Non-Gaussian State—Space Modeling of Nonstationary Time Series

Abstract A non-Gaussian state—space approach to the modeling of nonstationary time series is shown. The model is expressed in state—space form, where the system noise and the observational noise are not necessarily Gaussian. Recursive formulas of prediction, filtering, and smoothing for the state estimation and identification of the non-Gaussian state—space model are given. Also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas. Significant merits of non-Gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples. A typical application of this non-Gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes. Simple Gaussian state—space modeling is not adequate for this situation. Here the model with small system noise variance cannot detect jump, whereas the one with large system noise variance yields unfavorable wiggle. To work...

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

[2]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[3]  Genshiro Kitagawa,et al.  An approach to the prediction of time series with trends and seasonalities , 1982, 1982 21st IEEE Conference on Decision and Control.

[4]  Edmund Taylor Whittaker.  On a New Method of Graduation , 1922, Proceedings of the Edinburgh Mathematical Society.

[5]  G. Kitagawa,et al.  A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series , 1985, IEEE Transactions on Automatic Control.

[6]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[7]  M. Ishiguro,et al.  A bayesian approach to binary response curve estimation , 1983 .

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

[9]  R. Bucy,et al.  Digital synthesis of non-linear filters , 1971 .

[10]  H. Tong.  A note on using threshold autoregressive models for multi-step-ahead prediction of cyclical data , 1982 .

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

[12]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[13]  Hirotugu Akaike,et al.  Trend estimation with missing observations , 1980 .

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

[15]  G. Kitagawa,et al.  A Smoothness Priors–State Space Modeling of Time Series with Trend and Seasonality , 1984 .

[16]  Genshiro Kitagawa,et al.  Extraction of signal by a time series model and screening out micro earthquakes , 1985 .

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

[18]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[19]  A. M. Walker,et al.  A Survey of statistical work on the Mackenzie River series of annual Canadian lynx trappings for the , 1977 .

[20]  Hirotugu Akaike,et al.  Likelihood and the Bayes procedure , 1980 .