“Gaussian Processes” are stochastic processes/fields with jointly Gaussian distributions of over all finite sets of observation locations. The most familiar of these to finance and physics people is usually the Gauss-Markov process, a.k.a. the Wiener process, but there are many others. These processes are convenient due to certain useful properties of the multivariate Gaussian distribution e.g. being uniquely specified by first and second moments, nice behaviour under various linear operations, kernel tricks…. Especially famous applications include Gaussian process regression and spatial statistics. Check out Ti’s Interactive visualization for some examples.

Gaussian processes are, specifically, probabilistic distributions over random functions \(\mathcal{T}\to \mathbb{C}\) for some index (or argument) set \(\mathcal{T}\) often taken to be \(\mathcal{T}:=\mathbb{R}^d\).

We typically work with a mean-zero process, in which case for every finite set
\(\mathbf{f}:=\{f(t_k);k=1,\dots,K\}\)
of observations of that process, the joint distribution is mean-zero Gaussian,
\[\begin{aligned}
\mathbf{f}(t)
&\sim \operatorname{GP}\left(0, \kappa(t, t';\mathbf{\theta})\right)
\\
&\Rightarrow\\
p(\mathbf{f}) &=(2\pi )^{-{\frac {K}{2}}}\det({\boldsymbol {\mathrm{K} }})^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}\mathbf {f}^{\!{\mathsf {T}}}{\boldsymbol {\mathrm{K} }}^{-1}\mathbf {f}}\\
&=\mathcal{N}(\mathbf{f};0, \mathrm{K}).
\end{aligned}\]
where \(\mathrm{K}\) is the sample covariance matrix defined such that its
entries are given by \(\mathrm{K}_{jk}=\kappa(t_j,t_k).\)
That is, this is the covariance kernel that maps from function *argument* — \(t\) — to second moment of function *values*.
In this case, we are specifying *only* the second moments and this is giving us
all the remaining properties of the process.

## Simulation/generation

See GP simulation.

## Derivatives and integrals

### Integral of a Gaussian process

### Derivative of a Gaussian process

TBD.

For now, see these blog posts:

- Spatial derivatives of Gaussian process models
- Derivatives and Gaussian Processes – CDT Data Science Blog

I am using results from Adler (2010), Adler and Taylor (2007).

## Incoming

## References

*The Geometry of Random Fields*. SIAM ed. Philadelphia: Society for Industrial and Applied Mathematics.

*Random Fields and Geometry*. Springer Monographs in Mathematics 115. New York: Springer.

*arXiv:1509.07526 [Math]*, October.

*Lectures on Fourier Integrals*. Princeton University Press.

*Gaussian Processes, Function Theory, and the Inverse Spectral Problem*. Dover ed. Dover Books on Mathematics. Mineola, N.Y: Dover Publications.

*Journal of Machine Learning Research*.

*arXiv:1807.02582 [Cs, Stat]*, July.

*Mathematische Annalen*109 (1): 604–15.

*Stochastic Partial Differential Equations: An Introduction*. Springer.

*Transactions of the American Mathematical Society*353 (10): 3945–69.

*Heliyon*5 (2): e01136.

*Probability, Random Variables and Stochastic Processes*.

*Journal of Machine Learning Research*11 (Nov): 3011–15.

*Gaussian Processes for Machine Learning*. Adaptive Computation and Machine Learning. Cambridge, Mass: MIT Press.

*Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References*. Springer Series in Statistics. New York, NY: Springer Science & Business Media.

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