# Generalized Nataf Transformation¶

The Generalized Nataf transformation is an isoprobabilistic transformation which is used under the following context: the input random vector is with marginal cumulative density functions and copula . The copula is assumed to be elliptical.

## 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)¶

The Generalized Nataf transformation allows one to calculate . This mapping is a diffeomorphism from the support of into the standard space , where distributions are spherical, with zero mean, unit variance and unit correlation matrix. The type of the spherical distribution is the type of the elliptical copula .

The Generalized Nataf transformation presented here is a generalization of the traditional Nataf transformation (see [nataf1962]): the reference [lebrun2009a] shows that the Nataf transformation can be used only if the copula of is normal. The Generalized Nataf transformation (see [lebrun2009b]) extends the Nataf transformation to elliptical copulas.

Let us recall some definitions.

A random vector in has an *elliptical
distribution* if and only if there exists a deterministic vector
such that the characteristic function of
is a scalar function of the quadratic
form :

with a symmetric positive definite matrix of rank . As is symmetric positive, it can be written in the form , where is the diagonal matrix with and .

With a specific choice of normalization for , in the case of finite second moment, the covariance matrix of is and is then its linear correlation matrix. The matrix is always well-defined, even if the distribution has no finite second moment: even in this case, we call it the correlation matrix of the distribution. We note .

We denote by the cumulative distribution function of the elliptical distribution .

An

*elliptical copula*is the copula of an elliptical distribution .The

*generic elliptical representative*of an elliptical distribution family is the elliptical distribution whose cumulative distribution function is .The

*standard spherical representative*of an elliptical distribution family is the spherical distribution whose cumulative distribution function is .The family of distributions with marginal cumulative distribution functions and any elliptical copula is denoted by . The cumulative distribution function of this distribution is noted .

The random vector is supposed to be continuous and with full rank. It is also supposed that its cumulative marginal distribution functions are strictly increasing (so they are bijective) and that the matrix of its elliptical copula is symmetric positive definite.

## Generalized Nataf transformation¶

Let in
be a continuous random vector following the
distribution . The
*Generalized Nataf transformation* is defined
by:

where the three transformations , and are given by:

where is the cumulative distribution function of the standard 1-dimensional elliptical distribution with characteristic generator and is the inverse of the Cholesky factor of .

The distribution of is the generic elliptical representative associated to the copula of . The step maps this distribution into its standard representative, following exactly the same algebra as the normal copula. Thus, in the Generalized Nataf standard space, the random vector follows the standard representative distribution of the copula of the physical random vector .

If the copula of is normal, follows the standard normal distribution with independent components.