Re: Most negative multi-variate correlation matrix

On Sep 23, 6:12 am, Tom <chi....@xxxxxxxxx> wrote:
Hi all,

I want to find a multi-variate correlation matrix which make the S.D.
of the sum of RV as small as possible.

I don't understand the above. Please clarify.

It's found that I cannot simply put "-1" to any element with i =/= j,
because the matrix will not be +ve defined.
For example, I have a weight vector p [1,1,1] for 3 vectors.
I want to find a correlation matrix "M" such that p * M * p^T will
be minimized.

May I ask if you have any good reference on that? Many thanks.

For a given M, p^T * M * p / p^T *p will be minimized if p is
proportional to a minimal eigenvector of M. Given p,
I assume the same is true.

If M is a correlation matrix, 0 <= p^T * M * p. Therefore,
to minimize p^T * M * p for a given p, solve

0 = M * p

for M. Since M is a correlation matrix,

M = [1 a b; a 1 c; b c 1],
0 < = |a|,|b|,|c| <= 1

Consequently,

0 = M*p = [ p1+ap2+bp3; ap1+p2+cp3;bp1+cp2+p3]

or

Q * [a; b; c] = -p

where

Q = [ p2 p3 0; p1 0 p3; 0 p1 p2]

A unique solution satisfying

0 = det(M) = 1+2abc-(a^2+b^2+c^2)

will exist if

detQ = det(Q) = -2p1p2p3 ~= 0.

a = -det([ p1 p3 0; p2 0 p3; p3 p1 p2] / detQ
= (p3^2 - p1^2 - p2^2) / (2p1p2)

b = -det([ p2 p1 0; p1 p2 p3; 0 p3 p2] / detQ
= (p2^2 - p1^2 - p3^2) / (2p1p3)

c = -det([ p2 p3 p1; p1 0 p2; 0 p1 p3] / detQ
= (p1^2 - p2^2 - p3^2) / (2p2p3)

If p1 = p2 =p3,

a = b = c = -1/2

Note that if one or two components of p are zero,
a solution will exist. however, it will not be unique.

Hope this helps.

greg





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