> Now, lets say I specify a target correlation matrix as follows:
>
>
> A B C
> A 1
> B 1 1
> C 1 -1 1
>
> The problem with above matirx is that we want large values of 'A' to
> be paired with large values of 'B' and also large values of 'A' to
> be paired with large values of 'C'.
> BUT, we specify a (-1) corrleation between B and C, which means we
> want large values of
> 'B' to be paired with small values of 'C'. This might pose a problem
> because of earlier
> specified correlations between A,B and A,C.
>
> Is there any way of checking for validity of the target correlation
> matirx.
The problems you describe with conflicting values of A, B, C under this
correlation matrix arise because the matrix is not positive definite.
Therefore, it is not a possible correlation or covariance matrix.
Positive definiteness of a matrix M is the property that a'*M*a>=0 for
all (real) vectors a. Since the variance of a linear combination a'*X of
a random vector X is a'*Var(X)*a, a positive definite covariance
matrix means that the variance of any linear combination of the
components of X must be nonnegative. This is obviously a necessary
condition in order for M to be a covariance matrix. It can also be shown
to be sufficient-- if M is positive definite, you can construct random
variables A, B, C with covariance matrix M.
M is positive definite (or non-negative definite, to be more precise) iff
all of its eigenvalues are non-negative. The eigenvalues of your matrix
are 2, 2, and -1. So it's not positive definite, and can't be a
covariance (or correlation) matrix. Here's the proof: using that
correlation matrix, compute the variance of -A+B+C. It's negative.
In the particular case of rank correlations, I'm sure there are other
conditions too, but I don't know what they are right now.
A.
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