It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set.
That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold.
In its discrete version, a random field is a list of random numbers whose indices are identified with a discrete set of points in a space (for example, n-dimensional Euclidean space).
In quantum field theory the notion is generalized to a random functional, one that takes on random values over a space of functions (see Feynman integral).
In 1974, Julian Besag proposed an approximation method relying on the relation between MRFs and Gibbs RFs.
[citation needed] An MRF exhibits the Markov property for each choice of values
The probability of a random variable in an MRF[clarification needed] is given by where the sum (can be an integral) is over the possible values of k.[clarification needed] It is sometimes difficult to compute this quantity exactly.
When used in the natural sciences, values in a random field are often spatially correlated.
This is an example of a covariance structure, many different types of which may be modeled in a random field.
A common use of random fields is in the generation of computer graphics, particularly those that mimic natural surfaces such as water and earth.
Random fields have been also used in subsurface ground models as in [2] In neuroscience, particularly in task-related functional brain imaging studies using PET or fMRI, statistical analysis of random fields are one common alternative to correction for multiple comparisons to find regions with truly significant activation.
[3] More generally, random fields can be used to correct for the look-elsewhere effect in statistical testing, where the domain is the parameter space being searched.
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.
This leads to tensor-valued random fields[clarification needed] in which the key role is played by a statistical volume element (SVE), which is a spatial box over which properties can be averaged; when the SVE becomes sufficiently large, its properties become deterministic and one recovers the representative volume element (RVE) of deterministic continuum physics.
The second type of random field that appears in continuum theories are those of dependent quantities (temperature, displacement, velocity, deformation, rotation, body and surface forces, stress, etc.).