Tips

Cross wavelet analysis and wavelet coherence are powerful methods for testing proposed linkages between two time series.

• Check the histograms of the time series to ensure that they are not too far from normally distributed. Consider transforming the time series, if the pdf's of the time series are far from Gaussian. When choosing a transformation, it is preferable to choose an analytic transformation such as taking the logarithm if the data is log-normally distributed. In other cases a simple 'percentile' transformation might be useful. An advantage of using that particular transformation is that it does not have any outliers.
• Consider what the expectations are for the outcome of the analysis given the proposed linking mechanism. We caution against blindly applying these methods to randomly chosen data sets. Like other statistical tests some data set sets will display highly statistically significant links simply by chance.
• When a wavelet has been chosen the CWTs of both time series are calculated. We suggest a scale resolution of 10 scales per octave and use of the Morlet wavelet unless there are good grounds to do otherwise. For geophysical time series an AR1 red noise assumption is often suitable and can be used to calculate the significance level of the wavelet power. Remember to take special care not to misinterpret results inside the COI.
• From the two CWTs the XWT is calculated. The XWT exposes regions with high common power and further reveals information about the phase relationship. If the two series are physically related we would expect a consistent or slowly varying phase lag that can be tested against mechanistic models of the physical process. The circular mean of the phase angles can be used to quantify the phase relationship.
• Also, from two CWTs the WTC can be calculated which can be thought of as the local correlation between the time series in time frequency space. Where XWT unveils high common power, WTC finds locally phase locked behavior. The more desirable features of the WTC come at the price of being slightly less localized in time frequency space. The significance level of the WTC has to be determined using Monte Carlo methods.