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Gaussian random field

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Concept in statistics

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Instatistics, aGaussian random field (GRF) is arandom field involvingGaussian probability density functions of the variables. A one-dimensional GRF is also called aGaussian process. An important special case of a GRF is theGaussian free field.

With regard to applications of GRFs, the initial conditions ofphysical cosmology generated byquantum mechanical fluctuations duringcosmic inflation are thought to be a GRF with a nearlyscale invariant spectrum.^[1]

Construction

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One way of constructing a GRF is by assuming that the field is the sum of a large number of plane, cylindrical or spherical waves with uniformly distributed random phase. Where applicable, thecentral limit theorem dictates that at any point, the sum of these individual plane-wave contributions will exhibit a Gaussian distribution. This type of GRF is completely described by itspower spectral density, and hence, through theWiener–Khinchin theorem, by its two-pointautocorrelation function, which is related to the power spectral density through a Fourier transformation.

Supposef(x) is the value of a GRF at a pointx in someD-dimensional space. If we make a vector of the values off atN points,x₁, ..., x_N, in theD-dimensional space, then the vector (f(x₁), ..., f(x_N)) will always be distributed as a multivariate Gaussian.

References

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^Peacock, John (1999).Cosmological Physics. Cambridge University Press. p. 342.ISBN 0-521-41072-X – viaGoogle Books.

External links

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For details on the generation of Gaussian random fields using Matlab, seecirculant embedding method for Gaussian random field.

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