LMS-like AR modeling in the case of missing observations

This paper presents a new recursive algorithm for the time domain reconstruction and spectral estimation of uniformly sampled signals with missing observations. An autoregressive (AR) modeling approach is adopted. The AR parameters are estimated by optimizing a mean-square error criterion. The optimum is reached by means of a gradient method adapted to the nonperiodic sampling. The time-domain reconstruction is based on the signal prediction using the estimated model. The power spectral density is obtained using the estimated AR parameters. The development of the different steps of the algorithm is discussed in detail, and several examples are presented to demonstrate the practical results that can be obtained. The spectral estimates are compared with those obtained by known AR estimators applied to the same signals sampled periodically. We note that this algorithm can also be used in the case of nonstationary signals.

[1]  Yonina Rosen,et al.  Optimal ARMA parameter estimation based on the sample covariances for data with missing observations , 1989, IEEE Trans. Inf. Theory.

[2]  Emanuel Parzen.  Time Series Analysis of Irregularly Observed Data , 1984 .

[3]  P. Robinson,et al.  Estimation of Time Series Models in the Presence of Missing Data , 1981 .

[4]  Emanuel Parzen,et al.  ON SPECTRAL ANALYSIS WITH MISSING OBSERVATIONS AND AMPLITUDE MODULATION , 1962 .

[5]  William T. M. Dunsmuir,et al.  Asymptotic theory for time series containing missing and amplitude modulated observations , 1981 .

[6]  H. Sakai.  Fitting autoregression with regularly missed observations , 1980 .

[7]  E. Masry,et al.  Spectral estimation of continuous-time processes: Performance comparison between periodic and Poisson sampling schemes , 1978 .

[8]  Richard H. Jones.  FITTING A CONTINUOUS TIME AUTOREGRESSION TO DISCRETE DATA , 1981 .

[9]  W. M. Carey,et al.  Digital spectral analysis: with applications , 1986 .

[10]  P. Robinson,et al.  Estimation of a time series model from unequally spaced data , 1977 .

[11]  K. M. Tao.  Statistical averaging and PARTAN-some alternatives to LMS and RLS , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[12]  Frederick J. Beutler,et al.  Alias-free randomly timed sampling of stochastic processes , 1970, IEEE Trans. Inf. Theory.

[13]  Elias Masry,et al.  A Consistent Estimate of the Spectrum by Random Sampling of the Time Series , 1975 .

[14]  E. Masry,et al.  Discrete-time spectral estimation of continuous-parameter processes - A new consistent estimate , 1976, IEEE Trans. Inf. Theory.

[15]  Richard H. Jones,et al.  Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations , 1980 .

[16]  Elias Masry,et al.  Random Sampling and Reconstruction of Spectra , 1971, Inf. Control..

[17]  Elias Masry,et al.  Poisson sampling and spectral estimation of continuous-time processes , 1978, IEEE Trans. Inf. Theory.