A nonparametric-based approach on the propagation of imprecise probabilities due to small datasets
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Quantification of uncertainty (UQ) is typically done by the use of precise probabilities, which requires a very high level of precision and consistency of information for the uncertain sources, and is rarely available for actual engineering applications. For better accuracy in the UQ process, greater flexibility in accommodating distributions for uncertain sources is needed to base inferences on weaker assumptions and avoid introducing unwarranted information. Latest literatures proposed a parametric-based approach for the propagation of uncertainty created by lack of sufficient statistical data, yet still has some notable limitations and constraints. This work proposes a nonparametric-based approach that facilitates the propagation of uncertainty in the small dataset case. The first part of this work uses Kernel Density Estimation (KDE) and Bootstrap to estimate the probability density function of a random variable based on small datasets. As a result, two types of sampling densities for propagating uncertainty are generated: an optimal sampling density representing the best estimate of the true density, and a maximum variance density representing risk and uncertainty that is inherent in small datasets. The second part extends the first part, to generate two-dimensional nonparametric density estimates and capture dependencies among variables. After a process to confirm the correlation among the variables based on small datasets, Copulas and the Sklar's Theorem are used to link the marginal nonparametric densities and create joint densities. By propagating the joint densities for dependent variables, researchers can prevent uncertainty in the outputs from being underestimated or overestimated. The effectiveness of the nonparametric density estimation methods is tested by selected test cases with different statistical characteristics. A complete uncertainty propagation test through a complex systems model is also conducted. Finally, the nonparametric-based methods developed in this thesis are applied to a challenging problem in aviation environmental impact analysis.