Random field
Encyclopedia
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 number
s whose indices 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 (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.
,
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
where each
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
where
is a set of neighbours of the random variable Xi. 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
where Ω' is the same realization of Ω, except for random variable Xi. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag
in 1974.
, 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 that mimic natural surfaces such as water and earth.
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
is a generalization of a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
such that the underlying parameter need no longer be a simple real
Real space
Position space denominates the space of possible locations of an object in classical physics.The real space coordinates specify the position of an object. For instance,...
or integer valued "time", but can instead take values that are multidimensional vectors
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, or points on some manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
.
At its most basic, discrete case, a random field is a list of random number
Random number
Random number may refer to:* A number generated for or part of a set exhibiting statistical randomness.* A random sequence obtained from a stochastic process.* An algorithmically random sequence in algorithmic information theory....
s whose indices are mapped onto a space (of n dimensions
Dimensions
Dimensions is a French project that makes educational movies about mathematics, focusing on spatial geometry. It uses POV-Ray to render some of the animations, and the films are release under a Creative Commons licence....
). 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 (i.e. values with adjacent indices) do not differ as much as values that are further apart. This is an example of a covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
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 spaceProbability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
,an X-valued random field is a collection of X-valued
random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s indexed by elements in a topological space T. That is, a random field F is a collection
where each
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
Conditional random field
A conditional random field is a statistical modelling method often applied in pattern recognition.More specifically it is a type of discriminative undirected probabilistic graphical model. It is used to encode known relationships between observations and construct consistent interpretations...
(CRF), and Gaussian random field
Gaussian random field
A Gaussian random field is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process....
. An MRF exhibits the Markovian property
where
is a set of neighbours of the random variable Xi. 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
where Ω' is the same realization of Ω, except for random variable Xi. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag
Julian Besag
Julian Ernst Besag FRS was a British statistician known chiefly for his work in spatial statistics , and Bayesian inference .- Biography:Besag was born in Loughborough and was educated at Loughborough Grammar School...
in 1974.
Applications
Random fields are of great use in studying natural processes by the Monte Carlo methodMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
, 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 that mimic natural surfaces such as water and earth.
See also
- CovarianceCovarianceIn probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
- KrigingKrigingKriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
- VariogramVariogramIn spatial statistics the theoretical variogram 2\gamma is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z...
- Table of mathematical symbolsTable of mathematical symbolsThis is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
- ReselReselA resel is a resolution element - a concept used in image analysis. It describes the actual spatial image resolution in an image .The number of resels in the image will be lower or equal to the number of pixel/voxels in the image....
- Stochastic processStochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...