Abstract
The problem we consider here is the elimination of the redundant complexity in the modeling of stationary time series by means of stochastic realization theory. The question is posed in terms of approximate modeling of a gaussian process. It is shown that a minimal representation of the process can be obtained with a very simple algorithm of polynomial complexity from a nonminimal one, and that this algorithm can be extended to give an approximate realization of fixed degree k We also show that different realizations yield different approximation errors, and discuss how to choose the representation which gives the best approximant.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahren, P.R., and Clark, D.N., On functions orthogonal to invariant subspaces Acta. Math. 124 (1970), pp. 191–204.
Caines, P.E., On the scientific method and the foundations of system identification Modelling, Identification and Robust Control, C.I. Byrnes and A. Lindquist Eds., Elsevier Science Publisher B.V. North Holland, 1986.
Dewilde, P., and Dym, H., Schur recursion, error formulas, and convergence of rational estimators for stationary stochastic sequences IEEE Trans. Inform. Theory, IT-27 (1981), pp. 446–461.
Georgiou, T.T., and Khargonekar, P.P., Spectral factorization and the Nevanlinna-Pick interpolation SIAM J. Control and Optimization, Vol. 24 (1988), pp. 754–766.
Gohberg, I., Lancaster, P., and Rodman L., Invariant subspaces of matrices John Wiley & Sons (1986).
Gombani, A., Consistent approximations of linear stochastic models SIAM Journal on Control and Optimization, 27 (1989), pp. 83–107.
Lindquist, A., and Pavon, M., On the structure of state space models for discrete-time stochastic vector processes IEEE Trans. Automatic Control, AC-29 (1984), pp. 418–432.
Lindquist, A., and Picci, G., Realization Theory for Multivariate Stationary Gaussian Processes, SIAM J. Control and Optimization, Vol. 23, No. 6 (1985), pp. 809–857.
Ljung, L., and Söderstöm, T., Theory and practice of recursive identification MIT Press, Cambridge, Mass. (1989).
Nikol’skii, N.K., Treatise on the shift operator Springer-Verlag (1986).
Rozanov, Y.A., Stationary Random Processes, Holden-Day (1987).
Fuhrmann, P.A., Linear Systems and Operators in Hilbert Space, McGraw-Hill (1981).
Walsh, J.L., Interpolation and approximation by rational functions in the complex domain Amer. Math. Soc. Coll. Publ. 20 (1935).
Nagy, B.Sz., and Foias, C, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam (1970).
Baratchart, L., and Olivi, M., New tools in rational L 2 approximationPreprints of the 8th IFAC Symposium on Identification and System Parameter Estimation, Beijing China (1988), pp. 1014–1019.
Hannan, E.J., and Deistler, M., The Statistical Theory of Linear Systems, Wiley (1988).
Caines, P.E., Linear Stochastic Systems, Wiley (1988).
Gombani, A., On approximate stochastic realizations to appear in Mathematics of Control, Signals and Systems.
Gombani, A., On the multivariable approximate stochastic realization problem to appear in Stochastics.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Gombani, A., Polini, C. (1993). On Approximate Modeling of Linear Gaussian Processes. In: Brillinger, D., Caines, P., Geweke, J., Parzen, E., Rosenblatt, M., Taqqu, M.S. (eds) New Directions in Time Series Analysis. The IMA Volumes in Mathematics and its Applications, vol 46. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9296-5_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9296-5_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9298-9
Online ISBN: 978-1-4613-9296-5
eBook Packages: Springer Book Archive