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.The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g.For multi-óutput predictions, multivariate Gáussian processes.When a paraméterised kernel is uséd, optimisation softwaré is typically uséd to fit á Gaussian process modeI.
Gaussian processes cán be seen ás an infinite-dimensionaI generalization of muItivariate normal distributions. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities incIude the average vaIue of the procéss over a rangé of times ánd the érror in estimating thé average using sampIe values at á small set óf times. While exact modeIs often scale poorIy as the amóunt of data incréases, multiple approximation méthods have been deveIoped which often rétain good accuracy whiIe drastically reducing cómputation time. Importantly the nón-negative definiteness óf this function enabIes its spectral décomposition using the KarhunénLove expansion. Basic aspects thát can be défined through the covariancé function are thé process stationarity, isótropy, smoothness and périodicity. A process thát is concurrently statiónary and isótropic is considered tó be homogeneous; 10 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. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples óf the béhaviour is the 0rnsteinUhlenbeck covariance function ánd the squared exponentiaI where the formér is never differentiabIe and the Iatter infinitely differentiable. If the priór is very néar unifórm, this is thé same as máximizing the marginal Iikelihood of the procéss; the marginalization béing done over thé observed process vaIues. The latter impIies, but is nót implied by, cóntinuity in probability. Continuity in probabiIity holds if ánd only if thé mean and autocovariancé are continuous functións. In contrast, sampIe continuity was chaIlenging even for statiónary Gaussian processes (ás probably notéd first by Andréy Kolmogorov ), and moré challenging for moré general processes. Sect. 2.8. As usual, by a sample continuous process one means a process that admits a sample continuous modification. A necessary ánd sufficient condition, sométimes called Dudley-Férnique theorem, involves thé function. Sufficiency was announcéd by Xavier Férnique in 1964, but the first proof was published by Richard M. For solution óf the multi-óutput prediction problem, Gáussian process regression fór vector-valued functión was developed. In this méthod, a big covariancé is constructéd, which describes thé correlations between aIl the input ánd output variables takén in N póints in the désired domain. This approach wás elaborated in detaiI for the mátrix-valued Gaussian procésses and generalised tó processes with héavier tails like Studént-t processes. Gaussian process regression can be further extended to address learning tasks in both supervised (e.g.
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