Argumentwise invariant kernels for the approximation of invariant functions
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 3, pp. 501-527.

Nous considérons le problème d’approximation par méthodes à noyaux de fonctions invariantes sous l’action d’un groupe fini. Nous introduisons les noyaux doublement invariants, et montrons qu’ils caractérisent les champs aléatoires centrés de carré intégrable à trajectoires invariantes, ainsi que les espaces de Hilbert à noyau reproduisant de fonctions invariantes. Deux classes particulières de noyaux doublement invariants sont considérées, basées respectivement sur un domaine fondamental ou sur une double somme sur les orbites. Nous établissons ensuite des propriétés d’invariance pour les modèles de Krigeage et les simulations consitionnelles associés. L’applicabilité et les avantages de tels noyaux sont illustrés sur plusieurs exemples, incluant une fonction symétrique issue d’un problème de fiabilité des structures.

We consider the problem of designing adapted kernels for approximating functions invariant under a known finite group action. We introduce the class of argumentwise invariant kernels, and show that they characterize centered square-integrable random fields with invariant paths, as well as Reproducing Kernel Hilbert Spaces of invariant functions. Two subclasses of argumentwise kernels are considered, involving a fundamental domain or a double sum over orbits. We then derive invariance properties for Kriging and conditional simulation based on argumentwise invariant kernels. The applicability and advantages of argumentwise invariant kernels are demonstrated on several examples, including a symmetric function from the reliability literature.

DOI : 10.5802/afst.1343

David Ginsbourger 1 ; Xavier Bay 2 ; Olivier Roustant 2 ; Laurent Carraro 3

1 University of Bern, Institute of Mathematical Statistics and Actuarial Science, Alpeneggstrasse 22, CH-3012 Bern, Switzerland
2 École Nationale Supérieure des Mines, Fayol-EMSE, LSTI, 158 cours Fauriel, F-42023 Saint-Etienne, France
3 Télécom Saint-Etienne, 25 rue du Docteur Rémy Annino, F-42000 Saint-Etienne, France
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David Ginsbourger; Xavier Bay; Olivier Roustant; Laurent Carraro. Argumentwise invariant kernels for the approximation of invariant functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 3, pp. 501-527. doi : 10.5802/afst.1343. https://afst.centre-mersenne.org/articles/10.5802/afst.1343/

[1] Abrahamsen (P.).— A review of Gaussian random fields and correlation functions, second edition. Technical report, Norwegian Computing Center (1997).

[2] Adler (R.J.) and Taylor (J.E.).— Random Fields and Geometry. Springer, Boston (2007). | MR | Zbl

[3] Amari (S.) and Nagaoka (H.).— Transactions of mathematical monographs: methods of information geometry, volume 191. Oxford University Press (2000). | MR | Zbl

[4] Anderes (E.B.).— Estimating deformations of isotropic Gaussian random fields. PhD thesis, Univ. Chicago (2005). | MR

[5] Anderes (E.B.) and Chatterjee (S.).— Consistent estimates of deformed isotropic Gaussian random fields on the plane. The Annals of Statistics, 37, No. 5A, p. 2324-2350 (2009). | MR | Zbl

[6] Anderes (E.B.) and Stein (M.L.).— Estimating deformations of isotropic Gaussian random fields on the plane. The Annals of Statistics, Vol. 36, No. 2, p. 719-741 (2008). | MR | Zbl

[7] Bect (J.), Ginsbourger (D.), Li (L.), Picheny (V.) and Vazquez (E.).— Sequential design of computer experiments for the estimation of a probability of failure. Statistics and Computing, Vol. 22, 3, p. 773-793 (2012).

[8] Bekka (B.), de la Harpe (P.) and Valette (A.).— Kazhdan’s property (T). Cambridge University Press (2008). | MR | Zbl

[9] Berg (C.), Christensen (J.P.R.) and Ressel (P.).— Harmonic Analysis on Semigroups. Springer (1984). | MR | Zbl

[10] Berlinet (A.) and Thomas Agnan (C.).— Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Academic Publishers (2003). | MR | Zbl

[11] Cartan (H.) and Godement (R.).— Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts. Annales Scientifiques de l’Ecole Normale Supérieure, 64, p. 79-99 (1947). | Numdam | MR | Zbl

[12] Cramér (H.) and Leadbetter (M.R.).— Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. Dover (1967). | MR | Zbl

[13] Cressie (N.A.C.).— Statistics for spatial data. Wiley series in probability and mathematical statistics (1993). | MR | Zbl

[14] Curriero (F.C.).— On the use of non-euclidean isotropy in geostatistics. Technical report, Johns Hopkins University, 2005.

[15] Eaton (M.L.).— Group invariance applications in statistics. Regional Conference Series in Probability and Statistics Volume 1, (1989). | MR | Zbl

[16] Echard (B.), Gayton (N.) and Lemaire (M.).— Kriging-based monte carlo simulation to compute the probability of failure efficiently: AK-MCS method. In 6èmes Journées Nationales de Fiabilité, 24-26 mars, Toulouse, France (2010).

[17] Gaetan (C.) and Guyon (X.).— Modélisation et Statistique Spatiales. Springer (2008). | MR

[18] Genton (M.G.).— Classes of kernels for machine learning: A statistics perspective. Journal of Machine Learning Research, 2, p. 299-312 (2001). | MR | Zbl

[19] Gramacy (R.B.) and Lee (H.K.H.).— Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103, p. 1119-1130 (2008). | MR | Zbl

[20] Guttorp (P.) and Sampson (P.D.).— Nonparametric estimation of nonstationary spatial covariance structure. Journal of the American Statistical Association, 87, No. 417, p. 108-119 (1992).

[21] Hastie (T.), Tibshirani (R.) and Friedman (J.).— The Elements of Statistical Learning. Springer (2001). | MR | Zbl

[22] Hristopoulos (D.T.).— Spatial random field models inspired from statistical physics with applications in the geosciences. Physica A, (365), p. 211-216, June (2006).

[23] Istas (J.).— Manifold indexed fractional fields. ESAIM P&S (2011).

[24] Jones (D.R.), Schonlau (M.) and Welch (W.J.).— Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13, p. 455-492 (1998). | MR | Zbl

[25] Krige (D.G.).— A statistical approach to some basic mine valuation problems on the witwatersrand. J. of the Chem., Metal. and Mining Soc. of South Africa, 52 (6), p. 119-139 (1951).

[26] Lang (S.).— Algebra. Addison-Wesley, Reading, Mass. (1965). | MR | Zbl

[27] Lee (H.), Higdon (D.), Calder (K.) and Holloman (C.).— Spatial models via convolutions of intrinsic processes. Statistical Modelling, 5, p. 1-21 (2005).

[28] Lévy (P.).— Processus stochastiques et mouvement Brownien (1965).

[29] Matheron (G.).— La théorie des variables régionalisées et ses applications. Technical report, Centre de Morphologie Mathématique de Fontainebleau, École Nationale Supérieure des Mines de Paris (1970).

[30] O’Hagan (A.).— Bayesian analysis of computer code outputs: a tutorial. Reliability Engineering and System Safety, 91(91), p. 1290-1300 (2006).

[31] Paciorek (C.J.) and Schervish (M.J.).— Nonstationary covariance functions for Gaussian process regression. In Advances in Neural Information Processing Systems 16 (NIPS) (2004).

[32] Paciorek (C.J.).— Nonstationary Gaussian Processes for Regression and Spatial Modelling. PhD thesis, Carnegie Mellon University (2003). | MR

[33] Parthasarathy (K.R.) and Schmidt (K.).— Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory. Lecture Notes in Mathematics, Springer (1972). | MR | Zbl

[34] Rasmussen (C.E.) and Williams (K.I.).— Gaussian Processes for Machine Learning. M.I.T. Press (2006). | MR | Zbl

[35] Revuz (D.) and Yor (M.).— Continuous Martingales and Brownian Motion. Springer-Verlag (1991). | MR | Zbl

[36] Santner (T.J.), Williams (B.J.) and Notz (W.J.).— The Design and Analysis of Computer Experiments. Springer (2003). | MR | Zbl

[37] Schmidt (A.M.) and O’Hagan (A.).— Bayesian inference for non-stationary spatial covariance structure via spatial deformations. Journal of the Royal Statistical Society B, 65(3), p. 745-758 (2003). | MR | Zbl

[38] Schoenberg (I.J.).— Metric spaces and positive definite functions. Transactions of the American Mathematical Society, 44(3), p. 522-536 (1938). | MR | Zbl

[39] Schölkopf (B.).— The kernel trick for distances. In Neural Information Processing Systems (2000).

[40] Schueremans (L.) and Van Gemert (D.).— Benefit of splines and neural networks in simulation based structural reliability analysis. Structural safety, 27(3), p. 246-261 (2005).

[41] Stein (M.L.).— Interpolation of Spatial Data, Some Theory for Kriging. Springer (1999). | MR | Zbl

[42] Vapnik (V.N.).— Statistical Learning Theory. Wiley-Interscience (1998). | MR | Zbl

[43] Vazquez (E.).— Modélisation Comportementale de Systèmes Non-linéaires Multivariables par Méthodes à Noyaux et Applications. PhD thesis, Université Paris XI Orsay (2005).

[44] Vert (J.-P.).— Kernel Methods. Centre for Computational Biology, Ecole des Mines de Paris (2007).

[45] Waarts (P.H.).— Structural reliability using finite element methods: an appraisal of DARS. PhD thesis, Delft University of Technology (2000).

[46] Wahba (G.).— Spline Models for Observational Data. Siam (1990). | MR | Zbl

[47] Xiong (Y.), Chen (W.), Apley (D.), and Ding (X.).— A non-stationary covariance-based kriging method for metamodelling in engineering design. International Journal of Numerical Methods in Engineering, 71, p. 733-756 (2007). | Zbl

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