Estimation of the significance of the Foster’s wavelet spectrum by means of a permutation test and its application for paleoclimate records

Abstract

Here we propose a permutation test -a non-parametric computing-intensive test- to evaluate the statistical significance of the Foster’s wavelet spectrum by means of Monte Carlo simulations. A procedure (algorithm) is introduced in order to carry out this aim. The Foster’s wavelet is an adequate method to cope directly with unevenly spaced paleoclimatic time series. We have conducted time series simulations to study the performance of the Foster’s wavelet spectrum and applied the permutation test to localize periodic signals known a priori by randomly increasing the fraction of missing data (from 25% to 75% of the total amount of data of an evenly spaced time series). We found that the periodic signals are progressively lost as larger amounts of data are removed. This loss becomes noticeable at circa 50% of the missing data and gets more evident when 75% of the data is removed. The signal loss is more pronounced in the high-frequency range, due to an intrinsic bias in the wavelet spectrum. Notwithstanding, the Foster’s wavelet spectrum and the permutation test succeeded in locating the periodic signals, even with a moderate amount of missing data. Finally, we applied our procedure to the oxygen isotope ratio (δ18O) data of the GISP2 deep ice core (Greenland) and we were able to detect the main spectral signature of the unevenly spaced time series of the GISP2 δ18O record, i.e., the spectral peak around 1,470 years.