Generalized Nataf Transformation

The Generalized Nataf transformation is an isoprobabilistic transformation (refer to ) which is used under the following context : \vect{X} is the input random vector, F_i the cumulative density functions of its components and C its copula, which is supposed to be elliptical.

Let us denote by \vect{d} a deterministic vector, g(\vect{X}\,,\,\vect{d}) the limit state function of the model, \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} the event considered here and g(,) = 0 its boundary.

One way to evaluate the probability content of the event \cD_f:

(1)P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0}= \int_{\cD_f} \pdf\, d\vect{x}

is to use the Generalized Nataf transformation T which is a diffeomorphism from \supp{\vect{X}} into the standard space \Rset^n, 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 C.

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

Let us recall some definitions. A random vector \vect{X} in \Rset^n has an elliptical distribution if and only if there exists a deterministic vector \vect{\mu} such that the characteristic function of \vect{X} - \vect{\mu} is a scalar function of the quadratic form \vect{u}^t\mat{\Sigma}\, \vect{u}:

\begin{aligned} \varphi_{\vect{X}-\vect{\mu}}(\vect{u})=\psi(\vect{u}^t\,\mat{\Sigma}\, \vect{u}) \end{aligned}

with \mat{\Sigma} a symmetric positive definite matrix of rank p. As \mat{\Sigma} is symmetric positive, it can be written in the form \mat{\Sigma}=\mat{D}\,\mat{R}\,\mat{D}, where \mat{D} is the diagonal matrix \mat{\diag{\sigma_i}} with \sigma_i=\sqrt{\Sigma_{ii}} and R_{ij}=\frac{\Sigma_{ij}}{\sqrt{\Sigma_{ii}\Sigma_{jj}}}.

With a specific choice of normalization for \psi, in the case of finite second moment, the covariance matrix of \vect{X} is \mat{\Sigma} and \mat{R} is then its linear correlation matrix. The matrix \mat{R} 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 \vect{\sigma}=(\sigma_1,\dots,\sigma_n).

We denote by E_{\vect{\mu},\vect{\sigma},\mat{R},\psi} the cumulative distribution function of the elliptical distribution \cE_{\vect{\mu},\vect{\sigma}, \mat{R},\psi}.

An elliptical copula C^E_{\mat{R},\psi} is the copula of an elliptical distribution \cE_{\vect{\mu},\vect{\sigma},\mat{R},\psi}.

The generic elliptical representative of an elliptical distribution family \cE_{\vect{\mu},\vect{\sigma},\mat{R},\psi} is the elliptical distribution whose cumulative distribution function is E_{\vect{0},\vect{1},\mat{R},\psi}.

The standard spherical representative of an elliptical distribution family \cE_{\vect{\mu},\vect{\sigma},\mat{R},\psi} is the spherical distribution whose cumulative distribution function is E_{\vect{0},\vect{1},\mat{I}_n,\psi}.

The family of distributions with marginal cumulative distribution functions are F_1,\dots,F_n and any elliptical copula C^E_{\mat{R},\psi} is denoted by {\cD}_{F_1,\dots,F_n,C^E_{\mat{R},\psi}}. The cumulative distribution function of this distribution is noted D_{F_1,\dots,F_n,C^E_{\mat{R},\psi}}.

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

Generalized Nataf transformation: Let \vect{X} in \Rset^n be a continuous random vector following the distribution D_{F_1,\dots,F_n,C^E_{\mat{R},\psi}}. The Generalized Nataf transformation T_{Nataf}^{gen} is defined by:

\vect{u} = T_{Nataf}^{gen}(\vect{X})=T_3\circ T_2\circ T_1(\vect{X})

where the three transformations T_1, T_2 and T_3 are given by:

\begin{array}{l} \begin{array}{rcl} T_1 : \Rset^n & \rightarrow & \Rset^n\\ \vect{x} & \mapsto & \vect{w}=\Tr{(F_1(x_1),\dots,F_n(x_n))} \end{array}\\ \begin{array}{rcl} T_2 : \Rset^n & \rightarrow & \Rset^n\\ \vect{w} & \mapsto & \vect{v}=\Tr{(E^{-1}(w_1),\dots,E^{-1}(w_n))} \end{array}\\ \begin{array}{rcl} T_3 : \Rset^n & \rightarrow & \Rset^n\\ \vect{v} & \mapsto & \vect{u}=\mat{\Gamma}\,\vect{v} \end{array} \end{array}

where E is the cumulative distribution function of the standard 1-dimensional elliptical distribution with characteristic generator \psi and \mat{\Gamma} is the inverse of the Cholesky factor of \mat{R}.

The distribution of \vect{W}=T_2\circ T_1(\vect{X}) is the generic elliptical representative associated to the copula of \vect{X}. The step T_3 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 \vect{U} follows the standard representative distribution of the copula of the physical random vector \vect{X}.

If the copula of \vect{X} is normal, \vect{U} follows the standard normal distribution with independent components.