Using fminsearch with a function containing summation (via for loop)

Dear MATLAB community,

I would like to obtain the values of , γ, δ, and L in the following equations:

To obtain these values, I am optimizing the error function :

where is the experimental result.

The strategy that I would like to implement is:

(1) Generate a function that calculates both the and

(2) Use that function to optimize the values of , γ, δ, and L using fminsearch (since I want to use Nelder-Mead optimization).

The script that I have written for the function is:

function [f] = ErrorFunction(x)

% n_ave = x(1)
% gamma = x(2)
% delta = x(3)
% length = x(4)

c = 299792458; % Speed of light in m/s
n_air = 1.000293; % Index of refraction of air 
k = 1e-12; % ps to s coversion
cor_fac = 0.04;

M = 4;
fp_signal = 0.0; %fp_signal is E_FP(t)
for k=1:M
  fp_signal = fp_signal +...
      ((x(1)-1)/(x(1)+1))^(2*k).*...
      x(2)^(2*k).*...
      raw_signal_ref.*k... % raw_signal_ref is E_reference(t)
      (raw_time_ref.*(1e-12)-(x(1)*2*x(4)*M/c)-((x(1)-n_air)*x(4)/c))/...
      x(3)^(2*M+1);
end

signal_simulated = cor_factor*(4*x(1))/(x(1)+1)^2*x(2)*...
    ((raw_signal_ref.*k.*((raw_time_ref*(1e-12)-(x(1)-1)*(x(4)*(1e-3)/c))/x(3))) + fp_signal); % raw_time_ref = t 

f = sum(sim_signal_sam - raw_signal_sam); %raw_signal_sam is E_sample_(t)

To find , γ, δ, and L, I use fminsearch using the following separate script:

%% Optimization of paramters: n_ave , gamma, delta, length
 
 
 x0 = [n_ave_in;gamma_in;delta_in;thickness_in];
 x_optimized = fminsearch('ErrorFunction',x0);

The initial values for these parameters are , , , and .

When I run both scripts, however, I am getting this result:

Questions now are:

1. Should I use a different approach on implementing the summation (via for loop) for the FP effect, i.e. using symsum?
2. Can optimization using fminsearch be even done if a summation is inside the equation?
3. What should be changed in the function script for fminsearch to work?

Thanks in advanced for your help.

Answer 1

[Alan Weiss]

Youu have several typos in the problem. I corrected them as best I could, but please make sure that I didn't change anything that you meant to write.

Your main problem is not passing the extra data for the problem. Here is my version, with typos corrected as best I could and with data passed correctly:

function [f] = ErrorFunction(x,raw_signal_ref,raw_signal_sam,raw_time_ref)

% n_ave = x(1)
% gamma = x(2)
% delta = x(3)
% length = x(4)

c = 299792458; % Speed of light in m/s
n_air = 1.000293; % Index of refraction of air 
k = 1e-12; % ps to s coversion
cor_fac = 0.04;

M = 4;
fp_signal = 0.0; %fp_signal is E_FP(t)
for k=1:M
  fp_signal = fp_signal +...
      ((x(1)-1)/(x(1)+1))^(2*k)*...
      x(2)^(2*k)*...
      raw_signal_ref*k.*... % raw_signal_ref is E_reference(t)
      (raw_time_ref*(1e-12)-(x(1)*2*x(4)*M/c)-((x(1)-n_air)*x(4)/c))/...
      x(3)^(2*M+1);
end

sim_signal_sam = cor_fac*(4*x(1))/(x(1)+1)^2*x(2)*...
    ((raw_signal_ref*k.*((raw_time_ref*(1e-12)-(x(1)-1)*(x(4)*(1e-3)/c))/x(3))) + fp_signal); % raw_time_ref = t 

f = sum(sim_signal_sam - raw_signal_sam); %raw_signal_sam is E_sample_(t)

Here is how I called that function after placing x0 in the workspace:

fun = @(x)ErrorFunction(x,raw_signal_ref,raw_signal_sam,raw_time_ref);
x_optimized = fminsearch(fun,x0);

To learn about passing extra parameters this way, see Passing Extra Parameters.

When I run the problem fminsearch exits with this message:

Exiting: Maximum number of function evaluations has been exceeded

- increase MaxFunEvals option.

Current function value: -907777101105805751482258996129483335894703616332083863052416498711493108878399830640514887677380554503199361365622832602682026028412939674686787717791027291595548412048643393197842340492625295522296145364560679359874843053719749781643093886750418805308954509312.000000

So I am not sure that my corrections were complete or correct, but at least it now runs.

Alan Weiss

MATLAB mathematical toolbox documentation

[Alan Weiss]

Come to think of it, perhaps the last line should be

f = sum((sim_signal_sam - raw_signal_sam).^2); %raw_signal_sam is E_sample_(t)

Alan Weiss

MATLAB mathematical toolbox documentation

[Mark Justine Zapanta]

Dear Dr. Weiss,

Thanks for the help! The script now works after your suggested improvements. Indeed, better result is obtained when the error is squared.

Kind regards,

Mark