(284d) New and Efficient Interval Sampling Methods for P-Box Uncertainties

Risk analysts recognize two fundamentally distinct forms of uncertainty. The first is the variability that arises from environmental stochasticity, inhomogeneity of materials, fluctuations in time, variation in space, or heterogeneity or other differences among components or individuals. Variability is sometimes called Type I uncertainty, or less cryptically, aleatory uncertainty to emphasize its relation to the randomness in gambling and games of chance. It is also sometimes called irreducible uncertainty because, in principle, it cannot be reduced by further empirical study (although it may be better characterized). The second kind of uncertainty is the incertitude of scientific ignorance, measurement uncertainty, unobservability, censoring, or other lack of knowledge. This is sometimes called Type I1 uncertainty, or simply epistemic uncertainty. In contrast with aleatory uncertainty, epistemic uncertainty is sometimes called reducible uncertainty because an additional empirical effort can generally reduce it. For situations in which the uncertainty about quantities is purely aleatory, probability theory is usually preferred. However, when purely epistemic uncertainties, or both aleatory and epistemic uncertainties, are present several researchers have converged on essentially the same idea: that one can work with bounds on probability for this purpose. In the 1990s, researchers introduced interval-type bounds on cumulative distribution functions, which we call “probability boxes”, or “p-boxes” for short. Uncertainty propagation for uncertainties described by probability distributions is generally carried out using sampling techniques. For p-box uncertainties, researchers use interval Monte Carlo for propagation. However, this method is computationally intensive. In this paper, we develop and evaluate five new interval sampling techniques for P-boxes. These sampling techniques are derived from quasi-Monte Carlo sequences like Hammersley and Sobol sequences, uniformity theory, and interval mathematics. The sampling techniques are evaluated using various functions of two to 800 uncertain variables defined by p-box uncertainties. Heuristics for selecting an efficient sampling technique based on the number of uncertainties is presented. A real-world example of quality control of a CSTR where the variance is minimized is used to illustrate this approach's usefulness and efficiency.

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