Fitting a sum of exponentials to data (Least squares)

How do I find and given and in the model

which describes the decay of two materials. Nis the total amount of material remaining after t hours, and and is the amount of material at (B is just a background constant).

I have solved it according to this paper, however my answer is not optimal compared to what the answer is supposed to be (according to an exercise sheet).

Below is what I did, note that and and the matrix a contains the parameters wanted.

Answer in the exercise sheet: .

x=[0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0];
y=[5995 4930 3485 2550 1910 1500 1085 935 830 655 585];
p=-0.5;
q=-0.2;
G=[length(x), sum(exp(p.*x)), sum(exp(q.*x));
  sum(exp(p.*x)), sum(exp((2*p).*x)), sum(exp((p+q).*x));
  sum(exp(q.*x)), sum(exp((p+q).*x)), sum(exp((2*p).*x))];
h=[sum(y);sum(y.*exp(p.*x));sum(y.*exp(q.*x))];
a=(G)\h;

Answer 1

[Alan Stevens]

Use Matlab's best-fit matrix approach as follows:

t=[0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0];
N=[5995 4930 3485 2550 1910 1500 1085 935 830 655 585];

 p=-0.5;
 q=-0.2;
 
 E1 = exp(p*t);
 E2 = exp(q*t);
 
 % [E1 E1 1]*[N1; N2; B] = N
 M = [E1' E2' ones(numel(t),1)];

 % Least squares best-fit
 NB = M\N';
 
 N1 = NB(1);
 N2 = NB(2);
 B = NB(3);
 
 tt = 0.5:0.1:10;
 NN = N1*exp(p*tt) + N2*exp(q*tt) + B;
 
 plot(t,N,'o',tt,NN), grid
 xlabel('t'), ylabel('N')
 legend('data','fit')

This results in