Expectation-Maximization Algorithm for the Calibration of Complex Simulator Using a Gaussian Process Emulator.

Yun Am Seo,Jeong-Soo Park

doi:10.3390/e23010053

Abstract

The approximated non-linear least squares (ALS) tunes or calibrates the computer model by minimizing the squared error between the computer output and real observations by using an emulator such as a Gaussian process (GP) model. A potential defect of the ALS method is that the emulator is constructed once and it is no longer re-built. An iterative method is proposed in this study to address this difficulty. In the proposed method, the tuning parameters of the simulation model are calculated by the conditional expectation (E-step), whereas the GP parameters are updated by the maximum likelihood estimation (M-step). These EM-steps are alternately repeated until convergence by using both computer and experimental data. For comparative purposes, another iterative method (the max-min algorithm) and a likelihood-based method are considered. Five toy models are tested for a comparative analysis of these methods. According to the toy model study, both the variance and bias of the estimates obtained from the proposed EM algorithm are smaller than those from the existing calibration methods. Finally, the application to a nuclear fusion simulator is demonstrated.

Highlights

Modern researchers have attempted to develop and use simulation code instead of excessively expensive or infeasible physical experiments in many fields
An EM algorithm is an iterative method for determining the MLE or maximum a posteriori (MAP) estimates of parameters in statistical models [32], where the model usually depends on the unobserved latent variables
The calibration parameters of the simulation code are calculated by the conditional expectation (E-step), whereas the Gaussian process (GP) parameters are updated by maximum likelihood estimation (M-step)

Summary

Introduction

Modern researchers have attempted to develop and use simulation code instead of excessively expensive or infeasible physical experiments in many fields. A classic method for determining the universal constants in computer models is non-linear least squares estimation (NLSE) It makes the sum of the squared error between the real observations and the computer responses as minimum. The NLSE, will become too computationally expensive or infeasible in terms of time when the computer model is time-consuming to run In such cases, a statistical emulator can be used to determine the universal constants in the computer model so that the simulator or emulator can represent the real experiments effectively well. A statistical emulator can be used to determine the universal constants in the computer model so that the simulator or emulator can represent the real experiments effectively well This process is known as “code tuning” or “calibration” [1,2,3,4]

Methods

Discussion

Conclusion