Metamodel-based Variance Estimation and Global Sensitivity Analysis

We propose a multilevel Monte Carlo metamodeling approach for variance function estimation.  we show that, under some mild conditions, the proposed MLMC metamodeling approach for variance function estimation can achieve a computational complexity superior to the standard Monte Carlo while achieving a target accuracy level. Additionally, we establish the asymptotic normality of the MLMC metamodeling estimator under a set of sufficient conditions, providing valuable utility for uncertainty quantification. An application in global sensitivity analysis shows that the approach can achieve competitive performance in supporting Sobol' indices estimation.

Sobol' indices are widely used in global sensitivity analysis for assessing the input parameters' impact on the model output,  with successful applications in epidemiological modeling, defect detection in manufacturing, pollutant transport modeling, etc. We propose joint metamodel-based Sobol' index estimators  which rely on estimation of both the mean and variance functions implied by a stochastic simulation experiment. We prove the estimators' asymptotic normality, based on which asymptotic confidence intervals can be constructed for uncertainty quantification. 

We provide a comprehensive examination of conventional methods for computing Sobol's indices and introduce a nested simulation estimator. We investigate the asymptotically optimal allocation of computational resources for this nested estimator to optimize the convergence rate of its mean square error. Additionally, we propose two jackknife estimators: one that achieves unbiasedness and another that surpasses the nested simulation estimator in terms of the convergence rate. Through extensive numerical studies, we highlight the strengths and weaknesses of each estimator, providing a detailed comparison of their effectiveness.

This paper extends Sobol’ indices for global sensitivity analysis (GSA) of stochastic models with time-dependent outputs by introducing generalized Sobol’ indices that rigorously capture temporal effects while accounting for inherent model stochasticity. We develop and analyze three estimators for these indices: the nested pick-freeze (NPF), pick-freeze (PF), and partition-based (PAR) estimators. The PF estimator is shown to attain a faster mean squared error (MSE) convergence rate than NPF, whereas the MSE rate of PAR depends on the input-space dimensionality. Importantly, PAR eliminates the need for specifically designed experiments, thereby enabling the estimation of generalized Sobol’ indices from existing datasets. Numerical studies, including a synthetic example and an application to the enteric immunity simulator for modeling and analyzing Helicobacter pylori infection dynamics, corroborate the theoretical results. The findings reveal critical temporal variations in immune responses and demonstrate the effectiveness of generalized Sobol’ indices as a rigorous tool for GSA in stochastic dynamic models.

This paper investigates a model-free, partition-based method for estimating Sobol’ indices using existing datasets, addressing the limitations of traditional variance-based global sensitivity analysis (GSA) methods that rely on designed experiments. We provide a theoretical analysis of the bias, variance, and mean squared error (MSE) associated with the partition-based estimator, exploring the effects of the sample size of the dataset and the number of partition bins on its performance. Furthermore, we propose a data-driven approach for determining the optimal number of bins to minimize the MSE. Numerical experiments demonstrate that the proposed partition-based method outperforms state-of-the-art GSA techniques.