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 peut devenir démesuré lorsque la dimension du domaine de définition augmente. Afin de contourner le fléau de la dimension, une alternative bien connue est de se tourner vers des modèles simplifiés comme les modèles additifs. Nous présentons ici une famille de noyaux de covariance permettant de combiner les caractéristiques des modèles de krigeage et les avantages des modèles additifs puis nous décrivons certaines propriétés des modèles obtenus.
Gaussian Process models are often used for predicting and approximating expensive experiments. However, the number of observations required for building such models may become unrealistic when the input dimension increases. In oder to avoid the curse of dimensionality, a popular approach in multivariate smoothing is to make simplifying assumptions like additivity. The ambition of the present work is to give an insight into a family of covariance kernels that allows combining the features of Gaussian Process modeling with the advantages of generalized additive models, and to describe some properties of the resulting models.
Nicolas Durrande 1 ; David Ginsbourger 2 ; Olivier Roustant 3
@article{AFST_2012_6_21_3_481_0,
author = {Nicolas Durrande and David Ginsbourger and Olivier Roustant},
title = {Additive {Covariance} kernels for high-dimensional {Gaussian} {Process} modeling},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {481--499},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {3},
year = {2012},
doi = {10.5802/afst.1342},
mrnumber = {3076409},
zbl = {1266.60068},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1342/}
}
TY - JOUR AU - Nicolas Durrande AU - David Ginsbourger AU - Olivier Roustant TI - Additive Covariance kernels for high-dimensional Gaussian Process modeling JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 481 EP - 499 VL - 21 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1342/ DO - 10.5802/afst.1342 LA - en ID - AFST_2012_6_21_3_481_0 ER -
%0 Journal Article %A Nicolas Durrande %A David Ginsbourger %A Olivier Roustant %T Additive Covariance kernels for high-dimensional Gaussian Process modeling %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 481-499 %V 21 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1342/ %R 10.5802/afst.1342 %G en %F AFST_2012_6_21_3_481_0
Nicolas Durrande; David Ginsbourger; Olivier Roustant. Additive Covariance kernels for high-dimensional Gaussian Process modeling. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 3, pp. 481-499. doi : 10.5802/afst.1342. https://afst.centre-mersenne.org/articles/10.5802/afst.1342/
[1] Azaïs (J.M.) and Wschebor (M.).— Level sets and extrema of random processes and fields, Wiley Online Library (2009). | MR | Zbl
[2] Bach (F.).— Exploring large feature spaces with hierarchical multiple kernel learning, Arxiv preprint arXiv:0809.1493 (2008).
[3] Buja (A.), Hastie (T.) and Tibshirani (R.).— Linear smoothers and additive models, The Annals of Statistics, p. 453-510 (1989). | MR | Zbl
[4] Chilès (J.P.) and Delfiner (P.).— Geostatistics: modeling spatial uncertainty, volume 344, Wiley-Interscience (1999). | MR
[5] Cressie (N.).— Statistics for spatial data, Terra Nova, 4(5), p. 613-617 (1992). | MR | Zbl
[6] Fang (K.).— Design and modeling for computer experiments, volume 6. CRC Press (2006). | MR | Zbl
[7] Fortet (R.M.).— Les operateurs integraux dont le noyau est une covariance, Trabajos de estadística y de investigación operativa, 36(3), p. 133-144 (1985). | Zbl
[8] Gaetan (C.) and Guyon (X.).— Spatial statistics and modeling, Springer Verlag (2009). | MR
[9] Ginsbourger (D.), Dupuy (D.), Badea (A.), Carraro (L.) and Roustant (O.).— A note on the choice and the estimation of kriging models for the analysis of deterministic computer experiments, Applied Stochastic Models in Business and Industry, 25(2), p. 115-131 (2009). | MR | Zbl
[10] Gunn (S.R.) and Brown (M.).— Supanova: A sparse, transparent modelling approach, In Neural Networks for Signal Processing IX, 1999, Proceedings of the 1999 IEEE Signal Processing Society Workshop, p. 21-30. IEEE (1999).
[11] Hastie (T.).— gam: Generalized Additive Models, 2011, R package version 1.04.1.
[12] Hastie (T.J.) and Tibshirani (R.J.).— Generalized additive models, Chapman & Hall/CRC (1990). | MR | Zbl
[13] Loeppky (J.L.), Sacks (J.) and Welch (W.J.).— Choosing the sample size of a computer experiment: A practical guide, Technometrics, 51(4), p. 366-376 (2009). | MR
[14] Muehlenstaedt (T.), Roustant (O.), Carraro (L.) and Kuhnt (S.).— Data-driven Kriging models based on FANOVA-decomposition, to appear in Statistics and Computing.
[15] Newey (W.K.).— Kernel estimation of partial means and a general variance estimator, Econometric Theory, 10(02), p. 1-21 (1994). | MR
[16] Rasmussen (C.E.) and Williams (C.K.I.).— Gaussian processes for machine learning (2005). | MR
[17] Roustant (O.), Ginsbourger (D.) and Deville (Y.).— DiceKriging: Kriging methods for computer experiments, 2011, R package version 1.3.
[18] Saltelli (A.), Chan (K.), Scott (E.M.) et al.— Sensitivity analysis, volume 134, Wiley New York (2000). | MR | Zbl
[19] Santner (T.J.), Williams (B.J.) and Notz (W.).— The design and analysis of computer experiments, Springer Verlag (2003). | MR | Zbl
[20] Sobol (I.M.).— Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates, Mathematics and Computers in Simulation, 55(1-3), p. 271-280, (2001). | MR | Zbl
[21] Stone (C.J.).— Additive regression and other nonparametric models, The annals of Statistics, p. 689-705 (1985). | MR | Zbl
[22] R Team.— R: A language and environment for statistical computing, R Foundation for Statistical Computing Vienna Austria ISBN, 3(10) (2008).
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