Appendix A

Operative Procedures for the Methods of Uncertainty Propagation

This appendix reports the operative procedures for uncertainty propagation according to some of the methods described in Chapter 6.

A.1 Level 1 Hybrid Probabilistic–Possibilistic Framework

We assume that the uncertainty related to the first n input quantities, , of the function g is described using the probability distributions , whereas the uncertainty related to the remaining quantities is represented using the possibility distributions . The number of repetitions of the Monte Carlo sampling of the input quantities will be referred to as . The operative steps for the propagation of the hybrid uncertainty information through the function g are (see Figure 6.14)

1. Set .

2. Sample the j th realization of from the probability distributions .

3. Select a value and determine the corresponding α -cuts of the possibility distributions as intervals of possible values of the possibilistic quantities .

4. Calculate the smallest and largest values of , denoted by and respectively, considering the fixed values sampled for the random quantities and all values of the possibilistic quantities in the α -cuts of their possibility distributions .

5. Take the extreme values and found in step 4 as the lower and upper limits of the α -cut of .

6. Return to step 3 and repeat for another α -cut; the possibility distribution of is obtained as the collection of values and for each α -cut.

7. If set and go back to step 1, otherwise exit from the procedure.

At the end of the procedure, having Monte Carlo-sampled values of the probabilistic quantities, an ensemble of realizations of possibility distributions is obtained, that is, a set of possibility distributions . Then, according to the method described in Section 6.1.3 (Equations (6.16)–(6.19)), the belief and the plausibility for any set of the output quantity can be obtained.

A.2 Level 2 Purely Probabilistic Framework

We assume the presence of quantities whose uncertainty is characterized by frequentist probability distributions , , where is a vector of the (unknown) parameters of the corresponding probability distribution. The epistemic uncertainty on the parameters and the aleatory uncertainty on the input quantities are represented using subjective and frequentist probability distributions, respectively. In particular, we let denote the subjective probability density function describing the uncertainty of the parameter(s) . The number of repetitions of the Monte Carlo sampling of the quantities will be referred to as , whereas the number of repetitions of the Monte Carlo sampling of the quantities will be referred as .

The operative steps of the procedure for the propagation of the uncertainty through a function are:

1. Set j _e = 1.

2. Sample the j _e th realization of the epistemic parameter(s) from their respective distributions .

3. Set j _a = 1.

4. Sample the j _a th realization of the aleatory quantities from their respective distributions , sampled in step 2 for each .

5. Compute the model output Z corresponding to the realization of the input quantities:

6. If set and go back to step 4, otherwise go to step 7.

7. Estimate the cumulative distribution of the model output Z, conditioned by the sampled values of the epistemic quantities, from the obtained , .

8. If set and go back to step 2, otherwise exit the procedure.

The application of this procedure provides a set of a cumulative distributions, , , one for each repetition of the outer loop. The interpretation of these distributions is discussed in Section 6.2.