Generalized Nataf Transformation¶
The Generalized Nataf 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, which is supposed to be elliptical.
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 :
is to use the Generalized Nataf transformation which is a diffeomorphism from 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 generalisation 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 are 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.
See the available Nataf transformations.
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H.O. Madsen, Krenk, S., Lind, N. C., 1986, “Methods of Structural Safety,” Prentice Hall.