Centre for Research on Environmental Systems and Statistics Captain Toolbox

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Lancaster University
Systems and Control
Environmental Science

Analysis of a signal with sawtooth changing frequency.

Dynamic Autoregression (DAR) is particularly useful for evaluating signal spectra and time frequency analysis, since it provides the AR spectrum at each point in time, based on the locally optimum time variable AR parameters. In this example, we utilise the Captain Toolbox to analyse an artifical signal with sawtooth changing frequency.

Signal with sawtooth changing frequency.

We will fit a 2nd order autoregression model, with the two time variable parameters described by an integrated random walk. The first stage of the analysis is to optimise the Noise Variance Ratio (NVR) hyper-parameters of the model. In the Captain Toolbox, these parameters are optimised via Maximum Likelihood (ML) based on prediction error decomposition. Finally, Kalman Filtering and Fixed Interval Smoothing algorithms are employed to estimate the model, based on the optimsed NVR values. This is achieved with a straightforward call to the Captain Toolbox dar function. As illustrated in the graph below, the first AR parameter is relatively constant, while the second parameter varies over time to account for the changing frequency of the series. Finally, the analysis yields the time varying AR spectrum shown in the 3d plot.

Estimated parameters.

Estimated AR spectra plotted against time.

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c.taylor@lancaster.ac.uk 23-FEB-2011