Efficient Propagation with Line Sampling

where, the indicator function, I :R ÞÑ t0,1u equals one if the statement is true and zerootherwise. The image of the focal element is obtained via min-max propagation as

Gtsu “ inf xPγtsuGpxq, G tsu “ sup xPγtsu Gpxq. (5.2)

The propagation of focal elements is computationally very demanding, as a large number of samples, NS, and one min-max propagation, for each sample, is required, which

is usually completed invoking global optimisation (such as evolutionary or stochastic algorithms). Thus, efficient approaches are needed to reduce the number of samples and consequently the number of propagations. For example, Subset simulation can be used to efficiently produce samples in the aleatory space, Ω, as shown in [6]. Here, a

novel approach based on Line Sampling is proposed to produce samples in the aleatory space,Ω, and to significantly reduce the number of focal elements required to estimate

the failure probability bounds.

5.1.2 Efficient Propagation with Line Sampling

By means of Line Sampling, the number of required focal elements can be reduced by orders of magnitude. In order to exploit the feature of the advanced simulation method, points in the aleatory space α have to be mapped to the Standard Normal Space and vice versa. In the latter space, lines can be generated as explained in Chapter 3 to look for points at the border between safe, plausibility and belief domains. The method generates normal random pointsutsu_{in the SNS, maps them to the aleatory space and}

computes the corresponding focal elements. The procedure to produce a focal element and its image through G, starting from a point in the SNS, is

• Draw a sample,utsu

„Np0,1q, from the SNS;

• Map the sample back to the aleatory space as αtsu “C ´ Φput₁suq, . . . ,Φput_dsuq ¯ ; (5.3)

• Obtain the focal elementγtsu

“Γpαtsu

q;

• Obtain the image of the focal element through the system function,Gtsu;

The key feature about the aforementioned procedure is that each point in the SNS, u, is associated with a unique focal element γ (see Figure 5.2). Therefore, in the SNS, as well as in the aleatory domain Ω, it possible to identify three different regions: a

survival region where, γ XXF “ ∅, a plausibility region where, γ XXF ‰ ∅ and a

belief region where, γ ĎXF. These regions are fully characterised by the performance

Table 5.1: Survival, Plausibililty and Belief regions Survival γXXF “∅ Gą0

Plausibility γXXF ‰∅ Gď0

Belief γ ĎXF Gď0

in the SNS and in the aleatory space, Ω, that separate these three regions (see e.g.

Figure 5.2). In the SNS, these two boundaries provide information about the failure probability, as has been shown in Chapters 3 and 4. The procedure to compute lower and upper failure probability bounds using Line Sampling is:

• Set up an important direction,aPRd, in the SNS;

• Generate a number of lines, NL, normally spaced that are orthogonal to the

hyperplane identified by a. Points on the generated line, l, can be parametrised via the scalar,cPRmaking use of the coordinatesu˚of the hyperplane orthogonal

to the important direction, as utlu pcq “c¨a`u˚tlu ´ ´ u˚tlu ´a ¯ ¨a; (5.4)

where,u˚tluare the coordinates of the point where the line_lmeets the orthogonal

hyperplane.

• On each line, look for points at the border between survival, plausibility and belief regions. This step requires the point to be mapped back to the aleatory space, and subsequently requires the focal elements to be propagated through the system function,G;

• Denote byut_Pluthe point on linelat the border between survival and plausibility regions, and by ut_Blu the point on line l at the border between plausibility and belief regions;

• LetcP and cB be the scalars, such thatutlupcPq “u_Ptlu and utlupcBq “ut_Blu;

• Compute the probability lower and upper bounds on linel as ptlu

F “Φp´|cP|q; p

tlu

F “Φp´|cB|q; (5.5)

• Estimate lower and upper probability bounds, making the average over the number of lines as p_F “ 1 NL NL ÿ j“1 pt_Flu, p_F “ 1 NL NL ÿ j“1 pt_Flu. (5.6)

The proposed strategy can be used for the estimation of very small probabilities without compromising the accuracy. It is therefore particularly suited either in problems with

Table 5.2: Mean values and standard deviations for the definition of the p-box bounds

P-box µ σ Distribution

x r2.0, 2.5s r0.5, 0.6s Normal

y r´1.5, ´1s r0.5, 0.6s Normal

high levels of imprecision where the lower probability bound is quite small, or in prob- lems involving safety-critical systems with strict failure probability requirements. With this strategy the amount of computations required is independent of the magnitude of the probability target. For example, with a probability target of 10´6_{, more than ten}

million focal sets would be required using a plain Monte Carlo approach, which is clearly prohibitive. The efficiency of the strategy can be further enhanced by means of good programming, i.e. by selecting the lines to a specific order, and by efficiently searching for the bordering points.