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From: =?ISO-8859-1?Q?Hans-Bernhard_Br=F6ker?= <HBBroeker@t-online.de>
Newsgroups: comp.graphics.apps.gnuplot
Subject: Re: Fitting: How does gnuplot calculate the covariance matrix?
Date: Mon, 11 Apr 2011 23:27:57 +0200
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On 11.04.2011 21:42, Ingo Thies wrote:
> Am 2011-04-10 19:31, schrieb Hans-Bernhard Bröker:
>
>>> a = 9.482095E-01 -/+-1.947044E-01, 1.947043E-01
>>> b = 4.270960E-01 -/+-8.327496E-02, 8.327495E-02

>> Those errors are _way_ too big. Do yourself a favour and plot your data
>> along with the model, using parameters modified by those errors:
>
> Well, I plotted the enveloping contours for all (a,b) sets on the
> 1-sigma contour,

So, was a comparison of the model function's values for parameter pairs 
(a,b), (a+sigma_a, b+sigma_b) and (a-sigma_a, b-sigma_b) at x=4.0 to the 
data value of y=2.65 +/- 0.1 part of that study?  Did you notice that 
you get a model corridor [after correcting my a<-->b mixup] from 2.13 to 
3.18, more than 5 times wider than the error in the data?

Or, to keep it short: did you even look at the plot I described?  What's 
actually so wrong with the error corridor you get from gnuplot alone?

> And along this 1-sigma contour a and b walk through
> the error intervals above.

So which parameters of the error ellipse did you give above: horizontal 
and vertical section at the center, or overall size of the enclosed, 
axis-aligned bounding box?

>> [assuming 'set fit errorvar' active, and fit done:]
>>
>> gnuplot> p 'fit.dat' u 1:2:3 w err, a*x+b w l
>
> You mixed up a and b here; it is actually f(x)=a+b*x

Yep.  Sorry about that.  But even with that mixup repaired, the message 
is still the same.  a_thies and b_thies are similar enough to each other 
for that...

As to using opposite-sign combinations of deviations on the parameters 
--- that's just as bad, but in a different way, since it completely 
underestimates the data errors in the middle of the region.

The errors you get from gnuplot achieve a much more sensible error band 
with a whole lot less hassle.