How to reproduce the example figure using lsqcurvefit

Hi, I am trying to follow this example for non-linear curve fitting.

Looking at the section 'Splitting the Linear and Nonlinear Problems', I understood upto the point how to estimate two exponents lambda1 & 2 (line 51). However, I don't understand how to recover the estimate for 'c' to produce the 'Global fit' in the last figure in the following section.

More specifically, once I obtain the unknown lambdas, what should I do to obtain the unknown vector 'c' (in the function 'fitvector')? Which of the followings would be the right approach?

1. c=A \ ydata
2. c=A \ yEst, or
3. Once obtained the estimates for lambda, I follow the full parameter estimation using the lambdas.

Thanks in advance for your help!

Answer 1

[Mathieu NOE]

hello

well, if the question is only about how to use lsqcurvefit, my answer is not gotta help you because I'm using either fminsearch , or , much better and more robust to crude / bad initial guess : Fminspleas - File Exchange - MATLAB Central (mathworks.com)

see the two examples attached if it helps you

all the best

[yp78]

@Mathieu Thank you for the suggestions and examples. If you don't mind, may I ask you why you are including 1 in the funlist in your 'demo3_fminspleas.m' , when the function is y = c(1)*exp(-lam(1)*t) + c(2)*exp(-lam(2)*t)?

% your example
funlist = {1, @(c,xdata) exp(c(1)*xdata), @(c,xdata) exp(c(2)*xdata)};
% wouldn't it be like this?
funlist2 = {@(c,xdata) exp(-c(1)*xdata), @(c,xdata) exp(-c(2)*xdata)};

[Mathieu NOE]

hello yp78

you're right

i(s because the code was originally intended to include a constant term in the equation, which is not relevant here;

if this is neglected / not needed you can do as you suggest : there are two modifications to do :

beware to use the same postive or negative sign in the equations !

% funlist = {1, @(c,xdata) exp(c(1)*xdata), @(c,xdata) exp(c(2)*xdata)};
funlist = {@(c,xdata) exp(-c(1)*xdata), @(c,xdata) exp(-c(2)*xdata)};

and

% y_fit1 = ILP(1) + ILP(2)*exp(INLP(1)*x) + ILP(3)*exp(INLP(2)*x) ; % fit model
y_fit1 = ILP(1)*exp(-INLP(1)*x) + ILP(2)*exp(-INLP(2)*x) ; % fit model

[yp78]

Hi Mathieu,

Thanks for the clarification. If I understood correctly, the Fminspleas function is compatible for matrix operations, no? In my working problem, the x and y data look like

x =

    0.1190
    0.2381
    0.3571
    0.4762
    0.5952
    0.7143
    0.8333
    1.0714
    1.4286
    2.1429
y =

    0.3458    0.1737   -0.0417
    0.2848    0.0959    0.0261
    0.2545    0.0085    0.0599
    0.2334   -0.0521    0.0520
    0.2117   -0.0693    0.0225
    0.2023   -0.0863   -0.0028
    0.1952   -0.0878   -0.0185
    0.1836   -0.0662   -0.0465
    0.1689   -0.0378   -0.0483
    0.1543   -0.0618   -0.0279

and I want to simultaneouly estimate the parameters of the functions

where f_1 fits to the first column of y, f_2 for the second column of y, so on. I modified the funlist as follows, but it is not working (it is running, but the solutions are rather strange, the initial guesses stay as they are etc.).

funlist = {1, @(c,xdata) exp(c(1)*xdata), @(c,xdata) exp(c(2)*xdata); ...
    1, @(c,xdata) exp(c(1)*xdata), @(c,xdata) exp(c(2)*xdata);...
    1, @(c,xdata) exp(c(1)*xdata), @(c,xdata) exp(c(2)*xdata)};

I should probably ask the author of the function for the tips. But if you have any clues, would you mind sharing your expertise with me here?

[Mathieu NOE]

hello

I am not sure that is capable of doing the 3 functions optimization all together. You may have to ask the author for better support than me.

a "low cost" approach I did is to do the lambda1 and lambad2 optimization on the first column of y data (f1 fit) and reintoduce (hard copy) those values in the second and thrird function optimisation.

see attachements

but this is not really a global optimisation as requested

[yp78]

Hi Mathieu,

Thanks again for the helpful example files! I wrote the author of the function to ask about the multivariate usage of the function. When I had a reply, I will share the information with you :)

[Mathieu NOE]

Good idea , that would for sure interest me as well !