# Correlation matrix disturbances (shocks)

I need ideas on generating disturbances of the correlation matrix. Say, I have a n-by-n correlation matrix $C_{i,j}$. I'd like to shock it, and produce the disturbed matrix $C'_{i,j}$ and see what happens to my process after plugging it instead of the original.

UPDATE: I must clarify that that shocks don't need to be small. So, maybe the term disturbances is unfortunate in my question title. For instance, if I have the 2-by-2 matrix:

1 1
1 1


Then the "largest" in some sense shock would be

 1 -1
-1  1


In n>2 dimensional case it gets a bit complicated, because due to constraints, not every matrix can be considered a correlation matrix. So, I'm looking for a way to disturb my initial matrix in some regular fashion, where the shock can be "large" or "small".

• But if you allow arbitrarily large deviations, any correlation matrix could be considered a "disturbance" of your original correlation matrix. Commented Aug 21, 2015 at 3:06
• @A.Donda, yes. So, I need to be able to set how large is deviation somehow. Also, it's not that easy to generate a correlation matrix, because not every symmetric matrix is positive definite Commented Aug 21, 2015 at 3:33
• Have you found a good approach for this? I am in a somewhat similar situation where I need to transfer past changes in a correlation parameter to scenarios to be applied to current correlation level. I thought about using Fisher transformation, but it seems to give somewhat counter-intuitive behaviour. Could you share with us what you found? Thanks Commented Dec 17, 2018 at 10:06
• @Confounded, I haven't found a good way of doing this yet Commented Dec 17, 2018 at 13:41

You could consider drawing your disturbed matrices from the Wishart distribution $W(V, n)$ with your original correlation matrix as the parameter $V$, and then rescale to make sure the diagonal only contains 1s. The smaller $n$, the larger the deviations from $V$ that you can expect.

• This is interesting Commented Aug 21, 2015 at 3:35
• For now this is the path I'm taking. It doesn't answer my exact question, bit it provides me with the work around. Commented Aug 21, 2015 at 21:42
• @Aksakal, glad I could help. Commented Aug 22, 2015 at 2:07

I can't comment, so I will put this as an answer.

What's wrong with doing something like: $C^\prime = (1-p)\cdot C + pI$, where $p$ is some small-ish value and $I$ is the identity matrix?

You need a tranformation function of correlation coefficient that make it becomes normal distribution. May be starting from the statical testing of r would help.

• Welcome to the site, @W.W.. This is hard to follow & seems rather incomplete. At present, I think it is more of a comment than an answer. Can you elaborate on it? Since you're new here, you may want to take our tour, which has information for new users. Commented Aug 21, 2015 at 2:14