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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 f ( x ) {\displaystyle f(x)} that takes on a random value at each point x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} (or some other domain).
Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without considering "neighbouring" samples, a CRF can take context into account.
One notable variant of a Markov random field is a conditional random field, in which each random variable may also be conditioned upon a set of global observations . In this model, each function φ k {\displaystyle \varphi _{k}} is a mapping from all assignments to both the clique k and the observations o {\displaystyle o} to the nonnegative ...
In the domain of physics and probability, the filters, random fields, and maximum entropy (FRAME) model [1] [2] is a Markov random field model (or a Gibbs distribution) of stationary spatial processes, in which the energy function is the sum of translation-invariant potential functions that are one-dimensional non-linear transformations of linear filter responses.
One way of constructing a GRF is by assuming that the field is the sum of a large number of plane, cylindrical or spherical waves with uniformly distributed random phase. Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions will exhibit a Gaussian distribution.
But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions. [5] [28] [227] If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process ...
For example, the probability of the union of the mutually exclusive events and in the random experiment of one coin toss, (), is the sum of probability for and the probability for , () + (). Second, the probability of the sample space Ω {\displaystyle \Omega } must be equal to 1 (which accounts for the fact that, given an execution of the ...
In mathematics, an ambit field is a d-dimensional random field describing the stochastic properties of a given system. The input is in general a d-dimensional vector (e.g. d-dimensional space or (1-dimensional) time and (d − 1)-dimensional space) assigning a real value to each of the points in the field.