This paper presents the formulation of an uncertainty quantification challenge problem consisting of five subproblems. These problems focus on key aspects of uncertainty characterization, sensitivity analysis, uncertainty propagation, extreme-case analysis, and robust design.
This paper develops techniques for constructing metamodels that predict the range of an output variable given input–output data. We focus on models depending linearly on the parameters and arbitrarily on the input. This structure enables to rigorously characterize the range of the predicted output and the uncertainty in the model’s parameters. Strategies for calculating optimal interval predictor models (IPMs) that are insensitive to outliers are proposed. The models are optimal in the sense that they yield an interval valued function of minimal spread containing all (or, depending on the formulation, most) of the observations. Outliers are identified as the IPM is calculated by evaluating the extent by which their inclusion into the dataset degrades the tightness of the prediction. When the data generating mechanism (DGM) is stationary, the data are independent, and the optimization program (OP) used for calculating the IPM is convex (or when its solution coincides with the solution to an auxiliary convex program); the model’s reliability, which is the probability that a future observation would fall within the predicted range, is bounded tightly using scenario optimization theory. In contrast to most alternative techniques, this framework does not require making any assumptions on the underlying structure of the DGM.
Introduction. The purpose of this paper is to study a convexity property on normed linear spaces (NLS's) which we call P-convexity. Interest in P-convexity is generated by a theorem of Anatole Beck ([1] or [2]) which states that a Banach space 3£ is P-convex if and only if a certain strong law of large numbers is valid for X-valued random variables. Let 3£ be a NLS and (S, 2, m) a measure space. The Borel a-field 38 of X is the tr-field of sets generated by the subsets of BE open in the strong (norm) topology. A Borel set is an element of 38. A function X from 5 into 3t is called strongly measurable if for each Be Sä, X~ \B) = {se S : X(s) e B} e 2. X is called essentially separably valued if there is N eH, m(N) = 0, such that X(S-N) is separable. If js || X(s) || dm{s) < oo and X is essentially separably valued, then X is strongly (Bochner) integrable and there exists ye?), where ?) is the completion of 3£, such that for every x* e £)*, the conjugate space to 9), Js x*X(s) dm(s) = x*y. In this case we write y=js X(s) dm(s)=$s X dm. y is called the strong integral of X. A probability space is a measure space (Í2, 2, 3P) of total measure 1 (^(D)= 1). An 3t-valued random variable on Q. is a function X: Q.-*■ X which is strongly measurable and essentially separably valued. The expectation of X, E{X) = ja X d£? if this integral exists. For a random variable X with expectation, we define the variance of X by (20= f ixw-EvnvtWr). Ja var i We will call a finite set Xu..., Xn of 3t-valued random variables on Q. independent if for each choice of Borel sets Blt..., Bn e 38, we have d\Xi eB1&-&XneBn) = 3^{XX eB,)-3P{Xn e Bn); that is, n {o £ Q : XiUm) e B1 & • ■ & Xn(oj) e Bn) = Y\0>{ojeQ.: X¿w) e Bt}. i = i An infinite set of random variables is independent if each finite subset is.
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