Problems to fit with gnuplot and non-linear function

Question

I have this set of data that want to fit with gnuplot using the function f(x) = exp(A+ B/(x-x0)) where A,B and x0 are my set of parameters to fit

# x   f(x)
0.382 8.29023731095968 
0.509 6.36124122026352 
0.637 4.66938977764103 
0.764 3.3194714217965
0.891 2.15140777817893
1.019 1.15428884806615 
1.146 0.262232461832655 

I have try it with

fit log(f(x)) 'data.dat' using 1:(log($2)) via A, B, x0

also have defined the function as f(x) = A+ B/(x-x0) and tried with

fit f(x) 'data.dat' using 1:(log($2)) via A, B, x0

and then plot exp(f(x))

The code works but the fitted parameters are not fine, because when I plot the curve and the points together not make sense. Is this fit too complicate for gnuplot?

Answer 1

Fitting can fail if you have an inappropriate function or if you have starting values which might make it difficult for the fitting procedure to converge. In your case, I guess x0 is an important parameter. You should help the gnuplot fitting algorithm a little to have a chance to find reasonable values. Here, I guess x0=1.5 is a reasonable starting value. If this is not sufficient and if your model permits you might want to add additional variables or terms to get a better fit.

Code:

### fitting with appropriate starting values
reset session

$Data <<EOD
0.382 8.29023731095968
0.509 6.36124122026352
0.637 4.66938977764103
0.764 3.3194714217965
0.891 2.15140777817893
1.019 1.15428884806615
1.146 0.262232461832655
EOD

A = 1
B = 1
x0 = 1.5
f(x) = exp(A + B/(x-x0))
set fit nolog

fit f(x) $Data u 1:2 via A,B,x0

plot $Data u 1:2 w lp pt 7 ti "Data",\
     f(x) w l lc rgb "red" ti "Fit"
### end of code

Result:

Final set of parameters            Asymptotic Standard Error
=======================            ==========================
A               = 4.61445          +/- 0.3907       (8.466%)
B               = 3.57094          +/- 0.8876       (24.86%)
x0              = 1.80616          +/- 0.1371       (7.593%)