Random field
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as ). That is, it is a function that takes on a random value at each point (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 or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold.[1]
Formal definition
Given a probability space , 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
where each 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-
More generally, the values each 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
Several kinds of random fields exist, among them the
Example properties
An MRF exhibits the Markov property
for each choice of values . Here each is the set of neighbors of . 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[clarification needed] is given by
where the sum (can be an integral) is over the possible values of k.[clarification needed] It is sometimes difficult to compute this quantity exactly.
Applications
When used in the
A common use of random fields is in the generation of computer graphics, particularly those that mimic natural surfaces such as
In neuroscience, particularly in task-related functional brain imaging studies using PET or fMRI, statistical analysis of random fields are one common alternative to correction for multiple comparisons to find regions with truly significant activation.[3]
They are also used in machine learning applications (see graphical models).
Tensor-valued random fields
Random fields are of great use in studying natural processes by the
See also
- Covariance
- Kriging
- Variogram
- Resel
- Stochastic process
- Interacting particle system
- Stochastic cellular automata
References
- ISBN 978-9812563538.
- .
- PMID 1400644.
- ISBN 9781108429856.
Further reading
- Adler, R. J. & Taylor, Jonathan (2007). Random Fields and Geometry. Springer. ISBN 978-0-387-48112-8.
- Besag, J. E. (1974). "Spatial Interaction and the Statistical Analysis of Lattice Systems". .
- Griffeath, David (1976). "Random Fields". In ISBN 0-387-90177-9.
- ISBN 0-387-95459-7.