Abstract
The stability of time-varying autoregressive (AR) models is an important issue in such applications as time-varying spectrum estimation and electroencephalography simulation and estimation. In some cases, such as time-varying spectrum estimation, the models that exhibit roots near unit moduli are difficult to use. Thus a tighter stability condition such as stability with a positive margin is needed. A time-varying AR model is stable with a positive margin if the moduli of the roots of the time-varying characteristic polynomial are somewhat less than unity for every time instant. Recently, a new method for the estimation of the time-varying AR models was introduced. This method is based on the interpretation of the underdetermined time-varying prediction equations as an ill-posed inverse problem that is solved by Tikhonov regularization. The method is referred to as the deterministic regression smoothness priors (DRSP) scheme. In this paper, a stabilization method in which the DRSP scheme is augmented with nonlinear stability constrainst is proposed. The problem is formulated so that stability with a positive margin can also be achieved. The problem is solved iteratively with an exterior point algorithm. The performance of the algorithm is studied with a simulation. It is shown that the proposed approach is well suited to stable modeling of signals containing narrowband transitions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Bohlin, Analysis of EEG signals with changing spectra using a short-word Kalman estimator,Math. Biosci., 35, 221–259, 1977.
M. Boudaoud and L. Chaparro, Composite modeling of nonstationary signals,J. Franklin Inst., 324, 113–124, 1987.
S. N. Elaydi,An Introduction to Difference Equations, Springer-Verlag, New York, p. 175, 1996.
A. V. Fiacco and G. P. McCormick,Nonlinear Programming, SIAM, Philadelphia, PA, p. 57, 1990.
W. Gersch, Smoothness priors multichannel autoregressive time series modelling, inICASSP89, pp. 2158–2161, 1989.
W. Gersch, Smoothness priors, in:New Directions in Time Series Analysis, Part II, Springer-Verlag, Berlin, pp. 113–146, 1991.
W. Gersch. A. Gevins, and G. Kitagawa, A multivariate time varying autoregressive modeling of nonstationary covariance time series, inIEEE Conf. CDC-83, pp. 579–584, 1983.
W. Gersch and G. Kitagawa, Smoothness priors in transfer function estimation,Automatica, 25, 603–608, 1989.
C. W. Groetsch,Inverse Problems in the Mathematical Sciences, Friedr, Vieweg, Braunschweig, Germany, 1993.
M. Hanke and P. C. Hansen, Regularization methods for large-scale problems,Surv. Math. Ind., 3, 253–315, 1993.
P. C. Hansen,Rank-Deficient and Discrete Ill-Posed Problems. Numerical Aspects of Linear Inversion, SIAM, Philadelphia, PA, 1998.
J. Hiltunen, P. A. Karjalainen, J. Partanen, and J. P. Kaipio, Estimation of the dynamics of eventrelated synchronization changes in EEG,Med. Biol. Engrg. Comput., 37, 307–315, 1999.
D. M. Himmelblau,Applied Nonlinear Programming, McGraw-Hill, New York, 1972.
B. Jansen, Analysis of biomedical signals by means of linear modeling,CRC Crit. Rev. Biomed. Engrgrg., 12, 343–392, 1985.
M. Juntunen, Direct stabilization of autoregressive models, Ph.D. thesis, University of Kuopio, Finland, 1999.
M. Juntunen, J. Tervo, and J. P. Kaipio, Stabilization of stationary and time-varying autoregressive models, inICASSP98, pp. 2173–2176.
M. Juntunen, J. Tervo, and J. P. Kaipio, Root modulus constraints in autoregressive model estimation,Circuits Syst. Signal Process., 17, 709–718, 1998.
M. Juntunen, J. Tervo, and J. P. Kaipio, Stabilization of Subba Rao-Liporace models,Circuits Syst. Signal Process., 18, 395–406, 1999.
J. P. Kaipio, Simulation and estimation of nonstationary EEG, Ph.D. thesis, University of Kuopio, Finland, 1996.
J. P. Kaipio and M. Juntunen, Deterministic regression smoothness priors TVAR modelling, inICASSP99.
J. P. Kaipio and P. A. Karjalainen, Estimation of event related synchronization changes by a new TVAR method,IEEE Trans. Biomed. Engrg. 44(8), 649–656, 1997.
E. W. Kamen, P. P. Khargonekar, and K. R. Poolla, A transfer-function approach to linear time-varying discrete-time systems,SIAM J. Control. Optim., 23, 550–565, 1985.
G. Kitagawa and W. Gersch, A smoothness priors long AR model method for spectrum estimation,IEEE Trans. Automat. Control. 30, 57–65, 1985.
G. Kitagawa and W. Gersch,Smoothness Priors Analysis of Time Series, Springer-Verlag, Berlin 1996.
C. L. Lawson and R. J. Hanson,Solving Least Squares Problems, SIAM, Philadelpia, PA, pp. 158–169, 1995.
L. A. Liporace, Linear estimation of nonstationary signals,J. Acoust. Soc. Am., 58, 1288–1295, 1975.
G. Ludyk, Stability of time-variant discrete-time systems, in:Advances in Control Systems and Signal Processing, vol. 5, Vieweg, Braunschweig, Germany, p. 52, 1985.
S. L. Marple,Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, NJ, 1987.
G. Müller, R. A. Richter, S. Weisbrod, and F. Klingberg, Duration of EEG alpha wave blockade by tone stimulation is prolonged in early stage of presenile onset dementia of the Alzheimer type,Biomed. Biochim. Acta, 50(8), 987–991, 1991.
J. J. Rajan, Time series classification, Ph.D. thesis, University of Cambridge, England, 1994.
T. Subba Rao, The fitting of non-stationary time series models with time-dependent parameters,J. Roy., Statist. Soc. B, 32, 312–322, 1970.
B. L. van der Waerden,Algebra 1, Frederick Ungar, New York, p. 186, 1970.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Juntunen, M., Kaipio, J.P. Stabilization of smoothness priors time-varying autoregressive models. Circuits Systems and Signal Process 19, 423–435 (2000). https://doi.org/10.1007/BF01196156
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01196156