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G. Filippi and M. Vasile
The network’ topology is then exploited with Algorithm 2 to propagate epistemic
uncertainty and calculate at a reduced computational cost the Belief and Plausibility
curves corresponding to d opt . For this purpose, the Frame of Discernment as defined
in Sect. 17.2.1 is constructed assigning to each uncertain variable u two possible
intervals and their bpa. The total number of focal elements is then n f e = 2 n u,tot .
The number of optimisations required for exact quantification of the Belief curve is
n opt,exact = n f e while the number required by the decomposition approach follows
Eq. (17.60).
Table 17.1 collects the results of the simulations for the different network’
topology. It is shown the gain in computational cost offered by the decomposition
approach together with the generated error. The error has been evaluated as the
ratio (A exact − A dec )/A dec where A exact and A dec are the integral of the exact
and decomposition curve, respectively. For problems satisfying the conditions in
Sect. 17.4.4.1, the decomposition approach assures to quantify exactly the DST’
measures of probability when the partial curves are entirely sampled. For example,
considering the topology (c) and using 64 samples (all the combinations for the 4
samples for each of the 3 coupled curve), we obtain an error equal to zero running a
number of optimisations that is 19% of total number of focal elements. Furthermore,
the smaller is the number of samples, the lower is the computational cost but the
higher the error. For example, for the same problem, a single sample brings to a
cost that is 0.59% of the exact evaluation increasing the error, however, to 187%.
Between these two extreme positions we can make a trade-off between cost and
accuracy.
Figure 17.2 shows instead the plots of the partial curves and the final curves
calculated with the decomposition strategy and also the exact curves calculated
running an optimisation for each focal element for f applied to topology (c).
It has been noticed that sampling very close points in the space of the maxima
in Fig. 17.2a brings to negligible contribution to the error reduction in Fig. 17.2b.
Then, future works will show how a careful selection of the samples can improve
the algorithm performance mitigating both the error (possibly bringing it to zero)
and the cost under the condition that the user is not interested in the entire curve
but only in the value assumed by belief and plausibility at some specific thresholds.
This is particularly important when the decomposition strategy is nested within an
optimisation loop, for example, in the approach presented in Eq. (17.39).
17.5 Conclusions
In this chapter we described the importance of uncertainty quantification in real
world modern problems and its challenge due to the computational cost, an overview
of the uncertain probability theory and a deeper insight in Evidence Theory. Two
novel tools have been described to do a rigorous uncertainty quantification when the
G. Filippi and M. Vasile
The network’ topology is then exploited with Algorithm 2 to propagate epistemic
uncertainty and calculate at a reduced computational cost the Belief and Plausibility
curves corresponding to d opt . For this purpose, the Frame of Discernment as defined
in Sect. 17.2.1 is constructed assigning to each uncertain variable u two possible
intervals and their bpa. The total number of focal elements is then n f e = 2 n u,tot .
The number of optimisations required for exact quantification of the Belief curve is
n opt,exact = n f e while the number required by the decomposition approach follows
Eq. (17.60).
Table 17.1 collects the results of the simulations for the different network’
topology. It is shown the gain in computational cost offered by the decomposition
approach together with the generated error. The error has been evaluated as the
ratio (A exact − A dec )/A dec where A exact and A dec are the integral of the exact
and decomposition curve, respectively. For problems satisfying the conditions in
Sect. 17.4.4.1, the decomposition approach assures to quantify exactly the DST’
measures of probability when the partial curves are entirely sampled. For example,
considering the topology (c) and using 64 samples (all the combinations for the 4
samples for each of the 3 coupled curve), we obtain an error equal to zero running a
number of optimisations that is 19% of total number of focal elements. Furthermore,
the smaller is the number of samples, the lower is the computational cost but the
higher the error. For example, for the same problem, a single sample brings to a
cost that is 0.59% of the exact evaluation increasing the error, however, to 187%.
Between these two extreme positions we can make a trade-off between cost and
accuracy.
Figure 17.2 shows instead the plots of the partial curves and the final curves
calculated with the decomposition strategy and also the exact curves calculated
running an optimisation for each focal element for f applied to topology (c).
It has been noticed that sampling very close points in the space of the maxima
in Fig. 17.2a brings to negligible contribution to the error reduction in Fig. 17.2b.
Then, future works will show how a careful selection of the samples can improve
the algorithm performance mitigating both the error (possibly bringing it to zero)
and the cost under the condition that the user is not interested in the entire curve
but only in the value assumed by belief and plausibility at some specific thresholds.
This is particularly important when the decomposition strategy is nested within an
optimisation loop, for example, in the approach presented in Eq. (17.39).
17.5 Conclusions
In this chapter we described the importance of uncertainty quantification in real
world modern problems and its challenge due to the computational cost, an overview
of the uncertain probability theory and a deeper insight in Evidence Theory. Two
novel tools have been described to do a rigorous uncertainty quantification when the