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Hi there, a couple of graduate students bring up the following
questions. How are you sure that your nonlinear curve-fitting in the
modeling process converges to global minima not local minima. One of
methods to check if the results reach global minima seems to change
initial estimates to 10 or more times of the original values. It assumes
global minima achieve if the same results are obtained from different
initial estimates. The questions are what to do in the cases (1) the
initial estimates of first time reach global minima while the initial
estimates of second time reach local minima; and (2) the initial
estimates of first time reach local minima while the initial estimates
of second time reach another local minima. Also, how to select the
initial estimates of second time in a case where many parameters, say
25, need to be estimated, all or parts (how many) in a time change? What
about some parameter estimates in global minima and some in local
minima. Another question no related to the minimal issue is what=92s
minimal ratio of data (sample) points to the numbers of parameters to be
estimated for ideal curve-fitting. Thank you very much for your opinions
and comments.
Johnny
Emailto:jyzhu.at.usa.net
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First, be sure your model is not over-parameterized. Check the
parameter correlation matrix, and if you find a lot of 0.9 or higher
values in it, your model should be reduced. If you have the right
number of well-defined parameters in the model, you will hardly have
got any problems with local minima. Many curve fitting programs
produce the correlation matrix.
Then, each time you fit a model to your data, examine the plot of
residuals against the independent variable. If residuals are randomly
scattered around zero, most probably you've got a global minimum.
There are no common rules concerning the number of points needed per
parameter. This issue is only a part of the problem of study design.
The number of points depends not only on the complexity of your model
(linear or nonlinear, etc.) and on its sensitivity with respect to
changes in parameters, but also on the magnitude and the structure of
the random noise in your data.
Vladimir
vpiotrov&janbe.jnj.com
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There are methods known as global optimization which (in principle)
localize the
global minimum. Simulated annealing (SA) is one of them. It allows the
objective
function to "escape" from a local minimum with a probability computed
according to
Boltzman's law. There are several SA codes available on the Internet (see
for instance
Lester Ingber's web page at http://www.ingber.com/).
I am not aware of any application of these techniques in Pharmacokinetics,
however.
The only biological references that I have found dealt with the
minimization of
protein structures.
Sincerely,
--
Jean Debord
Laboratoire de Pharmacologie, Faculte de Medecine
2 Rue du Docteur Marcland, F-87025 Limoges, France
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Copyright 1995-2010 David W. A. Bourne (david@boomer.org)