IGSA: Imprecise global sensitivity analysis

CDF of imprecise first-order Sobol’ indices as a function of data set size from 100, 250, 500, 1000, 5000 to 10000.
CDF of imprecise first-order Sobol’ indices as a function of data set size from 100, 250, 500, 1000, 5000 to 10000.


Global Sensitivity Analysis (GSA) aims to understand the relative importance of uncertain input variables to model response. Conventional GSA involves calculating sensitivity (Sobol’) indices for a model with known model parameter distributions. However, model parameters are affected by aleatory and epistemic uncertainty, with the latter often caused by a lack of data. We propose a new framework to quantify uncertainty in probability model form and model parameters resulting from small datasets and integrate these uncertainties into Sobol’ index estimates. First, the process establishes, through Bayesian multimodel inference, a set of candidate probability models and their associated probabilities. Imprecise Sobol’ indices are calculated from these probability models using an importance sampling reweighting approach. This results in probabilistic Sobol’ indices, whose distribution characterizes uncertainty in the sensitivity resulting from small dataset size. The imprecise Sobol’ indices thus provide a measure of confidence in the sensitivity estimate and, moreover, can be used to inform data collection efforts targeted to minimize the impact of uncertainties. Through an example studying the parameters of a Timoshenko beam, we show that these probabilistic Sobol’ indices converge to the true/deterministic Sobol’ indices as the dataset size increases and hence, distribution-form uncertainty reduces. The approach is then applied to assess the sensitivity of the out-of-plane properties of an E-glass fiber composite material to its constituent properties. This second example illustrates the approach for an important class of materials with wide-ranging applications when data may be lacking for some input parameters.


Jiaxin Zhang*, Stephanie TerMaath, and Michael D. Shields. "Imprecise global sensitivity analysis using bayesian multimodel inference and importance sampling." Mechanical Systems and Signal Processing 148 (2020): 107162.


Significance and Impact

  • We derive a formulation for computing main effect and total sensitivity indices using importance sampling. 
  • We apply a Bayesian multimodel inference process in conjunction with the importance sampling-based GSA to estimate the distribution of imprecise Sobol’ indices whose uncertainty results from a lack of data quantifying input uncertainty. 
  • We demonstrate how to obtain the optimal sampling density to efficiently employ the proposed multimodel GSA for engineering systems. 
  • We illustrate how to use the results of the proposed imprecise GSA to assess confidence in computed sensitivity indices and inform future testing and data collection.

Last Updated: November 11, 2020 - 5:03 pm