generation of normally distributed random vector with covariance matrix
In matlab it is easy to generate a normally distributed random vector with a mean and a standard deviation. From the help randn:
Generate values from a normal distribution with mean 1 and standard deviation 2. r = 1 + 2.*randn(100,1);
Now I have a covariance matrix C and I want to generate N(0,C).
But how could I do this?
From the randn help: Generate values from a bivariate normal distribution with specified mean vector and covariance matrix. mu = [1 2]; Sigma = [1 .5; .5 2]; R = chol(Sigma); z = repmat(mu,100,1) + randn(100,2)*R;
But I don't know exactly what they are doing here.
Answer
This is somewhat a math question, not a programming question. But I'm a big fan of writing great code that requires both solid math and programming knowledge, so I'll write this for posterity.
You need to take the Cholesky decomposition (or any decomposition/square root of a matrix) to generate correlated random variables from independent ones. This is because if X is a multivariate normal with mean m and covariance D, then Y = AX is a multivariate normal with mean Am and covariance matrix ADA' where A' is the transpose. If D is the identity matrix, then the covariance matrix is just AA' which you want to be equal to the covariance matrix C you are trying to generate.
The Cholesky decomposition computes such a matrix A and is the most efficient way to do it.
For more information, see: http://web.as.uky.edu/statistics/users/viele/sta601s03/multnorm.pdf