I'm trying to extract 4 parameters by fitting one set of data with an ODE via NonlinearModelFit. I originally thought the process should be straightforward, but later on I found I was totally wrong.

The data shown below has a sigmoidal curve:


The model with 4 target parameters (n, m, Ea, LogZ) is below:

model65[n_?NumberQ,m_?NumberQ,Ea_?NumberQ,LogZ_?NumberQ]:=Module[{α,t},First[α /.NDSolve[{α'[t] == ((10^LogZ)/60) Exp[-Ea/(8.314 (65 + 273.15))] α[t]^m (1 - α[t])^n, α[0] == 0}, α, {t, 0, 2400}]]]

Using NonLinearModelFit to extract n, m ,Ea, LogZ (with initial guesses):

fit = NonlinearModelFit[data,model[n, m, Ea, LogZ][t], {{n, 1.5}, {m, 0.5},{Ea, 85000}, {LogZ, 10}}, t];

Below is the error message I'm having:

NonlinearModelFit::fmgz: Encountered a gradient that is effectively zero. The result returned may not be a minimum; it may be a maximum or a saddle point. >>

Can anyone please help me with this problem? Thank you in advance for the great help!


1 Answer 1


I think the system can be simplified. There is no t dependence in ((10^LogZ)/60) Exp[-Ea/(8.314 (65 + 273.15))]). Time can also be rescaled. Jittering initial condition "just off" zero.

After this,

s = y /. ParametricNDSolve[{y'[t] == a y[t]^b (1 - y[t])^c, 
     y[0] == 0.001}, y, {t, 0, 1}, {a, b, c}];
datar = Transpose[{Rescale[#1], #2} & @@ Transpose[data]];

Now fit model (rescaled and simplifying a=1):

nlm = NonlinearModelFit[datar,s[a, b, c][t], {{a, 6}, {b, 0.21}, {c, 1.1}}, t];
Show[ListPlot[datar, PlotMarkers -> {Automatic, 8}],Plot[nlm[t], {t, 0, 1}, PlotStyle -> Red]]

where b and c are exponents and a is constant. enter image description here

I hope this is helpful. Note: I used Manipulate to help with starting values. I have left the full aim to OP or others.

  • $\begingroup$ Thank you so much @ubpdqn !! Just wondering why the time has to be scaled to make the NonlinearModelFit works? Will these fitted parameters be applicable to the real time scale? I really appreciate your great help! $\endgroup$
    – DavidC
    Mar 28, 2016 at 20:56
  • $\begingroup$ @DavidC apologies for delay (time zones). I do not think it needed rescaling but it just made it easier for me. Just look at what rescale does and you will be able to work it out.:) $\endgroup$
    – ubpdqn
    Mar 28, 2016 at 22:14
  • $\begingroup$ @ubpdqnThank you! I really appreciate your great help :) $\endgroup$
    – DavidC
    Mar 29, 2016 at 15:05

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