In mathematics, in the area of statistical analysis, bicoherence 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. It is also known as bispectral coherency. 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 bispectrum is a statistic used to search for nonlinear interactions. 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 third order polyspectrum (bispectrum) is the easiest to compute, and hence the most popular.
The difference with measuring coherence (coherence analysis is an extensively used method to study the correlations in frequency domain, between two simultaneously measured signals) is the need for both input and output measurements by estimating two auto-spectra and one cross spectrum. On the other hand, bicoherence is an auto-quantity, i.e. it can be computed from a single signal. 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 assesement and widely in plasma physics (nonlinear energy transfer) and also for detection of gravitation 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.
Definition
The bispectrum is defined as the triple product
where
Suppose that the three Fourier components
There is some inconsistency with the definition of the bicoherence normalization constant. Some of the definitions that have been used are
which was provided in Sigl and Chamoun 1994, but does not appear to be correctly normalized. Alternatively, plasma physics typically uses
where the angle brackets denote averaging. Note that this is the same as using a sum, because
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. This quantity is large if there is phase coupling, and approaches 0 in the limit of random phases. The denominator, which normalizes the bispectrum, is given by calculating the bispectrum after setting all of the phases to 0. This corresponds to the case where there is perfect phase coupling, because all of the samples have zero phase. Therefore, the bicoherence has a value between 0 (random phases) and 1 (total phase coupling).