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Annales Geophysicae An interactive open-access journal of the European Geosciences Union
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Volume 25, issue 11
Ann. Geophys., 25, 2259–2269, 2007
https://doi.org/10.5194/angeo-25-2259-2007
© Author(s) 2007. This work is distributed under
the Creative Commons Attribution 3.0 License.
Ann. Geophys., 25, 2259–2269, 2007
https://doi.org/10.5194/angeo-25-2259-2007
© Author(s) 2007. This work is distributed under
the Creative Commons Attribution 3.0 License.

  29 Nov 2007

29 Nov 2007

Significance tests for the wavelet power and the wavelet power spectrum

Z. Ge Z. Ge
  • Research associate of the National Research Council; Ecosystems Research Division, NERL, USEPA, 960 College Station Road, Athens, Georgia 30605, USA

Abstract. Significance tests usually address the issue how to distinguish statistically significant results from those due to pure randomness when only one sample of the population is studied. This issue is also important when the results obtained using the wavelet analysis are to be interpreted. Torrence and Compo (1998) is one of the earliest works that has systematically discussed this problem. Their results, however, were based on Monte Carlo simulations, and hence, failed to unveil many interesting and important properties of the wavelet analysis. In the present work, the sampling distributions of the wavelet power and power spectrum of a Gaussian White Noise (GWN) were derived in a rigorous statistical framework, through which the significance tests for these two fundamental quantities in the wavelet analysis were established. It was found that the results given by Torrence and Compo (1998) are numerically accurate when adjusted by a factor of the sampling period, while some of their statements require reassessment. More importantly, the sampling distribution of the wavelet power spectrum of a GWN was found to be highly dependent on the local covariance structure of the wavelets, a fact that makes the significance levels intimately related to the specific wavelet family. In addition to simulated signals, the significance tests developed in this work were demonstrated on an actual wave elevation time series observed from a buoy on Lake Michigan. In this simple application in geophysics, we showed how proper significance tests helped to sort out physically meaningful peaks from those created by random noise. The derivations in the present work can be readily extended to other wavelet-based quantities or analyses using other wavelet families.

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