Gaussian Random Field

A Gaussian random field (GRF) within statistics, 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.

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, ''x''₁, ..., ''x''_''N'', in the ''D''-dimensional space, then the vector (''f''(''x''₁), ..., ''f''(''x''_''N'')) will always be distributed as a multivariate Gaussian.

References

External links

*For details on the generation of Gaussian random fields using Matlab, se
circulant embedding method for Gaussian random field

Spatial processes

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