Editing Random field

{{Short description|Mathematical function}}
In [[physics]] and [[mathematics]], a '''random field''' is a random function over an arbitrary domain (usually a multi-dimensional space such as <math>\mathbb{R}^n</math>). That is, it is a function <math>f(x)</math> that takes on a random value at each point <math>x \in \mathbb{R}^n</math>(or some other domain). 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 coordinate space|real]] or [[integer]] valued "time" but can instead take values that are multidimensional [[vector space|vectors]] or points on some [[manifold]].<ref>{{cite book | author=Vanmarcke, Erik | title=Random Fields: Analysis and Synthesis | publisher=World Scientific Publishing Company | year=2010 | isbn=978-9812563538}}</ref>

== Formal definition ==

Given a [[probability space]] <math>(\Omega, \mathcal{F}, P)</math>, an ''X''-valued random field is a collection of ''X''-valued [[random variable]]s indexed by elements in a [[topological space]] ''T''. That is, a random field ''F'' is a collection
: <math> \{ F_t : t \in T \}</math>
where each <math>F_t</math> is an ''X''-valued random variable.

== Examples ==
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]]). Suppose there are four random variables, <math>X_1</math>, <math>X_2</math>, <math>X_3</math>, and <math>X_4</math>, located in a 2D grid at (0,0), (0,2), (2,2), and (2,0), respectively. Suppose each random variable can take on the value of -1 or 1, and the probability of each random variable's value depends on its immediately adjacent neighbours. This is a simple example of a discrete random field.

More generally, the values each <math>X_i</math> can take on might be defined over a continuous domain. In larger grids, it can also be useful to think of the random field as a "function valued" random variable as described above. In [[quantum field theory]] the notion is generalized to a random [[Functional (mathematics)|functional]], one that takes on random values over a [[Function space|space of functions]] (see [[Feynman integral]]). 

Several kinds of random fields exist, among them the [[Markov random field]] (MRF), [[Gibbs random field]], [[conditional random field]] (CRF), and [[Gaussian random field]]. In 1974, [[Julian Besag]] proposed an approximation method relying on the relation between MRFs and Gibbs RFs.{{Citation needed|date=May 2019}}

=== Example properties ===

An MRF exhibits the [[Markov property]]

: <math>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), \,</math>

for each choice of values <math>(x_j)_j</math>. Here each <math>\partial_i</math> is the set of neighbors of  <math>i</math>. In other words, the probability that a random variable assumes a value depends on its immediate neighboring random variables.  The probability of a random variable in an MRF{{what|reason=Or in any probability measure, since the denominator is always 1.|date=October 2023}} is given by

:<math> P(X_i=x_i|\partial_i) = \frac{P(X_i=x_i, \partial_i)}{\sum_k P(X_i=k, \partial_i)},  </math>

where the sum (can be an integral) is over the possible values of k.{{what|reason=What is the content of this equation? The sum in the denominator is automatically 1 since P is a probability measure.|date=October 2023}} It is sometimes difficult to compute this quantity exactly.

== Applications ==
When used in the [[natural sciences]], values in a random field are often spatially correlated. For example, 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. One example is the [[Ising model]] where sometimes nearest neighbor interactions are only included as a simplification to better understand the model.

A common use of random fields is in the generation of computer graphics, particularly those that mimic natural surfaces such as [[Fluid simulation|water]] and [[Digital terrain model|earth]]. Random fields have been also used in subsurface ground models as in <ref>{{cite journal|last1= Cardenas |first1=IC|title= A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields|journal=Engineering Geology|date=2023|doi=10.1016/j.enggeo.2023.107001|doi-access=free}}</ref>

In [[neuroscience]], particularly in [[Functional neuroimaging|task-related functional brain imaging]] studies using [[Positron emission tomography|PET]] or [[Functional magnetic resonance imaging|fMRI]], statistical analysis of random fields are one common alternative to [[Multiple comparisons problem|correction for multiple comparisons]] to find regions with ''truly'' significant activation.<ref>{{Cite journal|last=Worsley|first=K. J.|last2=Evans|first2=A. C.|last3=Marrett|first3=S.|last4=Neelin|first4=P.|date=November 1992|title=A Three-Dimensional Statistical Analysis for CBF Activation Studies in Human Brain|journal=Journal of Cerebral Blood Flow & Metabolism|language=en-US|volume=12|issue=6|pages=900–918|doi=10.1038/jcbfm.1992.127|pmid=1400644|issn=0271-678X|doi-access=free}}</ref> 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.<ref>    {{cite journal |last1=Vitells |first1=Ofer |last2=Gross |first2=Eilam |date=2011 |title=Estimating the significance of a signal in a multi-dimensional search |journal=Astroparticle Physics |volume=35 |pages=230-234 |doi=10.1016/j.astropartphys.2011.08.005 |arxiv=1105.4355}}</ref>

They are also used in [[machine learning]] applications (see [[graphical model]]s).

== Tensor-valued random fields ==

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{{what|reason=There is no indication here or in the linked Wikipedia article what these tensors might be.|date=October 2023}} 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.).<ref>{{cite book | author1=Malyarenko, Anatoliy |author2= Ostoja-Starzewski, Martin|authorlink2=Martin Ostoja-Starzewski |title=Tensor-Valued Random Fields for Continuum Physics | publisher=Cambridge University Press | year=2019 | isbn=9781108429856}}</ref>{{what|reason=What does "dependent" mean in this context? Is there supposed to be a dichotomy here between "tensor-valued" and "dependent"? Can't something be both or neither?|date=October 2023}}

== See also == 
* [[Covariance]]
* [[Kriging]]
* [[Variogram]]
* [[Resel]]
* [[Stochastic process]]
* [[Interacting particle system]]
* [[Stochastic cellular automata]]

==References==
{{Reflist}}

==Further reading==
*{{cite book |last1=Adler |first1=R. J. |last2=Taylor |first2=Jonathan |name-list-style=amp | title=Random Fields and Geometry | publisher=Springer | year=2007 | isbn=978-0-387-48112-8}}
*{{cite journal |last=Besag |first=J. E. |year=1974 |title=Spatial Interaction and the Statistical Analysis of Lattice Systems |journal=[[Journal of the Royal Statistical Society]] |series=Series B |volume=36 |issue=2 |pages=192–236 |doi=10.1111/j.2517-6161.1974.tb00999.x }}
*{{cite book |first=David |last=Griffeath |year=1976 |chapter=Random Fields |title=Denumerable Markov Chains |url=https://archive.org/details/springer_10.1007-978-1-4684-9455-6 |edition=2nd |editor-link=John G. Kemeny |editor-first=John G. |editor-last=Kemeny |editor2-link=Laurie Snell |editor2-first=Laurie |editor2-last=Snell |editor3-first=Anthony W. |editor3-last=Knapp |publisher=Springer |isbn=0-387-90177-9 }}
*{{cite book|author1-link=Davar Khoshnevisan | author=Davar Khoshnevisan | title=Multiparameter Processes : An Introduction to Random Fields | publisher=Springer | year=2002 | isbn=0-387-95459-7}}

{{Stochastic processes}}

[[Category:Spatial processes]]