Importance factors from FORM method¶
Importance Factors are evaluated in the following context: let be a probabilistic input vector with
joint density probability
, let
be the
limit state function of the model and let
be
an event whose probability
is defined as:
(1)¶
In this context, the probability can often be
efficiently estimated by FORM or SORM approximations.
FORM importance factors rank the importance of the input components with respect to the realization of the event. They are often seen as indicators of the impact of modeling the input components as random variables rather than fixed values. FORM importance factors are defined as follows.
Let be the special orthogonal group which is the set of square matrices
such that
and with
determinant equal to 1.
The isoprobabilistic transformation
used in the FORM and
SORM approximation is a diffeomorphism from
into
, such that the
distribution of the random vector
has the
following properties:
and
have the same distribution for all rotations
.
In the standard space, the design point is the
point on the limit state boundary the that is colosest to the origin of the
standard space. The design point is
in the physical
space, where
. We note
the Hasofer-Lind reliability index:
.
When the
-space is normal, the literature proposes to
calculate the importance factor
of the variable
as the square of the co-factors of the design point in the
-space :
(2)¶
This definition implies that the sum of importance factors is equal to 1, i.e.:
This definition raises the following difficulties:
How can the
coefficients be interpreted when the
variables are correlated? In that case, the isoprobabilistic transformation does not associate
to
but
to a set of
.
In the case of dependence of the variables
, the shape of the limit state function in the
-space depends on the isoprobabilistic transformation and in particular on the order of the variables
within the random vector
. Thus, changing this order has an impact on the localization of the design point in the
-space and, consequently, on the importance factors … (see [lebrun2009c] to compare the different isoprobabilistic transformations).
It is possible to give another definition to the importance factors
which may be defined in the elliptical space of the iso-probabilistic
transformation (refer to Isoprobabilistic transformations), where the marginal distributions are all elliptical,
with cumulative distribution function noted , and not yet
uncorrelated. Let
be the cumulative distribution of the component
.
The importance factor writes:
(3)¶
This definition still guarantees the relation: