Kriging via variably scaled kernels

2026-03-16

AI Summary

This work proposes a nonstationary Gaussian process modeling approach based on a variable-scale kernel to overcome the limitations of traditional Gaussian processes and Kriging models, which rely on stationary kernels and struggle to capture data with heterogeneous correlation structures or abrupt changes. By explicitly incorporating a spatially varying scale function to modulate local correlations, the method more accurately fits target functions exhibiting discontinuities or sharp variations. The approach establishes theoretical connections to classical nonstationary kernels and leverages kernel power functions to analyze predictive uncertainty. Numerical experiments demonstrate that the proposed method significantly improves reconstruction accuracy, while its uncertainty estimates effectively reflect the intrinsic nonstationary structure of the underlying data.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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

Problem

• non-stationary kernels
• Gaussian processes
• Kriging
• heterogeneous correlation structures
• discontinuities

Innovation

• variably scaled kernels
• non-stationary Gaussian processes
• Kriging
• scaling function
• predictive uncertainty