Random field

A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold.

At its most basic, discrete case, a random field is a list of random numbers whose indices are mapped into a space (of n dimensions). When used in the natural sciences, values in a random field are often spatially correlated in one way or another. In its most basic form this might mean that adjacent values (i.e. values with adjacent indices) 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 modeled in a random field. More generally, the values might be defined over a continuous domain, and the random field might be thought of as a "function valued" random variable.

Definition and examples

Given a probability space $(\Omega, \mathcal{F}, P)$, an X-valued random field is a collection of X-valued random variables indexed by elements in a topological space T. That is, a random field F is a collection

$\{ F_t : t \in T \}$

where each $F_t$ is an X-valued random variable.

Several kinds of random fields exist, among them the Markov random field (MRF), Gibbs random field (GRF), conditional random field (CRF), and Gaussian random field. An MRF exhibits the Markovian property

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

where $\partial_i$ is a set of neighbours of the random variable X_i. In other words, the probability that a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbours. The probability of a random variable in an MRF is given by

$P(X_i=x_i|\partial_i) = \frac{P(\omega)}{\sum_{\omega'}P(\omega')},$

where Ω' is the same realization of Ω, except for random variable X_i. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.

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 of the scale of centimeters. This leads to tensor random fields in which the key role is played here a Statistical Volume Element (SVE); 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 fields that appear in continuum theories are those of dependent quantities (temperature, displacement, velocity, deformation, rotation, body and surface forces, stress, ...).

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

See also

• Covariance
• Kriging
• Variogram
• Resel
• Stochastic process
• Interacting particle system
• Stochastic cellular automata

References

• 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.