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Next: Adding models to XSPEC Up: Fitting with few counts/bin Previous: Theory

Practice

XSPEC uses a variant of Marquardt's algorithm described in §11.5 of ``Data Reduction and Error Analysis for the Physical Sciences" by Bevington. (The reader is advised that this description is designed to be read in conjunction with Bevington.) The algorithm turns on finding a matrix tex2html_wrap_inline12999 and a vector tex2html_wrap_inline13001 such that the equation :

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gives sensible values of the change in parameters, tex2html_wrap_inline13003 , for the fitting function. Bevington §11.4 gives the derivation of tex2html_wrap_inline13015 and tex2html_wrap_inline12589 and shows that tex2html_wrap_inline12589 is parallel to the gradient of tex2html_wrap_inline10235 .

Now the C statistic has a gradient with respect to the parameters of the fitting function of :

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So, following Bevington, expand tex2html_wrap_inline12977 about tex2html_wrap_inline13013 :

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substitute into C and minimize with respect to the changes in the parameters :

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so to first order in the parameter changes :

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or :

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where :

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These tex2html_wrap_inline13015 and tex2html_wrap_inline12589 then are substituted for those used in the tex2html_wrap_inline10235 case and the algorithm works as required.

There is one further difference in XSPEC between the tex2html_wrap_inline10235 and likelihood methods, which is caused by the fact that XSPEC uses an analytic formula for setting the model normalisation. In the tex2html_wrap_inline10235 case, this means multiplying the current model by :

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where tex2html_wrap_inline13019 is the error on tex2html_wrap_inline12975 . In the likelihood case the corresponding factor is :

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Keith Arnaud (kaa@genji.gsfc.nasa.gov)
Wed May 28 10:59:33 EDT 1997