Help with global optimization problem

So I'm trying to fit a model to data using global search (I also tried all of the global optimization solvers without a better result). Basically I am trying to fit a signal from an instrument. This signal is supposed to be composed of three overlapped peaks, and I am trying to deconvolve the signal into those 3 individual peaks so I can analyze them further one by one. The way I'm trying to do this is by fitting the signal to a sum of three functions, so called Fraser Suzuki functions (which I read somewhere are the same as the lognormal function), each of which has 4 parameters, so 12 parameters overall.

I created a toy version of my problem, in which instead of using real data, mock data is generated using said sum of these functions. I then try to use global optimization algorithms to minimize the sum of square differences to get a result and then compare it to the parameters found by the minimizations to the ones used to generate the data.

The problem is that the parameters determined by minimization are not the same as the original ones, and in theory one should be able to determine the original parameters by minimization, since the data were created with the same functions I am trying to fit to them. I got to get good results by imposing really stringent upper and lower bounds, but I have to analyze a lot of signals and trying to make this as automatic as possible is the reason I'm using code in the first place.

I'd be glad to read some of your input.

Thank you very much in advance.

PS: I ended up adding the pattern search solver since it returned a way better result that globalsearch.

Here's the code

%Parameters to create data
pa = [0.0363    0.0682  560.0000   55.1830    0.0597   -0.2543  632.3738   35.0086    0.0118    0.9939  609.8258  160.0000];

%independent variable vector
Temp = (400:1000)';

%create data
fs_data = fsm(Temp,pa);

%set lower and upper bounds
lb = [0,    -.5,   0,    0,...
      0,    -.5,   0,    0,...
      0,    -.5,   0,    0];
  
ub = [0.1,  0.5,    800,    200,...
      0.1,  0.5,    800,    200,...
      0.1,  0.5,    800,    400];
  
%Set initial guess
p0 = [0.0363    0.0682  560.0000   55.1830    0.0597   -0.2543  632.3738   35.0086    0.0118    0.9939  609.8258  160.0000];


%Set objective function
obj_fun =@(parameters) sum((fsm(Temp,parameters)-fs_data).^2);

%Set optmization problem

% PATTERN SEARCH
rng default
[fsfit,fval] = patternsearch(obj_fun,p0,[],[],[],[],lb,ub)

% % Set optmization problem
% rng default % For reproducibility
% opts = optimoptions(@fmincon,'Algorithm','sqp');
% gs = GlobalSearch;
% problem = createOptimProblem('fmincon','x0',p0,'objective',obj_fun,'lb',lb,'ub',ub,'options',opts);

% % fit model to data
% [fsfit,fval] = run(gs,problem)

difference = pa - fsfit


figure
hold on
plot(Temp,fs_data,'-')
plot(Temp,fsm(Temp,fsfit))
plot(Temp,fs(Temp,fsfit(1:4)))
plot(Temp,fs(Temp,fsfit(5:8)))
plot(Temp,fs(Temp,fsfit(9:12)))
plot(Temp,fs(Temp,pa(1:4)),'--')
plot(Temp,fs(Temp,pa(5:8)),'--')
plot(Temp,fs(Temp,pa(9:12)),'--')
hold off
legend('real','calc','1','2','3','1c','2c','3c')



%Define Fraser Suzuki Function
function y = fs(T,p)

 in = 2 * p(2) * ((T-p(3)) / p(4));
 out = - log(2) / p(2)^2;
 
 y = p(1) * exp(out .* (log(1+in)).^2);
 y(y~=real(y))=0;
end

%Define Fraser Suzuki mixture

function y = fsm(T,pm)

y = fs(T,pm(1:4)) + fs(T,pm(5:8)) + fs(T,pm(9:12));

end

Answer 1

[Star Strider]

The Golbal Optimization Toolbox functions (as much as I like them) may not be appropriate for this. See if the rica function (that performs an independent components analysis decomposition) will separate them. If so, it will then be possible to estimate the parameters describing the individual signals.

.

[adullak]

Thank you very much for your answer, I will look into the rica to see if I can get the results I want.

[Star Strider]

My pleasure!

Without your actual (or simulated) data, a more precise response is not possible.

.