Random field

A random field is a generalization of a stochastic process 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.^[1]

At its most basic, discrete case, a random field is a list of random numbers whose indices are mapped onto a space (of n dimensions). 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.

Definition and examples

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

\{ F_t : t \in T \}

where each $F_t$ is an X-valued random variable.

Several kinds of random fields exist, among them the Markov random field (MRF), Gibbs random field (GRF), conditional random field (CRF), and Gaussian random field. An MRF exhibits the Markovian property

P(X_i=x_i|X_j=x_j, i\neq j) =P(X_i=x_i|\partial_i), \,

where $\partial_i$ is a set of neighbours of the random variable X_i. In other words, the probability that a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbours. The probability of a random variable in an MRF is given by

P(X_i=x_i|\partial_i) = \frac{P(\omega)}{\sum_{\omega'}P(\omega')},

where Ω' is the same realization of Ω, except for random variable X_i. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.

Applications

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, or concrete strength of the scale of centimeters. This leads to tensor random fields in which the key role is played here a Statistical Volume Element (SVE); when the SVE becomes sufficiently large, its properties become deterministic and one recovers the Representative volume element (RVE) of deterministic continuum physics. The second type of random fields that appear in continuum theories are those of dependent quantities (temperature, displacement, velocity, deformation, rotation, body and surface forces, stress, ...). ^[2]

A further common use of random fields is in the generation of computer graphics, particularly those that mimic natural surfaces such as water and earth.

References

↑ Vanmarcke, Erik (2010). Random Fields: Analysis and Synthesis. World Scientific Publishing Company. ISBN 978-9812563538.
↑ Ostoja-Starzewski, Shen and Malyarenko

Besag, J. E. "Spatial Interaction and the Statistical Analysis of Lattice Systems", Journal of Royal Statistical Society: Series B 36, 2 (May 1974), 192-236.
Adler, RJ & Taylor, Jonathan (2007). Random Fields and Geometry. Springer. ISBN 978-0-387-48112-8.
Khoshnevisan (2002). Multiparameter Processes - An Introduction to Random Fields. Springer. ISBN 0-387-95459-7.
Ostoja-Starzewski, M.; Shen, L.; Malyarenko, A. (2013), "Tensor random fields in conductivity and classical or micropolar elasticity", Mathematics & Mechanics of Solids

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itō diffusion Itō process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market SABR volatility Vašíček

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itō integral Itō's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prohorov theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

List of topics Category

Random field

Definition and examples

Applications

See also

References