A Statistical Study of Temporally Smoothed Wavelet Coherence

The use of the wavelet coherence of two series in hypothesis testing relies on some sort of smoothing being carried out in order that the coherence estimator is not simply unity. A previous study considered averaging via the use of multiple Morse wavelets. Here we consider time-domain smoothing and use of a single Morlet wavelet. Since the Morlet wavelet is complex-valued, we derive analytic results for the case of wavelet coherence calculated from complex-valued, jointly stationary and Gaussian time series. The temporally smoothed wavelet coherence can be written in terms of Welch's overlapping segment averaging (WOSA) spectrum estimators, and by using multitaper equivalent representations for the WOSA estimators we show that Goodman's distribution is appropriate asymptotically, and readily derive the appropriate degrees of freedom. The theoretical results are verified via simulations and illustrated using solar physics data.