MATLAB: Nonlinear fit comparison – Matlab vs OriginLab

curve fittingdatanonlinearstatistics

Good afternoon, I'm posting this question because I'm not able to find a satisfactory result of a non-linear fit with Matlab. On the contrary, OriginLab returns a good fit together with errors for fit parameters. Here my x and y data: x = [177600,961200, 2504000, 4997000, 8884000]; y = [6.754, 24.416, 58.622, 107.980, 154.507]; yErr = sqrt(y);
The fitting function is the following: y = V + (A-V)*S*x*EXP(-S*x) where V, A and S are the parameters which I need to calculate from the fit. OriginLab gives me the following:
  • V = 1.978 ± 0.889
  • A = 585 ± 64
  • S = 4.392E-8 ± 6.771E-9
On Matlab I tried all the possible form of non-linear regression: fitnlm, lsqcurvefit, nlinfit etc… every time the same warning: The Jacobian is ill conditioned. Here is an example:
    Nonlinear regression model:
    y ~ a1 + (a2 - a1)*a3*x*exp( - a3*x)
Estimated Coefficients:
         Estimate         SE           tStat         pValue  
        __________    __________    ___________    __________
  a1       -0.6789    2.9104e-19    -2.3327e+18    2.0265e-73
  a2        530.78    8.0894e-20     6.5614e+21    3.2371e-87
  a3    5.2511e-08    5.1865e-10         101.25    5.7063e-08
Of course such small standard errors are not reliable, but still I'm not able to reproduce the Origin results. Any suggestions?

Best Answer

Well, I guess I would credit OriginLab with knowing how to pre-scale the x,y data appropriately. If you do this manually, then lsqcurvefit works fine, and gives a slightly better fit (according to resnorm) than OriginLab,
c=10*max(x); %pre-scaling factor
a0=[1.9780 ,585, 4.3920e-08*c];
F=@(a,x) a(1)+(a(2)-a(1)).*a(3).*x/c.*exp(-a(3).*x/c);
[a,resnorm,~,~,~,~,J]=lsqcurvefit(F,[1,100,1],x,y,[],[],...
                       optimoptions('lsqcurvefit','Display','iter'));
solution=[a(1), a(2), a(3)/c]
resnorm %10.1985
condJ = cond(full(J))  %510.8393
plot(x,y,'x',x,F(a,x),'--'); xlabel 'x', xlabel 'y';
Related Question