# Computing equivalent vector of random variables from covarience matrix

Given a covariance matrix, how can I construct a vector of expressions of randomly distributed variables whose covariance matrix is equal to the given one?

EDIT: All variables are normally distributed.

I have an algorithm that gets the covariances correct, but not the variances on the diagonal:

a = *len(r)
for x, row in enumerate(cov_matrix(r)):
for y, item in enumerate(row):
if x > y: continue
v = noise(math.sqrt(abs(item)))
a[x] += v
if item > 0:
a[y] += v
else:
a[y] -= v


I feel like this should be simple ...

• Meta-question: Why did you write that as a comment instead of an answer? – Forrest Sep 4 '10 at 14:59
• Good point. I was originally planning to leave a short comment, but I ended up giving more detail than I had thought I would. I deleted the comment and reposted it as an answer (and added a paragraph generalizing the construction slightly). – Darsh Ranjan Sep 4 '10 at 23:41

If $A$ is your target covariance matrix and $LL^T = A$, and $x = (x_1, \ldots, x_n)$ is a vector of independent random variables with mean zero and variance 1, then $y = Lx$ has the required covariance. Here $L$ is a matrix and $L^T$ is its transpose. $L$ can just be the Cholesky factor of $A$. ((Check: $\mathrm{cov}(y) = E[yy^T] = E[(Lx)(Lx)^T] = E[Lxx^TL^T] = LE[xx^T]L^T$ (by linearity of expectation) $= L\mathrm{cov}(x)L^T = LIL^T = LL^T = A$. $\mathrm{cov}(y) = E[yy^T]$ because $y$ has mean 0, and likewise for $\mathrm{cov}(x)$.)
That's not too far from a "complete" solution, actually. If you start with a vector $y$ of random variables with mean zero and covariance matrix $A$, then if $A = LL^T$ and $x = L^{-1}y$, then $\mathrm{cov}(x) = I$. That doesn't necessarily imply that the components of $x$ are independent; it means they are uncorrelated. So the most general construction is to begin with a vector $x$ of uncorrelated random variables with mean zero and variance 1 and let $y=Lx$. (I only mean that every example can theoretically be obtained that way, not that it's necessarily the best or most computationally efficient way to do it.)