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**Nonstationary data analysis by time deformation.**
*(English)*
Zbl 1062.62211

Summary: We discuss methodology for analyzing nonstationary 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 contents 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 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 time-varying 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 article can be extended to processes whose frequencies change like \(at^{\alpha}\), \(a > 0\), \(-\infty < \alpha <-\infty\).

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 time-varying 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 article can be extended to processes whose frequencies change like \(at^{\alpha}\), \(a > 0\), \(-\infty < \alpha <-\infty\).

### MSC:

62M15 | Inference from stochastic processes and spectral analysis |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62P35 | Applications of statistics to physics |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

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\textit{H. L. Gray} et al., Commun. Stat., Theory Methods 34, No. 1, 163--192 (2005; Zbl 1062.62211)

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### References:

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[8] | Liu , L. , Gray , H. L. , Woodward , W. A. ( 2004 ). On the analysis of linear and quadratic chirp processes using time deformation. Southern Methodist University Department of Statistical Science Technical Report SMU-TR-317. |

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[10] | Vijverberg , C. P. , Gray , H. L. ( 2003 ). Time deformation and the M-spectrum—a new tool for cyclical analysis. Southern Methodist University Department of Statistical Science Technical Report SMU-TR-314. |

[11] | DOI: 10.1109/10.871405 |

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