17 Introduction to Evidence-Based Robust Optimisation
557
three categories: sampling methods, dimensionality reduction methods and outer
approximation methods.
17.4.1 Belief Estimation by Sampling
In the literature one can find a handful of sampling-based methods for the
computation and estimation of Bel [22, 79–81]. As a representative example, in
[22], the authors suggest the use of the density function:
d j (u j ) =
n(j )
k=1
δ(u j |I jk )bpa jk (I jk )/(b jk − a jk )
(17.43)
for the j -th dimension of the uncertain space, if the intervals for the uncertain
parameters are given in the form I jk = {u j : a ik ≤ u j ≤ b jk }. Here n(j ) is
the number of intervals in the j -th dimension and
δ(x j |I jk ) =
1 if x j ∈ I jk
0 otherwise;
(17.44)
then the sampling distribution is
d(x) =
nU
j =1
d j (u j )
(17.45)
Distribution (17.45) explores adequately the uncertain space giving more importance to the focal elements with higher bpa and then sampling uniformly inside
them. Samples can be generated with a Latin Hypercube scheme and propagated
through the system model in order to build a response surface, for example, through
a non-parametric regression model, that can either directly approximate the Bel or
the quantity of interest from which one can calculate the Bel.
The main problem with sampling is that a correct calculation of the Bel requires
the maximisation of the quantity of interest within each focal element. If sampling
is used to directly estimate the Bel one obtains a potentially efficient approach
but an approximation that can be significantly poor. Furthermore, this type of
approximation provides estimated Bel values that are always better than the actual
ones, leading to an overconfidence in the realisation of an event. A possible
mitigation of this problem was recently proposed in [82] to address the solution
of optimal control problem under epistemic uncertainty. In [82] the value of the Bel
was approximated with the surrogate of a weighted integral obtained by sampling
the space of the focal elements. The integral was elevated to an exponent factor
k, the higher k the more the integral was resembling the actual Bel. Furthermore,
557
three categories: sampling methods, dimensionality reduction methods and outer
approximation methods.
17.4.1 Belief Estimation by Sampling
In the literature one can find a handful of sampling-based methods for the
computation and estimation of Bel [22, 79–81]. As a representative example, in
[22], the authors suggest the use of the density function:
d j (u j ) =
n(j )
k=1
δ(u j |I jk )bpa jk (I jk )/(b jk − a jk )
(17.43)
for the j -th dimension of the uncertain space, if the intervals for the uncertain
parameters are given in the form I jk = {u j : a ik ≤ u j ≤ b jk }. Here n(j ) is
the number of intervals in the j -th dimension and
δ(x j |I jk ) =
1 if x j ∈ I jk
0 otherwise;
(17.44)
then the sampling distribution is
d(x) =
nU
j =1
d j (u j )
(17.45)
Distribution (17.45) explores adequately the uncertain space giving more importance to the focal elements with higher bpa and then sampling uniformly inside
them. Samples can be generated with a Latin Hypercube scheme and propagated
through the system model in order to build a response surface, for example, through
a non-parametric regression model, that can either directly approximate the Bel or
the quantity of interest from which one can calculate the Bel.
The main problem with sampling is that a correct calculation of the Bel requires
the maximisation of the quantity of interest within each focal element. If sampling
is used to directly estimate the Bel one obtains a potentially efficient approach
but an approximation that can be significantly poor. Furthermore, this type of
approximation provides estimated Bel values that are always better than the actual
ones, leading to an overconfidence in the realisation of an event. A possible
mitigation of this problem was recently proposed in [82] to address the solution
of optimal control problem under epistemic uncertainty. In [82] the value of the Bel
was approximated with the surrogate of a weighted integral obtained by sampling
the space of the focal elements. The integral was elevated to an exponent factor
k, the higher k the more the integral was resembling the actual Bel. Furthermore,