Gaussian process

In the mathematical theory of probability, a Gaussian process is a stochastic process {X_t}_{t ∈T} for which any finite linear combination of samples will be normally distributed (or, more generally, any linear functional applied to the sample function X_t will give a normally distributed result).

Some authors also assume the random variables X_t have mean zero.

History

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.

Alternative definitions

Alternatively, a process is Gaussian if and only if for every finite set of indices t₁, ..., t_k in the index set T

${\vec {\mathbf {X} }}_{t_{1},\ldots ,t_{k}}=(\mathbf {X} _{t_{1}},\ldots ,\mathbf {X} _{t_{k}})$

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 if and only if for every finite set of indices t₁, ..., t_k there are positive reals σ_{l j} and reals μ_j such that

$\operatorname {E} \left(\exp \left(i\ \sum _{\ell =1}^{k}t_{\ell }\ \mathbf {X} _{t_{\ell }}\right)\right)=\exp \left(-{\frac {1}{2}}\,\sum _{\ell ,j}\sigma _{\ell j}t_{\ell }t_{j}+i\sum _{\ell }\mu _{\ell }t_{\ell }\right).$

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

Important Gaussian processes

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.

The fractional Brownian motion is a Gaussian process whose covariance function is a generalisation of Wiener process.

Applications

A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. (Given any set of $N$ points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.) Inference of continuous values with a Gaussian process prior is known as Gaussian process regression, or Kriging . Gaussian processes are also useful as a powerful non-linear interpolation tool.

External links

• The Gaussian Processes Web Site
• www.GaussianProcess.com
• A gentle introduction to Gaussian processes