Gaussian process

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

Some authors ^[1] also assume the random variables X_t have mean zero.

Contents 1 History 2 Alternative definitions 3 Important Gaussian processes 4 Uses 5 Notes 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.

[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

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

The numbers σ_{l j} and μ_j can be shown to be the covariances and means of the variables in the process. ^[2]

[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. ^[3] (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 ^[4].

[edit] Notes

1. ^ Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press. 
2. ^ Dudley, R.M. (1989). Real Analysis and Probability. Wadsworth and Brooks/Cole. 
3. ^ Rasmussen, C.E.; Williams, C.K.I (2006). Gaussian Processes for Machine Learning. MIT Press. ISBN 0-262-18253-X. 
4. ^ Stein, M.L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer. 

[edit] External links

• The Gaussian Processes Web Site