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.
In recent years the wavelet coherence measure has been used to analyse
correlations between a pair of processes in the time-scale domain, typically in
hypothesis testing scenarios, but it has proven resistant to analytic study with
resort to simulations for statistical properties. As part of the null hypothesis
being tested, such simulations invariably assume joint stationarity of the
series. In this thesis two methods of calculating wavelet coherence have been
developed and distributional properties of the wavelet coherence estimators
have been fully derived.
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. Since both sets of wavelets are complex-valued, we consider the case of
wavelet coherence calculated from discrete-time complex-valued and stationary
time series. Under Gaussianity, the Goodman distribution is shown, for large
samples, to be appropriate for wavelet coherence. The true wavelet coherence
value is identified in terms of its frequency domain equivalent and degrees
of freedom can be readily derived. The theoretical results are verified via
simulations.
The notion of a spectral function is considered for the nonstationary case.
Particular focus is given to Priestley’s evolutionary process and a Wold-Cramér
nonstationary representation where time-varying spectral functions can be
clearly defined. Methods of estimating these spectra are discussed, including
the continuous wavelet transform, which when performed with a Morlet
wavelet and temporal smoothing is shown to bear close resemblance to Priestley’s
own estimation procedure.
The concept of coherence for bivariate evolutionary nonstationary processes
is discussed in detail. In such situations it can be shown that the coherence
function, as in the stationary case, is invariant of time. It is shown that
for spectra that vary slowly in time the derived statistics of the temporally
smoothed wavelet coherence estimator are appropriate. Further to this the
similarities with Priestleys spectral estimator are exploited to derive distributional
properties of the corresponding Priestley coherence estimator.
A well known class of the evolutionary and Wold-Cramér nonstationary
processes are the modulated stationary processes. Using these it is shown that
bivariate processes can be constructed that exhibit coherence variation with
time, frequency, and time-and-frequency. The temporally smoothed Morlet
wavelet coherence estimator is applied to these processes. It is shown that
accurate coherence estimates can be achieved for each type of coherence, and
that the distributional properties derived under stationarity are applicable.