An approach to time series analysis and ARMA spectral estimation

This paper presents an approach to time series analysis and ARMA spectral estimation from only the output data corrupted by noise. It is shown that the generalized (not well-known) modified Yule-Walker (MYW) equations hold when the residual is some correlated noise. To solve such equations, a new version of the generalized least squares (GLS) method is proposed, yielding AR parameter estimates with higher accuracy. This GLS method can also be used to enhance the estimation accuracy of AR parameters for short and noisy data in case of the MYW equation holding theoretically. Furthermore, a simple procedure for improving MA parameter estimates is studied. Our approach, as Cadzow's singular value decomposition (SVD) method, has provided significantly higher performance spectral estimates for a low-order ARMA model than those obtained via usual techniques which are based upon direct solution of the MYW equations and which use a high-order model without considering an additive noise.

[1]  V. Strejc,et al.  On Least Squares and Regression Methods in Identification; A Survey of Progresses and Trends , 1978 .

[2]  J. Cadzow,et al.  Spectral estimation: An overdetermined rational model equation approach , 1982, Proceedings of the IEEE.

[3]  J. Bee Bednar,et al.  Fast algorithms for lpdeconvolution , 1985, IEEE Trans. Acoust. Speech Signal Process..

[4]  T. Söderström.  Convergence properties of the generalised least squares identitication method , 1974, Autom..

[5]  R. Sivan,et al.  On Least Squares Algorithms for System Parameter Identification , 1976 .

[6]  M. W. Sage,et al.  Recursive generalised-least-squares procedure for online identification of process parameters , 1969 .

[7]  H. Takeda,et al.  An order recursive generalized least squares algorithm for system identification , 1985 .

[8]  David Q. Mayne,et al.  Linear identification of ARMA processes , 1982, Autom..

[9]  J. Cadzow,et al.  High performance spectral estimation--A new ARMA method , 1980 .

[10]  B. Friedlander,et al.  The Modified Yule-Walker Method of ARMA Spectral Estimation , 1984, IEEE Transactions on Aerospace and Electronic Systems.

[11]  R. Isermann,et al.  Comparison of six on-line identification and parameter estimation methods , 1974, Autom..

[12]  Yiu-Tong Chan,et al.  Spectral estimation via the high-order Yule-Walker equations , 1982 .

[13]  Steven Kay,et al.  A new ARMA spectral estimator , 1980 .

[14]  H. Akaike.  Canonical Correlation Analysis of Time Series and the Use of an Information Criterion , 1976 .

[15]  M. Kaveh.  High resolution spectral estimation for noisy signals , 1979 .

[16]  A. A. Beex,et al.  Covariance sequence approximation for parametric spectrum modeling , 1981 .

[17]  R. Kumaresan,et al.  Singular value decomposition and improved frequency estimation using linear prediction , 1982 .

[18]  J. Cadzow,et al.  Singular-value decomposition approach to time series modelling , 1983 .

[19]  R. Kumaresan,et al.  Improved spectral resolution III: Efficient realization , 1980, Proceedings of the IEEE.

[20]  R. Yarlagadda,et al.  Fast Algorithms for lp Deconvolution , 1985 .

[21]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[22]  E. J. Hannan,et al.  Linear estimation of ARMA processes , 1983, Autom..

[23]  C. Woodside.  Estimation of the order of linear systems , 1971 .

[24]  T. Hsia,et al.  On least squares algorithms for system parameter identification , 1976 .

[25]  I. Introductiok.  On the Estimation of the Moving-Average Parameters , 1972 .

[26]  Tien C. Hsia,et al.  System identification: Least-squares methods , 1977 .

[27]  L. Scharf,et al.  Statistical design of autoregressive-moving average digital filters , 1979 .