- Psych 1924
Title: A theoretical framework of the scaled Gaussian stochastic process for calibrating imperfect mathematical models
Model calibration or data inversion involves using experimental or field data to estimate the unknown parameters of a mathematical model. This task is complicated by the discrepancy between the model and reality, and by possible bias in field data. The model discrepancy is often modeled by a Gaussian stochastic process (GaSP), but it was observed in many studies that the calibrated mathematical model can be far from the reality. Here we show that modeling the discrepancy function via a GaSP often leads to an inconsistent estimation of the calibration parameters even if one has an infinite number of repeated experiments and an infinite number of observations in each experiment. In this work, we develop the scaled Gaussian stochastic process (S-GaSP), a new stochastic process to model the discrepancy function in calibration. We establish the explicit connection between the GaSP and S-GaSP through the orthogonal series representation. We show the predictive mean estimator in the S-GaSP calibration model converges to the reality at the same rate as the one by the GaSP model, and the calibrated mathematical model in the S-GaSP calibration converges to the one that minimizes the L2 loss between the reality and mathematical model, whereas the GaSP calibration model does not have this property. The scientific goal of this work is to use multiple interferometric synthetic-aperture radar (InSAR) interferograms to calibrate a geophysical model for Kilauea Volcano, Hawaii. Analysis of both simulated and real data confirms that our approach is better than other approaches in prediction and calibration. Both the GaSP and S-GaSP calibration models are implemented in the "RobustCalibration" R Package on CRAN.
Mengyang Gu is an Assistant Research Professor at Johns Hopkins University.