# Importance factors from FORM method¶

(1)¶

(2)¶

Which signification for 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).

The importance factor writes:

(3)¶

This definition still guarantees the relation: .

Here, the event considered is established directly from the limit state function : this is the classical structural reliability formulation. However, if the event is a threshold exceedance, it is useful to explicit the variable of interest , evaluated from the model . In that case, the event considered, associated to the threshold has the formulation: and the limit state function is : . is the threshold exceedance probability, defined as : . Thus, the FORM importance factors offer a way to rank the importance of the input components with respect to the threshold exceedance by the quantity of interest . They can be seen as a specific sensitivity analysis technique dedicated to the quantity Z around a particular threshold rather than to its variance.