# 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.