxford University Press.
Dyrholm, M., Makeig, S., & Hansen, L. K. (2007). Model selection for convolutive ICA with an application to spatiotemporal analysis of EEG. Neural Comput., 19, 934–955.
Engle, R. F., & Watson, M. (1981). A one-factor multivariate time series model of metropolitan wage rates. J. Am. Stat. Assoc., 76, 774–781.
Galka, A., Yamashita, O., & Ozaki, T. (2004). GARCH modelling of covariance in dynamical estimation of inverse solutions. Phys. Lett. A, 333, 261–268.
Galka, A., Ozaki, T., Bosch-Bayard, J., & Yamashita, O. (2006). Whitening as a tool for estimating mutual information in spatiotemporal data sets. J. Stat. Phys., 124, 1275–1315.
Galka, A., Wong, K., & Ozaki, T. (2010). Generalized state space models for modeling non-stationary EEG time series. In A. Steyn-Ross & M. Steyn-Ross (Eds.), Springer series in computational neuroscience. Modeling phase transitions in the brain (pp. 27–52). Berlin: Springer.
Gevers, M. (2006). A personal view of the development of system identification. IEEE Control Syst. Mag., 26, 93–105.
Gnedenko, B. V. (1969). The theory of probability. Moscow: Mir Publishers.
Grewal, M. S., & Andrews, A. P. (2001). Kalman filtering: theory and practice using MATLAB. New York: Wiley-Interscience.
Harman, H. H. (1976). Modern factor analysis (3rd ed.). Chicago: University of Chicago Press.
Harvey, A., Koopman, S. J., & Shephard, N. (Eds.) (2004). State space and unobserved component models. Cambridge: Cambridge University Press.
Hyvärinen, A. (1999). Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw., 10, 626–634.
Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent component analysis. New York: Wiley.
James, C., & Hesse, C. (2005). Independent component analysis for biomedical signals. Physiol. Meas., 26, R15–R39.
Jung, A., & Kaiser, A. (2003). Considering temporal structures in independent component analysis. In: Proc. 4th int. symp. ICA BSS, ICA 2003 (pp. 95–100). Nara, Japan, Apr. 2003.
Jung, T.-P., Makeig, S., McKeown, M., Bell, A., Lee, T.-W., & Sejnowski, T. (2001). Imaging brain dynamics using independent component analysis. IEEE Proc., 88, 1107–1122.
Kailath, T. (1968). An innovations approach to least-squares estimation—Part I: linear filtering in additive white noise. IEEE Trans. Autom. Control, 13, 646–655.
Kailath, T. (1980). Information and system sciences series. Linear systems. Englewood Cliffs: Prentice-Hall.
Kallenberg, O. (2002). Foundations of modern probability. Berlin: Springer.
Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. J. Basic Eng., 82, 35–45.
Kalman, R. E., Falb, P. L., & Arbib, M. A. (1969). International series in pure and applied mathematics. Topics in mathematical system theory. New York: McGraw-Hill.
Ljung, L. (1999). System identification: theory for the user (2nd ed.). Englewood Cliffs: Prentice-Hall.
Mehra, R. K. (1971). Identification of stochastic linear systems using Kalman filter representation. AIAA J., 9, 28–31.
Mehra, R. K. (1974). Identification in control and econometrics: similarities and differences. Ann. Econ. Soc. Meas., 3, 21–47.
Meinecke, F., Ziehe, A., Kawanabe, M., & Müller, K.-R. (2002). A resampling approach to estimate the stability of one- or multidimensional independent components. IEEE Trans. Biomed. Eng., 49, 1514–1525.
Miwakeichi, F., Martínez-Montes, E., Valdés-Sosa, P., Nishiyama, N., Mizuhara, H., & Yamaguchi, Y. (2004). Decomposing EEG data into space-time-frequency components using parallel factor analysis. NeuroImage, 22, 1035–1045.
Molenaar, P. C. (1985). A dynamic factor model for the analysis of multivariate time series. Psychometrika, 50, 181–202.
Molgedey, L., & Schuster, H. G. (1994). Separation of a mixture of independent signals using time delayed correlations. Phys. Rev. Lett., 72, 3634–3637.
Negishi, M., Abildgaard, M., Nixon, T., & Constable, R. (2004). Removal of time-varying gradient artifacts from EEG data acquired during continuous fMRI. Clin. Neurophysiol., 115, 2181–2192.
Neumaier, A., & Schneider, T. (2001). Estimation of parameters and eigenmodes of multivariate autoregressive models. ACM Trans. Math. Softw., 27, 27–57.
Niazy, R., Beckmann, C., Iannetti, D., Brady, J., & Smith, S. (2005). Removal of FMRI environment artifacts from EEG data using optimal basis sets. NeuroImage, 28, 720–737.
Otter, P. (1986). Dynamic structural systems under indirect observation: identifiability and estimation aspects from a system theoretic perspective. Psychometrika, 51, 415–428.
Ozaki, T., & Iino, M. (2001). An innovation approach to non-Gaussian time series analysis. J. Appl. Probab., 38, 78–92.
Pagan, A. R. (1975). A note on the extraction of components from time series. Econometrica, 43, 163–168.
Pearlmutter, B. A., & Parra, L. C. (1997). Maximum likelihood blind source separation: a context-sensitive generalization of ICA. In M. C. Mozer, M. I. Jordan & T. Petsche (Eds.), Advances in neural information processing systems (Vol. 9, pp. 613–619). Cambridge: MIT Press.
Protter, P. (1990). Stochastic integration and differential equations. Berlin: Springer.
Rauch, H. E., Tung, G., & Striebel, C. T. (1965). Maximum likelihood estimates of linear dynamic systems. AIAA J., 3, 1445–1450.
Schwarz, G. (1978). Estimating the dimension of a model. Ann. Stat., 6, 461–464.
Schweppe, F. (1965). Evaluation of likelihood functions for Gaussian signals. IEEE Trans. Inf. Theory, 11, 61–70.
Sorenson, H. W. (1970). Least-squares estimation: from Gauss to Kalman. IEEE Spectr., 7, 63–68.
Stögbauer, H., Kraskov, A., Astakhov, S. A., & Grassberger, P. (2004). Least-dependent-component analysis based on mutual information. Phys. Rev. E, 70, 066123.
Tong, L., Liu, R., Soon, V. C., & Huang, Y. (1991). Indeterminacy and identifiability of blind separation. IEEE Trans. Circuits Syst., 38, 499–509.
Vigário, R., Sarela, J., Jousmiki, V., Hamalainen, M., & Oja, E. (2000). Independent component approach to the analysis of EEG and MEG recordings. IEEE Trans. Biomed. Eng., 47, 589–593.
Waheed, K., & Salem, F. M. (2005). Linear state space feedforward and feedback structures for blind source recovery in dynamic environments. Neural Process. Lett., 22, 325–344.
Wong, K. F. K., Galka, A., Yamashita, O., & Ozaki, T. (2006). Modelling non-stationary variance in EEG time series by state space GARCH model. Comput. Biol. Med., 36, 1327–1335.
Zhang, L., & Cichocki, A. (2000). Blind deconvolution of dynamical systems: a state space approach. J. Signal Process., 4, 111–130.
Ziehe, A., & Müller, K.-R. (1998). TDSEP—an efficient algorithm for blind separation using time structure. In L. Niklasson, M. Bodén & T. Ziemke (Eds.), Proc. 8th int. conf. artificial neural networks, ICANN’98 (pp. 675–680). Berlin: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Galka, A., Wong, K.F.K., Ozaki, T. et al. Decomposition of Neurological Multivariate Time Series by State Space Modelling. Bull Math Biol 73, 285–324 (2011). https://doi.org/10.1007/s11538-010-9563-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-010-9563-y