RE: Unexpected influence of parameter order on estimation results
Both the BFGS optimization of the objective function in the $EST step and the
inversion of the numerical
Hessian matrix in the $COV step involve a Cholesky decomposition of a
(hopefully) positive definite matrix whose rows and columns correspond to the
individual parameters. If the order of the parameters is changed, the rows and
columns of the matrix being decomposed are permuted. The Cholesky
decomposition is numerically sensitive to such permutations since no pivoting
is done in the standard implementations. This sensitivity is particularly acute
if the matrix is poorly conditioned or, even worse, indefinite. So indeed it
is to be expected that changing the order of the parameters will affect the
results. For well conditioned problems, this effect is minimal. But it is
quite possible, for example, that an $EST or $COV step that fails with one
ordering will succeed with another.
Quoted reply history
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From: [email protected] [mailto:[email protected]] On
Behalf Of Sebastien Bihorel
Sent: Wednesday, June 23, 2010 9:53 AM
To: Nick Holford
Cc: [email protected]
Subject: Re: [NMusers] Unexpected influence of parameter order on estimation
results
I am aware of the issues associated numerical representation in computer memory
but I must say that it is more than a bit surprising (disturbing) that the
order of the parameters results in these pseudo-random outcomes in NONMEM
computations. As far as I know, this is not the case in R, despite the same
issues of numerical representation. That being said, I don't want to re-start
the old debate on the value of the covariance step, but some people would
consider that the two versions of my model gave significantly different
results, simply based upon the objective function (at least a 10-point
difference) and the (lack of) success of the covariance step.
Nick Holford wrote:
Welcome to the world of 'real' numbers i.e. the limited representation of
numbers in computer arithmetic that leads to unexpected (pseudo-random) results.
Both versions of your model are giving the same answer. The apparent
differences are due to pseudo-random chance.
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