Random field

At its most basic a random field is a list of random numbers whose values are mapped onto a space (of n dimensions). Values in a random field are usually spatially correlated in one way or another, in its most basic form this might mean that adjacent values do not differ as much as values that are further apart. This is an example of a covariance structure, many different types of which may be modelled in a random field.

Mathematically
In probability theory, let S = {X1, ..., Xn}, with the Xi in {0, 1, ..., G &minus; 1}, be a set of random variables on the sample space &Omega; = {0, 1, ..., G &minus; 1}n. A probability measure &pi; is a random field if


 * $$\pi(\omega)>0\,$$

for all &omega; in &Omega;. Several kinds of random fields exist, among them Markov random fields (MRF), Gibbs random fields (GRF), conditional random fields(CRF), and Gaussian random fields. A MRF exhibits the Markovian property


 * $$\pi (X_i=x_i|X_j=x_j, i\neq j) = \pi (X_i=x_i|\partial_i), \,$$

where $$\partial_i$$ is a set of neighbours of the random variable Xi. In other words, the probability a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbors. A probability of a random variable in a MRF is shown by equation 1, &Omega;' is the same realization of &Omega;, except for random variable Xi. It is easy to see that it is difficult to calculate with this equation. The solution to this problem was proposed by Besag in 1974, when he made a relation between MRF and GRF.


 * $$ \pi (X_i=x_i|\partial_i) = \frac{\pi(\omega)}{\sum_{\omega'}\pi(\omega')} \;\;\;\;(1) $$

Applications
Random fields are of great use in studying natural processes by the Monte Carlo method, in which the random fields correspond to naturally spatially varying properties, such as soil permeability over the scale of meters, or concrete strength on the scale of centimeters.

A further common use of random fields is in the generation of computer graphics, particularly those which mimic natural surfaces such as water, earth and sky.

Reference

 * Besag, J. E. "Spatial Interaction and the Statistical Analysis of Lattice Systems", Journal of Royal Statistical Society: Series B 36, 2 (May 1974), 192-236.