among the factors have a significant impact on the uncertainty in the reliability index. If interaction effects were a concern, the variance decomposition analysis could be extended to quantify partial variances between specific factor combinations, as outlined by Saltelli et al (2004). 5 Conclusions The use of probabilistic approaches is becoming more and more common in engineering analyses. This allows the probabilistic response of a deterministic model to be computed, given probabilistic descriptions of the model inputs. Reliability studies are often performed in this fashion, in which the probability of a response quantity exceeding a specified threshold is of interest. This work presents several ideas regarding the distinction between aleatory (irreducible) and epistemic (reducible) uncertainty in the context of probabilistic analysis. Previous work has demonstrated approaches for quantifying uncertainty in model output statistics due to epistemic uncertainty in the characterization of the model input distributions. This work proposes a variance decomposition approach for then identifying which uncertainty sources have the largest contribution to the overall epistemic uncertainty. The primary motivation is that the resulting information can be of significant value when making decisions related to resource allocation, such as whether it would be more effective to collect additional data on material properties or load conditions. The approach is demonstrated via the reliability analysis for the deflection of a statically indeterminate beam. The example problem illustrates the distinction between aleatory and epistemic uncertainty, as the aleatory distributions for the model inputs are estimated based on limited sample data, which introduces epistemic uncertainty about the distribution parameters such as the means and standard deviations. It is shown that the variance decomposition approach can successfully identify a data-rich input as having a negligible contribution to variance, even though the deterministic model is highly sensitive to that input. Several possible avenues for further work on this topic exist. One area for research is the extent to which stochastic probabilistic methods such as Monte Carlo simulation may be used in conjunction with epistemic variance decomposition: such an approach was avoided here for fear that the stochastic method would add additional variance to the model output statistic, potentially masking the effects of the other epistemic uncertainty sources being considered. Another area for further study is the impact of the parametrization of the model output statistic whose variance is to be analyzed. For the reliability analysis example given here, we chose to analyze the reliability index as opposed to the failure probability: whereas both capture the same information, it was felt that the highly skewed distribution of failure probability might adversely impact the variance decomposition. However, further study would be needed to better understand the role of model output parametrization in variance decomposition. Variance Decomposition in the Presence of Epistemic and Aleatory Uncertainty 427
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