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 a deterministic vector, let be the limit state function of the model and let be an event whose probability is defined as:
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.
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 :
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:
This definition still guarantees the relation: