A Gaussian process is a stochastic process {X_t}_{t ∈T} such that every finite linear combination of the X_t (or, more generally, any linear functional of the sample function X_t) is normally distributed. The concept is named after Carl Friedrich Gauss simply because the normal distribution is sometimes called the Gaussian distribution, although Gauss was not the first to study that distribution. Note that some authors (for example B. Simon in the reference cited below) also assume the variables X_t have mean zero. Alternatively, a process is Gaussian iff for every finite set of indices t₁, ..., t_k in the index set T In the mathematics of probability, a stochastic process is a random function. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

is a vector-valued Gaussian random variable. Using characteristic functions of random variables, we can formulate the Gaussian property as follows:{X_t}_{t ∈ T} is Gaussian iff for every finite set of indices t₁, ..., t_k there are positive reals σ_{l j} and reals μ_j such that In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ...

The numbers σ_{l j} and μ_j can be shown to be the covariances and means of the variables in the process.

The Wiener process is perhaps the most widely studied Gaussian process. It is not stationary, but it has stationary increments. The Ornstein-Uhlenbeck process is a stationary Gaussian process. The Brownian bridge is a Gaussian process whose increments are not independent. In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ... In mathematics, the Ornstein-Uhlenbeck process (also known as the Mean reverting process in probability) is a stochastic process given by the following stochastic differential equation where, θ, μ and σ are parameters. ... This article may be too technical for most readers to understand. ...

A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. Inference of continuous values with a Gaussian process prior is known as Gaussian process regression, or Kriging. A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... In mathematics, a function returns a unique output for a given input. ... Bayesian inference is a statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ... Kriging is a regression technique used in geostatistics. ... Kriging is a regression technique used in geostatistics. ...

References

• R. M. Dudley, Real Analysis and Probability, Wadsworth and Brooks/Cole, 1989.

• B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979.

• C E Rasmussen, C K I Williams, Gaussian Processes for Machine Learning, MIT Press, 2005. ISBN 026218253X

MIT Press Books The MIT Press is a university publisher affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts. ...

External link

• The Gaussian Processes Web Site