Abstract
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.
Highlights
After two decades of fast development, the wavelet analysis has become a powerful and effective tool for analysing nonlinear and especially nonstationary time series in many disciplines (Addison, 2002)
TC98 is one of the earliest works that gives a guide for conducting significance tests for the wavelet power, auto, and cross-spectrum, which is pertinent to the present work
For brevity, only continuous wavelet transform (CWT) using the Morlet wavelet is considered in the present paper, the methodologies described here can be readily extended to the discrete wavelet transform and the wavelet analysis based on other wavelet families, such as the Mexican hat and the derivative of a Gaussian (DOG) wavelets
Summary
After two decades of fast development, the wavelet analysis has become a powerful and effective tool for analysing nonlinear and especially nonstationary time series in many disciplines (Addison, 2002). The implementation of the wavelet analysis has become easy for many researchers, owing to the work of Torrence and Compo (1998) (referred to TC98 hereafter) In this widely-acknowledged paper, which has been cited over 1000 times as of June of 2007 (ISI Web of knowledge), many practical issues for the wavelet analysis are discussed, accompanied by source codes in Fortran and Matlab posted on their website (http://atoc.colorado.edu/ research/wavelets/). The significance test for the wavelet analysis is undoubtedly important due to the simple fact that there always seem to be some patterns (e.g. peaks) in the wavelet scalogram even if the analysed signal is pure noise. In this case, a bottom line must be drawn below which no conclusion can be made based on the results. For brevity, only CWT using the Morlet wavelet is considered in the present paper, the methodologies described here can be readily extended to the discrete wavelet transform and the wavelet analysis based on other wavelet families, such as the Mexican hat and the derivative of a Gaussian (DOG) wavelets
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