Details of Algorithms
Nearest Correlation Matrix
This algorithm is trying to solve the optimization problem
\[\begin{aligned} \mathrm{min}\quad & \frac{1}{2} \Vert G - X \Vert^2 \\ \mathrm{s.t.}\quad & X_{ii} = 1, \quad i = 1, \ldots , n, \\ & X \in S_{+}^{n} \end{aligned}\]
Pearson Matching
NORTA
Given:
- A target correlation matrix, $\rho$
- A list of marginal distributions, $F$
Do:
- Generate $Z_{n \times d} = \mathcal{N}(0, 1)$ IID standard normal samples
- Transform $Y = ZC$ where $C$ is the upper Cholesky factor of $\rho$
- Transform $U = \Phi(Y)$ where $\Phi(\cdot)$ is the CDF of the standard normal distribution
- Transform $X_i = F_{i}^{-1}(U_i)$