Gaussian random field
{{Short description|Concept in statistics}}
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In statistics, 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.{{cite book |last=Peacock |first=John |title=Cosmological Physics |location= |publisher=Cambridge University Press |year=1999 |isbn=0-521-41072-X |page=342 |url=https://books.google.com/books?id=t8O-yylU0j0C&pg=PA342 |via=Google Books }}
Construction
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.
See also
References
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External links
- For details on the generation of Gaussian random fields using Matlab, see [http://www.mathworks.com.au/matlabcentral/fileexchange/38880-circulant-embedding-method-for-generating-stationary-gaussian-field circulant embedding method for Gaussian random field].
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