A Gaussian random field (GRF) within statistics, is a

random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other ...

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 In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version ...

. With regard to applications of GRFs, the initial conditions of

physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...

generated by quantum mechanical fluctuations during cosmic inflation are thought to be a GRF with a nearly

scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical ter ...

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 In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...

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

, and hence, through the

Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...

, by its two-point

autocorrelation function 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 variabl ...

, 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 {{probability-stub