Gaussian Random Field

A Gaussian Random Field is a random field involving [[Gaussian multivariate probability density functions of the random variables.

• • …
• Counter-Example(s):
  • a Markov Random Field.
• See: Gaussian Mixture Model, Autocorrelation Function, Gaussian Free Field, Gaussian Process, Power Spectral Density, Scale Invariant.

---

---

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Gaussian_random_field Retrieved:2015-5-16.
  • A Gaussian random field (GRF) is a random field involving [[multivariate normal distribution|Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process.

    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. For details on the generation of Gaussian random fields using Matlab, see

    circulant embedding method for Gaussian random 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.

    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, ..., x_N, in the D-dimensional space, then the vector (f(x₁), ..., f(x_N)) will always be distributed as a multivariate Gaussian.

    An important special case of a GRF is the Gaussian free field.

---