Uncertainty quantification with copula dependence modeling

Probabilistic dependent input models
Probabilistic dependent input models


This paper addresses the problem of quantification and propagation of uncertainties associated with dependence modeling when data for characterizing probability models are limited. Practically, the system inputs are often assumed to be mutually independent or correlated by a multivariate Gaussian distribution. However, this subjective assumption may introduce bias in the response estimate if the real dependence structure deviates from this assumption. In this work, we overcome this limitation by introducing a flexible copula dependence model to capture complex dependencies. A hierarchical Bayesian multimodel approach is proposed to quantify uncertainty in dependence model-form and model parameters that result from small data sets. This approach begins by identifying, through Bayesian multimodel inference, a set of candidate marginal models and their corresponding model probabilities, and then estimating the uncertainty in the copula-based dependence structure, which is conditional on the marginals and their parameters. The overall uncertainties integrating marginals and copulas are probabilistically represented by an ensemble of multivariate candidate densities. A novel importance sampling reweighting approach is proposed to efficiently propagate the overall uncertainties through a computational model. Through an example studying the influence of constituent properties on the out-of-plane properties of transversely isotropic E-glass fiber composites, we show that the composite property with copula-based dependence model converges to the true estimate as data set size increases, while an independence or arbitrary Gaussian correlation assumption leads to a biased estimate.


Jiaxin Zhang and Michael Shields. "On the quantification and efficient propagation of imprecise probabilities with copula dependence." International Journal of Approximate Reasoning, 2020 (122):24-46.

Significance and Impact

  • Propose a hierarchical multimodel approach to investigates the effect of uncertainties associated with small data sets for quantifying and propagating probabilistic model inputs with dependencies. The joint CDF of the probabilistic model inputs is composed of marginal distributions and copulas, which are modeled separately. T
  • The proposed approach is set in a hierarchical Bayesian multimodel inference framework, where the model-form and model parameter uncertainties associated with marginals are first quantified, and uncertainties associated with the copula are conditioned on specified marginal pairs. This results in an ensemble of joint probability densities that represent the imprecise probabilities in the assignment of probability model inputs with statistical dependence.
  • A novel importance sampling reweighting algorithm is derived to efficiently propagate the imprecise probabilities through a mathematical or physical model, which is often computationally intensive. The proposed approach, therefore, estimates the uncertainty in the quantity of interest given multiple candidate model input distributions at a low computational cost when compared with the typical nested Monte Carlo simulations.
  • The results show that the assumption of independent and arbitrary Gaussian correlated marginals in the imprecise UQ modeling both underestimates the uncertainty in predictions of the modulus and yields biased statistical estimates. When copula-based dependence is integrated into the multimodel UQ framework, the model achieves more realistic bounds on the uncertainty and more accurate probabilistic predictions

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