The application of the Kalman filter to nonstationary time series through time deformation

Abstract

An increasingly valuable tool for modelling a nonstationary time series, X(t), is time deformation. In this procedure time, t, is transformed to a 'time' scale, u = g(t), on which the process Y(u) = X(g(t)) is stationary. However, since the time scale is transformed, equally spaced data on the original time scale data become unequally spaced data in transformed time. In practice interpolation in the original time scale is currently used to obtain equally spaced data in transformed time which can then be modelled using the classical autoregressive moving average modelling techniques. In this article, the need for interpolation is eliminated by employing the continuous time autoregressive model and estimating the parameters using the Kalman filter. The resulting improvements include more accurate estimation of the spectrum, and the separation of the data into its time-varying latent components. The technique is applied to simulated and real data for illustrations. © 2009 Blackwell Publishing Ltd.