Thomas Fetz

Propagation of uncertainty through multivariate functions in the framework of sets of probability measures

Abstract This paper proposes a solution to Challenge Problem 1 posed by Oberkampf et al. [Reliab. Engng Syst. Safety (this issue)] and focuses on questions of dependence or independence of variables. The specification of uncertainties is consistently reduced to sets of probability measures. For whatever interpretation, the results essentially depend on the choice of combination rule in defining the joint distributions. The approach adopted here is shown to be consistent with interval analysis, random sets and fuzzy sets, and admits a clear description of the influence of the concept of independence used. A MATLAB-tool has been developed that allows to compute the lower and upper probability distributions for all choices of framework and combination rule.

Fuzzy models in geotechnical engineering and construction management

This article is devoted to a variety of applications of fuzzy models in civil engineering, presenting current work of a group of researchers at the University of Innsbruck. With fuzzy methods and possibility theory as an encompassing framework, the following areas are addressed: uncertainties in geotechnical engineering, fuzzy finite element computation of a foundation raft, fuzzy dynamical systems, processing uncertainty in project scheduling and cost planning.

Applications of possibility and evidence theory in civil engineering

This article is devoted to applications of fuzzy set theory, possibility theory and evidence theory in civil engineering, presenting current work of a group or researchers at the University of Innsbruck. We argue that these methods are well suited for analyzing and processing the parameter uncertainties arising in soil mechanics and construction management. We address two specific applications here: finite element computations in foundation engineering and a queueing model in earth work.

Imprecise random variables, random sets, and Monte Carlo simulation

The paper addresses the evaluation of upper and lower probabilities induced by functions of an imprecise random variable. Given a function g and a family X λ of random variables, where the parameter λ ranges in an index set ź, one may ask for the upper/lower probability that g ( X λ ) belongs to some Borel set B. Two interpretations are investigated. In the first case, the upper probability is computed as the supremum of the probabilities that g ( X λ ) lies in B. In the second case, one considers the random set generated by all g ( X λ ) , λ ź ź , e.g. by transforming X λ to standard normal as a common probability space, and computes the corresponding upper probability. The two results are different, in general. We analyze this situation and highlight the implications for Monte Carlo simulation. Attention is given to efficient simulation procedures and an engineering application is presented. Functions of an imprecise random variable defined by a family of random variables are studied.Lower and upper probabilities are defined in two different ways and compared.Cost saving computational methods for simulating families of random variables and random sets are presented.Monte Carlo reweighting and polynomial chaos expansions are employed.The results are applied to estimating the failure probability of a beam under uncertain bedding.

MULTIVARIATE MODELS AND VARIABILITY INTERVALS: A LOCAL RANDOM SET APPROACH

This article is devoted to the propagation of families of variability intervals through multivariate functions comprising the semantics of confidence limits. At fixed confidence level, local random sets are defined whose aggregation admits the calculation of upper probabilities of events. In the multivariate case, a number of ways of combination is highlighted to encompass independence and unknown interaction using random set independence and Frechet bounds. For all cases we derive formulas for the corresponding upper probabilities and elaborate how they relate. An example from structural mechanics is used to exemplify the method.