A Statistical Study of Temporally Smoothed
Edward A. K. Cohen, Andrew T. Walden · 2010
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.