Gaussian process

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

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

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 iff 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.

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.

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.

References

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

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

External link

• The Gaussian Processes Web Site