Inprobability theory andstatistics, aGaussian process is astochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has amultivariate normal distribution. The distribution of a Gaussian process is thejoint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

The concept of Gaussian processes is named afterCarl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions.

Gaussian processes are useful instatistical modelling, benefiting from properties inherited from the normal distribution. For example, if arandom process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times. While exact models often scale poorly as the amount of data increases, multipleapproximation methods have been developed which often retain good accuracy while drastically reducing computation time.

A time continuousstochastic process${\displaystyle \left\{X_t;t\in T\right\}}$ is Gaussianif and only if for everyfinite set ofindices${\displaystyle t_1,\dots ,t_k}$ in the index set${\displaystyle T}$

$${\displaystyle {\mathbf{X}}_{t_1,\dots ,t_k}=(X_{t_1},\dots ,X_{t_k})}$$

is amultivariate Gaussianrandom variable.^[1] As the sum of independent and Gaussian distributed random variables is again Gaussian distributed, that is the same as saying every linear combination of${\displaystyle (X_{t_1},\dots ,X_{t_k})}$ has a univariate Gaussian (or normal) distribution.

Usingcharacteristic functions of random variables with${\displaystyle i}$ denoting theimaginary unit such that${\displaystyle i^2=-1}$, the Gaussian property can be formulated as follows:${\displaystyle \left\{X_t;t\in T\right\}}$ is Gaussian if and only if, for every finite set of indices${\displaystyle t_1,\dots ,t_k}$, there are real-valued${\displaystyle {\sigma }_{\ell j}}$,${\displaystyle {\mu }_{\ell }}$ with${\displaystyle {\sigma }_{jj}>0}$ such that the following equality holds for all${\displaystyle s_1,s_2,\dots ,s_k\in \mathbb{R}}$,

$${\displaystyle \mathbb{E}\left[\exp \left(i\sum _{\ell =1}^k{s}_{\ell }{\mathbf{X}}_{t_{\ell }}\right)\right]=\exp \left(-\frac{1}{2}\sum _{\ell ,j}{\sigma }_{\ell j}{s}_{\ell }{s}_j+i\sum _{\ell }{\mu }_{\ell }{s}_{\ell }\right),}$$

or${\displaystyle \mathbb{E}\left[{\mathrm{e}}^{i\mathbf{s}({\mathbf{X}}_t-\mu )}\right]={\mathrm{e}}^{-\mathbf{s}\sigma \mathbf{s}/2}}$. The numbers${\displaystyle {\sigma }_{\ell j}}$ and${\displaystyle {\mu }_{\ell }}$ can be shown to be thecovariances andmeans of the variables in the process.^[2]

The variance of a Gaussian process is finite at any time${\displaystyle t}$, formally^{[3]: p. 515}

For general stochastic processesstrict-sense stationarity implieswide-sense stationarity but not every wide-sense stationary stochastic process is strict-sense stationary. However, for a Gaussian stochastic process the two concepts are equivalent.^{[3]: p. 518}

A Gaussian stochastic process is strict-sense stationary if and only if it is wide-sense stationary.

There is an explicit representation for stationary Gaussian processes.^[4] A simple example of this representation is

$${\displaystyle X_t=\cos (at){\xi }_1+\sin (at){\xi }_2}$$

where${\displaystyle {\xi }_1}$ and${\displaystyle {\xi }_2}$ are independent random variables with thestandard normal distribution.

A key fact of Gaussian processes is that they can be completely defined by their second-order statistics.^[5] Thus, if a Gaussian process is assumed to have mean zero, defining thecovariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using theKarhunen–Loève expansion. Basic aspects that can be defined through the covariance function are the process'stationarity,isotropy,smoothness andperiodicity.^[6][7]

Stationarity refers to the process' behaviour regarding the separation of any two points${\displaystyle x}$ and${\displaystyle x^{\prime }}$. If the process is stationary, the covariance function depends only on${\displaystyle x-x^{\prime }}$. For example, theOrnstein–Uhlenbeck process is stationary.

If the process depends only on${\displaystyle |x-x^{\prime }|}$, the Euclidean distance (not the direction) between${\displaystyle x}$ and${\displaystyle x^{\prime }}$, then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to behomogeneous;^[8] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.

Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function.^[6] If we expect that for "near-by" input points${\displaystyle x}$ and${\displaystyle x^{\prime }}$ their corresponding output points${\displaystyle y}$ and${\displaystyle y^{\prime }}$ to be "near-by" also, then the assumption of continuity is present. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable.

Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input${\displaystyle x}$ to a two dimensional vector${\displaystyle u(x)=\left(\cos (x),\sin (x)\right)}$.

There are a number of common covariance functions:^[7]

• Constant :${\displaystyle K_{\mathrm{C}}(x,x^{\prime })=C}$
• Linear:${\displaystyle K_{\mathrm{L}}(x,x^{\prime })=x^{\mathsf{T}}{x}^{\prime }}$
• white Gaussian noise:${\displaystyle K_{\mathrm{GN}}(x,x^{\prime })={\sigma }^2{\delta }_{x,x^{\prime }}}$
• Squared exponential:${\displaystyle K_{\mathrm{SE}}(x,x^{\prime })=\exp \left(-\frac{d^2}{2{\ell }^2}\right)}$
• Ornstein–Uhlenbeck:${\displaystyle K_{\mathrm{OU}}(x,x^{\prime })=\exp \left(-\frac{d}{\ell }\right)}$
• Matérn:${\displaystyle K_{\mathrm{Matern}}(x,x^{\prime })=\frac{2^{1-\nu }}{\mathrm{\Gamma }(\nu )}{\left(\frac{\sqrt{2\nu }d}{\ell }\right)}^{\nu }{K}_{\nu }\left(\frac{\sqrt{2\nu }d}{\ell }\right)}$
• Periodic:${\displaystyle K_{\mathrm{P}}(x,x^{\prime })=\exp \left(-\frac{2}{{\ell }^2}{\sin }^2(d/2)\right)}$
• Rational quadratic:${\displaystyle K_{\mathrm{RQ}}(x,x^{\prime })={\left(1+d^2\right)}^{-\alpha },\alpha \ge 0}$

Here${\displaystyle d=|x-x^{\prime }|}$, which is a result of the stationary process property, that is, for any stationary process the covariance function will only depend on${\displaystyle d}$. The parameter${\displaystyle \ell }$ is the characteristic length-scale of the process (practically, "how close" two points${\displaystyle x}$ and${\displaystyle x^{\prime }}$ have to be to influence each other significantly),${\displaystyle \delta }$ is theKronecker delta and${\displaystyle \sigma }$ thestandard deviation of the noise fluctuations. Moreover,${\displaystyle K_{\nu }}$ is themodified Bessel function of order${\displaystyle \nu }$ and${\displaystyle \mathrm{\Gamma }(\nu )}$ is thegamma function evaluated at${\displaystyle \nu }$. Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand.

The inferential results are dependent on the values of the hyperparameters${\displaystyle \theta }$ (e.g.${\displaystyle \ell }$ and${\displaystyle \sigma }$) defining the model's behaviour. A popular choice for${\displaystyle \theta }$ is to providemaximum a posteriori (MAP) estimates of it with some chosen prior. If the prior is very near uniform, this is the same as maximizing themarginal likelihood of the process; the marginalization being done over the observed process values${\displaystyle y}$.^[7] This approach is also known asmaximum likelihood II,evidence maximization, orempirical Bayes.^[9]

For a Gaussian process,continuity in probability is equivalent tomean-square continuity^{[10]: 145 : 91 "Gaussian processes are discontinuous at fixed points." [11]}andcontinuity with probability one is equivalent tosample continuity.^[12]The latter implies, but is not implied by, continuity in probability.Continuity in probability holds if and only if themean and autocovariance are continuous functions. In contrast, sample continuity was challenging even forstationary Gaussian processes (as probably noted first byAndrey Kolmogorov), and more challenging for more general processes.^{[13]: Sect. 2.8 [14]: 69, 81 [15]: 80 [16]}As usual, by a sample continuous process one means a process that admits a sample continuousmodification.^{[17]: 292 [18]: 424}

For a stationary Gaussian process${\displaystyle X=(X_t{)}_{t\in \mathbb{R}},}$ some conditions on its spectrum are sufficient for sample continuity, but fail to be necessary. A necessary and sufficient condition, sometimes called Dudley–Fernique theorem, involves the function${\displaystyle \sigma }$ defined by$${\displaystyle \sigma (h)=\sqrt{\mathbb{E}\left[{\left(X(t+h)-X(t)\right)}^2\right]}}$$(the right-hand side does not depend on${\displaystyle t}$ due to stationarity). Continuity of${\displaystyle X}$ in probability is equivalent to continuity of${\displaystyle \sigma }$ at${\displaystyle 0.}$ When convergence of${\displaystyle \sigma (h)}$ to${\displaystyle 0}$ (as${\displaystyle h\to 0}$) is too slow, sample continuity of${\displaystyle X}$ may fail. Convergence of the following integrals matters:$${\displaystyle I(\sigma )={\int }_0^1\frac{\sigma (h)}{h\sqrt{\log (1/h)}}dh={\int }_0^{\mathrm{\infty }}2\sigma (e^{-x^2})dx,}$$these two integrals being equal according tointegration by substitution$h=e^{-x^2},$$x=\sqrt{\log (1/h)}.$ The first integrand need not be bounded as${\displaystyle h\to 0+,}$ thus the integral may converge () or diverge (${\displaystyle I(\sigma )=\mathrm{\infty }}$). Taking for example$\sigma (e^{-x^2})=\frac{1}{x^a}$ for large${\displaystyle x,}$ that is,$\sigma (h)=(\log (1/h){)}^{-a/2}$ for small${\displaystyle h,}$ one obtains when${\displaystyle a>1,}$ and${\displaystyle I(\sigma )=\mathrm{\infty }}$ whenIn these two cases the function${\displaystyle \sigma }$ is increasing on${\displaystyle [0,\mathrm{\infty }),}$ but generally it is not. Moreover, the condition

(*)   there exists${\displaystyle \epsilon >0}$ such that${\displaystyle \sigma }$ is monotone on${\displaystyle [0,\epsilon ]}$

does not follow from continuity of${\displaystyle \sigma }$ and the evident relations${\displaystyle \sigma (h)\ge 0}$ (for all${\displaystyle h}$) and${\displaystyle \sigma (0)=0.}$

Some history.^[18]: 424Sufficiency was announced byXavier Fernique in 1964, but the first proof was published byRichard M. Dudley in 1967.^{[17]: Theorem 7.1}Necessity was proved by Michael B. Marcus andLawrence Shepp in 1970.^[19]: 380

There exist sample continuous processes${\displaystyle X}$ such that${\displaystyle I(\sigma )=\mathrm{\infty };}$ they violate condition(*). An example found by Marcus and Shepp^[19]: 387 is a randomlacunary Fourier series$${\displaystyle X_t=\sum _{n=1}^{\mathrm{\infty }}{c}_n({\xi }_n\cos {\lambda }_{n}t+{\eta }_n\sin {\lambda }_{n}t),}$$where${\displaystyle {\xi }_1,{\eta }_1,{\xi }_2,{\eta }_2,\dots }$ are independent random variables withstandard normal distribution; frequencies are a fast growing sequence; and coefficients${\displaystyle c_n>0}$ satisfy The latter relation implies

whence almost surely, which ensures uniform convergence of the Fourier series almost surely, and sample continuity of${\displaystyle X.}$

Its autocovariation function$${\displaystyle \mathbb{E}[X_t{X}_{t+h}]=\sum _{n=1}^{\mathrm{\infty }}{c}_n^2\cos {\lambda }_{n}h}$$is nowhere monotone (see the picture), as well as the corresponding function${\displaystyle \sigma ,}$$${\displaystyle \sigma (h)=\sqrt{2\mathbb{E}[X_t{X}_t]-2\mathbb{E}[X_t{X}_{t+h}]}=2\sqrt{\sum _{n=1}^{\mathrm{\infty }}{c}_n^2{\sin }^2\frac{{\lambda }_{n}h}{2}}.}$$

AWiener process (also known as Brownian motion) is the integral of awhite noise generalized Gaussian process. It is notstationary, but it hasstationary increments.

TheOrnstein–Uhlenbeck process is astationary Gaussian process.

TheBrownian bridge is (like the Ornstein–Uhlenbeck process) an example of a Gaussian process whose increments are notindependent.

Thefractional Brownian motion is a Gaussian process whose covariance function is a generalisation of that of the Wiener process.

Let${\displaystyle f}$ be a mean-zero Gaussian process${\displaystyle \left\{X_t;t\in T\right\}}$ with a non-negative definite covariance function${\displaystyle K}$ and let${\displaystyle R}$ be a symmetric and positive semidefinite function. Then, there exists a Gaussian process${\displaystyle X}$ which has the covariance${\displaystyle R}$. Moreover, thereproducing kernel Hilbert space (RKHS) associated to${\displaystyle R}$ coincides with theCameron–Martin theorem associated space${\displaystyle R(H)}$ of${\displaystyle X}$, and all the spaces${\displaystyle R(H)}$,${\displaystyle H_X}$, and${\displaystyle \mathcal{H}(K)}$ are isometric.^[20] From now on, let${\displaystyle \mathcal{H}(R)}$ be areproducing kernel Hilbert space with positive definite kernel${\displaystyle R}$.

Driscoll's zero-one law is a result characterizing the sample functions generated by a Gaussian process:where${\displaystyle K_n}$ and${\displaystyle R_n}$ are the covariance matrices of all possible pairs of${\displaystyle n}$ points, implies$${\displaystyle Pr[f\in \mathcal{H}(R)]=1.}$$

Moreover,$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mathrm{tr}[K_n{R}_n^{-1}]=\mathrm{\infty }}$$implies^[21]$${\displaystyle Pr[f\in \mathcal{H}(R)]=0.}$$

This has significant implications when${\displaystyle K=R}$, as$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mathrm{tr}[R_n{R}_n^{-1}]=\underset{n\to \mathrm{\infty }}{lim}\mathrm{tr}[I]=\underset{n\to \mathrm{\infty }}{lim}n=\mathrm{\infty }.}$$

As such, almost all sample paths of a mean-zero Gaussian process with positive definite kernel${\displaystyle K}$ will lie outside of the Hilbert space${\displaystyle \mathcal{H}(K)}$.

For many applications of interest some pre-existing knowledge about the system at hand is already given. Consider e.g. the case where the output of the Gaussian process corresponds to a magnetic field; here, the real magnetic field is bound by Maxwell's equations and a way to incorporate this constraint into the Gaussian process formalism would be desirable as this would likely improve the accuracy of the algorithm.

A method on how to incorporate linear constraints into Gaussian processes already exists:^[22]

Consider the (vector valued) output function${\displaystyle f(x)}$ which is known to obey the linear constraint (i.e.${\displaystyle {\mathcal{F}}_X}$ is a linear operator)$${\displaystyle {\mathcal{F}}_X(f(x))=0.}$$Then the constraint${\displaystyle {\mathcal{F}}_X}$ can be fulfilled by choosing${\displaystyle f(x)={\mathcal{G}}_X(g(x))}$, where${\displaystyle g(x)\sim \mathcal{G}\mathcal{P}({\mu }_g,K_g)}$ is modelled as a Gaussian process, and finding${\displaystyle {\mathcal{G}}_X}$ such that$${\displaystyle {\mathcal{F}}_X({\mathcal{G}}_X(g))=0\mathrm{\forall }g.}$$Given${\displaystyle {\mathcal{G}}_X}$ and using the fact that Gaussian processes are closed under linear transformations, the Gaussian process for${\displaystyle f}$ obeying constraint${\displaystyle {\mathcal{F}}_X}$ becomes$${\displaystyle f(x)={\mathcal{G}}_{X}g\sim \mathcal{G}\mathcal{P}({\mathcal{G}}_X{\mu }_g,{\mathcal{G}}_X{K}_g{\mathcal{G}}_{X^{\prime }}^{\mathsf{T}}).}$$Hence, linear constraints can be encoded into the mean and covariance function of a Gaussian process.

A Gaussian process can be used as aprior probability distribution overfunctions inBayesian inference.^[7][24] Given any set ofN points in the desired domain of the functions, take amultivariate Gaussian whose covariancematrix parameter is theGram matrix of thoseN points with some desiredkernel, andsample from that Gaussian. For solution of the multi-output prediction problem, Gaussian process regression for vector-valued function was developed. In this method, a 'big' covariance is constructed, which describes the correlations between all the input and output variables taken inN points in the desired domain.^[25] This approach was elaborated in detail for the matrix-valued Gaussian processes and generalised to processes with 'heavier tails' likeStudent-t processes.^[26]

Inference of continuous values with a Gaussian process prior is known as Gaussian process regression, orkriging; extending Gaussian process regression tomultiple target variables is known ascokriging.^[27] Gaussian processes are thus useful as a powerful non-linear multivariateinterpolation tool. Kriging is also used to extend Gaussian process in the case of mixed integer inputs.^[28]

Gaussian processes are also commonly used to tackle numerical analysis problems such as numerical integration, solving differential equations, or optimisation in the field ofprobabilistic numerics.

Gaussian processes can also be used in the context of mixture of experts models, for example.^[29][30] The underlying rationale of such a learning framework consists in the assumption that a given mapping cannot be well captured by a single Gaussian process model. Instead, the observation space is divided into subsets, each of which is characterized by a different mapping function; each of these is learned via a different Gaussian process component in the postulated mixture.

When concerned with a general Gaussian process regression problem (Kriging), it is assumed that for a Gaussian process${\displaystyle f}$ observed at coordinates${\displaystyle x}$, the vector of values⁠${\displaystyle f(x)}$⁠ is just one sample from a multivariate Gaussian distribution of dimension equal to number of observed coordinates⁠${\displaystyle n}$⁠. Therefore, under the assumption of a zero-mean distribution,⁠${\displaystyle f(x^{\prime })\sim N(0,K(\theta ,x,x^{\prime }))}$⁠, where⁠${\displaystyle K(\theta ,x,x^{\prime })}$⁠ is the covariance matrix between all possible pairs⁠${\displaystyle (x,x^{\prime })}$⁠ for a given set of hyperparametersθ.^[7]As such the log marginal likelihood is:

$${\displaystyle \log p(f(x^{\prime })\mid \theta ,x)=-\frac{1}{2}\left(f(x{)}^{\mathsf{T}}K(\theta ,x,x^{\prime }{)}^{-1}f(x^{\prime })+\log det(K(\theta ,x,x^{\prime }))+n\log 2\pi \right)}$$

and maximizing this marginal likelihood towardsθ provides the complete specification of the Gaussian processf. One can briefly note at this point that the first term corresponds to a penalty term for a model's failure to fit observed values and the second term to a penalty term that increases proportionally to a model's complexity. Having specifiedθ, making predictions about unobserved values⁠${\displaystyle f(x^{\ast })}$⁠ at coordinatesx* is then only a matter of drawing samples from the predictive distribution${\displaystyle p(y^{\ast }\mid x^{\ast },f(x),x)=N(y^{\ast }\mid A,B)}$ where the posterior mean estimateA is defined as$${\displaystyle A=K(\theta ,x^{\ast },x)K(\theta ,x,x^{\prime }{)}^{-1}f(x)}$$and the posterior variance estimateB is defined as:$${\displaystyle B=K(\theta ,x^{\ast },x^{\ast })-K(\theta ,x^{\ast },x)K(\theta ,x,x^{\prime }{)}^{-1}K(\theta ,x^{\ast },x{)}^{\mathsf{T}}}$$where⁠${\displaystyle K(\theta ,x^{\ast },x)}$⁠ is the covariance between the new coordinate of estimationx* and all other observed coordinatesx for a given hyperparameter vectorθ,⁠${\displaystyle K(\theta ,x,x^{\prime })}$⁠ and⁠${\displaystyle f(x)}$⁠ are defined as before and⁠${\displaystyle K(\theta ,x^{\ast },x^{\ast })}$⁠ is the variance at pointx* as dictated byθ. It is important to note that practically the posterior mean estimate of⁠${\displaystyle f(x^{\ast })}$⁠ (the "point estimate") is just a linear combination of the observations⁠${\displaystyle f(x)}$⁠; in a similar manner the variance of⁠${\displaystyle f(x^{\ast })}$⁠ is actually independent of the observations⁠${\displaystyle f(x)}$⁠. A known bottleneck in Gaussian process prediction is that the computational complexity of inference and likelihood evaluation is cubic in the number of points |x|, and as such can become unfeasible for larger data sets.^[6][31] Works on sparse Gaussian processes, that usually are based on the idea of building arepresentative set for the given processf, try to circumvent this issue.^[32][33][34] Thekriging method can be used in the latent level of anonlinear mixed-effects model for a spatial functional prediction: this technique is called the latent kriging.^[35] Other classes of scalable Gaussian process for analyzing massive datasets have emerged from theVecchia approximation and Nearest Neighbor Gaussian Processes (NNGP).^[36][31]

Often, the covariance has the form$K(\theta ,x,x^{\prime })=\frac{1}{{\sigma }^2}\tilde{K}(\theta ,x,x^{\prime })$, where${\displaystyle {\sigma }^2}$ is a scaling parameter. Examples are the Matérn class covariance functions. If this scaling parameter${\displaystyle {\sigma }^2}$ is either known or unknown (i.e. must be marginalized), then the posterior probability,${\displaystyle p(\theta \mid D)}$, i.e. the probability for the hyperparameters${\displaystyle \theta }$ given a set of data pairs${\displaystyle D}$ of observations of${\displaystyle x}$ and${\displaystyle f(x)}$, admits an analytical expression.^[37]

Bayesian neural networks are a particular type ofBayesian network that results from treatingdeep learning andartificial neural network models probabilistically, and assigning aprior distribution to theirparameters. Computation in artificial neural networks is usually organized into sequential layers ofartificial neurons. The number of neurons in a layer is called the layer width. As layer width grows large, many Bayesian neural networks reduce to a Gaussian process with aclosed form compositional kernel. This Gaussian process is called the Neural Network Gaussian Process (NNGP) (not to be confused with the Nearest Neighbor Gaussian Process^[36]).^[7][38][39] It allows predictions from Bayesian neural networks to be more efficiently evaluated, and provides an analytic tool to understanddeep learning models.

Gaussian processes have found increasing applications in many domains of the natural sciences due to their statistical modelling properties.Molecular property prediction has employed these process models in small molecular datasets due to their inference capabilities and computational costs.^[40][41] They are also being increasingly used as surrogate models for force field optimization.^[42]

Gaussian processes have also found extensive use inastrophysical and astronomical settings. Gaussian processes can model correlated noise, a specific type of non-Gaussian noise dependent on some underlying unknown distribution correlated with observed values. This type of noise is often present in astronomical signals as instrumental systematics or as intrinsic to the observed object as a result of physical processes. Correlated noise is often a consideration for exoplanettransit events, and Gaussian processes have been used to de-trend these transitlight curves (at timescales greater than that of the transit) to allow detection of weaker, more short-lived signals.^[43] These processes have also been used to disentangle planetary signals from stellar activity indicators insideradial velocity data, another method of exoplanet detection. This is done by training the Gaussian process model to optimize the hyperparameters of the kernel until it accurately recreates the noise components of the radial velocity data, which ultimately allows it to determine which signals are best defined as strictly periodic (which the planet should be) and which signals are best represented by the evolving, quasi-periodic kernel (which the star should be).^[44] Correlated noise produced by active regions on a star'sphotosphere (as a result ofmagnetic field interactions) can be of similar timescales as transit events, and Gaussian process models which handle sparsely sampled data are used to confirm exoplanet detections especially around young stars.^[45][46]

The variability of rotating, magnetically active Sun-like stars can be modeled fairly accurately using Gaussian processes.^[45] This quasi-periodic variability is often represented by a covariance function given as^[47][48]$${\displaystyle {\text{K}}_{\text{QP}}(x,x^{\prime })={\alpha }^2\exp \left(-\frac{d^2}{2{\lambda }_1^2}-\mathrm{\Gamma }{\sin }^2\left[\frac{\pi d}{{\lambda }_2}\right]\right)}$$where parameter${\displaystyle \alpha }$ is amplitude,${\displaystyle {\lambda }_1}$ is period, and${\displaystyle {\lambda }_2}$ is decoherence timescale. This covariance function allows for the limited but feasible determination of stellar periods as a result of parameter${\displaystyle {\lambda }_1}$ but lacks physical information about where these active regions are on the star observed.^[45][49]

Gaussian processes are also used in the analysis of individual and populations ofactive galactic nuclei (AGNs) due to their stochastic variability in the optical and radio parts of theelectromagnetic spectrum.^[45] Damped random walk kernels in particular have previously been used to identify the extent ofbroad-line emission regions aroundsupermassive black holes usingreverberation mapping, and these kernels can also be used to characterize light curve variations for large AGN populations.^[50][51]

Other astrophysical applications of Gaussian processes include models forpulsar timing and dispersion measure,gravitational wave structure and detector uncertainty (notably in theLIGO-Virgo-KAGRA collaboration), transient classification, and quasi-periodic oscillations.^{[45][52][53][54][55]}

In practical applications, Gaussian process models are often evaluated on a grid leading to multivariate normal distributions. Using these models for prediction or parameter estimation using maximum likelihood requires evaluating a multivariate Gaussian density, which involves calculating the determinant and the inverse of the covariance matrix. Both of these operations have cubic computational complexity which means that even for grids of modest sizes, both operations can have a prohibitive computational cost. This drawback led to the development of multipleapproximation methods.^[31]

• Bayes linear statistics
• Bayesian interpretation of regularization
• Kriging
• Gaussian free field
• Gauss–Markov process
• Gradient-enhanced kriging (GEK)
• Student's t-process

Wikibooks has a book on the topic of:Gaussian process

• The Gaussian Processes Web Site, including the text of Rasmussen and Williams' Gaussian Processes for Machine Learning
• Ebden, Mark (2015). "Gaussian Processes: A Quick Introduction".arXiv:1505.02965 [math.ST].
• A Review of Gaussian Random Fields and Correlation Functions
• Efficient Reinforcement Learning using Gaussian Processes

• GPML: A comprehensive Matlab toolbox for GP regression and classification
• STK: a Small (Matlab/Octave) Toolbox for Kriging and GP modeling
• Kriging module in UQLab framework (Matlab)
• CODES Toolbox: implementations of Kriging, variational kriging and multi-fidelity models (Matlab)^{[permanent dead link]}
• Matlab/Octave function for stationary Gaussian fields^{[permanent dead link]}
• Yelp MOE – A black box optimization engine using Gaussian process learning
• ooDACEArchived 2020-08-09 at theWayback Machine – A flexible object-oriented Kriging Matlab toolbox.
• GPstuff – Gaussian process toolbox for Matlab and Octave
• GPy – A Gaussian processes framework in Python
• GSTools - A geostatistical toolbox, including Gaussian process regression, written in Python
• Interactive Gaussian process regression demo
• Basic Gaussian process library written in C++11
• scikit-learn – A machine learning library for Python which includes Gaussian process regression and classification
• SAMBO Optimization library for Python supports sequential optimization driven by Gaussian process regressor fromscikit-learn.
• [1] - The Kriging toolKit (KriKit) is developed at the Institute of Bio- and Geosciences 1 (IBG-1) of Forschungszentrum Jülich (FZJ)

• Gaussian Process Basics by David MacKay
• Learning with Gaussian Processes by Carl Edward Rasmussen
• Bayesian inference and Gaussian processes by Carl Edward Rasmussen

• Stochastic processes
• Kernel methods for machine learning
• Nonparametric Bayesian statistics
• Normal distribution

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