Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments

In terms of uncertainty propagation, it is typically achieved for an orthorectangular uncertainty structure using interval propagation for each focal element, using the methods mentioned in section 1.5for interval analysis.

Figure 1.12. Illustrations of (a) a BPA function m(x); (b) the corresponding cumulative plausibility function (CPlF) and cumulative belief function (CBF)

Evidence theory is based on the definition of a structure of uncertainties in the form of masses attributed to different focal elements. The definition of this uncertainty structure comes from experts. Analogously, as in the framework of possibility theory, rules have been defined in order to be able to aggregate different and sometimes divergent sources of uncertainty, for example, from different experts.

Let m1 and m2 denote two BPA functions with the respective sets Ci and Cj as focal elements. These functions can be aggregated according to the following rules:

– the Dempster rule, defined by:[1.22]The denominator reflects the conflict between the two sources through normalization. Note that therefore only the agreement between sources is retained in the aggregate uncertainty structure. A disadvantage of this rule is in the way it deals with divergent opinions and it is preferable to apply this rule only in cases where there is reasonable agreement between sources;

– the Yager rule, defined by:[1.23][1.24][1.25][1.26]The Yager rule attributes the conflict to the entire universe Ω, thus increasing the degree of ignorance associated with it;

– the weighted mean rule:

[1.27]

In this rule, the conflict is simply averaged with a weighting based on the confidence attributed to each expert. Other rules, as well as variations around the three presented, have been proposed and the reader can refer to Sentz and Ferson (2002) for more information.

The quantification of epistemic uncertainties is a major challenge for achieving an integral framework for the quantification and propagation of uncertainties in industrial problems. Depending on the nature of the lack of knowledge, these uncertainties can be modeled in different ways. In some situations, epistemic uncertainties may be due to insufficient sampling, in which case the probabilistic framework, through the use of probability boxes ( p-boxes ), is probably the most relevant framework.

In other situations, no samples may be available and obtaining them would be too time consuming or costly, so only expert opinion can provide information on the uncertainties involved. Some consider that such advice cannot be objective and does not need to be mixed with rigorous, sample-based uncertainty quantification within a probabilistic framework. Others consider that, although imperfect, because it is subjective, expert opinions can contain useful information for uncertainty quantification and, rather than ignoring them, have focused on developing theories in order to be able to model the uncertainties expressed in this way. This is particularly the case with the theories of possibility and evidence that we have reviewed. Basically, both theories have the same objective, which is to bound an unknown probability distribution function by the construction of an uncertainty structure based on expert opinion. It is worth noting, however, that possibility theory is based on axioms which are quite different from those of probability theory, which can sometimes lead to counterintuitive results. These axioms also imply a relatively wide bounding for the unknown probability distribution function. Evidence theory, on the other hand, allows a narrower bounding for the distribution function, provided that it is possible to construct BPA functions with a very fine granularity. It also does not involve the more constraining axioms of possibility theory. To conclude, let us also note that in most real problems, epistemic and aleatory uncertainty modeling must be simultaneously considered. This combination of different approaches remains a challenge, especially from the point of view of the associated numerical costs.

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