A comparison of uncertainty and sensitivity analysis results obtained with random and Latin hypercube sampling
Introduction
The identification and representation of the implications of uncertainty is widely recognized as a fundamental component of analyses of complex systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The study of uncertainty is usually subdivided into two closely related activities referred to as uncertainty analysis and sensitivity analysis, where (i) uncertainty analysis involves the determination of the uncertainty in analysis results that derives from uncertainty in analysis inputs and (ii) sensitivity analysis involves the determination of relationships between the uncertainty in analysis results and the uncertainty in individual analysis inputs.
At an abstract level, the analysis or model under consideration can be represented as a function of the formwhereis a vector of uncertain analysis inputs andis a vector of analysis results. Further, a sequence of distributionsis used to characterize the uncertainty associated with the elements of x, where Di is the distribution associated with xi for i=1, 2,…,nX. Correlations and other restrictions involving the elements of x are also possible. The goal of uncertainty analysis is to determine the uncertainty in the elements of y that derives from the uncertainty in the elements of x characterized by the distributions D1,D2,…,DnX and any associated restrictions. The goal of sensitivity analysis is to determine relationships between the uncertainty associated with individual elements of x and the uncertainty associated with individual elements of y.
A variety of approaches to uncertainty and sensitivity analysis are in use, including (i) differential analysis, which involves approximating a model with a Taylor series and then using variance propagation formulas to obtain uncertainty and sensitivity analysis results [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], (ii) response surface methodology, which is based on using classical experimental designs to select points for use in developing a response surface replacement for a model and then using this replacement model in subsequent uncertainty and sensitivity analyses based on Monte Carlo simulation and variance propagation [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], (iii) the Fourier amplitude sensitivity test (FAST) and other variance decomposition procedures, which involve the determination of uncertainty and sensitivity analysis results on the basis of the variance of model predictions and the contributions of individual variables to this variance [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], (iv) fast probability integration, which is primarily an uncertainty analysis procedure used to estimate the tails of uncertainty distributions for model predictions [56], [57], [58], [59], [60], [61], [62], and (v) sampling-based (i.e. Monte Carlo) procedures, which involve the generation and exploration of a probabilistically based mapping from analysis inputs to analysis results [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73]. Additional information on uncertainty and sensitivity analysis is available in a number of reviews [69], [70], [74], [75], [76], [77], [78], [79], [80]. The primary focus of this presentation is on sampling-based methods for uncertainty and sensitivity analysis.
Sampling-based approaches for uncertainty and sensitivity analysis are very popular [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96]. Desirable properties of these approaches include conceptual simplicity, ease of implementation, generation of uncertainty analysis results without the use of intermediate models, and availability of a variety of sensitivity analysis procedures [67], [69], [76], [97], [98]. Despite these positive properties, concern is often expressed about using these approaches because of the computational cost involved. In particular, the concern is that the sample sizes required to obtain meaningful results will be so large that analyses will be computationally impracticable for all but the most simple models. At times, statements are made that 1000 to 10,000s of model evaluations are required in a sampling-based uncertainty/sensitivity analysis.
In this presentation, results obtained with a computationally demanding model for two-phase fluid flow are used to illustrate that robust uncertainty and sensitivity analysis results can be obtained with relatively small sample sizes. Further, results are obtained and compared for replicated random and Latin hypercube samples (LHSs) [63], [73]. For the problem under consideration, random and LHSs of size 100 produce similar, stable results.
The presentation is organized as follows. The analysis problem is described in Section 2. Then, the following topics are considered: stability of uncertainty analysis results (Section 3), stability of sensitivity analysis results based on stepwise rank regression (Section 4), use of coefficients of concordance in comparing replicated sensitivity analyses (Section 5), sensitivity analysis based on replicated samples and the top down coefficient concordance (Section 6), sensitivity analysis with reduced sample sizes (Section 7), and sensitivity analysis without regression analysis (Section 8). Finally, the presentation ends with a concluding discussion (Section 9).
Section snippets
Analysis problem
The analysis problem under consideration comes from the 1996 performance assessment (PA) for the Waste Isolation Pilot Plant (WIPP) [99], [100]. This PA was the core analysis that supported the successful Compliance Certification Application (CCA) by the US Department of Energy (DOE) to the US Environmental Protection Agency (EPA) for the operation of the WIPP [101]. With the certification of the WIPP by the EPA for the disposal of transuranic waste in May 1998 [102], the WIPP became the first
Uncertainty analysis results
The time-dependent results in Fig. 2 display the uncertainty in solutions to Eqs. (2.1), (2.2), (2.3), (2.4), (2.5), (2.6) that results from uncertainty in the 31 variables in Table 1. The goal of this presentation is to illustrate the robustness of such uncertainty representations with respect to the type and size of the sample in use. As previously indicated, results at 1000, 10,000–1000, and 10,000 yr will be used for illustration.
One way to compare uncertainty analysis results is to present
Stepwise results
A sensitivity analysis based on stepwise regression analysis with rank-transformed data [118] was carried out for the replicated samples summarized in Fig. 4 (Table 3, Table 4, Table 5, Table 6). This analysis required α-values of 0.02 and 0.05 for variables to enter and to be retained in a given analysis, respectively, and was carried out with the stepwise program [119]. The summary tables (Tables 3–6) present results for both the individual replicates and for the three replicates of a given
Coefficients of concordance
Inspection of the results in Table 3, Table 4, Table 5, Table 6 suggests that the individual replicates are producing similar results. Kendall's coefficient of concordance (KCC) provides a way to formally assess this similarity (p. 305, Ref. [120]). This coefficient is based on the consideration of arrays of the formwhere x1,x2,…,xnX are the variables under consideration (i.e. nX=29 with the exclusion of
Sensitivity analysis with the TDCC
Replicated samples and the TDCC provide the basis for a sensitivity analysis procedure to identify important sets of variables that does not depend on direct testing of the statistical significance of sensitivity measures (e.g. the significance of the coefficients in a stepwise regression model as defined by an α-value for entry into the model). Rather, important variables are identified by the similarity of outcomes in analyses performed for the individual replicated samples.
The procedure
Sensitivity analysis with small samples
The sensitivity analysis results obtained with random and LHSs of size 100 are very similar and thus indicate that a sample size of 100 is adequate for the problem under consideration. The question naturally arises if smaller sample sizes would also be adequate.
To partially address this question, the random samples were pooled to produce 300 observations, and then three samples of size 50 were obtained by randomly sampling from these 300 observations. Each new sample of size 50 was produced by
Sensitivity analysis without regression
The regression analyses summarized in Table 3, Table 4, Table 5, Table 6 exhibit various levels of success. Some analyses are quite good, with R2 values above 0.9. Other analyses are not quite so good, with R2 values in the range from 0.6 to 0.8. The analyses for WAS_PRES at 10,000 yr are effectively failures, with R2 values in the vicinity of 0.2.
An important aspect of the analyses in Table 3, Table 4, Table 5, Table 6 is that the identification of dominant variables tends to remain the same
Discussion
Uncertainty and sensitivity analysis results obtained with replicated random and LHSs are compared. In particular, uncertainty and sensitivity analyses were performed for a large model for two-phase fluid flow with three independently generated random samples of size 100 each and also three independently generated LHSs of size 100 each.
For the outcomes under consideration, analyses with random and LHSs produced similar results. Specifically, there is little difference in the uncertainty and
Acknowledgements
Work performed for Sandia National Laboratories (SNL), which is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. Review provided at SNL by M. Chavez, J. Garner, and S. Halliday. Editorial support provided by F. Puffer, J. Ripple, and K. Best of Tech Reps, Inc.
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