Estimation of the covariance function for Gaussian processes

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Table of contents

1 Introduction
I Kriging models
2 Kriging models with known covariance function
2.1 Gaussian processes
2.1.1 Denition and properties of Gaussian processes
2.1.2 The relationship between the covariance function and the trajectories of a Gaussian process
2.2 Prediction and conditional simulation for Gaussian processes
2.2.1 Ordinary, simple and universal Kriging models
2.2.2 Point-wise prediction
2.2.3 Conditional simulation of Gaussian processes
2.2.4 Cross Validation formulas
2.2.5 Alternative RKHS formulation
3 Covariance function estimation for Kriging models
3.1 Introduction to parametric estimation
3.1.1 Denition and properties for parametric estimation
3.1.2 Classical asymptotic results for parametric estimation
3.2 Estimation of the covariance function for Gaussian processes
3.2.1 Parametric estimation of the covariance function
3.2.2 Maximum Likelihood for estimation
3.2.3 Cross Validation for estimation
3.2.4 Gradients of the dierent criteria
3.2.5 The challenge of taking into account the uncertainty on the covariance function
4 Asymptotic results for Kriging
4.1 Two asymptotic frameworks
4.2 Asymptotic results for prediction with xed covariance function
4.2.1 Consistency
4.2.2 Asymptotic inuence of a misspecied covariance function
4.3 Asymptotic results for Maximum Likelihood
4.3.1 Expansion-domain asymptotic results
4.3.2 Fixed-domain asymptotic results
II Cross Validation and Maximum Likelihood for covariance hyperparameter estimation
5 Cross Validation and Maximum Likelihood with well-specied family of covariance functions
5.1 Introduction
5.2 Expansion-domain asymptotic framework with randomly perturbed regular grid .
5.3 Consistency and asymptotic normality for Maximum Likelihood and Cross Validation
5.3.1 Consistency and asymptotic normality
5.3.2 Closed form expressions of the asymptotic variances in dimension one
5.4 Study of the asymptotic variance
5.4.1 Small random perturbations
5.4.2 Large random perturbations
5.4.3 Estimating both the correlation length and the smoothness parameter .
5.4.4 Discussion
5.5 Analysis of the Kriging prediction
5.5.1 Asymptotic inuence of covariance hyper-parameter misspecication on prediction
5.5.2 Inuence of covariance hyper-parameter estimation on prediction .
5.5.3 Analysis of the impact of the spatial sampling on the Kriging prediction .
5.6 Conclusion
5.7 Proofs
5.7.1 Proofs for subsection 5.3.1
5.7.2 Proofs for subsection 5.3.2
5.7.3 Proofs for section 5.5
6 Cross Validation and Maximum Likelihood with misspecied family of covariance functions
6.1 Introduction
6.2 Estimation of a single variance parameter
6.2.1 Theoretical framework
6.2.2 Numerical results
6.3 Estimation of variance and correlation hyper-parameters
6.3.1 Procedure
6.3.2 Results and discussion
6.4 Discussion
III Applications to Uncertainty Quantication for Computer Experiments
7 Probabilistic modeling of discrepancy between computer model and experiments
7.1 Framework for computer models and experiments
7.2 Errors modeled by a variability of the physical system
7.2.1 The general probabilistic model
7.2.2 Non-linear methods
7.2.3 Methods based on a linearization of the computer model
7.3 Errors modeled by a model error process
7.3.1 The general probabilistic model
7.3.2 Non-linear methods
7.3.3 Methods based on a linearization of the computer model
8 Calibration and improved prediction of the thermal-hydraulic code FLICA
8.1 Presentation of FLICA 4 and of the experimental results
8.1.1 The thermal-hydraulic code FLICA 4
8.1.2 The experimental results
8.2 Description of the procedure for the Gaussian process modeling
8.2.1 Objectives for the universal Kriging procedure
8.2.2 Exponential, Matérn and Gaussian covariance functions considered
8.2.3 K-folds Cross Validation for Kriging model validation
8.3 Results
8.3.1 Results in the isothermal regime
8.3.2 Results in the single-phase regime
8.3.3 Inuence of the linear approximation
9 Kriging meta-modeling of the GERMINAL computer model
9.1 Introduction
9.2 Presentation and context for the GERMINAL computer model
9.2.1 A nuclear reactor core design problem
9.2.2 Inputs and outputs considered
9.2.3 Setting for the Kriging model
9.3 Results of the Kriging model
9.3.1 Interpretation of the estimated covariance hyper-parameters
9.3.2 Prediction results
9.3.3 Detection of computation failures for the « Fusion_Margin » output
10 Conclusion and perspectives
A Notation
B Reference