如何在 R 中生成多元正态数据？

Question

I'm completing an assignment, in which I have to generate a sample X = (X1, X2) from a bivariate normal in which each marginal is N(0,1) and the correlation between X1 and X2 is 0.5.

I think the way to approach this is to use the mvrnorm function, but I'm not quite sure how to proceed after that. Any advice? Thanks in advance!

Answer 1

Indeed, the mvrnorm function from the MASS package is probably your best bet. This function can generate pseudo-random data from multivariate normal distributions.

Examining the help page for this function ( ??mvrnorm ) shows that there are three key arguments that you would need to simulate your data based your given parameters, ie:

• n - the number of samples required (an integer);
• mu - a vector giving the means of the variables - here, your distributions are standard normal so it will be a vector of zeros; and
• Sigma - a positive-definite symmetric matrix specifying the covariance matrix of the variables - ie, in your case, a matrix with variance on the diagonal of ones and covariance on the off-diagonals of 0.5.

Have a look at the examples in this help page, which should help you put these ideas together!

Answer 2

Here are some options:

1. mvtnorm::rmvnorm and MASS::mvrnorm work the same way, although the mvtnorm::rmvnorm function does not require that you specify the means (ie, the default is 0). Giving names to the mu vector will specify the names of the simulated variables.

n <- 100
R <- matrix(c(1, 0.5,
              0.5, 1), 
            nrow = 2, ncol = 2, byrow = TRUE)
            
mu <- c(X = 0, Y = 0)
mvtnorm::rmvnorm(n, mean = mu, sigma = R)
MASS::mvrnorm(n, mu = mu, Sigma = R)

2. simstandard::sim_standardized will make standardized data only, but will do so with less typing:

simstandard::sim_standardized("X ~~ 0.5 * Y", n = 100)

Answer 3

Using base R (no package needed) and a bit of statistics:

Sigma = matrix(c(1,0.5,0.5,1), ncol=2)
R = chol(Sigma) # Sigma == t(R)%*%  R
n = 1000
X = t(R) %*% matrix(rnorm(n*2), 2)

X %*% t(X)/n # test