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st: RE: RE: correlation between two correlation matrices (more)


From   "Feiveson, Alan H. (JSC-SK311)" <alan.h.feiveson@nasa.gov>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: RE: correlation between two correlation matrices (more)
Date   Wed, 9 May 2007 12:46:35 -0500

Sorry - I left out that the Wishart distributional properties I stated
depend on the assumptions that (x1,...,xn) is distributed n-multivariate
normal with covariance matrix V1 and (z1,...,zm) is distributed
m-multivariate normal with covariance matrix V2. Also for the combined
case, that (x1,,..xn,z1,..,zm) is distributed (m+n)-multivariate normal
with covariance matrix(V1, H \ H', V2). Actually, if the latter
assumption is true, the first two follow (but not vice-versa).

Al Feiveson


-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Feiveson,
Alan H. (JSC-SK311)
Sent: Wednesday, May 09, 2007 11:40 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: RE: correlation between two correlation matrices

It depends on where those correlation matrices came from. For example,
suppose S1 is a sample covariance matrix of variables x1,..,xn (N1
observations) and S2 is a sample covariance matrix of other variables
z1,..,zm (N2 observations). Then (N1-1)S1 ~ W_n(V1, N1-1) and
(N2-1)S2~W_m(V2, N2-1) where W_p(V,k) is a pxp-dimenisonal Wishart
distribution with population covariance matrix V and k degrees of
freedom. 

If N1 = N2, then one can think of an overall sample (m+n) x (m+n)
covariance matrix S formed form all m+n variables.In this case S1 is the
upper n x n diagonal block and S2 is the lower m x m diagonal block.
Expressions for the covariances between elements of S1 and elements of
S2 can be found in Chapter 7 of TW Andersen's book "An Introduction to
Multivariate Statistical Analysis" (Wiley).

If N1 ~= N2 then those expressions for the correaltions would have to be
modified.


On the other hand if all you have is two matrices without knowing where
they came from, you can't say anything about their correlation any more
than asking whether two numbers are "correlated".

Al Feiveson

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of
chichi@pob.huji.ac.il
Sent: Wednesday, May 09, 2007 10:44 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: correlation between two correlation matrices

I have two correlation matrices. How could I find the correlation
between them? In other words, how is one explained by the other?

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