Importance factors from FORM method

Importance Factors are evaluated in the following context: let \inputRV be a probabilistic input vector with joint density probability \pdf, let \model be the limit state function of the model and let \cD_f = \{\vect{x} \in \Rset^{\inputDim} \, / \, \model(\vect{x}) \le 0\} be an event whose probability P_f is defined as:

(1)P_f = \int_{\model(\inputRV) \le 0} \pdf\, d\vect{x}

In this context, the probability P_f 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 \mat{R} \in {\cS\cO}_{\inputDim}(\Rset) be the special orthogonal group which is the set of square matrices \mat{M} \in \cM_{\inputDim}(\Rset) such that \mat{M} \Tr{\mat{M}} = \mat{I}_{\inputDim} and with determinant equal to 1. The isoprobabilistic transformation T used in the FORM and SORM approximation is a diffeomorphism from \supp{\inputRV} into \Rset^{\inputDim}, such that the distribution of the random vector \RVU=T(\inputRV) has the following properties: \RVU and \mat{R}\,\RVU have the same distribution for all rotations \mat{R}\in{\cS\cO}_d(\Rset).

In the standard space, the design point \vect{u}^* is the point on the limit state boundary the that is colosest to the origin of the standard space. The design point is \vect{x}^* in the physical space, where \vect{x}^* = T^{-1}(\vect{u}^*). We note \beta the Hasofer-Lind reliability index: \beta = ||\vect{u}^{*}||. When the \bdU-space is normal, the literature proposes to calculate the importance factor \alpha_i^2 of the variable X_i as the square of the co-factors of the design point in the \bdU-space :

(2)\alpha_i^2 = \displaystyle \frac{(u_i^{*})^2}{\beta_{HL}^2}

This definition implies that the sum of importance factors is equal to 1, i.e.:

\sum_{i=1}^{\inputDim} \alpha_i^2 = 1

This definition raises the following difficulties:

  • How can the \alpha_i coefficients be interpreted when the X_i variables are correlated? In that case, the isoprobabilistic transformation does not associate U_i to X_i but U_i to a set of X_i.

  • In the case of dependence of the variables X_i, the shape of the limit state function in the \bdU-space depends on the isoprobabilistic transformation and in particular on the order of the variables X_i within the random vector \inputRV. Thus, changing this order has an impact on the localization of the design point in the \bdU-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 E , and not yet uncorrelated. Let F_i be the cumulative distribution of the component i.

\begin{aligned} \boldsymbol{Y}^* = \left( \begin{array}{c} E^{-1}\circ F_1(X_1^*) \\ E^{-1}\circ F_2(X_2^*) \\ \vdots \\ E^{-1}\circ F_d(X_{\inputDim}^*) \end{array} \right).\label{varY10} \end{aligned}

The importance factor \alpha_i^2 writes:

(3)\alpha_i^2 = \displaystyle \frac{(y_i^{*})^2}{||\vect{y}^{*}||^2}

This definition still guarantees the relation:

\sum_{i=1}^{\inputDim} \alpha_i^2 = 1