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Abstract
Wavelet transform, wavelet spectra, and coherence are popular tools for studying fluctuations in time series in the form of a bidimensional time and scale representation. We discuss two aspects of wavelet analysis—namely the significance and stochastic/deterministic character of the wavelet spectra. Real-time series of discharge, sodium, and sulfate concentrations in the alpine Rhône River, Switzerland, are used to illustrate these issues. First, the consequences of using an arbitrary stochastic process (usually, AR (1)) instead of the best-fitted general ARMA process in the evaluation of the significance of wavelet spectra are analyzed. Using a general ARMA instead of AR (1) decreases the significance level of the differences in wavelet power spectra (WPS) of ARMA and AR (1) compared to the WPS of the time series in all cases studied and points to a possible systematic overestimation of significance in many published studies. Besides, the significance of particular patches in the spectra is affected by multiple testing. A (conservative) way to circumvent this problem, using global wavelet spectra and global coherence spectra, is evaluated. Finally, we discuss the issue of causality and investigated it in the three measured time series mentioned above. Even if the use of the best fitted ARMA pointed to no deterministic features being present in the corrected series studied (i.e., stochastic processes are dominant in the three data series), coherence spectra between variables allowed to reveal cause-effect relationships between two “coherent” variables and/or the existence of a common effect on both variables. Therefore, such type of analysis provides a useful tool to better understand data causal relationships.
Highlights
When studying time series, it is usual to perform a spectral analysis (e.g., Fourier transform) alongside the time analysis
Even if the use of the best fitted ARMA pointed to no deterministic features being present in the corrected series studied, coherence spectra between variables allowed to reveal cause-effect relationships between two “coherent” variables and/or the existence of a common effect on both variables
Our results clearly show that very differentprocess conclusions concerning can beMoreover, obtained using more realistic processes instead of gives less significant results in all the systems in wavelet analysis depending on the stochastic process chosen to derive that significance
Summary
It is usual to perform a spectral (frequency) analysis (e.g., Fourier transform) alongside the time analysis. Significance in WCSs can be defined in the same way, that is, adjusting an AR (1) process to each of the two series we are comparing, obtaining a distribution of coherences at each point by using pairs of realizations of both processes, and comparing the coherence of the two-original series’ with the distribution of coherence at each point This is the method used in three of the four software packages cited above. Once the most suitable stochastic model is selected, it is possible to apply a significance test at each (time, scale) point of the WPS to look for deterministic effects. We explore the consequences of failing to use the most suitable stochastic process when assessing WPS and WCS significance through the evaluation of real-time series of discharge, sodium, and sulfate in the alpine catchment of Rhône River in Switzerland. The related question of deterministic effects (causality) on those time series is addressed by studying the global coherence spectra (GCSs)
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