Non-linear curve fitting using GNU plot

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Table 1: Example Data

Fitting Using GNU Plot

There are several ways to do this fit using GNU Plot. I have found the one shown here to be the easiest. You will need to create two files: a script file and a data file. Both can be created using a simple text editor. The script file and the data file are shown in Figures 1 and 2 respectively. Figure 1 is configured to do a weighted curve fit. The function shown on the second line is the inverse of the Sakuma-Hattori Equation (1). The code to do a non-weighted curve fit is shown as well and is commented out. The text in Figure 2 can be separated by tabs or spaces.

# Blog Article Example
f(x) = c2 / A / log(C/x + 1) - B / A
c2 = 14387.752
A = 10
B = 100
C = 1e6
# weighted curve fit
fit f(x) 't_s.txt' using 2:1:3 via A, B, C
plot 't_s.txt' using 1:($f($$2) - $$1):3 title "" with yerrorbars
set title "SH Curve Fit - Weighted - Final"
replot
pause -1
# no weight curve fit
# fit f(x) 't_s.txt' using 2:1 via A, B, C
# plot 't_s.txt' using 1:($f($$2) - $$1) title ""
# set title "SH Curve Fit - Non Weighted - Final"
# replot
# pause -1

Figure 1. Example Script File
 

Figure 2. Example Text File  

To do the curve fit, simply run GNU Plot, change to your local directory, and use the call function followed by the script file name in quotes. The fitted parameters will be outputted in a text window. The example in Figure 1 plots a graph of the fitting error and is shown in Figure 3.

Figure 3. GNU Plot Graph  

Using Microsoft Excel
 

Using Excel, one may write a VBA macro to perform a curve fit. That is not a simple task. Microsoft provides an easier solution in the form of an add-in to Excel called Solver. It is not installed with a standard Excel installation, but a web search for “Solver Excel Install Add in” should give instructions for your specific version of Excel. (Note: we are not discussing ‘Open Solver’ here).

My initial data is shown in Table 2. The parameters are listed on top. Columns 1, 2, and 3 contain the reference temperature, the signal, and the expanded uncertainty of temperature (k = 2). Column 4 contains the result with the given parameters calculated using the inverse of the Sakuma-Hattori Equation (1). Column 5 contains the curve fit error divided by the uncertainty. This is done for a weighted curve fit. If the curve fit were not weighted, we would not divide by the uncertainty. Column 6 is Column 5 squared. Finally, we sum all of the squares of errors which is shown above Column 6. This is what we are trying to minimize. The chi squared result is shown above. Since we are using expanded uncertainties (k = 2), it is the sum of the errors squared multiplied by 4 (2 squared). This would not be the case for a non-weighted curve fit.

Table 2. Initial Excel Input

For my version of Excel, I followed these steps to perform the curve fit. I opened Solver. I placed the sum of error squared into the ‘Set Objective’ box. I picked the ‘To: Min’ radio button. Then under ‘By Changing Variable Cells’, I picked the cells with parameters A, B, and C. Do not pick c2, as it is a physical constant. Under ‘Select a Solving Method’, I picked ‘GRG-Nonlinear’. Under Options -> GRG-Nonlinear, I used 0.000000000001 for convergence. Then I clicked ‘Solve’.

Comparison of Answers
 

Both Excel and GNU Plot gave the same answers. Table 3 shows the results. The real differentiation between the two software packages is which is more convenient for your own use. GNU Plot is free. However, it is cumbersome to use. Microsoft Excel is not free, but in the modern day work place, most people have it installed on their computer.

Table 2. Curve Fit Results