Markovian Representation of Stochastic Processes and Its Application to the Analysis of Autoregressive Moving Average Processes

  • Hirotugu Akaike
Part of the Springer Series in Statistics book series (SSS)


The problem of identifiability of a multivariate autoregressive moving average process is considered and a complete solution is obtained by using the Markovian representation of the process. The maximum likelihood procedure for the fitting of the Markovian representation is discussed. A practical procedure for finding an initial guess of the representation is introduced and its feasibility is demonstrated with numerical examples.


Stochastic System Canonical Correlation Canonical Correlation Analysis Factor Analysis Model State Space Representation 
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.


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  1. [1]
    Akaike, H. (1971). Autoregressive model fitting for control, Ann. Inst. Statist. Math., 23, 163–180.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Akaike, H. (1972). Use of an information theoretic quantity for statistical model identification, Proc. 5th Hawaii International Conference on System Sciences, 249–250.Google Scholar
  3. [3]
    Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, Proc. 2nd International Symposium on Information Theory, (B. N. Petrov and F. Csaki eds.), Akademiai Kiado, Budapest, 267–281.Google Scholar
  4. [4]
    Akaike, H. (1973). Block Toeplitz matrix inversion, SIAM J. Appl. Math., 24, 234–241.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Akaike, H. (1973). Maximum likelihood identification of Gaussian autoregressive moving average models, Biometrika, 60, 255–265.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Akaike, H. (1973). Markovian representation of stochastic processes by canonical variables, to be published in SIAM J. Control.Google Scholar
  7. [7]
    Akaike, H. (1973). Stochastic theory of minimal realizations, to be published in IEEE Trans. Automat. Contrl.Google Scholar
  8. [8]
    Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis, Wiley, New York.MATHGoogle Scholar
  9. [9]
    Hannan, E. J. (1969). The identification of vector mixed autoregressive-moving average systems, Biometrika, 56, 223–225.MathSciNetMATHGoogle Scholar
  10. [10]
    Kalman, R. E., Falb, P. L. and Arbib, M. A. (1969). Topics in Mathematical System Theory, McGraw-Hill, New York.MATHGoogle Scholar
  11. [11]
    Kullback, S. and Leibler, R. A. (1951). On information and sufficiency, Ann. Math. Statist., 22, 79–86.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Lawley, D. N. and Maxwell, A. E. (1971). Factor Analysis as a Statistical Method, Butterworths, London.MATHGoogle Scholar
  13. [13]
    Priestley, M. B., Subba Rao, T. and Tong, H. (1972). Identification of the structure of multivariable stochastic systems, to appear in Multivariate Analysis III, Ed. P. R. Krishnaiah, Academic Press.Google Scholar
  14. [14]
    Rissanen, J. (1972). Estimation of parameters in multi-variate random processes, unpublished.Google Scholar
  15. [15]
    Rissanen, J. (1973). Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factorising positive matrix polynomials, Mathematics of Computation, 27, 147–154.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Whittle, P. (1953). The analysis of multiple stationary time series, J. R. Statist. Soc., B, 15, 125–139.MathSciNetMATHGoogle Scholar
  17. [17]
    Whittle, P. (1962). Gaussian estimation in stationary time series, Bulletin L’Institut International de Statistique, 39, 2e Livraison, 105–129.Google Scholar
  18. [18]
    Whittle, P. (1963). On the fitting of multivariate autoregressions, and the approximate factorization of a spectral density matrix, Biometrika, 50, 129–134.MathSciNetMATHGoogle Scholar
  19. [19]
    Whittle, P. (1969). A view of stochastic control theory, J. R. Statist. Soc., A, 132, 320–334.MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 1998

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  • Hirotugu Akaike

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