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Gaussian random field

From Wikipedia, the free encyclopedia

A Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process. An important special case of a GRF is the Gaussian free field.

With regard to applications of GRFs, the initial conditions of physical cosmology generated by quantum mechanical fluctuations during cosmic inflation are thought to be a GRF with a nearly scale invariant spectrum.[1]

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  • ✪ 6.2 Gaussian Markov Random Fields (GMRF) | Image Analysis Class 2013
  • ✪ 15.1 Gaussian Markov Random Fields | Image Analysis Class 2015
  • ✪ 6.1 Markov Random Fields (MRFs) | Image Analysis Class 2013



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, the central 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 its power spectral density, and hence, through the Wiener-Khinchin theorem, by its two-point autocorrelation function, which is related to the power spectral density through a Fourier transformation.

Suppose f(x) is the value of a GRF at a point x in some D-dimensional space. If we make a vector of the values of f at N points, x1, ..., xN, in the D-dimensional space, then the vector (f(x1), ..., f(xN)) will always be distributed as a multivariate Gaussian.


  1. ^ Peacock, John. Cosmological Physics, Cambridge University Press, 1999. ISBN 0-521-41072-X[page needed]

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This page was last edited on 9 February 2019, at 08:08
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