Bad Exponential Fit of Data: How to solve a problem with bad parameter estimation

Hello to all.

I need to fit data which should be well fitted by one or two exponential model. I tried to use fit function and non-linear fit.

I am interested only in a decay part after the peaks:

First I tried fit function:

[fitresult1, gof1] = fit(time', y_peak_raw_decay', 'exp1');

[fitresult2, gof2] = fit(time', y_peak_raw_decay', 'exp2');

Here is no explicit error notifiation and looks like 2-exp fit works better.

General model Exp2:

fitresult2(x) = a*exp(b*x) + c*exp(d*x)

Coefficients (with 95% confidence bounds):

a = 2.816e+04 (2.27e+04, 3.362e+04)

b = -0.07659 (-0.09724, -0.05593)

c = 2.428e+04 (1.834e+04, 3.023e+04)

d = -0.007086 (-0.01101, -0.003164)

I suppose that my data has an offset, but fit with exp2 does not take into account and return offset constant. So I tried fitnlm function:

X = time;

Y = y_peak_raw_decay;

tbl = table(X', Y');

modelfun = @(b,x) b(1) * exp(-b(2)*x(:, 1)) + b(3) * exp(-b(4)*x(:, 1))+ b(5);

beta0 = zeros(1,5);

mdl_peak = fitnlm(tbl, modelfun, beta0);

b1 = 8038.37302712328

b2 = 0.394548525519020

b3 = 33291.0317256818

b4 = 0.0433576771723344

b5 = 13612.4854074245

And here I have a warning:

"Warning: Rank deficient, rank = 3, tol = 1.044552e-13.

Warning: The Jacobian at the solution is ill-conditioned, and some model parameters may not be estimated well (they are not identifiable). Use caution in making

predictions. "

I read here that it means that model really does not describe data well. But my signal is quite standard and it normally can be modeled with exp function. So I suggest I do mistake, but have no idea how to fix it.

I tried to rescale data, because the difference between X and Y is 3-order, that can cause a problem for fitting. It didn't help and I still have the same notification.

Could you please help me to understand where is a problem can lie.

Answer 1

[Mathieu NOE]

hello

a very simple approach can give you the right answer :

load('Qdata.mat')

% model : y = b*exp(m*x);
P = polyfit(X, log(Yrescaled), 1);
m = P(1);
b = exp(P(2));

t2 = linspace(min(X),max(X),100);
c2 = b*exp(t2*m);
plot(X, Yrescaled, '*r', t2, c2, '-b');

TE = sprintf(' C = %0.6f * e^{%0.6f * t}',b, m);
text(t2(4)+1, c2(4), TE);

[Mathieu NOE]

Hello

Problem solved ?

[Olga Se]

Late big thanks. It works well.

[Mathieu NOE]

better late than nevr :)

thanks !