Random field

A random field is a generalization of a 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 numbers whose indices are identified with a discrete set of points in a space (for example, n-dimensional Euclidean space). When used in the natural sciences, values in a random field are often 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.

Given a probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$, 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 collection

where each ${\displaystyle F_{t}}$ is an X-valued random variable.

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