Gaussian process

A Gaussian process is a stochastic process which generates samples over time {Xt}t ∈T such that no matter which finite linear combination of the Xt ones takes (or, more generally, any linear functional of the sample function Xt), that linear combination will be normally distributed.

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.

Note that some authors (for example B. Simon in the reference cited below) also assume the variables Xt have mean zero.

Alternative definitions
Alternatively, a process is Gaussian if and only if for every finite set of indices t1, ..., tk 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:{Xt}t ∈ T is Gaussian if and only if for every finite set of indices t1, ..., tk there are positive reals σl j and reals μj 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 σl j and  μj 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.

Uses
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.