.. _rosenblatt_transformation: Rosenblatt Transformation ------------------------- The Rosenblatt transformation is an :ref:`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 `. .. 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.