Rapid surrogate testing of wavelet coherences

Background. The use of wavelet coherence methods enables the identification of frequency-dependent relationships between the phases of the fluctuations found in complex systems such as medical and other biological timeseries. These relationships may illuminate the causal mechanisms that relate the variables under investigation. However, computationally intensive statistical testing is required to ensure that apparent phase relationships are statistically significant, taking into account the tendency for spurious phase relationships to manifest in short stretches of data. Methods. In this study we revisit Fourier transform based methods for generating surrogate data, with which we sample the distribution of coherence values associated with the null hypothesis that no actual phase relationship between the variables exists. The properties of this distribution depend on the cross-spectrum of the data. By describing the dependency, we demonstrate how large numbers of values from this distribution can be rapidly generated without the need to generate correspondingly many wavelet transforms. Results. As a demonstration of the technique, we apply the efficient testing methodology to a complex biological system consisting of population timeseries for planktonic organisms in a food web, and certain environmental drivers. A large number of frequency dependent phase relationships are found between these variables, and our algorithm efficiently determines the probability of each arising under the null hypothesis, given the length and properties of the data. Conclusion. Proper accounting of how bias and wavelet coherence values arise from cross spectral properties provides a better understanding of the expected results under the null hypothesis. Our new technique enables enormously faster significance testing of wavelet coherence.