Whether is it possible to plot normal probability distribution in gnuplot - linux

Question

My data file is as-
2 3 4 1 5 2 0 3 4 5 3 2 0 3 4 0 5 4 3 2 3 4 4 0 5 3 2 3 4 5 1 3 4
My requirement is to plot normal PDF in gnuplot.
I could do it by calculating f(x)
f(x) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
for each x using shell script.
Then I plot it in gnuplot using the command-
plot 'ifile.txt' using 1:2 with lines
But whether is it possible to plot directly in gnuplot?

Answer 1

gnuplot provides a number of processing options under the smooth keyword (try typing help smooth for more info). For your specific case, I would recommend a fit though.
First, note that your data points are in a row, you need to convert it to columns for gnuplot to use it. You can do it with awk:
awk '{for (i=1;i<=NF;i++) print $i}' datafile
which can be invoked from within gnuplot:
plot "< awk '{for (i=1;i<=NF;i++) print $i}' datafile" ...
Now assume that datafile has the right format for simplicity.
You can use the smooth frequency option to see how many occurrences of each value you have:
plot "datafile" u 1:(1.) smooth frequency w lp pt 7
To get the normalized distribution, you divide by the number of values. This can be done automatically within gnuplot with stats:
stats "datafile"
This will store the number of values in variable STATS_records, which in you case has value 33:
gnuplot> print STATS_records
33.0
So the normalized distribution (the probability of getting a value at x) is:
plot "datafile" u 1:(1./STATS_records) smooth frequency w lp pt 7
As you can see, your distribution doesn't really look like a normal distribution, but anyway, let's go on. Create a Gaussian for fitting and fit to your data, and plot it. You need to fit to the probability, rather than to the data itself. To do so, we plot to a table to extract the data generated by smooth frequency:
# Non-normalized Gaussian
f(x)= A * exp(-(x-x0)**2/2./sigma**2)
# Save probability data to table
set table "probability"
plot "datafile" u 1:(1./STATS_records) smooth frequency not
unset table
# Fit the Gaussian to the data, exclude points from table with grep
fit f(x) "< grep -v 'u' probability" via x0, sigma, A
# Normalize the gaussian
g(x) = 1./sqrt(2.*pi*sigma**2) * f(x) / A
# Plot
plot "datafile" u 1:(1./STATS_records) smooth frequency w lp pt 7, g(x)
set table generates some points which you should exclude, that's why I used grep to filter the file. Also, the Gaussian needs to be normalized after the fitting is done with a variable amplitude. If you want to retrieve the fitting parameters:
gnuplot> print x0, sigma
3.40584703189268 1.76237558717934
Finally note that if the spacing between data points is not homogeneous, e.g. instead of x = 0, 1, 2, 3 ... you have values at x = 0, 0.1, 0.5, 3, 3.2 ... then you'll need to use a different way to do this, for example defining bins of regular size to group data points.