Random field

A random field is another term for stochastic process in modern mathematics[1] with some restriction on its index set. The modern definition of a random field or a stochastic process is a generalization of the classic naive definition of "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.[2]

At its most basic, discrete case, 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). 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 ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ , 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

${\displaystyle \{F_{t}:t\in T\}}$

where each ${\displaystyle F_{t}}$  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

${\displaystyle P(X_{i}=x_{i}|X_{j}=x_{j},i\neq j)=P(X_{i}=x_{i}|X_{j}=x_{j},j\in \partial _{i}),\,}$

for each choice of values ${\displaystyle (x_{j})_{j}}$ , and for each ${\displaystyle i}$ , ${\displaystyle \partial _{i}}$  is a designated set of "neighbours" of the index point ${\displaystyle i}$ . 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

${\displaystyle P(X_{i}=x_{i}|\partial _{i})={\frac {P(\omega )}{\sum _{\omega '}P(\omega ')}},}$

where ω' is a subset of the parameter space Ω, valid for Xi.[clarification needed] It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.[clarification needed]

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

Tensor-Valued Random Fields

Random fields are of great use in studying natural processes by the Monte Carlo method,[3] in which the random fields correspond to naturally spatially varying properties, such as soil permeability over the scale of meters, concrete strength over the scale of centimeters or graphite stiffness over the scale of millimeters.[4] This leads to tensor-valued random fields in which the key role is played by 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, ...).[5]