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.
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Random field in the context of Spatial statistics
Spatial statistics is a field of applied statistics dealing with spatial data.It involves stochastic processes (random fields, point processes), sampling, smoothing and interpolation, regional (areal unit) and lattice (gridded) data, point patterns, as well as image analysis and stereology.
View the full Wikipedia page for Spatial statisticsRandom field in the context of Covariance function
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z(x) on a domain D, a covariance function C(x, y) gives the covariance of the values of the random field at the two locations x and y:
View the full Wikipedia page for Covariance function
![{\displaystyle C(x,y):=\operatorname {cov} (Z(x),Z(y))=\mathbb {E} {\Big [}{\big (}Z(x)-\mathbb {E} [Z(x)]{\big )}{\big (}Z(y)-\mathbb {E} [Z(y)]{\big )}{\Big ]}.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b0780ed4ee402030228c6e7a9b2b1dbc13c246)