In this section, a simple numerical example is presented to show limitations and advan- tages of the proposed strategy. The system performance function, G :X ÞÑ g, has the expression
g“x2 y`ex; (5.7)
where, x and y are p-boxes obtained from a Normal distribution with interval hyper- parameters, as shown in Table 5.2. The example function has been selected to simulate a common situation in systems analysis, where several disconnected failure regions may appear. The failure region is defined as XF “ tx, y:Gpx, yq ď0u, and the failure
probability is expressed as the intervalpF “PΓrgď0s.
5.2.1 Random Set approach
In this case, given that the performance is monotonic and increasing with respect toy, an initial important direction can be set as a “ t0, ´1u. Realisations in the aleatory
space,Ω, and in the SNS are shown in Figure 5.2a and in Figure 5.2b. A bundle of lines
parallel to the set direction, a, are, subsequently, generated in the SNS, as shown in Figure 5.3. On each line, the pointsutPluandutBluare identified using a Newton-Raphson algorithm for root finding. The scalars ctPlu and ctBlu are obtained on line l respectively as ctPlu“ ! cPR:Gtluď0 ) , ctBlu“ ! cPR:Gtluď0 ) . (5.8)
Each candidate solution selected by the root finder corresponds to one focal element propagation of Eq. (5.2). The fact that the performance function is monotonic, sensi-
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Aleatory Space α1 α2 survival plausibility belief
(a)5000 focal elements in Ω
−4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 u 1 u2
Sandard Normal Space
survival plausibility belief
(b) 5000 focal elements in SNS
Figure 5.2: Realisations in the Aleatory and Standard Normal Space in the survival, plausi-
bility and belief regions
RS approach NS pˆF pˆF CoVp%q
Monte Carlo 30985 0.016 0.250 4.4
Line Sampling 436 0.012 0.257 9.3
Table 5.3: Results obtained with direct Monte Carlo and Line Sampling for the Random Set
forward propagation
bly eases the focal element propagation, as the search for minimum and maximum is restricted to thex-dimension only. Results from a direct Monte Carlo are compared to results obtained with the proposed strategy, as shown in Table ??. The chance to hit
the belief region is much smaller than the plausibility region, and may require several thousands of runs in a direct Monte Carlo approach. In this particular example, more than104 samples are needed, as shown in Table ??, to obtain an estimate of the lower
probability bound. On the other hand, with the proposed approach, only a few samples are needed to estimate both probability bounds. Note that in Table ??, the number
of samples, NS, coincides with the number of focal elements, and results are obtained
setting the minimum hits in the belief region to 500. Line Sampling requires signifi-
cantly fewer focal elements than Monte Carlo to achieve the same accuracy, as shown in Table ??.
The successful implementation of Line Sampling, in problems involving Random Sets, follows from the flexibility of the adaptive algorithm described in Chapter 3. This would have not been possible in a standard implementation of Line Sampling. Accu- rate probability computations are achieved using Line Sampling with optimal direction, which can be achieved by means of the adaptive algorithm introduced in previous chap- ters. Moreover, on the same problem two optimal directions may be needed as there may be two optimal directions, one for the plausibility and one for the belief border. Figure 5.3 shows a standard implementation of Line Sampling where only one direction is used for the simulation. With the proposed strategy only a rough initial direction is
−4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 u1 u 2 survival plausibility belief
a
samplesFigure 5.3: Standard implementation of Line sampling operating in a transformed aleatory
Standard Normal Space, superimposed to 5000 focal sets obtained with Monte Carlo
needed to start the analysis, which is subsequently adapted as more points are collected near to the domains borders.
5.2.2 Double Loop approach
The Double Loop approach is performed searching for the probability bounds by pick- ing one value in the intervals of Table 5.2 and considering the corresponding normal distributions. All the normal distributions of Table 5.2 should be searched for, how- ever, in this case the system function is monotonic with respect toy, thus the search is restricted to the intervals µx andσx only.
Line Sampling has been used to estimate the probability of failure for each candidate solution. Failure probability estimations are shown in Figure 5.4. The upper bound of the failure probability corresponds to the maximum value ofσxandσy, to the minimum
value of µy and to a value of µx “ 2.05, as shown in Figure 5.4a. The lower bound
corresponds to the maximum value of µy and to the maximum value of σx,σy and to
the maximum value µx “ 2.5, as shown in Figure 5.4b. The double loop approach
enables the identification of the combinations of epistemic parameters that produce the extreme responses of the system. A summary of the obtained results is shown in Table ??, where it can be appreciated that the obtained probability intervals are
2 2.1 2.2 2.3 2.4 2.5 0.15 0.16 0.17 0.18 0.19 0.2 0.21 µx pF Upper Bound
(a)Upper Bound: µy“µy“ ´1.5
2 2.1 2.2 2.3 2.4 2.5 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 µx pF Lower Bound (b) Lower Bound: µy“µy“ ´1
Figure 5.4: Failure probability values with Line Sampling fixing the mean value µy to µy “
´1.5 (a) and toµy“ ´1(b)
Double Loop approach NS pˆminF pˆmaxF CoVp%q
Monte Carlo 106 0.0245 0.187 14.2
Line Sampling 23943 0.0237 0.184 10.3
Table 5.4: Results with a Double Loop approach
narrower than in the Random Set approach, as the Double Loop approach only looks for distributions belonging to the parental probability model.