# Rosenblatt Transformation¶

The Rosenblatt transformation is an isoprobabilistic transformation which is used under the following context: the input random vector is with marginal cumulative density functions and copula . Nothing special is assumed about the copula.

## Introduction¶

Let be a deterministic vector, let be the limit state function of the model and let be an event whose probability is defined as:

(1)¶

One way to evaluate the probability is to use the Rosenblatt transformation which is a diffeomorphism from the support of the distribution into the Rosenblatt standard space , where distributions are normal, with zero mean, unit variance and unit correlation matrix (which is equivalent in that normal case to independent components).

Let us recall some definitions.
The *cumulative distribution function* of the
-dimensional random vector is
defined by its marginal distributions and the copula
through the relation:

with

(2)¶

The *cumulative distribution function* of the conditional variable
is defined by:

## Rosenblatt transformation¶

Let in
be a continuous random vector defined by its marginal cumulative
distribution functions and its copula . The
*Rosenblatt transformation* of is
defined by:

(3)¶

where both transformations , and are given by:

where is the cumulative distribution function of the conditional random variable and is the cumulative distribution function of the standard -dimensional Normal distribution.