not so FAQ

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• After Plotting
• Miscellaneous Stuff
  • 
    ternary operator

  • 
    broken function

  • 
    loop

  • 
    least-squares fitting (I)

  • 
    least-squares fitting (II)

  • 
    dumb terminal

  • 
    many kinds of lines

  • 
    write a value

Miscellaneous Stuff (No.2)

Least-Squares Fitting with non-linear functions (I)

Gnuplot3.7 has a provision of a least-squares fitting. It searches
automatically for parameters of an arbitrary function by fitting it
to data in a file. The function can be linear or non-linear, but it
should be expressed by gnuplot’s expression manner. Here is an
example for data fitting — fit a Lorentzian function
f(x)=a/(1+b*x*x) to data. The variables a and b
are the parameters.

The following sample data were prepared by calculating this function
with a=2 and b=1, to show that gnuplot gives the
correct answer.

-10.000000    0.019802
-8.947370    0.024674
-7.894740    0.031582
-6.842110    0.041828
-5.789470    0.057941
-4.736840    0.085333
-3.684210    0.137236
-2.631580    0.252359
-1.578950    0.572561
-0.526316    1.566160
0.526316    1.566160
1.578950    0.572561
2.631580    0.252359
3.684210    0.137236
4.736840    0.085333
5.789470    0.057941
6.842110    0.041828
7.894740    0.031582
8.947370    0.024674
10.000000    0.019802

Those numerical data are stored into the data-file
“exp.dat”. Firstly the fitting function is defined. The parameters
to be searched for are arbitrary except for “x” or “t” those
gnuplot uses. A data fitting begins by the command
fit . The parameter names are
given by the via option.

At the data fitting, one can use using, index, every
options to specify or to skip data or
data block in the file. An uncertainty of the data point is treated
as a weight of it. When the data point has an error of sigma, the
weigh becomes sigma^{-2}. To do the weighted least-squares fitting,
the errors are prepared at the third column in your data-file, then
you have to tell that those are the uncertainties, like
fit f(x) "exp.dat" using 1:2:3 via a,b
. When the data error is omitted, all data points have
the equivalent weight.

gnuplot> f(x)=a/(1+b*x*x)
gnuplot> fit f(x) "exp.dat" via a,b


Iteration 0
WSSR        : 1.43879           delta(WSSR)/WSSR   : 0
delta(WSSR) : 0                 limit for stopping : 1e-05
lambda      : 0.201184

.......

Final set of parameters            Asymptotic Standard Error
=======================            ==========================

a               = 2                +/- 5.236e-07    (2.618e-05%)
b               = 0.999998         +/- 7.397e-07    (7.397e-05%)


correlation matrix of the fit parameters:

a      b
a               1.000
b               0.837  1.000

During the data fitting, you can see changes in the parameters on your
display. The same stuff you can find in a file “fit.log” which is generated
in the current directory. If succeeded, you get the final values of the
parameter as well as those covariance matrix. In the above case, the correct
parameters a=2 and b=1 were obtained.

The final results are stored in the variables, a and
b, so that you can compare the fitted function and the data by
plotting this function with the fitted parameters.

gnuplot> plot f(x) with lines, "exp.dat" using 1:2 w points

Least-Squares Fitting with non-linear functions (II)

The next example is more realistic one. We have experimental data like below.
A function f(x)=c*(x-a)**b is fitted to the data by adjusting the variables
a, b, and c.

# X        Y     Z
6.295    8.4   0.3
6.795   23.9   0.7
7.826  104.0   3.0
8.830  255.0   8.0
9.841  421.0  13.0
10.860  566.0  18.0
11.864  690.0  22.0

Each data point has an error (Z), then the weight of the point is 1/Z*Z.

At first, plot the experimental data only to see their tendency.

gnuplot> set log y
gnuplot> plot "exp.dat" using 1:2:3 w yerrorbars

Next, choose initial values of those variables, a,
b, and c. When the initial values are not given at the
parameter search, gnuplot assumes that those values are 1. Since this
choice is insufficient in some case, you should give better initial
values. Our fitting function is f(x)=c*(x-a)**b, then one can guess
that a is near 6 because f(x)=0 at x=a, c is in the
order of 10 (see log(f(x))), and b is 1 or 2 as f(x) becomes 10
to 100 times when x is increased by 1. Those values are used for the
initial.

gnuplot> a=6
gnuplot> b=1
gnuplot> c=10
gnuplot> f(x)=c*(x-a)**b
gnuplot> fit f(x) "exp.dat" using 1:2:3 via a,b,c

The calculation converges after 17 times iteration, the final
values with those covariance are displayed. The values obtained were
a=5.77, b=1.89, c=25.8.

After 17 iterations the fit converged.
final sum of squares of residuals : 90.8196
rel. change during last iteration : -9.71916e-09

degrees of freedom (ndf) : 4
rms of residuals      (stdfit) = sqrt(WSSR/ndf)      : 4.76497
variance of residuals (reduced chisquare) = WSSR/ndf : 22.7049

Final set of parameters            Asymptotic Standard Error
=======================            ==========================

a               = 5.76731          +/- 0.176        (3.051%)
b               = 1.89369          +/- 0.2127       (11.23%)
c               = 25.7956          +/- 9.807        (38.02%)


correlation matrix of the fit parameters:

a      b      c
a               1.000
b              -0.944  1.000
c               0.975 -0.975  1.000

Those results were compared with the experimental data.

gnuplot> plot f(x),"exp.dat" using 1:2:3 w yerrorbars