# PharmPK Discussion - Where are Global Minima in Curve-Fitting?

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• On 3 Mar 1997 at 12:19:55, "J. Zhu" (jyzhu.aaa.usa.net) sent the message
`Hi there, a couple of graduate students bring up the followingquestions. How are you sure that your nonlinear curve-fitting in themodeling process converges to global minima not local minima. One ofmethods to check if the results reach global minima seems to changeinitial estimates to 10 or more times of the original values. It assumesglobal minima achieve if the same results are obtained from differentinitial estimates. The questions are what to do in the cases (1) theinitial estimates of first time reach global minima while the initialestimates of second time reach local minima; and (2) the initialestimates of first time reach local minima while the initial estimatesof second time reach another local minima. Also, how to select theinitial estimates of second time in a case where many parameters, say25, need to be estimated, all or parts (how many) in a time change? Whatabout some parameter estimates in global minima and some in localminima. Another question no related to the minimal issue is what=92sminimal ratio of data (sample) points to the numbers of parameters to beestimated for ideal curve-fitting. Thank you very much for your opinionsand comments.JohnnyEmailto:jyzhu.at.usa.net`
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• On 4 Mar 1997 at 13:05:55, "Vladimir Piotrovskij" (VPIOTROV.-at-.JANBELC1.SSW.JNJ.COM) sent the message
`     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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• On 5 Mar 1997 at 11:38:18, Jean Debord (jdebord.-at-.MicroNet.fr) sent the message
`There are methods known as global optimization which (in principle)localize theglobal minimum. Simulated annealing (SA) is one of them. It allows theobjectivefunction to "escape" from a local minimum with a probability computedaccording toBoltzman's law. There are several SA codes available on the Internet (seefor instanceLester 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 theminimization ofprotein structures.Sincerely,--Jean DebordLaboratoire de Pharmacologie, Faculte de Medecine2 Rue du Docteur Marcland, F-87025 Limoges, France`
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