CRITERIOS PARA LA DEFINICIÓN DE LAS FALTAS

The failure probability bounds are obtained by generating a large number of samples or focal elements, and subsequently constructing the associated D-S structure of the response. The procedure for constructing the D-S structure of the response is briefly summarized as,

1. Draw a uniform random number,αtsu, for each p-box, between₀ and ₁;

2. Get the sample endpoints xtsu

“ xtsu

ˆytsu _{using the inverse bounding CDFs,}

F´1 as; xtsu “F´1 ´ αtsu¯_{; x}tsu “F´1´_αtsu¯_{; (5.11)}

3. Identify minimum, gtsu_{, and maximum response,g}tsu_{, within the search domain,}

xtsu. This step is also referred to as min-max propagation;

4. Repeat the above steps for s“1, . . . , NS;

5. Collect samples and corresponding response extrema, x˜t_minsu and x˜tmaxsu .

Once the D-S structure of the response is obtained, the failure probability bounds are obtained from the D-S plausibility and belief as

p_F “ lim NsÑ8 1 Ns Ns ÿ s“1 Irgtsuă0s; (5.12) p_F “ lim NsÑ8 1 Ns Ns ÿ s“1 Irgtsuă0s; (5.13)

where, I : R Ñ t0,1u is the indicator function. Most of the attention, in the above

procedure, is usually given to the min-max propagation step. In fact, this can be troublesome, especially if the response is the output of a black-box model, where the propagation is performed by invoking global optimisation algorithms.

5.3.3.1 The min-max propagation

The performance function of Eq. (5.7) is monotonically increasing with respect to y, which is a great advantage as it excludes the presence of relative minima and maxima. Moreover, it implies that, for every value ofx, as the variabley decreases/increases so does the performance function. This leads to the following relationships

g“x2_min y`exmin_{; g}

“x2_max y`exmax_; (5.14) wherexminandxmaxare yet to be determined. On the other side, the performance func-

tion is not monotonic with respect tox. The sign of the first and second derivatives ofg, says that the function is monotonically increasing with respect to xonly in the portion where y P r´1.36,0s. Whereas, for y P p´8,´1.36q Y p0,8q the function may have a

minimum or maximum. Within the latter portion of domain, the minimum/maximum is identified solving forx the partial derivativeBg{Bx, and subsequently checking if the obtained value is smaller/greater than the values at the endpoints x andx.

5.3.3.2 Solution to the back-propagation problem

The solution consists in collecting all those realisations in the input space that corre- spond to the response extrema. First the min-max propagation problem is solved and then all of the argument minima and maxima are collected back to the input space. These are also referred to as extreme realisations. The failure probability bounds,

p_F “ rp_F, p_Fs “ r0, 0.754s,

are computed by means of direct Monte Carlo, i.e. using Eqs. (5.12) and (5.13) with

105 MC samples.

The strategy makes use the distributions of minima and maxima to reconstruct the corresponding upper and lower bounds of the failure probability. In Figures 5.8a and 5.8b the distribution of minima, corresponding to the upper failure probability, are compared with the extreme normal distributions of Table 5.6 for the maximum failure probability, obtained by solving the optimisation problem. In Figure 5.8b the normal distributions obtained using the parametric approach, that corresponds to the maximum failure probability of Table 5.6, are superimposed to the distributions of collected minima. The collected minima (Random Set approach) are not normally distributed and for the p-box of variable y (Figure 5.8b) they coincide with the left bounding CDF.

The solution to the back propagation problem can be found by searching the space of parental (normal) distribution functions for those hyper-parameters corresponding to the min/max failure probability. Within the non-parametric approach, this is done selecting the parental distribution functions that provide the best fit to the collected

x -6 -4 -2 0 2 4 6 F(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Random Set approach Double Loop approach

(a)x-CDFs y -8 -6 -4 -2 0 2 4 F(y) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Random Set approach Double Loop approach

Bounding CDF

(b) y-CDFs

Figure 5.8: Extreme realisations for the failure probability upper bound corresponding to

Table 5.6 (Double Loop approach) and collected argument minima (Random Set approach)

minp˚ F maxp˚F P-box 1.8 10´5 _0.468 x pµxqmin˚“ 4.97 pµxqmax˚ “ 2.43 x pσxqmin˚ “ 0.23 pσxqmax˚ “ 0.60 y pµyqmin˚ “ -0.51 pµyqmax˚ “ -1.98 y pσyqmin˚ “ 1.10 pσyqmax˚“ 1.07

Table 5.7: Failure probability bounds and corresponding extreme normal distributions ob-

tained with the non-parametric approach

distributions of minima and maxima. Here, the normal distribution that best fits the extreme realisations is obtained using the Kolmogorov-Smirnov test, by minimizing the statistic (k-s distance) DNs “supx|FNspxq ´Fpxq|. The results from the k-s distance

minimisation are shown in Table??. Figures 5.9a and 5.9b show the x and y extreme normal distributions that best fit the collected minima, which correspond to the failure probability upper bound. Note that the two normal distribution functions of extreme values obtained using the two approaches are quite similar. The failure probability bounds solution of the Random Set back propagation problem are shown in in Table

??. These bounds appear to be included in the bounds of Table 5.6, therefore, they do

not correspond to the normal distributions responsible for the minimum and maximum failure probability. By comparing the argument optima of Table 5.6 and??, variable x seems to be quite close to the optimal distribution, while for variabley only the mean value corresponding to maxp˚

F is close to the target of Table 5.6. Note, from Figure

5.9b, that the extreme realisations of variableyare distributed as the upper CDF, since the model is monotonic with respect to this variable.

−40 −2 0 2 4 0.2 0.4 0.6 0.8 1 x F(x) extreme realizations normal best−fit

(a) x-normal distribution; best-fit obtained with a k-s distance of 0.11 −80 −6 −4 −2 0 2 4 0.2 0.4 0.6 0.8 1 y F(y) extreme realizations normal best−fit

(b) y-normal distribution; best-fit obtained with a k-s distance of 0.15

Figure 5.9: Extreme distributions for the failure probability upper bound obtained by fitting

normal distributions to the collected minima