State Space Modeling of Nonstationary Time Series and Smoothing of Unequally Spaced Data

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A smoothness prior and state space approach to the modeling of nonstationary or irregular time series is shown and various applications obtained so far are reviewed. The smooth change of parameters that characterizes the stochastic structure of the time series is modeled by stochastic linear difference or differential equation. A simple state space representation of the overall process is derived to facilitate the Kalman filter methodology for state estimation. The integrated likelihood of the model is used to determine the parameters contained in the state space model. Given the best choice of parameters that control the smoothness of the structral change, the smoothing algorithm then yields the estimates of the state vector. Detrending, seasonal adjustment, estimation of gradually changing spectrum, decomposition of time series into several series and outlier problem are shown as examples of the use of this approach. Discrete representation of the sampled continuous process is also shown and smoothing of unequally spaced data is shown for illustration.