mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the area of

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

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

Definitions

The

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

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 ha ...

, i.e., the

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

function, is the traditional

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

. 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 pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g. ...

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 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 ...

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 Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in the neocortex ...

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 and the propagation of elastic waves through the Earth or through other ...

, 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

filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...

s. By combining and removing two filters, much can be gleaned from only two filters. Through modern computerized

interpolation In the 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 often has a n ...

, 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 Bispectral index (BIS) is one of several technologies used to monitor depth of anesthesia. BIS monitors are used to supplement Guedel's classification system for determining depth of anesthesia. Titrating anesthetic agents to a specific bispectr ...

is applied to

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.

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 (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

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 sampler, which converts a continuous function or signal into a discrete sequence. In units of cycles per second ( Hz), it ...

. (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 d ...

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

kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...

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.

Definitions

Calculation

Applications

Generalizations

Trispectrum

References

Further reading