Bicoherence
Encyclopedia
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 http://www.iop.org/EJ/abstract/0741-3335/30/5/005.
Bicoherence measurements have been carried out for EEG
signals monitoring in sleep
, wakefulness
and seizures http://cogprints.org. Search for papers by Bullock and Achimowicz.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the area of statistical analysis, bicoherence is a squared normalised version of the bispectrum
Bispectrum
In mathematics, in the area of statistical analysis, 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 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
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the second-order cumulant
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
, i.e., the autocorrelation
Autocorrelation
Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
function, is the traditional power spectrum. The Fourier transform of C3(t1,t2) (third-order cumulant
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
) 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 http://www.iop.org/EJ/abstract/0741-3335/30/5/005.
Bicoherence measurements have been carried out for EEG
Electroencephalography
Electroencephalography is the recording of electrical activity along the scalp. EEG measures voltage fluctuations resulting from ionic current flows within the neurons of the brain...
signals monitoring in sleep
Sleep
Sleep is a naturally recurring state characterized by reduced or absent consciousness, relatively suspended sensory activity, and inactivity of nearly all voluntary muscles. It is distinguished from quiet wakefulness by a decreased ability to react to stimuli, and is more easily reversible than...
, wakefulness
Awake
Wakefulness is a daily recurring brain state and state of consciousness in which an individual is conscious and engages in a coherent cognitive and behavior responses to the external world such as communication, ambulation, nutritional ingestion and procreation...
and seizures http://cogprints.org. Search for papers by Bullock and Achimowicz.