Random field Did You Mean Random?

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Overview

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 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, w/., X_n}, with the X_i in {0, 1, w/., G − 1}, be a set of random variables on the sample space Ω = {0, 1, w/., G − 1}ⁿ. A probability measure π is a random field if

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

for all ω in Ω. 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 X_i. 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 showed by the equation 1, Ω' is the same realization of Ω, except for random variable X_i. 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 are used as alagories of natural spatially varying properties, such as soil permeability over the scale of meters, or concrete strength on the scale of centimeters.

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.

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Overview

Mathematically

Applications

Reference

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