# Isoprobabilistic transformations¶

The isoprobabilistic transformation is used in the following context: is the input random vector, the cumulative density functions of its components and its copula. Let us denote by a deterministic vector, the limit state function of the model, the event considered here.

One way to evaluate the probability content of the event :

(1)¶

is to introduce an isoprobabilistic transformation which is a diffeomorphism from the support of the distribution into , such that the distribution of the random vector has the following properties : and have the same distribution for all rotations . Such transformations exist and the most widely used are:

the Generalized Nataf transformation (refer to [lebrun2009b]),

the Rosenblatt transformation (refer to [rosenblatt1952]).

If we suppose that the numerical model has suitable properties of differentiability, then (1) can be written as:

(2)¶

where is a -diffeomorphism called an
*isoprobabilistic transformation*, the
probability density function of and
.
The vector is said to be in the *standard space*,
whereas 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 has a spherical distribution, which means that the density function is a function of . Thus, without loss of generality, it is possible to map the general failure domain to a domain for which the design point (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 :

the Nataf transformation (see [nataf1962], [lebrun2009a]): must have a normal copula,

the Generalized Nataf transformation (see [lebrun2009b]): must have an elliptical copula,

the Rosenblatt transformation (see [rosenblatt1952], [lebrun2009c]): there is no condition on the copula of .

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