Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times (or of space) such that each such random variable has a normal distribution. Moreover, every finite collection of those random variables has a multivariate normal distribution.

Gaussian processes are important in statistical modelling because of properties inherited from the normal. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include: the average value of the process over a range of times; the error in estimating the average using sample values at a small set of times.

Definition

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: see history of the normal/Gaussian 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, the Gaussian property can be formulated as follows:{ X_t ; t ∈ T } is Gaussian if and only if, for every finite set of indices t₁, ..., t_k, there are reals σ_{l j} with σ_{i i} > 0 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.

See also

• Kriging
• Bayes linear statistics

External links

• www.GaussianProcess.com
• The Gaussian Processes Web Site, including the text of Rasmussen and Williams' Gaussian Processes for Machine Learning
• A gentle introduction to Gaussian processes
• A Review of Gaussian Random Fields and Correlation Functions

Video tutorials

• Gaussian Process Basics by David MacKay
• Learning with Gaussian Processes by Carl Edward Rasmussen
• Bayesian inference and Gaussian processes by Carl Edward Rasmussen