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# Rosenblatt Transformation

## Overview

The Rosenblatt Transformation is another widely utilized isoprobabilistic transformation in structural reliability analysis. Similar to the Nataf Transformation, its purpose is to transform random vectors with correlated non-normal marginals $\vec{X}$ into random vectors with uncorrelated standard normal marginals $\vec{U}$. This transformation was first introduced by Murray Rosenblatt in 1952 (Rosenblatt, 1952).

The Nataf Transformation $\vec{U} = T^{R}(\vec{X})$ is composed of two transformations:

$$$\vec{U} = T^{R}(\vec{X}) = (T_{2}^{R} \circ T_{1}^{R})(\vec{X})$$$
• The first transformation $\vec{Z} = T_{1}^{R}(\vec{X})$ transforms random vector with correlated non-normal marginal random variables $\vec{X}$ (with correlation matrix $\rho^{X}$) into random vector with uncorrelated uniform marginal random variables $\vec{Z}$.
$$$\vec{Z} = T_{1}^{R}(\vec{X}) = \begin{bmatrix} F_{X_{1}}(X_{1}) \\ F_{X_{2} | X_{1}}(X_{2} | X_{1}) \\ \vdots \\ F_{X_{n} | X_{n - 1}, \dots, X_{1}}(X_{n} | X_{n - 1}, \dots, X_{1}) \end{bmatrix}$$$
• The second transformation $\vec{U} = T_{2}^{R}(\vec{Z})$ transforms random vector with uncorrelated uniform marginal random variables $\vec{Z}$ into random vector with uncorrelated standard normal marginal random variables $\vec{U}$.
$$$\vec{U} = T_{2}^{R}(\vec{Z}) = \begin{bmatrix} \Phi^{-1}(Z_{1}) \\ \vdots \\ \Phi^{-1}(Z_{n}) \end{bmatrix}$$$