Rosenblatt Transformation

The Rosenblatt transformation is an isoprobabilistic transformation which is used under the following context: the input random vector is \vect{X} with marginal cumulative density functions F_i and copula C. Nothing special is assumed about the copula.

Introduction

Let \vect{d} be a deterministic vector, let g(\vect{X}\,,\,\vect{d}) be the limit state function of the model and let \cD_f = \{\vect{X} \in \Rset^n \,/ \, g(\vect{X}\,,\,\vect{d}) \le 0\} be an event whose probability P_f is defined as:

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

One way to evaluate the probability P_f is to use the Rosenblatt transformation T which is a diffeomorphism from the support of the distribution f_{\vect{X}} into the Rosenblatt 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}) = \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 \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:

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 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 is the cumulative distribution function of the standard 1-dimensional Normal distribution.