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Kriging via variably scaled kernels

Gianluca Audone, Francesco Marchetti, Emma Perracchione, Milvia Rossini

Abstract

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Kriging via variably scaled kernels

Abstract

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data
Paper Structure (16 sections, 5 theorems, 97 equations, 14 figures, 3 tables)

This paper contains 16 sections, 5 theorems, 97 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Statistics and approximation with kernels
  3. Kernels in deterministic settings
  4. Kernels in statistical settings
  5. Non-stationary kernels
  6. Variably scaled kernels in a statistical perspective
  7. Connections between VSKs and classical non-stationary kernels
  8. A note on the VSK Kriging variance
  9. Numerical examples
  10. Approximating a discontinuous function
  11. Prior-driven behavior in the variably scaled setting
  12. Comparison with MLE optimized hyperparameters
  13. Reconstruction of a truncated Weierstrass function with increasing VSK complexity
  14. Approximating a function with a corner
  15. VSKs and other non-stationary kernels
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $d,q \geq 1$ and consider the squared exponential (Gaussian) kernel $\kappa_\ell$ built on $\varphi_\ell(r) = \exp\!\left(-\frac{1}{2\ell^2}r^2\right)$. Then, letting $\phi^\Psi_{\ell,f,n}(x,x')=\sigma_f^2 \kappa^{\Psi}_\ell(x,x') + \sigma_n^2 \delta_{xx'}$ for $x,x'\in\Omega$, we have that where and

Figures (14)

  • Figure 1: Stationary case (left) and VSK approach (right). The training data consist of $N=6$ equispaced nodes.
  • Figure 2: Realization of the GPs using the prior (left) and posterior (right) covariance: stationary case (top) and VSK approach (bottom).
  • Figure 3: Training matrices in the stationary (left) and VSK (right) case.
  • Figure 4: Approximation errors (left) and uncertainty (right) varying $N$ from $10$ to $790$ Halton nodes.
  • Figure 5: Stationary case (left) and VSK approach (right). The training data consist of $N=27$ Halton nodes.
  • ...and 9 more figures

Theorems & Definitions (12)

  • Remark 2.1: Universal kriging
  • Proposition 3.1: Gaussian VSKs and amplitude modulation
  • proof
  • Theorem 3.2: Local symmetric expansion of the VSK covariance
  • Corollary 3.3
  • Corollary 3.4: Local equivalence of VSKs with Paciorek--Schervish kernels
  • Remark 3.5: Discontinuous scaling functions
  • Proposition 3.6
  • proof
  • proof
  • ...and 2 more