that interaction effects play a role for that factor, in which case such effects may be explored further. A variety of methods have been proposed for the calculation of the sensitivity indices, with emphasis being placed on efficient methods requiring fewer deterministic model evaluations. Saltelli et al (2004) describe a structured Monte Carlo approach, which involves holding constant certain factors while varying others. Another notable approach is the Fourier Amplitude Sensitivity Test (FAST; Cukier et al, 1973, 1978; Koda et al, 1979), which involves a Fourier decomposition and discretization of the inputs over probability space. The main hypothesis for this work is that variance decomposition can be used to address epistemic uncertainty associated with amodel output statistic. Note that traditionally, variance decomposition would be applied to the output of the deterministic performance function. However, such an approach may not prove as useful when the inputs to the deterministic performance function are subject to irreducible uncertainty. While the inputs to the deterministic performance function may be subject to aleatory uncertainty, there may still be other sources of epistemic uncertainty that impact the probabilistic solution, as discussed in Section 2. Thus, this work demonstrates an approach for decomposing the variance of the model output statistic with respect to the epistemic variables, such as probability distribution parameters. The motivation for such an analysis is that it helps identify which factors are contributing the most to uncertainty in the probabilistic solution. For example, once epistemic uncertainty is taken account of, a reliability analysis may demonstrate confidence bounds on the reliability that are too wide to draw a meaningful conclusion. A variance decomposition analysis can help identify where additional resources should be allocated to reduce this uncertainty: this may mean collecting more data to characterize model input distributions or reducing the uncertainty associated with the deterministic performance model. Variance decomposition of a model output statistic introduces several challenges. Most notably, treatment of uncertain distribution parameters can be difficult. We will assume that the (epistemic) probability distributions for these parameters are obtained as the posterior distributions from a Bayesian analysis. Corresponding to each deterministic model input xi, we have probability distribution parameters θi. Depending on the form of the probability distribution model for xi, θi will be a vector of parameters. For example, if xi is normally distributed, then θi contains the mean and variance parameters. It is important to note that it is not necessary to treat the parameters within θi separately: as more data are collected about the variable xi, the uncertainty about all parameters in θi is reduced together. Moreover, the parameters within θi are not independent, which would be problematic for the variance decomposition if they were to be treated separately. As a result, we want to decompose the variance of the model output statistic with respect to each factor group θi. This immediately rules out certain variance decomposition methods, such as FAST, which are not amenable to treatment of variables by group. Furthermore, in practice the Bayesian posterior distribution of θi will not be constructed analytically but will instead be approximated via random sampling using Markov Chain Monte Carlo (MCMC) methods. Clearly, the availJohn McFarland and David Riha 422
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