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From: Ingo Thies <ingo.thies@gmx.de>
Newsgroups: comp.graphics.apps.gnuplot
Subject: Re: Fitting: How does gnuplot calculate the covariance matrix?
Date: Thu, 14 Apr 2011 15:07:19 +0200
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On 10.04.2011 19:31, Hans-Bernhard Bröker wrote:

> gnuplot> p 'fit.dat' u 1:2:3 w err, a*x+b w l
> gnuplot> rep (a+a_err)*x+b+b_err, (a-a_err)*x+b-b_err
> gnuplot> a_thies=0.195
> gnuplot> b_thies=.083
> gnuplot> rep (a+a_thies)*x+b+b_thies, (a-a_thies)*x+b-b_thies

I have modified the model function to minimize the correlation of a and
b. This has been done by substituting x by x-x0, where x0 is the mean of
all x (2 in this case). Here is the full script, to be applied to the
same data table ('fit.tab'). To enhance visibility, I have given
different colors for the different error types (blue for gnuplot's, red
for mine). Please look again at the errors. Are they still _way_ too
large? Or aren't gnuplot's errors now _way_ too small?

set fit errorvar
#x0=0.; a_thies=0.195; b_thies=0.083 # old fit
x0=2.; a_thies=0.101; b_thies=0.083 # new fit
f(x)=a+b*(x-x0)
set xrange [0:4]
set yrange [0.5:3]
fit [0:4] f(x) xfile u 1:2:3 via a,b
set key left
p a+b*(x-x0) w l lc rgb '#0000ff',\
  a-a_err+(b-b_err)*(x-x0) lc rgb '#007fff',\
  a+a_err+(b+b_err)*(x-x0) lc rgb '#007fff',\
  a-a_thies+(b-b_thies)*(x-x0) lc rgb '#ff0000',\
  a+a_thies+(b+b_thies)*(x-x0) lc rgb '#ff0000',\
  'fit.dat' u 1:2:3 w err lc rgb '#000000'

I would find it highly confusing if the error corridors depend on a
simple substitution. If one plotts the a,b error contours, and uses them
for the error estimate (i.e. walks around the 1-sigma contour in the a-b
space and creates the enveloping curve for all possible resulting
functions), then the "way-too-large" error for the slope b remains
constant. Only the error for the offset a changes, as expected.

-- 
Gruß,
      Ingo