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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".

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    $\begingroup$ But if you allow arbitrarily large deviations, any correlation matrix could be considered a "disturbance" of your original correlation matrix. $\endgroup$
    – A. Donda
    Commented Aug 21, 2015 at 3:06
  • $\begingroup$ @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 $\endgroup$
    – Aksakal
    Commented Aug 21, 2015 at 3:33
  • $\begingroup$ 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 $\endgroup$
    – Confounded
    Commented Dec 17, 2018 at 10:06
  • $\begingroup$ @Confounded, I haven't found a good way of doing this yet $\endgroup$
    – Aksakal
    Commented Dec 17, 2018 at 13:41

3 Answers 3

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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.

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

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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.

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    $\begingroup$ 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. $\endgroup$ Commented Aug 21, 2015 at 2:14

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