Gaussian process

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 [cite book |last=Simon |first=Barry |title=Functional Integration and Quantum Physics |year=1979 |publisher=Academic Press] 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.

Alternative definitions

Alternatively, a process is Gaussian if and only if for every finite set of indices "t"₁, ..., "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 "t"₁, ..., "t"_"k" there are positive reals σ_{"l j"} and reals μ_"j" such that

:$operatorname\{E\}left(expleft(i\; sum\_\{ell=1\}^k\; t\_ell\; mathbf\{X\}\_\{t\_ell\}\; ight)\; ight)\; =\; exp\; left(-frac\{1\}\{2\}\; ,\; sum\_\{ell,\; j\}\; sigma\_\{ell\; j\}\; t\_ell\; t\_j\; +\; i\; sum\_ell\; mu\_ell\; t\_ell\; ight).$

The numbers σ_{"l j"} and μ_"j" can be shown to be the covariances and means of the variables in the process. [cite book |last=Dudley |first=R.M. |title=Real Analysis and Probability |year=1989 |publisher=Wadsworth and Brooks/Cole]

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. [cite book |last=Rasmussen |first=C.E. |coauthors=Williams, C.K.I |title=Gaussian Processes for Machine Learning |year=2006 |publisher=MIT Press |isbn=0-262-18253-X] (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 [cite book |last=Stein |first=M.L. |title=Interpolation of Spatial Data: Some Theory for Kriging |year=1999 |publisher=Springer] .

Notes

External links

* [http://www.GaussianProcess.org The Gaussian Processes Web Site]
* [http://www.robots.ox.ac.uk/~mebden/reports/GPtutorial.pdf A gentle introduction to Gaussian processes]