Fitting multiple exponential function .

I am trying to fit my experimental data with the following function, where T is temperature and rest all are fitting parameter.

I am using following matlab code for fitting.

fun = @(x)shibita_fitting(x,data)

x0=[8000 2000 0.1 10^3 0.5 10^3 0.1];

bestx = fminsearch(fun,x0)

where fitting function is defined as

function sse = shibita_fitting(x,data)
a = x(1);
d=x(2);
e1=x(3);
c1=x(4);
E1=x(5);
c2=x(6);
E2=x(7);
T=data(:,1);
I=data(:,2);
I_new=a* (1+d*exp(-e1./T))./(1+c1*exp(-E1./T)+c2*exp(-E2./T));
sse = sum((I - I_new).^2);

The result of fitting is displayed below. The fitting is following the trend but it is not able to reproduce the kink near T=50.

Please suggest.

[Matt J]

I am skeptical that your equation model can capture the kink. Your equation model is smooth, differentiable function. The trend near the kink is not.

In any case, it might be better to use lsqcurvefit, if you have the appropriate toolbox. You have a number of unknowns that strains what fminsearch was designed for.

[Mathieu NOE]

are you sure your model is capable of reproducing that kink ?

also , on complex models, fminsearch is very sensitive to x0 values. If you are a bit too off, the convergence is poor

you may have to use a more robust tool.

[Saroj Poudyal]

Dear Mathieu,

I have taken the model from literature and it has been used to fit the similar data.

Answer 1

[Alex Sha]

@Saroj Poudyal, the result you obtained is not the best one, refer to the global optimization solution below:

Sum Squared Error (SSE): 1481189.88030936
Root of Mean Square Error (RMSE): 145.464275452741
Correlation Coef. (R): 0.996415848858336
R-Square: 0.992844543856079

Parameter	Best Estimate   
---------	-------------   
x1       	6850.52907094882
x2       	602297.690775148
x3       	565.832984121765
x4       	8348759.66056058
x5       	679.004219138599
x6       	9.35098845248168
x7       	70.2264432550839

[Saroj Poudyal]

@Alex Sha The fitted values looks fine.

Can you please guide me on how you used global optimization.

I am new to optimization/fitting.

[John D'Errico]

An important point is here, you really do need a global optimizer tool, as good starting values are absolutely crucial for any form of multiple exponential model.

[Alex Sha]

@Saroj Poudyal The global optimization algorithm does not need to guess the initial start-value of each parameter, at least in theory, but can obtain the global optimization solution, although in practical applications it is often far from the ideal result. However, for complex fitting problems, it is still helpful to choose the global optimization algorithms.

Matlab contains a global optimization toolbox, you can try various global optimization algorithms included, such as genetic algorithms, etc., but personal experience, the power of Matlab's global optimization toolbox is still relatively weak, and needs continuous improvement.

[Saroj Poudyal]

@Alex Sha Thanks. I could reproduce the similar result as you have posted.