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Kriging measure-valued data with sparse observations: application to nuclear safety studies

Florian Gossard, François Bachoc, Jean Baccou, Thibaut Le Gouic, Jacques Liandrat, Tony Glantz

TL;DR

This paper develops a Kriging framework for interpolating probability measure-valued data by embedding measures in Wasserstein space via quantile functions. It replaces the classical real-valued Kriging predictor with a Wasserstein barycenter in the quantile-function space, enabling linear predictions in $L_2([0,1])$ and a semivariogram tailored to measures. To address semivariogram estimation under sparse observations, it introduces cross-validation techniques, including virtual cross-validation formulas for quantile functions, and demonstrates that CV-based semivariogram estimation improves predictive performance, especially for extreme quantiles. The approach is validated on a toy example and a computationally costly nuclear-safety LOCA test case, where the measure-valued Kriging with CV outperforms traditional real-valued Kriging, offering a practical tool for uncertainty quantification in safety-critical simulations.

Abstract

This work addresses the interpolation of probability measures within a spatial statistics framework. We develop a Kriging approach in the Wasserstein space, leveraging the quantile function representation of the one-dimensional Wasserstein distance. To mitigate the inaccuracies in semivariogram estimation that arise from sparse datasets, we combine this formulation with cross-validation techniques. In particular, we introduce a variant of the virtual cross-validation formulas tailored to quantile functions. The effectiveness of the proposed method is demonstrated on a controlled toy problem as well as on a real-world application from nuclear safety.

Kriging measure-valued data with sparse observations: application to nuclear safety studies

TL;DR

This paper develops a Kriging framework for interpolating probability measure-valued data by embedding measures in Wasserstein space via quantile functions. It replaces the classical real-valued Kriging predictor with a Wasserstein barycenter in the quantile-function space, enabling linear predictions in and a semivariogram tailored to measures. To address semivariogram estimation under sparse observations, it introduces cross-validation techniques, including virtual cross-validation formulas for quantile functions, and demonstrates that CV-based semivariogram estimation improves predictive performance, especially for extreme quantiles. The approach is validated on a toy example and a computationally costly nuclear-safety LOCA test case, where the measure-valued Kriging with CV outperforms traditional real-valued Kriging, offering a practical tool for uncertainty quantification in safety-critical simulations.

Abstract

This work addresses the interpolation of probability measures within a spatial statistics framework. We develop a Kriging approach in the Wasserstein space, leveraging the quantile function representation of the one-dimensional Wasserstein distance. To mitigate the inaccuracies in semivariogram estimation that arise from sparse datasets, we combine this formulation with cross-validation techniques. In particular, we introduce a variant of the virtual cross-validation formulas tailored to quantile functions. The effectiveness of the proposed method is demonstrated on a controlled toy problem as well as on a real-world application from nuclear safety.
Paper Structure (24 sections, 3 theorems, 65 equations, 7 figures, 2 tables)

This paper contains 24 sections, 3 theorems, 65 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview on Kriging of real-valued data
  3. Kriging problem
  4. Identification of spatial structure
  5. Experimental semivariogram estimation
  6. Maximum likelihood estimator under Gaussian process assumption
  7. Kriging of probability measure-valued processes
  8. Probability measures on $\mathbb{R}$
  9. Kriging extension to probability measure-valued data
  10. Semivariogram for probability measures
  11. Cross validation techniques for Kriging of probability measures and extension of virtual cross validation formulas
  12. LOO Cross Validation
  13. Virtual cross validation formulas for quantile functions
  14. Scale parameter
  15. Numerical applications
  16. ...and 9 more sections

Key Result

Proposition 4.1

Let $(\mu(x_1),...,\mu(x_n))$ be $n$ observed measures of $\mathcal{P}_2(\mathbb{R})$. Assume that the associated observed quantile functions $(Q_{\mu(x_1)},...,Q_{\mu(x_n)})$ come from a $\mathcal{Q}$-valued spatially correlated stochastic process $\{\mathbf{Q}_{\mu(x)}, x \in \mathcal{D}\}$ where with $\tilde{\Gamma} = \Gamma_{\text{\tiny W}}^{-1} - \Gamma_{\text{\tiny W}}^{-1}\mathbb{1}_n(\mat

Figures (7)

  • Figure 1: Matérn semivariogram models with $\sigma^2=1$,$l=0.5$ and $\nu = \frac{1}{2},\frac{3}{2},\frac{5}{2}$.
  • Figure 2: Examples of predicted quantile functions and comparison with true quantile functions.
  • Figure 3: Verification of virtual cross validation formulas.
  • Figure 4: An example of a temperature map generated by DRACCAR (in K).
  • Figure 5: Boxplots of $RMSE_{mean}$ for each model
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 1
  • Definition 3.4
  • Proposition 4.1
  • ...and 7 more