Rosenblatt Transformation¶
The Rosenblatt transformation is an isoprobabilistic transformation which
is used under
the following context: the input random vector is with marginal cumulative
density functions
and copula
. Nothing special is assumed about the
copula.
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)¶
One way to evaluate the probability is to use the Rosenblatt transformation
which is a
diffeomorphism from the support of the distribution
into the Rosenblatt 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:
with
(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:
where is the cumulative distribution function of the conditional
random variable
and
is the cumulative distribution
function of the standard
-dimensional Normal distribution.