Advertisement

System Identification

  • E. J. Hannan
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 78)

Abstract

In the first section the introduction of parameters to describe multivariate ARMAX systems is described and some topological properties of the parameter spaces are considered that relate to parameter estimation. In particular the analytic manifold, M(n), of all ARMAX structures of given order (McMillan degree) is considered. A very general discussion is given, in the second section, of the properties of maximum likelihood estimation of the point on M(n), emphasising both minimal conditions on the inputs and the linear innovations and avoiding assumptions of a too restrictive nature concerning the true stochastic structure. In the third section the estimation of the order is discussed and a number of new results are presented. Other topics briefly considered are the construction of initial estimates for an iterative solution of the likelihood equations, time varying parameter models and non-linear models.

Keywords

Canonical Correlation Hankel Matrix Analytic Manifold Dynamical Index Coordinate Neighbourhood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    Akaike, H.: 1969, “Fitting autoregressive models for prediction”, Ann. Inst. Statist. Math. 21, pp. 243–247.MathSciNetMATHCrossRefGoogle Scholar
  2. (2).
    Akaike, H.: 1976, “Canonical correlation analysis of time series and the use of an information criterion”. In System Identification: Advances and Case Studies ( R.K. Mehra and D.G. Lainiotis, eds) pp. 27–96. Academic Press, New York.CrossRefGoogle Scholar
  3. (3).
    Andel, J.: 1976, “Autoregressive series with random parameters”, Math. Operationsforsch. u. Statist. 7, pp. 735–741.MathSciNetCrossRefGoogle Scholar
  4. (4).
    Cameron, M.A.: 1978 “The prediction variance and related statistics for stationary time series”, Biometrika 65, pp. 283–296.MathSciNetMATHCrossRefGoogle Scholar
  5. (5).
    Clark, J.M.C.: 1976, “The consistent selection of parameterizations in system identification”. Paper presented at JACC, Purdue University.Google Scholar
  6. (6).
    Deistler, M., and Hannan, E.J.: 1979, “Some properties of the parameterization of ARMA systems with unknown order”. To be published.Google Scholar
  7. (7).
    Forney, D.G.: 1975, “Minimal bases of rational vector spaces with applications to multivariable linear systems”. SIAM J. Control 13, pp. 493–520.MathSciNetMATHCrossRefGoogle Scholar
  8. (8).
    Fuller, W.A., Hasza, D.P., and Goebel, J.J.: 1979, “Estimation of the parameters of stochastic difference equations”, Res. Report. Dept of Stats. Iowa State Univ.Google Scholar
  9. (9).
    Granger, C.W.J., and Anderson, A.D.: 1978, An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht. Göttingen.MATHGoogle Scholar
  10. (10).
    Hannan, E. J.: 1969, “The identification of vector mixed autoregressive-moving average systems”, Biometrika 57, pp. 223–225.MathSciNetGoogle Scholar
  11. (11).
    Hannan, E.J.: 1970, Multiple Time Series. Wiley, New York.MATHCrossRefGoogle Scholar
  12. (12).
    Hannan, E.J.: 1975, “The estimation of ARMA models”, Ann. Statist. 3, pp. 975–981.MathSciNetMATHCrossRefGoogle Scholar
  13. (13).
    Hannan, E.J.: 1979, “The central limit theorem for time series regression”, Stoch. Proc. and their Appn. 9, pp. 281–289.MathSciNetMATHCrossRefGoogle Scholar
  14. (14).
    Hannan, E.J.: 1980, “The estimation of the order of an ARMA process”, Ann. Statist. 8Google Scholar
  15. (15).
    Hannan, E.J.: 1980, “Estimating the dimension of a linear system”. To be published.Google Scholar
  16. (16).
    Hannan, E.J., Dunsmuir, W.T., and Deistler, M.: 1980, “Estimation of vector ARMAX models”, J. Multivariate Anal. To appear.Google Scholar
  17. (17).
    Jones, D.A.: 1978, “Non-linear autoregressive processes”, Proc. Roy. Soc. London A, 360, pp. 71–95.MATHCrossRefGoogle Scholar
  18. (18).
    Pagan, A.R.: 1980, “Some identification and estimation results for regression models with stochastically varying coefficients”, J. Econometrics 13Google Scholar
  19. (19).
    Pham, Tuan D. and Tran, Lanh T.: 1980, “Quelques résultats sur les modèles bilinéaires de séries chronologiques”, C.R. Acad. Sc. Paris, 290, Série A, pp. 330–338.MathSciNetGoogle Scholar
  20. (20).
    Quinn, B.G. and Nicholls, D.F.: 1981, “The stability of autoregressive models with random coefficients”, J. Multivariate Anal. 11Google Scholar
  21. (21).
    Rissanen, J.: 1976, “Minimax entropy estimation of models for vector processes”. In Systems Identifieation: Advances and Case Studies, Eds. R.K. Mehra and D.G. Lainiotis. Academic Press, New York, pp. 97–120.CrossRefGoogle Scholar
  22. (22).
    Rissanen, J.: 1978, “Modeling by shortest data description”, Automatica 14, pp. 468–471.CrossRefGoogle Scholar
  23. (23).
    Rosenbrock, H.H.: 1970, State Space and Multivariable Theory. Nelson, London.MATHGoogle Scholar
  24. (24).
    Shibata, R.: 1980, Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Ann. Statist. 8, pp. 147–164.MathSciNetMATHCrossRefGoogle Scholar
  25. (25).
    Tong, H., and Lim, K.S.: 1980, Threshold autoregression, limit cycles and cyclical data. J. Roy. Statist. Soc. B, 42, pp.Google Scholar
  26. (26).
    Young, P., Jakeman, A., and McMurtrie, R.: 1980, “An instrumental variable method for model order identification”, Automatica l6Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • E. J. Hannan
    • 1
  1. 1.The Australian National UniversityAustralia

Personalised recommendations