Importance factors from FORM method

Importance Factors are evaluated in the following context : \vect{X} denotes a random input vector, representing the sources of uncertainties, \pdf its joint density probability, \vect{d} a deterministic vector, representing the fixed variables g(\vect{X}\,,\,\vect{d}) the limit state function of the model, \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} the event considered here and {g(\vect{X}\,,\,\vect{d}) = 0} its boundary (also called limit state surface).
The probability content of the event \cD_f is P_f:

     P_f =       \int_{g(\vect{X}\,,\,\vect{d}) \le 0}  \pdf\, d\vect{x}.

In this context, the probability P_f can often be efficiently estimated by FORM or SORM approximations.
The FORM importance factors offer a way to rank the importance of the input components with respect the realization of the event. They are often interpreted also as indicators of the impact of modeling the input components as random variables rather than fixed values. The FORM importance factors are defined as follows.
The isoprobabilistic transformation T used in the FORM and SORM approximation is a diffeomorphism from \supp{\vect{X}} into \Rset^n, such that the distribution of the random vector \vect{U}=T(\vect{X}) has the following properties: \vect{U} and \mat{R}\,\vect{U} have the same distribution for all rotations \mat{R}\in{\cS\cO}_n(\Rset).
In the standard space, the design point \vect{u}^* is the point on the limit state boundary the nearest 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_{HL} the Hasofer-Lind reliability index: \beta_{HL} = ||\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 guarantees the relation : \Sigma_i \alpha_i^2 = 1.
Let’s note that this definition arises the following difficulties :
  • Which signification for \alpha_i when the variables X_i 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 \vect{X}. 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, where the marginal distributions are all elliptical, with cumulative distribution function noted E, and not yet uncorrelated.

    \boldsymbol{Y}^* =  \left(
      E^{-1}\circ F_1(X_1^*) \\
      E^{-1}\circ F_2(X_2^*) \\
      \vdots \\
      E^{-1}\circ F_n(X_n^*)

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: \Sigma_i \alpha_i^2 = 1.

Here, the event considered is established directly from the limit state function g(\vect{X}\,,\,\vect{d}): this is the classical structural reliability formulation. However, if the event is a threshold exceedance, it is useful to explicit the variable of interest Z=\tilde{g}(\vect{X}\,,\,\vect{d}), evaluated from the model \tilde{g}(.). In that case, the event considered, associated to the threshold z_s has the formulation: \cD_f = \{ \vect{X} \in \Rset^n \, / \, Z=\tilde{g}(\vect{X}\,,\,\vect{d}) > z_s \} and the limit state function is : g(\vect{X}\,,\,\vect{d}) = z_s - Z = z_s - \tilde{g}(\vect{X}\,,\,\vect{d}). P_f is the threshold exceedance probability, defined as : P_f     =       P(Z \geq z_s) = \int_{g(\vect{X}\,,\,\vect{d}) \le 0}  \pdf\, d\vect{x}. 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 Z. They can be seen as a specific sensitivity analysis technique dedicated to the quantity Z around a particular threshold rather than to its variance.