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

possibility. The triplet (Ω, E, Π) is called the possibility space EPos.

      Note that the possibility measure (or PoDF) Π provides a measure of the likelihood of each element of E. The membership functions introduced for fuzzy sets (for example, triangular, trapezoidal functions) are typically used as PoDFs.

      DEFINITION 1.18.– Let ξ be an uncertain variable to which the possibility distribution Π is associated. Let

and

be the lower and upper bounds of the α-cut of ξ. Let f be a non-negative weighting function f : [0,1] → ℝ , monotonically decreasing and verifying the condition of normalization in the integral sense. The f-weighted possibilistic mean operator is:

      [1.11]

This operator is the equivalent of expectation in the context of probability theory. By introducing various weighting functions, different levels of importance can be assigned to the different α-cuts of the possibility distribution. There are also similar expressions existing for higher order moments in a possibility context.

      In order to be able to compare the likelihood of different events, two quantities, the possibility and necessity of an event, are introduced.

      DEFINITION 1.19.– Let (Ω, E, Π) be a space of possibility. We call the possibility and necessity of an event e ∈ E, respectively:

      [1.12]

[1.13]

      PROPERTY 1.2.– Let (Ω, E, Π) be a space of possibility and an event e ∈ E. We have the following properties:

      [1.14]

      [1.15]

      [1.16]

[1.17]

      [1.18]

      More intuitively, we can say that possibility measures the degree to which the facts do not contradict the assumption that an event can happen. If an event has a possibility of 1, it means that there is no reason to believe that the event cannot happen. It does not mean, however, that the event will certainly happen. In order for an event to be certain, both its possibility and its necessity must be equal to 1. On the other hand, if we consider that an event cannot happen, then it must be assigned a possibility of 0.

It should be noted that, similarly to what has been done in probability theory, a cumulative possibility function (CPoF), a cumulative necessity function (CNeF), a cumulative complementary possibility function (CCPoF) and a cumulative complementary necessity function (CCNeF) can be associated with a PoDF. Figure 1.7 illustrates these different functions with an example.

Figure 1.7. Illustrations of (a) a possibility distribution function (PoDF); (b) the cumulative possibility function (CPoF) and the corresponding cumulative necessity function (CNeF)

Note that for the example shown in Figure 1.7, the most likely values are those in the interval [5,7], which have a possibility of 1. The larger intervals encompassing this latter then have progressively smaller possibility values, until they reach 0 for the interval [1,10]. Values below 1 or above 10 are thus considered impossible. All the properties of equations [1.13]–[1.17] can be verified on these curves. Note also that the stepwise nature of the cumulative functions is typical of the modeling of epistemic uncertainties using alternative approaches. Indeed, the quantification of uncertainty is, most of the time, based on experts who would assign likelihood values to the different intervals, thus allowing the construction of the PoDF. The more finely the expert can elicit the uncertainty (by assigning likelihood values to many successively larger nested intervals), the smaller the steps will be. Finally, in terms of uncertainty propagation, given that the PoDFs are, most of the time, based on fuzzy set theory membership functions, they also make use of the same propagation techniques discussed in section 1.6. Propagation is thus typically done by interval propagation for different α-cuts.

      1.7.2. Comparison between probability theory and possibility theory

      This section aims to highlight the commonalities and differences between probability and possibility theories. First of all, in terms of axioms, the main difference lies in terms of σ-additivity for probability theory and subadditivity for possibility theory (see the definitions given in sections 1.3.1 and 1.7.1). Let us recall that the probability of the union of disjoint events is equal to the sum of the probabilities of the events. Thus, if {A₁, …, An} is a partition of the universe, then the sum of the probabilities of the A_i must be equal to 1. There is no similar constraint in terms of possibilities of events Ai. For example, the sum of the probabilities of the events “tomorrow it will snow” and “tomorrow it will not snow” must be equal to 1. On the other hand, if we assign a possibility of 0.7 to “it will snow tomorrow”, we must assign a possibility of 1 to the event “it will not snow tomorrow”. This is because the maximum possibility in the universe must be equal to 1. Thus, since the possibility of the universe Ω is the maximum possibility of the events in the universe, this implies that the possibility of at least one event in the partition must be equal to 1.

Both probability theory and possibility theory work with distribution functions, called PDF in the probabilistic framework and PoDF within the possibility framework. However, despite visual similarities, these two functions account for significantly different modeling. Figure 1.8 illustrates some of these differences.

      We can see the following differences in Figure 1.8:

       – the area under the probability density curve provides the probability of the corresponding event while the area under the curve of a PoDF has no significance;

       – in the case of (absolutely) continuous random variables the probability of the variable taking a specific value is zero, while the possibility of the same event can be any value between 0 and 1;

       – the area under the curve of a PDF must be 1 while its maximum can be any value. The converse is