Advances in Engineering Advances in Engineering features breaking research judged by Advances in Engineering advisory team to be of key importance in the Engineering field. Papers are selected from over 10,000 published each week from most peer reviewed journals.
Significance Statement
Modeling the dependence between uncertainties in decision and risk analyses is an important part of the problem structuring process. In practice, these dependencies are often neglected in modeling in order to simplify the analysis. For situations where such negligence could cause significant errors, the incorporation of these dependencies into the decision and risk probability models becomes important.
In this paper, we focus on situations where correlated uncertainties are discrete, and extend the concept of the copula-based approach for modeling correlated continuous uncertainties to the representation of correlated discrete uncertainties. This approach can further be extended to model the dependence between discrete and continuous uncertainties in the same event tree.
Event trees expand rapidly in terms of the number of uncertainties and the number of endpoints to be calculated grows even more quickly, particularly when the event tree is part of a decision tree. This increases the computational burden, but it is also difficult and time consuming to obtain the required conditional probabilities for dependent discrete uncertainties. However, with the correlated event tree methods used here, we only need to assess the marginal distributions and a lower order measure of dependence such as correlation, and we can then calculate the conditional probabilities in closed-form. This approach reduces the required number of probability assessments significantly compared to approaches requiring direct estimates of conditional probabilities. It also allows the use of multiple dependence measures, including product moment correlation, rank order correlation and tail dependence, and parametric families of copulas such as normal copulas, t-copulas and Archimedean copulas.
The proposed copulas-based approach provides some advantages over the optimization-based ME and AC approaches in the literature. First, the proposed copulas-based approach provides a closed-form solution for the joint and conditional probabilities that can be solved numerically. In comparison, the calculation of the maximum entropy distribution of correlated variables with pre-specified marginal distributions requires the solution of nonlinear coupled integral equations subject to local optima, with no exact solution for the discrete marginal distributions. Second, the proposed approach allows relatively easy incorporation of nonlinear measures of association, such as rank order correlation and tail dependence, in the same general framework, while the ME and AC will require customized coding for each application. In addition, the use of the multivariate normal copula provides a computationally more efficient approximation to the ME method that shares the benefits of a “near-maximum entropy” result while reducing its practical limitations. Another important insight from our work is a novel demonstration of the difficulties arising from discretizing continuous distributions before accounting for dependence among them.
Journal Reference
Risk Anal. 2016 Feb;36(2):396-410.
Wang T1, Dyer JS2, Butler JC2.
[expand title=”Show Affiliations”]- College of Business, Colorado State University, Fort Collins, CO, USA.
- McCombs School of Business, University of Texas at Austin, Austin, TX, USA.
Abstract
Modeling the dependence between uncertainties in decision and risk analyses is an important part of the problem structuring process. We focus on situations where correlated uncertainties are discrete, and extend the concept of the copula-based approach for modeling correlated continuous uncertainties to the representation of correlated discrete uncertainties. This approach reduces the required number of probability assessments significantly compared to approaches requiring direct estimates of conditional probabilities. It also allows the use of multiple dependence measures, including product moment correlation, rank order correlation and tail dependence, and parametric families of copulas such as normal copulas, t-copulas, and Archimedean copulas. This approach can be extended to model the dependence between discrete and continuous uncertainties in the same event tree.
© 2015 Society for Risk Analysis.
Go To Risk Anal
