Generate Correlated Normal Random Variables

Question

I know that for the $2$-dimensional case: given a correlation $\rho$ you can generate the first and second values, $ X_1 $ and $X_2$, from the standard normal distribution. Then from there make $X_3$ a linear combination of the two $X_3 = \rho X_1 + \sqrt{1-\rho^2}\,X_2$ then take $$ Y_1 = \mu_1 + \sigma_1 X_1, \quad Y_2 = \mu_2 + \sigma_2 X_3$$

So that now $Y_1$ and $Y_2$ have correlation $\rho$.

How would this be scaled to $n$ variables? With the condition that the end variables satisfy a given correlation matrix? I'm guessing at least n variables will need to be generated then a reassignment through a linear combination of them all will be required... but I'm not sure how to approach it.

Answer 1

If you need to generate $n$ correlated Gaussian distributed random variables $$ \bf Y \sim \mathcal N(\bf \mu, \Sigma) $$ where $\textbf{Y} = (Y_1,\dots,Y_n)$ is the vector you want to simulate, $\mu =(\mu_1,\dots, \mu_n)$ the vector of means and $\Sigma$ the given covariance matrix,

1. you first need to simulate a vector of uncorrelated Gaussian random variables, $\bf Z $
2. then find a square root of $\Sigma$, i.e. a matrix $\bf C$ such that $\bf C \bf C^\intercal = \Sigma$.

Your target vector is given by $$ \bf Y = \bf \mu + \bf C \bf Z. $$

A popular choice to calculate $\bf C$ is the Cholesky decomposition.

Answer 2

$\newcommand{\Cov}{\mathrm{Cov}}$

I wanted to add that comment to the previous answer :

Why is it true ?

Define $Y = \mu + C Z$, and recall that each component of $Z$ is i.i.d. normal :

$$ \Cov( Y ) = \Cov(CZ + \mu ) = \Cov(CZ ) = C \cdot \Cov (Z) \cdot C^T = C I C^T = \Sigma $$ which is the desired result : $$ \implies Y \sim N (\mu, \Sigma) $$