Trispectrum
Encyclopedia
In mathematics, in the area of statistical analysis, the trispectrum 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 C4 (t1, t2, t3) (fourth-order cumulant-generating function) is called the trispectrum or trispectral density.
The trispectrum T(f1,f2,f3) falls into the category of higher-order spectra, or polyspectra, and provides supplementary information to the power spectrum. The trispectrum is a three dimensional construct. The symmetries
of the trispectrum allow a much reduced support set to be defined, contained within the following verticies, where 1 is the Nyquist frequency
. (0,0,0) (1/2,1/2,-1/2) (1/3,1/3,0) (1/2,0,0) (1/4,1/4,1/4). The plane containing the points (1/6,1/6,1/6) (1/4,1/4,0) (1/2,0,0) divides this volume into an inner and an outer region. A stationary signal will have zero strength (statistically) in the outer region. The trispectrum support is divide into regions by the plane identified above, and by the (f1,f2) plane. Each region has different requirements in terms of the bandwidth of signal required for non-zero values.
In the same way that the bispectrum
identifies contributions to a signal's skewness
as a function of frequency triples, the trispectrum identifies contributions to a signal's kurtosis
as a function of frequency quadruplets.
The trispectrum has been used to investigate the domains of applicability of maximum kurtosis phase estimation used in the deconvolution of seismic data to find layer structure.
The trispectrum is the non-zero stationary support for the four-dimensional non-stationary trispectrum.
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 function, is the traditional power spectrum. The Fourier transform of C4 (t1, t2, t3) (fourth-order cumulant-generating function) is called the trispectrum or trispectral density.
The trispectrum T(f1,f2,f3) falls into the category of higher-order spectra, or polyspectra, and provides supplementary information to the power spectrum. The trispectrum is a three dimensional construct. The symmetries
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
of the trispectrum allow a much reduced support set to be defined, contained within the following verticies, where 1 is the Nyquist frequency
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...
. (0,0,0) (1/2,1/2,-1/2) (1/3,1/3,0) (1/2,0,0) (1/4,1/4,1/4). The plane containing the points (1/6,1/6,1/6) (1/4,1/4,0) (1/2,0,0) divides this volume into an inner and an outer region. A stationary signal will have zero strength (statistically) in the outer region. The trispectrum support is divide into regions by the plane identified above, and by the (f1,f2) plane. Each region has different requirements in terms of the bandwidth of signal required for non-zero values.
In the same way that 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...
identifies contributions to a signal's skewness
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...
as a function of frequency triples, the trispectrum identifies contributions to a signal's kurtosis
Kurtosis
In probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable...
as a function of frequency quadruplets.
The trispectrum has been used to investigate the domains of applicability of maximum kurtosis phase estimation used in the deconvolution of seismic data to find layer structure.
The trispectrum is the non-zero stationary support for the four-dimensional non-stationary trispectrum.