Re: [Gnuplot-info] Outputs of fit: details of how they're defined

[Hans-Bernhard B. <HBB...@t-...>]

Am 28.11.2017 um 00:34 schrieb Dan via gnuplot-info:

> - in the variables L and M, the values l and m of the L and M
> parameters respectively, which, among all the (L,M) pairs that the
> Marquardt-Levenberg algorithm has tried, produce the lowest value of
> the \chi^2 statistic;

Yes.

> - in the variable FIT_WSSR, the value \Chi^2 of the \chi^2 statistic
> that is achieved by setting L = l and M = m;

Yes.

> - in the variable L_err, the value (presumably calculated in a
> finite-difference way) of
> \sqrt{\frac{2}{\left(\partial\!^{2}\!\left(\chi^2\right)/\partial 
> L\!^{2}\right)_{L = l,M = 
> m}-\left(\left(\partial\!^{2}\!\left(\chi^2\right)/\left(\partial\!\!L\!\partial\!\!M\right)\right)_{L 
> = l,M = m}\right)^2/\left(\partial\!^{2}\!\left(\chi^2\right)/\partial 
> M\!^{2}\right)_{L = l,M = m}}}

The algorithm never actually computes it just like that.  What actually 
happens is that the code builds a matrix of all the second partial 
derivatives of \chi^2 with resepect to the given parameters.  This 
matrix is inverted, to yield the covariance matrix.  The roots of the 
diagonal elements of that matrix form the error values.  These get 
divided out of each column and row, to form the correlation matrix that 
is printed out.

For the special case of just two parameters the formulae you found may 
well be correct... I didn't compute it through to check one way or the 
other.

There's really nothing special about that method.  It's basically the 
definition of what all fitting tools do for asymptotic error estimation. 
  They're "asymptotic" because they're derived from the second 
derivatives, i.e. just lowest-order term describing the shape of the 
\chi^2 function "landscape" near the discovered minimum, which is only 
truly accurate asymptotically close to the point of evaluation.

If memory serves, the non-asymptotic approach would try to actually 
trace the contour of \chi^2 at a value of <minimum> + 1, instead.