Small Deviations Asymptotics for Matérn Processes and Fields under Weighted Quadratic Norm

Pusev, R. S.

doi:10.1137/s0040585x97984723

Cited by 6 publications

(4 citation statements)

References 9 publications

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“…where K ν is the modified Bessel function [22]. The eigenvalues of this kernel satisfy the following asymptotic behavior [23]:…”

Section: Asymptotic Normality With D-tensorised Matérn-ν Kernelsmentioning

confidence: 99%

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Asymptotic normality of a Sobol index estimator in Gaussian process regression framework

Gratiet¹

2013

Preprint

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Stochastic simulators such as Monte-Carlo estimators are widely used in science and engineering to study physical systems through their probabilistic representation. Global sensitivity analysis aims to identify the input parameters which have the most important impact on the output. A popular tool to perform global sensitivity analysis is the variance-based method which comes from the Hoeffding-Sobol decomposition. Nevertheless, this method requires an important number of simulations and is often unfeasible under reasonable time constraint. Therefore, an approximation of the input/output relation of the code is built with a Gaussian process regression model. This paper provides conditions which ensure the asymptotic normality of a Sobol's index estimator evaluated through this surrogate model. This result allows for building asymptotic confidence intervals for the considered Sobol index estimator. The presented method is successfully applied on an academic example on the heat equation.

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“…where K ν is the modified Bessel function [22]. The eigenvalues of this kernel satisfy the following asymptotic behavior [23]:…”

Section: Asymptotic Normality With D-tensorised Matérn-ν Kernelsmentioning

confidence: 99%

“…In practice, we want to minimize the budget allocated to the simulator and thus consider the case α tends to zero. As a consequence, for applications we will consider the allocation of the critical point (23).…”

Section: Asymptotic Normality With D-tensorised Matérn-ν Kernelsmentioning

confidence: 99%

Asymptotic normality of a Sobol index estimator in Gaussian process regression framework

Gratiet¹

2013

Preprint

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“…4]), derived from W (t) by alternate operations of centering and integration; 12) s-times centered-integrated Brownian bridge (see [9,Sec. 3]); 13) the Matern process M (s+1) (see [35], [36]) with covariance…”

Section: Small Ball Asymptotics Examplesmentioning

confidence: 99%

“…(11) s-times-centered-and-integrated Wiener process (see [20,Section 4]), derived from W (t) by alternate operations of centering and integration; (12) s-times-centered-and-integrated Brownian bridge (see [20,Section 3]); (13) the Matern process M (s+1) (see [26,27]) with covariance…”

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confidence: 99%

Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

Nazarov

Sheipak

2011

Bulletin of the London Mathematical Society

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Asymptotic analysis of the learning curve for Gaussian process regression

Gratiet

Garnier

2014

Mach Learn

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This paper deals with the learning curve in a Gaussian process regression framework. The learning curve describes the generalization error of the Gaussian process used for the regression. The main result is the proof of a theorem giving the generalization error for a large class of correlation kernels and for any dimension when the number of observations is large. From this theorem, we can deduce the asymptotic behavior of the generalization error when the observation error is small. The presented proof generalizes previous ones that were limited to special kernels or to small dimensions (one or two). The theoretical results are applied to a nuclear safety problem. Keywords Gaussian process regression • Asymptotic mean squared error • Learning curves • Generalization error • Convergence rate 1 Introduction Gaussian process regression is a useful tool to approximate an objective function given some of its observations (Laslett 1994). It has originally been used in geostatistics to interpolate a random field at unobserved locations (Wackernagel 2003; Berger et al. 2001; Gneiting et al. 2010), it has been developed in many areas such as environmental and atmospheric sciences.

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Small Deviations Asymptotics for Matérn Processes and Fields under Weighted Quadratic Norm

Cited by 6 publications

References 9 publications

Asymptotic normality of a Sobol index estimator in Gaussian process regression framework

Asymptotic normality of a Sobol index estimator in Gaussian process regression framework

Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

Asymptotic analysis of the learning curve for Gaussian process regression

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