A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data

TL;DR

This paper introduces a new method for Gaussian process interpolation that ensures computational stability and reduces over-smoothing, specifically designed for deterministic computer simulation data.

Contribution

It proposes a lower bound on the nugget parameter and an iterative regularization approach to improve interpolation accuracy and stability.

Findings

The method reduces over-smoothing caused by the nugget.

The approach ensures convergence to the GP interpolator.

It enhances stability in fitting Gaussian process models.

Abstract

For many expensive deterministic computer simulators, the outputs do not have replication error and the desired metamodel (or statistical emulator) is an interpolator of the observed data. Realizations of Gaussian spatial processes (GP) are commonly used to model such simulator outputs. Fitting a GP model to $n$ data points requires the computation of the inverse and determinant of $n \times n$ correlation matrices, $R$, that are sometimes computationally unstable due to near-singularity of $R$. This happens if any pair of design points are very close together in the input space. The popular approach to overcome near-singularity is to introduce a small nugget (or jitter) parameter in the model that is estimated along with other model parameters. The inclusion of a nugget in the model often causes unnecessary over-smoothing of the data. In this paper, we propose a lower bound on the nugget that minimizes the over-smoothing and an iterative regularization approach to construct a predictor that further improves the interpolation accuracy. We also show that the proposed predictor converges to the GP interpolator.