When analysing time series it is often useful to consider the frequency domain (the spectrum) where underlying oscillations driving the process become apparent and can be attached to some physical meaning. The coherence function measures the correlation between a pair of random processes in the frequency domain. It is a well studied and understood concept, and the distributional properties of conventional coherence estimators for stationary processes have been derived and applied in a number of physical settings. Such concepts only have theoretical grounding when dealing with (second-order) stationary processes. In recent years the wavelet coherence measure has been used to analyse correlations in the time-scale/frequency domain between a pair of processes that may be exhibiting non-stationary behaviour, typically in hypothesis testing scenarios. In order to obtain meaningful estimates of the wavelet coherence smoothing procedures need to be applied to the individual wavelet spectra. Understanding how to smooth spectra and the repercussions these smoothing procedures have on the distribution of the wavelet coherence estimator under common null models has until now been unobtainable.
In this talk, after an introduction to spectral analysis of time series and wavelet coherence, I will present two methods of calculating the wavelet coherence estimator that are amenable to statistical analysis. With the first method, in an analogous framework to multitapering, wavelet coherence is estimated using multiple orthogonal Morse wavelets. The second coherence estimator proposed uses time-domain smoothing and a single Morlet wavelet. For both methods it will be shown how distributions for the wavelet coherence estimator can be calculated and how they can be used to test for correlation between a pair of processes. Extending these wavelet methods to point processes is ongoing research that will be briefly discussed.