Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments. Группа авторов

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СКАЧАТЬ distributions). One of the fundamental questions that then arises is how to deal with divergent expert opinions. For this purpose, different rules for combining possibility distributions have been established.

      Let Π₁ and Π₂ be two possibility distributions. These distributions can be aggregated according to the following rules:

       – the conjunctive mode: this is the equivalent of the intersection of events. It corresponds to retaining only the consensus (the common area under the two distributions). This consensus is typically renormalized in order to satisfy the possibilistic axioms;

       – the disjunctive mode: this is the equivalent of the union of events. It corresponds to the union of the two distributions;

       – the intermediate mode (proposed by Dubois and Prade 1992): as its name indicates, this is an intermediate mode between the two previous ones. By defining the consensus, that is, the upper possibility bound following the intersection of the two distributions, by h, the distribution of the intermediate mode is defined by:

      [1.19]

      These three modes of combination usually make it possible to combine distributions of possibility from different sources (experts, for example). If there is good agreement between the sources (for example, trapezoidal distributions with overlapping cores), then either the connective or disjunctive modes are both well suited. The choice between the two depends on whether one wishes to consider only consensus or whether one wishes to integrate divergent views as well.

In the absence of good agreement between sources (for example, trapezoidal distributions with non-overlapping kernels), the conjunctive and disjunctive modes are less well suited. The disjunctive mode introduces a multimodal distribution (two peaks), which is generally not desirable in terms of quantifying and propagating uncertainties. The conjunctive mode by itself is too limiting, as it only considers consensus, which can be very limited or even null. The intermediate mode is a proposed solution to resolve these two problems associated with the conjunctive and disjunctive modes. It should be noted that variations on the intermediate mode have also been proposed in order to give more confidence to one distribution (in other words, an expert) than to another. The reader may refer to Dubois and Prade (1992) for more information.

1.8. Evidence theory

      The theory of belief functions (or evidence theory) is another theory for modeling uncertainties of an epistemic nature. It was developed by Dempster (1967) and Shafer (1976) and is consequently sometimes known as the Demspter–Shafer theory. This approach is similar in spirit to the probability box theory and possibility theory, in that it seeks to obtain a bounding for a CDF.

      1.8.1. Theoretical context

      Formally, evidence theory is defined on a belief space EE defined as follows:

      DEFINITION 1.20.– Let Ω be a set and E the set of the subsets of Ω. A function m : E → ℝ is called belief mass function if it satisfies the following axioms:

       – the function has values between 0 and 1: ∀A ∈ E,0 ≤ m(A) ≤ 1;

       – the image of the empty set is 0: m(∅) = 0;

       – the image of all the events in the universe is 1:m(Ω) = 1;

       – the sum of the masses of the events of E is 1: ∀An ∈ E,

      DEFINITION 1.21.– Let Ω be a set, called universe, E the set of subsets of Ω and m a belief mass function. The triplet (Ω, E, m) is called the belief space EE.

      Note that the belief mass function m provides, as was the case for the possibility distribution, a measure of the relative likelihood of each element of E. In the terminology of the theory of belief functions, an element of E that has a non-zero mass is called a focal element, denoted Ω_i. Note that these focal elements are, most of the time, defined such as to form a set of disjoint elements. The quantity mi = m(Ω_i) is then called mass (or sometimes basic probability assignment [BPA]) associated with this focal element.

      In order to be able to compare the likelihood of different events, two quantities, plausibility and belief of the event, are introduced.

      DEFINITION 1.22.– Let (Ω, E, m) be a belief space. We call belief (noted Bel) and plausibility (denoted Pl) of an event e ∈ E, respectively:

      [1.20]

      [1.21]

      From a conceptual point of view, the mass of a focal element m (Ω_i) provides the true likelihood that is attributed to that focal element, but without specifying how this likelihood is distributed among the subsets of Ωi. Therefore, the belief Bel(e) of an event e represents the minimal likelihood that can be associated with e, while the plausibility Pl(e) represents the maximum likelihood that can be associated with e.

In the vast majority of applications of evidence theory, focal elements are defined such as to form a set of adjacent polytopes (or even orthotopes). To illustrate this, consider a universe with nine focal elements Ω1 to Ω9 of mass m1 to m9 (see Figure 1.11). Let us now consider the belief and plausibility of the event e, shaded in the figure. According to the definition, the plausibility Pl(e) is obtained by adding the masses of the focal elements that have a non-zero intersection with e (in our case Ω2, Ω3, Ω5, Ω6, Ω8 and Ω9). This means that each focal point included even partially in e contributes to the plausibility of e. The belief Bel(e) is calculated by summing only the masses of the focal elements entirely included in e (in our case, Ω3 and Ω6). Belief and plausibility can thus be seen as a bounding for the likelihood of an event, reflecting the fact that the distribution of likelihood within a focal element is not known, due to epistemic uncertainty. By considering Ω8 we can, for example, consider that the likelihood of the small gray triangle included in Ω8 is very low and that the mass m8 thus contributes almost nothing to the likelihood of e. This hypothesis comes closer to the calculation of belief. On the other hand, we can also assume that all the likelihood of Ω8 is concentrated in this small gray triangle and all the rest of Ω8 is not at all likely. In this case, m8 must actually contribute in full to the likelihood of e. This assumption is thus close to the plausibility calculation. Both cases are thus extreme cases, bounding the true likelihood.

Figure 1.11. Example of an uncertainty structure (focal elements Ωi of mass mi) in the theory of belief functions. An event e is grayed out here

Note that, in a similar way to what has been done in probability СКАЧАТЬ