Wavelet Coherence Method

The coherence function is a direct measure of thecorrelation between the spectra of two random processes. Fourier method can provide accurate estimates of stationary signals. However, Most signals ( including our signals from brains) are non-stationary waveforms.

Wavelet coherence method was first used in physics to estimate interactions among non-stationary signals, now it has been applied in investigating neuroelectric waveforms such as EEG signals.

First of all, signals are decomposed along the Morlet wavelet family, the advantage of which is that it is simple and well suited for spectral estimations. It is defined for frequency f and time τ by:

The wavelet transform of a signal x(u) is a function of time (τ) and frequency ( f ) given by the convolution of x with this wavelet family:


From the wavelet transforms of two signals x and y, we can define the wavelet cross-spectrum between x and y around time t and frequency f

Whereδis an important parameter which depends on the frequency

Finally, analogous to the Fourier-based coherence, the wavelet coherence WCo(t, f) is defined at time t and frequency f by:

WCo(t, f) takes its values between 0 and 1, the bigger this value, the more dependence between x and y around time t and frequency f.

NOTE: It is a brief introduction of Wavelet Coherence Method. For further information about this method, please refer to relevant papers.

References of this article:
Lachaux J-P, Lutz A, Rudrauf D, Cosmelli D, Quyen MLV, Martinerie J,Varela F. Estimating the time-course of coherence between single-trial brain signals: an introduction to wavelet coherence. Neurophysiol Clin., 2002, 32: 157-174.
Li X, Yao X, G J, Jefferys,Fox J. Interaction Dynamics of Neuronal Oscillations Analysed Using Wavelet Transforms. J. Neurosci. Methods, 2007, 160(1): 178-185.