Nonstationary Data Analysis by Time Deformation

In this paper we discuss methodology for analyzing non-stationary time series whose periodic nature changes approximately linearly with time. We make use of the M-stationary process to describe such data sets, and in particular we use the discrete Euler(p) model to obtain forecasts and estimate the spectral characteristics. We discuss the use of the M-spectrum for displaying linear time-varying periodic content in a time series realization in much the same way that the spectrum shows periodic content within a realization of a stationary series. We also introduce the instantaneous frequency and instantaneous spectrum of an M-stationary process for purposes of describing how frequency changes with time. To illustrate our techniques we use one simulated data set and two bat echolocation signals that show time varying frequency behavior. Our results indicate that for data whose periodic content is changing approximately linearly in time, the Euler model serves as a very good model for spectral analysis, filtering and forecasting. Additionally, the instantaneous spectrum is shown to provide better representation of the timevarying frequency content in the data than window-based techniques such as the Gabor and wavelet transforms. Finally, it is noted that the results of this paper can be extended to processes whose frequencies change like , 0, . at a α α > −∞ < <−∞