Bayesian modeling and significant features exploration in wavelet power spectra

This study proposes and justifies a Bayesian ap- proach to modeling wavelet coefficients and finding statis- tically significant features in wavelet power spectra. The approach utilizes ideas elaborated in scale-space smoothing methods and wavelet data analysis. We treat each scale of the discrete wavelet decomposition as a sequence of independent random variables and then apply Bayes' rule for construct- ing the posterior distribution of the smoothed wavelet coef- ficients. Samples drawn from the posterior are subsequently used for finding the estimate of the true wavelet spectrum at each scale. The method offers two different significance test- ing procedures for wavelet spectra. A traditional approach assesses the statistical significance against a red noise back- ground. The second procedure tests for homoscedasticity of the wavelet power assessing whether the spectrum deriva- tive significantly differs from zero at each particular point of the spectrum. Case studies with simulated data and climatic time-series prove the method to be a potentially useful tool in data analysis.