mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in the area of

statistical analysis Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of ...

, the bispectrum is a statistic used to search for nonlinear interactions.

Definitions

The

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

of the second-order

cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...

, i.e., the

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...

function, is the traditional

power spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...

. The Fourier transform of ''C''₃(''t''₁, ''t''₂) (third-order

-generating function) is called the bispectrum or bispectral density.

Calculation

Applying the

convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time dom ...

allows fast calculation of the bispectrum:

B(f_1,f_2)=F(f_1)\cdot F(f_2)\cdot F^*(f_1+f_2)

, where

F

denotes the Fourier transform of the signal, and

F^*

its conjugate.

Applications

Bispectrum and bicoherence may be applied to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension. Bispectral measurements have been carried out for EEG signals monitoring. It was also shown that bispectra characterize differences between families of musical instruments. In

seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...

, signals rarely have adequate duration for making sensible bispectral estimates from time averages. Bispectral analysis describes observations made at two wavelengths. It is often used by scientists to analyze elemental makeup of a planetary atmosphere by analyzing the amount of light reflected and received through various color filters. By combining and removing two filters, much can be gleaned from only two filters. Through modern computerized

interpolation In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...

, a third virtual filter can be created to recreate true color photographs that, while not particularly useful for scientific analysis, are popular for public display in textbooks and fund raising campaigns. Bispectral analysis can also be used to analyze interactions between wave patterns and tides on Earth. A form of bispectral analysis called the bispectral index is applied to EEG waveforms to monitor depth of anesthesia. Biphase (phase of polyspectrum) can be used for detection of phase couplings, noise reduction of polharmonic (particularly, speech ) signal analysis.

A physical interpretation

The bispectrum reflects the energy budget of interactions, as it can be interpreted as a covariance defined between energy-supplying and energy-receiving parties of waves involved in an nonlinear interaction. On the other hand, bicoherence has been proven to be the corresponding correlation coefficient. Just as correlation cannot sufficiently demonstrate the presence of causality, spectrum and bicoherence also cannot sufficiently substantiate the existence of a nonlinear interaction.

Generalizations

Bispectra 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. A statistic defined analogously is the ''bispectral coherency'' or ''bicoherence''.

Trispectrum

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 () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

of the trispectrum allow a much reduced support set to be defined, contained within the following vertices, where 1 is the

Nyquist frequency In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a Sampling (signal processing), sampler, which converts a continuous function or signal into a discrete sequence. For a given S ...

. (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 divided 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 In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...

as a function of frequency triples, the trispectrum identifies contributions to a signal's

kurtosis In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtos ...

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.