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Random field

In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as R n {displaystyle mathbb {R} ^{n}} ). That is, it is a function f ( x ) {displaystyle f(x)} that takes on a random value at each point x ∈ R n {displaystyle xin mathbb {R} ^{n}} (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.Given a probability space ( Ω , F , P ) {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 collectionIn 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). 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 as described above. In quantum field theory the notion is even generalized to a random functional, one that takes on random value over a space of functions (see path integral).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.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, such as soil permeability over the scale of meters, concrete strength over the scale of centimeters or graphite stiffness over the scale of millimeters.

[ "Statistics", "Statistical physics", "Mathematical analysis", "random field ising model", "discriminative random fields", "homogeneous random fields", "random field theory", "Random element" ]
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