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Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices

Published online by Cambridge University Press:  17 February 2009

N. M. Spencer  and
V. V. Anh
N. M. Spencer
Affiliation:
School of Mathematics, Queensland Institute of Technology, Brisbane Q 4001, Australia.
V. V. Anh
Affiliation:
School of Mathematics, Queensland Institute of Technology, Brisbane Q 4001, Australia.
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Abstract

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This paper considers discrete multivariate processes with time-dependent rational spectral density matrices and gives a solution to the spectral factorisation problem. As a result, the corresponding state space representation for the process is obtained. The relationship between multivariate processes with time-dependent rational spectral density matrix functions and multivariate ARMA processes with time-dependent coefficients is discussed. Solutions for the prediction problem are given for the case when only finite data is available and the case when the whole history of the process is known.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 2 , October 1991 , pp. 192 - 210
Copyright
Copyright © Australian Mathematical Society 1991

References

[1] Anderson, B. D. O. and Moore, J. B., Optimal filtering (Prentice Hall, 1979).Google Scholar
[2] Brockwell, P. J. and Davis, R. A., Time series: theory and methods (Springer-Verlag, 1987).CrossRefGoogle Scholar
[3] Caines, P., Linear stochastic systems (Wiley, 1988).Google Scholar
[4] Hajek, J., “On linear statistical problems in stochastic processes”, Czechoslovak Math. J. 12 (1962) 404–444.CrossRefGoogle Scholar
[5] Jordan, C., The calculus of finite differences (Chelsea, 1965).Google Scholar
[6] Melard, G. and Herteleer-de-Schutter, A., “Contributions to evolutionary spectral theory”, J. Time Series Analysis 10 (1989) 41–63.CrossRefGoogle Scholar
[7] Miller, K. S., “Nonstationary autoregressive processes”, IEEE Trans. Form. Theory 15 (1969) 315–316.CrossRefGoogle Scholar
[8] Parzen, E., “Statistical inference on time series by Hilbert space methods I”, Technical Report No. 23 (1959). App. Math. & Stat. Laboratory, Stanford University; reprinted in Time Series Analysis Papers (Holden Day, 1976).Google Scholar
[9] Peiris, M. S., “On the prediction of multivariate ARMA processes with a time-dependent covariance structure”, Commun. Statist. Theory Meth. 17 (1988) 27–37.CrossRefGoogle Scholar
[10] Priestley, M. B., “Evolutionary spectra and non stationary processes”, J. R. Statist. Soc. B. 27 (1965) 204–237.Google Scholar
[11] Priestley, M. B., “Parameter adaptive prediction and control”, Technical Report No. 7, U.M.I.S.T. (1969).Google Scholar
[12] Priestley, M. B., “Control systems with time-dependent parameters”, Bull. Inst. Internal. Statist. 37 (1969) 359–361.Google Scholar
[13] Rissanen, J., “A fast algorithm for optimum linear predictors”, IEEE Trans. Autom. Control AC-18 (1973) 555.CrossRefGoogle Scholar
[14] Rissanen, J. and Barbosa, L., “Properties of infinite covariance matrices and stability of optimum predictors”, Information Sciences 1 (1969) 221–236.CrossRefGoogle Scholar
[15] Rissanen, J. and Barbosa, L., “A factorization on problem and the problem of predicting nonstationary vector-valued stochastic processes”, Z. Wahrsch. Verw. Geb. 12 (1969) 255–266.CrossRefGoogle Scholar
[16] Rozanov, Y. A., Stationary random processes (Holden-Day, 1967).Google Scholar
[17] Singh, N. and Peiris, M. S., “A note on the properties of some nonstationary ARMA processes”, Stochastic Processes and their Applications 24 (1987) 151–155.CrossRefGoogle Scholar
[18] Spencer, N. M. and Anh, V. V., “Markovian representation and prediction of random sequences with time-dependent rational spectrum”, QUT Working Paper in Mathematics 12/89.Google Scholar
[19] Whittle, P., “Recursive relations for predictors”, J. R. Statist. Soc, Ser B 28 (1965) 523–532.Google Scholar