Rapid Approximation Prediction for Kriging

2026-05-27

AI Summary

This study addresses the prohibitive computational cost of traditional kriging when applied to large datasets and dense prediction grids, which hinders efficient uncertainty quantification. The authors propose a fast kriging method based on a local L-order neighborhood sparse approximation. By leveraging the covariance structure of stationary Gaussian processes and properties of the Matérn covariance function, the approach approximates off-grid covariance vectors as sparse linear combinations of nearby grid points. This yields substantial reductions in computational complexity while maintaining theoretically controlled approximation error. The method integrates seamlessly into conditional simulation frameworks, enabling efficient uncertainty propagation. In experiments with North American summer precipitation data, it achieves over 150× speedup compared to exact kriging, with prediction errors as low as approximately 10⁻⁵ inches—visually indistinguishable from exact results—and outperforms existing methods such as Vecchia approximation and LatticeKrig.

Abstract

Exact Kriging and conditional simulation (CS) for uncertainty quantification are computationally infeasible for modern spatial analyses with large numbers of observations and dense prediction grids. We present a rapid approximation to the Kriging prediction step for stationary Gaussian processes for a regular prediction grid by approximating each off-grid covariance vector by a sparse linear combination of on-grid covariances within a local $L$-order neighborhood of $M = (2L)^2$ neighboring grid points. This reformulation reduces complexity from $O(N n^3)$ to $O(N \log N + nM + M^3)$ while preserving accuracy. A factorial study shows that approximation error decreases systematically with increased Matérn smoothness, neighbor order $L$, and grid resolution, aligning with bounds from kernel approximation theory. In a North American summer-rainfall application ($n=1368$), our method produces predictions visually indistinguishable from exact Kriging with point-wise errors on the order of $10^{-5}$ inches and achieves more than $150$ times speedups at a $350\times350$ grid, also outperforming Vecchia and LatticeKrig predictions. Embedded in a fast CS scheme, the approach reproduces Kriging standard errors and scales favorably with both $n$ and $N$. We recommend a practical workflow that uses a fast method for parameter estimation followed by our rapid predictor for fine-grid mapping and uncertainty quantification.

Problem

• Kriging
• conditional simulation
• uncertainty quantification
• large spatial data
• computational infeasibility

Innovation

• fast Kriging
• covariance approximation
• sparse linear combination
• computational scalability
• conditional simulation