Why & how to rectify the bias in the wavelet power spectrum?
For a time series comprised of sine waves with the same amplitude but different frequencies the widely adopted wavelet method [e.g., Torrence and Compo, 1998] does not produce a spectrum with identical peaks (see the middle panels of the figure to the right and the FAQs of the wavelet toolbox), in contrast to a Fourier analysis. This wavelet bias problem is addressed in Liu et al. [2007]. It is demonstrated that a physically consistent definition of energy for the wavelet power spectrum should be the transform coefficient squared divided by the scale it associates. Thus, a bias rectification is proposed, i.e., the wavelet power spectrum should be divided by its scales.
Such adjusted wavelet power spectrum results in a substantial improvement in the spectral estimate (see the bottom panels of the figure to the left), allowing for a comparison of the spectral peaks across the scales/frequencies/periods. This rectification is validated with an artificial time series and a real coastal sea level record at St. Petersburg, Florida.
Also re-examined is the example of the wavelet analysis of the Niño-3 SST data. Compared to the original wavelet spectrum, there is a slight adjustment of the relative magnitude of the wavelet spectrum across the frequency domain in the rectified spectrum. The adjusted wavelet spectrum overlays better with the 95% significance contours. The spectral line of the time averaged spectrum also becomes smoother. However, the major peak is still located between the 2- and 8-yr periods. Thus the main conclusion on the Niño-3 SST wavelet analysis in Torrence and Compo [1998] is still correct. That may be the main reason why the bias problem had been overlooked for decades. For details, see Liu et al. [2007].
References:
- Liu, Y., X.S. Liang, and R.H. Weisberg, 2007: Rectification of the bias in the wavelet power spectrum. Journal of Atmospheric and Oceanic Technology, 24(12), 2093-2102, doi:10.1175/2007JTECHO511.1.
- Torrence, C., and G.P. Compo, 1998: A practical guide to wavelet analysis. Bulletin of American Meteorological Society, 79, 61-78. (The wavelet toolbox).
Artificial time series (top) comprising of five sine waves, with a unit amplitude and five different periods (1, 8, 32, 128, and 365 days). The original (middle) and rectified (bottom) wavelet power spectra (left column) and time-averaged wavelet power spectra (right column) of the artificial time series. Red and blue indicate high and low wavelet power spectra, respectively. (Addapted from Figure 2 of Liu et al. [2007])
