# Rosenblatt Transformation¶

The Rosenblatt transformation is an isoprobabilistic transformation
(refer to ) which is used under the following context :
is the input random vector, the
cumulative density functions of its components and its
copula, without no condition on its type.

Let us denote by a deterministic vector,
the limit state function of the
model,
the event considered here and g(,) = 0 its boundary.

One way to evaluate the probability content of the event
:

(1)¶

is to use the Rosenblatt transformation which is a diffeomorphism from into the 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:(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:

(4)¶

(5)¶

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

API:

- See the available Rosenblatt transformations.

References:

- Ditlevsen and H.O. Madsen, 2004, “Structural reliability methods,” Department of mechanical engineering technical university of Denmark - Maritime engineering, internet publication.

- Goyet, 1998,”Sécurité probabiliste des structures - Fiabilité d’un élément de structure,” Collège de Polytechnique.

- Der Kiureghian, P.L. Liu, 1986,”Structural Reliability Under Incomplete Probabilistic Information”, Journal of Engineering Mechanics, vol 112, no. 1, p85-104.

- [Lebrun2009a]
- [Lebrun2009b]
- [Lebrun2009c]
- H.O. Madsen, Krenk, S., Lind, N. C., 1986, “Methods of Structural Safety,” Prentice Hall.
- [Nataf1962]
- Rosenblatt, “Remarks on a Multivariat Transformation”, The Annals of Mathematical Statistics, Vol. 23, No 3, pp. 470-472.