# 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:

References:

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

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

1. 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]

1. Rosenblatt, “Remarks on a Multivariat Transformation”, The Annals of Mathematical Statistics, Vol. 23, No 3, pp. 470-472.