Rosenblatt Transformation

The Rosenblatt transformation is an isoprobabilistic transformation (refer to ) which is used under the following context : \vect{X} is the input random vector, F_i the cumulative density functions of its components and C its copula, without no condition on its type.
Let us denote by \vect{d} a deterministic vector, g(\vect{X}\,,\,\vect{d}) the limit state function of the model, \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 \cD_f:

(1)P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0}= \int_{\cD_f} \pdf\, d\vect{x}

is to use the Rosenblatt transformation T which is a diffeomorphism from \supp{\vect{X}} into the standard space \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 F_{1,k} of the k-dimensional random vector (X_1, \dots, X_k) is defined by its marginal distributions F_i and the copula C_{1,k} through the relation:

F_{1,k}(x_1,\dots, x_k) = C_{1,k}(F_1(x_1),\dots, F_k(x_k)) with

(2)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 X_k|X_1, \dots, X_{k-1} is defined by:

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 \vect{X} in \Rset^n be a continuous random vector defined by its marginal cumulative distribution functions F_i and its copula C. The Rosenblatt transformation T_{Ros} of \vect{X} is defined by:

(3)\vect{U} = T_{Ros}(\vect{X})=T_2\circ T_1(\vect{X})

where both transformations T_1, and T_2 are given by:

(4)\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}

(5)\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 F_{k|1, \dots, k-1} is the cumulative distribution function of the conditional random variable X_k|X_1, \dots, X_{k-1} and \Phi the cumulative distribution function of the standard 1-dimensional Normal distribution.