RE: FW: OMEGA HAS A NONZERO BLOCK
From:Leonid Gibiansky
Subject:RE: FW: [NMusers] OMEGA HAS A NONZERO BLOCK
Date:Fri, 04 Oct 2002 12:34:36 -0400
Ken,
Given the degenerate OMEGA matrix, we always can find degenerate direction
(the one that corresponds to zero eigenvalue, such that OMEGA*U=0) and find
the relationship that need to be introduced to remove ill-conditioning. In
this case
U ~ (sqrt(OMEGA44),0,0,-sqrt(OMEGA11))
leading to the conclusion that eta4 ~ eta1. In a more general case, this
will be a relationship of the type
eta4~ coeff1*eta1+coeff2*eta2+coeff3*eta3
The problem, as I see it, is that you cannot trust the matrix that you
received from the computations, if it is ill-conditioned. Therefore, you
cannot find this degenerate direction (or at least, can not be sure in this
relation). Parametrization that I use may provide a useful tool to look
for this direction. Moreover, if the estimation step with the full OMEGA
matrix fails (that may happen with ill-conditioned problem), this
parametrization allows to go not only from the full degenerate matrix to
the simpler one, but also from the diagonal matrix to the correlated one,
even if the computation with the full matrix fails. This is similar to
covariate model building step, where covariates can be correlated: instead
of starting from the full matrix (full covariate model), one can try to
increase complexity step by step.
Leonid