Nonlinear least square minimization using 2 variables

Hello,

I am trying to fit computed data (dIdV) with experimental data (STSavg). I would like write a function where A and B is varied such that the sum of the squares of the function (f) is minimized. Ideally, I would be able to see the minimum value and also the values for A and B.

f = ((STSavg - B)/ A - 10^10*dIdV)^2

%STSavg is a 300x1 array. (Values ~ 10^-1) 
%dIdV is also a 300x1 array. (Values ~ 10^-9) 
%dIdV by a factor of 10^10, to have comparable magnitude as STSavg

I have written this in Mathematica, but would like to convert it to Matlab. Here is the Mathematica script, which may provide an idea of what my goal is:

1. Minimize X^2 between STS and dI/dV, assuming linear transformation. 
2. Cut off points near edge, as they may be noisy.
cutpts = 10;
OM = FindMinimum[Total[((STSavg2[[cutpts ;; TotPts - cutpts]] - B)/A - 10^10 didv[[cutpts ;; TotPts - cutpts]])^2],{{A, 0.1}, {B, 1}}]
OFFSET = B /. OM[[2]] ,  MAGNITUDE = 10^10 A /. OM[[2]]
Output: {639.834, {A -> 0.0389278, B -> -0.174553}}
Output: OFFSET = -0.174553
Output: MAGNITUDE = 3.89278 * 10^8

I've read about "fminsearch" and using anonymous functions, but I'm slightly confused as to what the input arguments should be when creating the function for two variables (A and B).

Thanks, Chris

Answer 1

[Matt J]

Since it's a linear transformation, it is as simple as

   sol=[10^10*dIdV, ones(300,1)]\STSavg;

   A=sol(1); B=sol(2);

[Christian Greenhill]

Matt, Are you suggesting an alternative method than minimizing the sum of the squares? Here is my script, based somewhat on your suggestions. A and B are the input variables.

function b = two_var(sol)

global dIdV 
global STSavg

A= sol(1); B = sol(2);

a = (STSavg-A)/(B-10^10*dIdV); 
b = sum(a).^2;

end

My error reads that there aren't enough input arguments. This function was built based on Optimizing Nonlinear Functions

In a separate script, I hope to execute the function as follows:

sol = [0.1, 1];
fminsearch(@two_var,sol)

[Matt J]

The least squares problem you have posed can be solved analytically. It doesn't require fminsearch or any other iterative solver. What I posted for you above constitutes a complete solution.

The solution I posted for you actually minimizes

f = ( A* 10^10*dIdV+B - STSavg )^2

This is slightly different from your original formulation, but better in my opinion. It avoids error-in-variable issues by applying the linear transformation to the theoretical data instead of the empirical data.

If you really insist on your original formulation, you can also do that analytically as follows,

     sol=[STSavg, ones(300,1)]\(10^10*dIdV);
     A=1./sol(1); B=-sol(2)*A;

[Christian Greenhill]

It works! Thanks Matt!