Random field

A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real, but can instead be a multidimensional vector space or even a manifold.

At its most basic, discrete case, 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.

Definition and Examples

In probability theory, let "S" = {"X"₁, ..., "X"_"n"}, with the "X"_"i" in {0, 1, ..., "G" − 1} being a set of random variables on the sample space Ω = {0, 1, ..., "G" − 1}^"n". A probability measure π is a random field if, for all ω in Ω,

: $pi(omega)>0.$

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

: $pi\; (X\_i=x\_i|X\_j=x\_j,\; i\; eq\; j)\; =\; pi\; (X\_i=x\_i|partial\_i),\; ,$

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

:$pi\; (X\_i=x\_i|partial\_i)\; =\; frac\{pi(omega)\}\{sum\_\{omega\text{'}\}pi(omega\text{'})\},$

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 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 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 and earth.

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.
#cite book | author=Adler, RJ & Taylor, Jonathan | title=Random Fields and Geometry | publisher=Springer | year=2007 | id=ISBN 978-0-387-48112-8
#cite book | author=Khoshnevisan | title=Multiparameter Processes - An Introduction to Random Fields | publisher=Springer | year=2002 | id=ISBN 0-387-95459-7

See also

* Covariance
* Kriging
* Variogram
* Table of mathematical symbols
* Resel
* Stochastic Process