[?] Perform a nonlinear model fit?

Questions
,
1

Dear All,
I have some “x, y” data values, and I am trying to use non-linear model fit command to fit Inverse Gaussian Distribution to the data. Yet (perhaps due to the negativity in a square root), I am getting an error. Can you please look at the attached code, and kindly let me know how can I avoid such errors and solve the particular problem?
Thanks a lot.

2

This is a small set of data and does not really require an attached notebook. It can all be placed in the question (less clicking and downloading that way).

3

The error message gives a strong hint as to what the problem is:

errormessage.png
errormessage.png907×121 4.98 KB

One can see that NonlinearModelFit tries \lambda=-1.13888 but you want to restrict \lambda to be greater than zero.

modelinvgauss = PDF[InverseGaussianDistribution[\[Mu], \[Lambda]]];
invgauss = 
  NonlinearModelFit[
   datanew, {modelinvgauss[x], \[Lambda] > 0}, {{\[Mu], 
     10}, {\[Lambda], 10}}, x];
invgauss["BestFitParameters"]
(* {\[Mu] -> 11.098760475474151, \[Lambda] -> 35.72763989840714} *)
Show[ListPlot[datanew, PlotRange -> {{0, 20}, {0, 0.12}}],
 Plot[invgauss[x], {x, 0, 20}]]

inversegaussianfit.png
inversegaussianfit.png540×339 4.91 KB

The fit isn’t so great. I have to wonder why you want to fit a probability density function using a regression. Usually a probability density function is fit with a random sample from that distribution. If what you have is really a regression and you’re just interested in the form of the density function being an inverse Gaussian, then usually translation and/or scaling parameters are usually added.

4

The error message gives a strong hint as to what the problem is:

errormessage.png
errormessage.png907×121 4.98 KB

One can see that NonlinearModelFit tries \lambda=-1.13888 but you want to restrict \lambda to be greater than zero.

modelinvgauss = PDF[InverseGaussianDistribution[\[Mu], \[Lambda]]];
invgauss = 
  NonlinearModelFit[
   datanew, {modelinvgauss[x], \[Lambda] > 0}, {{\[Mu], 
     10}, {\[Lambda], 10}}, x];
invgauss["BestFitParameters"]
(* {\[Mu] -> 11.098760475474151, \[Lambda] -> 35.72763989840714} *)
Show[ListPlot[datanew, PlotRange -> {{0, 20}, {0, 0.12}}],
 Plot[invgauss[x], {x, 0, 20}]]

inversegaussianfit.png
inversegaussianfit.png540×339 4.91 KB

The fit isn’t so great. I have to wonder why you want to fit a probability density function using a regression. Usually a probability density function is fit with a random sample from that distribution. If what you have is really a regression and you’re just interested in the form of the density function being an inverse Gaussian, then usually translation and/or scaling parameters are usually added.

5

Thank you all for your great and detailed replies. Alex.