Introducing inequality constraints in Gaussian process (GP) models can lead to more realistic uncertainties in learning a great variety of real-world problems. We consider the finite-dimensional Gaussian approach from Maatouk and Bay (2017) which can satisfy inequality conditions everywhere (either boundedness, monotonicity or convexity). Our contributions are threefold. First, we extend their approach in order to deal with general sets of linear inequalities. Second, we explore several Markov Chain Monte Carlo (MCMC) techniques to approximate the posterior distribution. Third, we investigate theoretical and numerical properties of the constrained likelihood for covariance parameter estimation. According to experiments on both artificial and real data, our full framework together with a Hamiltonian Monte Carlo-based sampler provides efficient results on both data fitting and uncertainty quantification.
Gaussian process models -also called Kriging models-are often used as mathematical approximations of expensive experiments. However, the number of observation required for building an emulator becomes unrealistic when using classical covariance kernels when the dimension of input increases. In oder to get round the curse of dimensionality, a popular approach is to consider simplified models such as additive models. The ambition of the present work is to give an insight into covariance kernels that are well suited for building additive Kriging models and to describe some properties of the resulting models.Résumé. -La modélisation par processus gaussiens -aussi appelée krigeage-est souvent utilisée pour obtenir une approximation mathémathique d'une fonction dont l'évaluation est coûteuse. Cependant, le nombre d'évaluations nécessaires pour construire un modèle basé sur des noyaux de covariance usuels devient démesuré lorsque la dimension des variables d'entrée augmente. Afin de contourner le fléau de la dimension, une alternative bien connue est de considérer des modèles simplifiés comme les modèles additifs. Nous présentons ici une classe de noyaux de covariance adaptéè a la construction de modèles de krigeage additifs et nous décrivons certaines propriété des modèles obtenus.
Given a reproducing kernel Hilbert space (H, ., . ) of real-valued functions and a suitable measure µ over the source space D ⊂ R, we decompose H as the sum of a subspace of centered functions for µ and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the effect of each (group of) variable(s) and computing sensitivity indices without recursivity.
This work falls within the context of predicting the value of a real function at some input locations given a limited number of observations of this function. The Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such a problem but the method suffers from its computational burden when the number of observation points is large. We introduce in this article nested Kriging predictors which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. Contrarily to some other methods it can be shown that the proposed aggregation method is consistent. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with 10 4 observations in a 6-dimensional space.
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