Penalized Projected Kernel Calibration for Computer Models

TL;DR

This paper introduces a penalized projected kernel calibration method that improves parameter estimation accuracy in computer models by addressing the limitations of traditional projected kernel calibration, supported by theoretical proofs and empirical results.

Contribution

It proposes a penalized version of projected kernel calibration, providing theoretical guarantees and demonstrating improved performance over existing methods.

Findings

The penalized method is as efficient as the original PK calibration.

It accurately estimates calibration parameters in simulations and real-world data.

Performance surpasses other calibration methods across various sample sizes.

Abstract

Projected kernel calibration is a newly proposed frequentist calibration method, which is asymptotic normal and semi-parametric. Its loss function is usually referred to as the PK loss function. In this work, we prove the uniform convergence of PK loss function and show that (1) when the sample size is large, any local minimum point and local maximum point of the $L_2$ loss between the true process and the computer model is a local minimum point of the PK loss function; (2) all the local minima of the PK loss function converge to the same value. These theoretical results imply that it is extremely hard for the projected kernel calibration to identify the global minimum of the $L_2$ loss, i.e. the optimal value of the calibration parameters. To solve this problem, a frequentist method which we term penalized projected kernel calibration method is suggested and analyzed in detail. We prove that the proposed method is as efficient as the projected kernel calibration method. Through an extensive set of numerical simulations, and a real-world case study, we show that the proposed calibration method can accurately estimate the calibration parameters. We also show that its performance compares favorably to other calibration methods regardless of the sample size.