Abstract
Step 3 provided a random sample of values indirectly drawn according to the unknown subjective probability distribution for the model result. This distribution expresses the state of knowledge of the model result. It follows, in a logical consistent way, from the propagation of the state of knowledge quantifications at the level of uncertain parameters, model formulations and input data through the arithmetic and logic instructions of the model. Uncertainty measures are obtained from this subjective probability distribution as either one- or two-sided intervals containing subjective probability u/100, with u generally chosen as 90, 95 or higher, or as the subjective probability for values below or above a specified limit or from a specified range.
Estimates of the above measures are derived from the random samples obtained in Step 3. Computationally demanding models permit only small (≤100) to medium (several times 100) sample sizes. Therefore, one- or two-sided statistical tolerance limits need to be computed. The difference between confidence and tolerance confidence limits is explained, and the minimum sample sizes, required for specified tolerance and confidence percentages, are derived. The sample sizes depend only on the two chosen percentages and not on the number of uncertainties considered in the analysis.
A Latin Hypercube sample is expected to provide estimates with less variability. However, confidence statements are not available unless the estimates from several Latin Hypercube samples are used.
The chapter ends with a collection of graphical ways to present the uncertainty measures.
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Notes
- 1.
In what follows, uncertain parameters, models and input data are simply referred to as uncertain data since model uncertainties are represented by uncertain parameters and uncertain parameters are categorized as uncertain data.
References
Conover, W. J. (1980). Practical nonparametric statistics (2nd ed.). New York: Wiley.
Guenther, W. C. (1985). Two-sided distribution-free tolerance intervals and accompanying sample size problems. Journal of Quality Technology, 17(1), 40–43.
Hanna, S. R., Chang, J. C., & Fernau, M. E. (1998). Monte Carlo estimates of uncertainties in predictions by a photochemical grid model (UAM-IV) due to uncertainties in input variables. Atmospheric Environment, 32(21), 3619–3628.
Hansen, C. W., Helton, J. C., & Sallaberry, C. J. (2012). Use of replicated Latin hypercube sampling to estimate sampling variance in uncertainty and sensitivity analysis results for the geologic disposal of radioactive waste. Reliability Engineering and System Safety, 107, 139–148.
Heinhold, J., & Gaede, K.-W. (1968). Ingenieur-Statistik. Wien: R. Oldenbourg München.
Ibrekk, H., & Morgan, M. G. (1987). Graphical communication of uncertain quantities to nontechnical people. Risk Analysis, 7(4), 519–529.
McKay, M. D., Beckman, R. J., & Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(2), 239–245.
Morgan, M. G., & Henrion, M. (1990). Uncertainty—A guide to dealing with uncertainty in quantitative risk and policy analysis. Cambridge: Cambridge University Press.
Scheffé, H., & Tukey, J. W. (1944). A formula for sample sizes for population tolerance limits. The Annals of Mathematical Statistics, 15, 217.
Wald, A. (1943). An extension of Wilks method for setting tolerance limits. The Annals of Mathematical Statistics, 14, 45–55.
Wilks, S. S. (1941). Determination of sample sizes for setting tolerance limits. The Annals of Mathematical Statistics, 12, 91–96.
Winkler, R. L., & Hays, W. L. (1975). Statistics: Probability, inference, and decision (2nd ed.). New York: Holt, Rinehart and Winston.
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Hofer, E. (2018). Step 4: Estimate Uncertainty. In: The Uncertainty Analysis of Model Results. Springer, Cham. https://doi.org/10.1007/978-3-319-76297-5_5
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