Approximate representations of random intervals for hybrid uncertainty quantification in engineering modeling

Abstract

We review our approach to the representation and propagation of hybrid uncertainties through high-complexity models, based on quantities known as random intervals. These structures have a variety of mathematical descriptions, for example as interval-valued random variables, statistical collections of intervals, or Dempster-Shafer bodies of evidence on the Borel field. But methods which provide simpler, albeit approximate, representations of random intervals are highly desirable, including p-boxes and traces. Each random interval, through its cumulative belief and plausibility measures functions, generates a unique p-box whose constituent CDFs are all of those consistent with the random interval. In turn, each p-box generates an equivalence class of random intervals consistent with it. Then, each p-box necessarily generates a unique trace which stands as the fuzzy set representation of the p-box or random interval. In turn each trace generates an equivalence class of p-boxes. The heart of our approach is to try to understand the tradeoffs between error and simplicity introduced when p-boxes or traces are used to stand in for various random interval operations. For example, Joslyn has argued that for elicitation and representation tasks, traces can be the most appropriate structure, and has proposed a method for the generation of canonical random intervals from elicited traces. But alternatively, models built as algebraic equations of uncertainty-valued variables (in our case, random-interval-valued) propagate uncertainty through convolution operations on basic algebraic expressions, and while convolution operations are defined on all three structures, we have observed that the results of only some of these operations are preserved as one moves through these three levels of specificity. We report on the status and progress of this modeling approach concerning the relations between these mathematical structures within this overall framework.

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