.. _rosenblatt_transformation: Rosenblatt Transformation ------------------------- | The Rosenblatt transformation is an isoprobabilistic transformation (refer to ) which is used under the following context : :math:\vect{X} is the input random vector, :math:F_i the cumulative density functions of its components and :math:C its copula, without no condition on its type. | Let us denote by :math:\vect{d} a deterministic vector, :math:g(\vect{X}\,,\,\vect{d}) the limit state function of the model, :math:\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 :math:\cD_f: .. math:: :label: PfX2 P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0}= \int_{\cD_f} \pdf\, d\vect{x} is to use the Rosenblatt transformation :math:T which is a diffeomorphism from :math:\supp{\vect{X}} into the 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}) = \displaystyle \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:: :label: usualRosDetailed \begin{array}{rcl} 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) \end{array} .. math:: :label: usualRosDetailed2 \begin{array}{rcl} 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) \end{array} | 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 the cumulative distribution function of the standard :math:1-dimensional Normal distribution. .. topic:: API: - See the available :ref:Rosenblatt 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.