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 $\vec{U}$.

\[\vec{U} = T_{2}^{R}(\vec{Z}) = \begin{bmatrix} \Phi^{-1}(Z_{1}) \\ \vdots \\ \Phi^{-1}(Z_{n}) \end{bmatrix}\]

API

RosenblattTransformation <: AbstractTransformation