In mathematics and statistical analysis, bicoherence (also known as bispectral coherency) is a squared normalised version of the bispectrum.
The bicoherence takes values bounded between 0 and 1, which make it a convenient measure for quantifying the extent of phase coupling in a signal.
The prefix bi- in bispectrum and bicoherence refers not to two time series xt, yt but rather to two frequencies of a single signal.
The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum.
The Fourier transform of C3(t1,t2) (third-order cumulant) is called bispectrum or bispectral density.
They fall in the category of Higher Order Spectra, or Polyspectra and provide supplementary information to the power spectrum.
The coherence function provides a quantification of deviations from linearity in the system which lies between the input and output measurement sensors.
The bicoherence measures the proportion of the signal energy at any bifrequency that is quadratically phase coupled.
It is usually normalized in the range similar to correlation coefficient and classical (second order) coherence.
It was also used for depth of anasthesia assessment and widely in plasma physics (nonlinear energy transfer) and also for detection of gravitational waves.
Bispectrum and bicoherence may be applied to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension.
[1] Bicoherence measurements have been carried out for EEG signals monitoring in sleep, wakefulness and seizures.
Detecting phase coupling requires summation over a number of independent samples- this is the first motivation for defining the bicoherence.
Finally, one of the most intuitive definitions comes from Hagihira 2001 and Hayashi 2007, which is The numerator contains the magnitude of the bispectrum summed over all of the time series segments.
Among the three normalizations above, the second one can be interpreted as a correlation coefficient defined between energy-supplying and energy-receiving parties in a second order nonlinear interaction, whereas the bispectrum has been proven to be the corresponding covariance.