A Statistical Study of Temporally Smoothed Wavelet Coherence

• Published in: IEEE transactions on signal processing Vol. 58; no. 6; pp. 2964 - 2973
• Main Authors: Cohen, E A K, Walden, A T
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Asymptotic properties; Coherence; Continuous wavelet transforms; Estimators; Exact sciences and technology; Goodman's distribution; Information, signal and communications theory; Mathematical analysis; Miscellaneous; Morlet wavelet; Physics; Signal processing; Smoothing; Smoothing methods; Telecommunications and information theory; temporal smoothing; Testing; Time domain analysis; Time series; Time series analysis; Wavelet; Wavelet analysis; wavelet coherence; Wavelet domain; Wavelet transforms; WOSA; Averaging method; Time series; Theoretical study; wavelet coherence; WOSA; Complex variable method; Statistical method; Hypothesis test; temporal smoothing; Time domain method; Simulation; Morlet wavelet; Coherence; Analytical method; Signal processing; Goodman's distribution
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2010.2043139

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.