2010b), A Statistical Study of Temporally Smoothed Wavelet Coherence

Abstract—The use of the wavelet coherence of two series in hy-pothesis testing relies on some sort of smoothing being carried out in order that the coherence estimator is not simply unity. A pre-vious 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 calcu-lated 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) spec-trum estimators, and by using multitaper equivalent representa-tions for the WOSA estimators we show that Goodman’s distribu-tion is appropriate asymptotically, and readily derive the appro-priate degrees of freedom. The theoretical results are verified via simulations and illustrated using solar physics data. Index Terms—Coherence, Goodman’s distribution, Morlet wavelet, temporal smoothing, wavelet coherence, WOSA