Significance tests for the wavelet power and the wavelet power spectrum

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.

FPGA Based Serial and Single-Clock Cycle Pipelined Fast Fourier Transforms in a Radio Detection of Cosmic Rays

“…The global wavelet spectrum, defined as the time average over a series of p-wavelet powers, can be expressed as [8]:…”

Significance tests for the wavelet cross spectrum and wavelet linear coherence

“…with δt being the sampling period or the reciprocal of the sampling frequency F s (Ge, 2007), their product…”

“…Torrence and Webster, 1999;Grinsted et al, 2004;Lundstedt et al, 2006;Rigozo et al, 2008) in the past ten years, especially in the fields of geophysics and engineering. Ge (2007) obtained analytical results of the sampling distributions of the wavelet power and wavelet power spectrum using prescribed background noise series. That was an improvement on TC98's results a large portion of which was based on Monte Carlo simulations.…”

“…The null hypothesis (H 0 ) in the present work will be based on two independent Gaussian White Noise (GWN) time series and the associated sampling distributions of the interested wavelet-based quantities (statistics), which is a conventional approach in many statistical studies (e.g. Jenkins and Watts, 1969;TC98;Ge, 2007). In this approach, it is important to bear two points in mind.…”

“…That was an improvement on TC98's results a large portion of which was based on Monte Carlo simulations. The present work was aimed to be a sequel to Ge (2007), continuing to develop significance tests for two other frequently used quantities (statistics), the wavelet cross spectrum and the wavelet linear coherence. Involving two time series instead of only one, these two wavelet-based quantities are often used to assess time-varying spectral relations between two signals, which are often nonstationary.…”