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.. _rosenblatt_transformation:
Rosenblatt Transformation
-------------------------
The Rosenblatt transformation is an :ref:`isoprobabilistic transformation <isoprobabilistic_transformation>` which
is used under
the following context: the input random vector is :math:`\vect{X}` with marginal cumulative
density functions :math:`F_i` and copula :math:`C`. Nothing special is assumed about the
copula.
Introduction
~~~~~~~~~~~~
Let :math:`\vect{d}` be a deterministic vector, let :math:`g(\vect{X}\,,\,\vect{d})` be the
limit state function of the model and let :math:`\cD_f = \{\vect{X} \in \Rset^n \,/ \,
g(\vect{X}\,,\,\vect{d}) \le 0\}` be an event whose probability :math:`P_f` is defined as:
.. math::
:label: PfX2
P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0}= \int_{\cD_f} \pdf\, d\vect{x}
One way to evaluate the probability :math:`P_f` is to use the Rosenblatt transformation :math:`T` which is a
diffeomorphism from the support of the distribution :math:`f_{\vect{X}}` into the Rosenblatt standard
space :math:`\Rset^n`, 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* :math:`F_{1,k}` of the
:math:`k`-dimensional random vector :math:`(X_1, \dots, X_k)` is
defined by its marginal distributions :math:`F_i` and the copula
:math:`C_{1,k}` through the relation:
.. math::
F_{1,k}(x_1,\dots, x_k) = C_{1,k}(F_1(x_1),\dots, F_k(x_k))
with
.. math::
:label: subCopula
C_{1,k}(u_1, \dots, u_k) = C(u_1, \dots, u_k, 1, \dots, 1)
The *cumulative distribution function* of the conditional variable
:math:`X_k|X_1, \dots, X_{k-1}` is defined by:
.. math::
F_{k|1, \dots, k-1} (x_k|x_1, \dots, x_{k-1})
= \frac{ \frac{\partial^{k-1} F_{1,k}(x_1, \dots, x_k)}{\partial x_1 \dots
\partial x_{k-1}} }{ \frac{\partial^{k-1} F_{1,k-1}(x_1, \dots, x_{k-1})}
{\partial x_1 \dots \partial x_{k-1}}}
Rosenblatt transformation
~~~~~~~~~~~~~~~~~~~~~~~~~
Let :math:`\vect{X}` in :math:`\Rset^n`
be a continuous random vector defined by its marginal cumulative
distribution functions :math:`F_i` and its copula :math:`C`. The
*Rosenblatt transformation* :math:`T_{Ros}` of :math:`\vect{X}` is
defined by:
.. math::
:label: usualRos
\vect{U} = T_{Ros}(\vect{X})=T_2\circ T_1(\vect{X})
where both transformations :math:`T_1`, and :math:`T_2` are given by:
.. math::
T_1 : \Rset^n & \rightarrow \Rset^n\\
\vect{X} & \mapsto \vect{Y}=
\left(
\begin{array}{l}
F_1(X_1)\\
\dots \\
F_{k|1, \dots, k-1}(X_k|X_1, \dots, X_{k-1})\\
\dots \\
F_{n|1, \dots, n-1}(X_n|X_1, \dots, X_{n-1})
\end{array}
\right) \\
T_2 : \Rset^n & \rightarrow \Rset^n\\
\vect{Y} & \mapsto \vect{U}=
\left(
\begin{array}{l}
\Phi^{-1}(Y_1)\\
\dots \\
\Phi^{-1}(Y_n)
\end{array}
\right)
where :math:`F_{k|1, \dots, k-1}` is the cumulative distribution function of the conditional
random variable :math:`X_k|X_1, \dots, X_{k-1}` and :math:`\Phi` is the cumulative distribution
function of the standard :math:`1`-dimensional Normal distribution.
.. topic:: API:
- See :ref:`Rosenblatt transformation <isoprobabilistic_transformations>`.
.. topic:: References:
- O. Ditlevsen and H.O. Madsen, 2004, "Structural reliability methods,"
Department of mechanical engineering technical university of Denmark -
Maritime engineering, internet publication.
- J. Goyet, 1998,"Sécurité probabiliste des structures -
Fiabilité d'un élément de structure," Collège de Polytechnique.
- A. 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]_
- M. Rosenblatt, "Remarks on a Multivariat Transformation",
The Annals of Mathematical Statistics, Vol. 23, No 3, pp. 470-472.
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