Isoprobabilistic transformations

The isoprobabilistic transformation is used in the following context: \vect{X} is the input random vector, F_i the cumulative density functions of its components and C its copula. Let us denote by \vect{d} a deterministic vector, 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(,) = 0 its boundary. One way to evaluate the probability content P_f of the event \cD_f:

(1)P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0}= \int_{\cD_f} \pdf\, d\vect{x}

is to introduce an isoprobabilistic transformation T which 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). Such transformations exist and the most widely used are:

If we suppose that the numerical model g has suitable properties of differentiability, the evaluation of the probability (1) can be transformed in the evaluation of the probability:

(2)P_f = \Prob{G(\vect{U}\,,\,\vect{d})\leq 0} = \int_{\Rset^n} \boldsymbol{1}_{G(\vect{u}\,,\,\vect{d}) \leq 0}\,f_{\vect{U}}(\vect{u})\,d\vect{u}

where T is a C^1-diffeomorphism called an isoprobabilistic transformation, f_{\vect{U}} the probability density function of \vect{U}=T(\vect{X}) and G=f\circ T^{-1}. The vector \vect{U} is said to be in the standard space, whereas \vect{X} is in the physical space. The interest of such a transformation is the rotational invariance of the distributions in the standard space : the random vector \vect{U} has a spherical distribution, which means that the density function f_{\vect{U}} is a function of \|\vect{u}\|. Thus, without loss of generality, it is possible to map the general failure domain {\cD} to a domain {\cD}' for which the design point {\vect{u}^{*}}' (the point of the event boundary at minimal distance from the center of the standard space) is supported by the last axis.

The following transformations verify that property, under some specific conditions on the dependence structure of the random vector \vect{X} :

The Generalized Nataf transformation is automatically used when the copula is elliptical and the Rosenblatt transformation for any other case.