A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series

A smoothness priors time varying AR coefficient model approach for the modeling of nonstationary in the covariance time series is shown. Smoothness priors in the form of a difference equation constraint excited by an independent white noise are imposed on each AR coefficient. The unknown white noise variances are hyperparameters of the AR coefficient distribution. The critical computation is of the likelihood of the hyperparameters of the Bayesian model. This computation is facilitated by a state-space representation Kalman filter implementation. The best difference equation order-best AR model order-best hyperparameter model locally in time is selected using the minimum AIC method. Also, an instantaneous spectral density is defined in terms of the instantaneous AR model coefficients and a smoothed estimate of the instantaneous time series variance. An earthquake record is analyzed. The changing spectral analysis of the original data and of simulations from a time varying AR coefficient model of that data are shown.

[1]  C. Page.  Instantaneous Power Spectra , 1952 .

[2]  P. Whittle.  Recursive Relations for Predictors of Non‐Stationary Processes , 1965 .

[3]  M. Priestley.  Evolutionary Spectra and Non‐Stationary Processes , 1965 .

[4]  M. B. Priestley,et al.  Power spectral analysis of non-stationary random processes , 1967 .

[5]  G. Wahba,et al.  A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines , 1970 .

[6]  K. Åström,et al.  Problems of Identification and Control , 1971 .

[7]  D. Lindley,et al.  Bayes Estimates for the Linear Model , 1972 .

[8]  R. Shiller.  A DISTRIBUTED LAG ESTIMATOR DERIVED FROM SMOOTHNESS PRIORS , 1973 .

[9]  Peter A. Tinsley,et al.  Linear prediction and estimation methods for regression models with stationary stochastic coefficients , 1980 .

[10]  T. Bohlin.  Four Cases of Identification of Changing Systems , 1976 .

[11]  D. Tj⊘stheim.  Spectral generating operators for non-stationary processes , 1976, Advances in Applied Probability.

[12]  Genshiro Kitagawa,et al.  A procedure for the modeling of non-stationary time series , 1978 .

[13]  R. Shibata.  Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process , 1980 .

[14]  F. Kozin,et al.  The order determination problem for linear time-varying AR models , 1980 .

[15]  G. Wahba.  Automatic Smoothing of the Log Periodogram , 1980 .

[16]  B. G. Quinn,et al.  THE ESTIMATION OF RANDOM COEFFICIENT AUTOREGRESSIVE MODELS. I , 1980 .

[17]  T. Kailath,et al.  Ladder-form filters for nonstationary processes , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[18]  George E. P. Box,et al.  Sampling and Bayes' inference in scientific modelling and robustness , 1980 .

[19]  I. Good,et al.  Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data , 1980 .

[20]  R. Shibata.  An optimal selection of regression variables , 1981 .

[21]  Will Gersch,et al.  A data analytic approach to the smoothing problem and some of its variations , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[22]  Genshiro Kitagawa,et al.  A NONSTATIONARY TIME SERIES MODEL AND ITS FITTING BY A RECURSIVE FILTER , 1981 .

[23]  Genshiro Kitagawa,et al.  An approach to the prediction of time series with trends and seasonalities , 1982, CDC 1982.

[24]  Graham Goodwin,et al.  Convergence and properties of the solutions of the Ricatti difference equation , 1982, 1982 21st IEEE Conference on Decision and Control.

[25]  W. Gersch,et al.  Estimation of stationary structural system parameters from non-stationary random vibration data: A locally stationary model method , 1982 .

[26]  C. Morris.  Parametric Empirical Bayes Inference: Theory and Applications , 1983 .

[27]  Carl N. Morris,et al.  Parametric Empirical Bayes Inference: Theory and Applications: Rejoinder , 1983 .

[28]  Simon Braun,et al.  Nearest Neighbor-Time Series Analysis Classification of Faults in Rotating Machinery , 1983 .

[29]  G. Kitagawa.  Changing spectrum estimation , 1983 .

[30]  W. Gersch,et al.  A smoothness priors long AR model method for spectral estimation , 1985, IEEE Transactions on Automatic Control.