Gaussian process

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A Gaussian process is a stochastic process which generates samples over time {X_t}_{t ∈T} such that nomatter which finite linear combination of the X_t ones takes (or, more generally, any linear functional of the sample function X_t), that linear combination will be normally distributed.

1 History
2 Alternative definitions
3 Important Gaussian processes
4 Uses
5 References
6 External links

[edit] 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 X_t have mean zero.

[edit] 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.

[edit] 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.

[edit] 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.