be characterized as an epistemic uncertainty), or others. Model verification and/or validation is often used to address and/or quantify these errors. The third type of epistemic uncertainty we have listed is related to the probabilistic method itself. One of the simplest cases is when the model output statistic of interest is a reliability and Monte Carlo simulation is used. In this case, random sampling error affects the result, and the behavior of this error is well understood (see Haldar and Mahadevan, 2000). If no other epistemic uncertainties are considered, it is a simple matter to develop confidence intervals on a reliability calculation based on sampling error. 3 Variance Decomposition of Epistemic Variables Variance decomposition is an approach for global sensitivity analysis, where the objective is to identify important model inputs by assessing how uncertainty associated with each input contributes to uncertainty associated with the model output. The approach is typically used to address questions such as, if the uncertainty associated with one of multiple model inputs could eliminated, what would be the most effective way to reduce the model output uncertainty? Saltelli et al (2000, 2004) give comprehensive overviews of both theory and practice of variance decomposition and other sensitivity analysis methods. Based on a functional analysis of variance, and under the assumption that the model inputs are independent (Sobol’, 1993), the model output variance can be decomposed as V = k ∑i=1 Vi + k ∑ i1=1 k ∑ i2=i1+1 Vi1i2 +···+V1 ,...,k, (2) where V is the total model output variance, k is the number of model inputs, and the subscripted V terms are partial variances corresponding to the various model inputs and combinations thereof. Each global sensitivity index is then defined as the contribution of a partial variance to the total variance: Si1,...,is =Vi1,...,is/V. (3) Note that sensitivity indices can be computed for individual variables or groups of variables being treated as a single factor. A sensitivity index Si corresponding to a single factor is called the first-order effect or main sensitivity index for that factor. Sensitivity indices corresponding to two or more factors capture interaction effects and are known as higher-order effects. The total influence of each individual factor is quantified by the total effects index, denotedSTi, which is the sum of all indices involving the factor of interest. In practice, it is common to compute only the first-order and total effects. A significant difference between the total effect and first-order effect for a factor would indicate Variance Decomposition in the Presence of Epistemic and Aleatory Uncertainty 421
RkJQdWJsaXNoZXIy MTMzNzEzMQ==