Least square surface fitting from first principles - matlab

Question

I want to make this "by hand" rather than using a surface fitting tool, because depending on the data I have, the surface fitting may vary. So, I first read the data in an excel sheet, then initialize some coefficients, calculate a 3D surface (f(x,y)) and then calculate the total least squares sum, which I'd like to minimise. Every time I run the script it tells me that I'm at the local minimum, even when I change the initial values. Changing the tolerance doesn't affect the result either.
This is the code:
% flow function in a separate .m file (approximation, it’s a negative paraboloid, maybe if required, this function may vary):
function Q = flow(P1,P2,a,b,c,d,e,f)
Q1 = a-b.*P1-c.*P1.^2;
Q2 = d-e.*P2-f.*P2.^2;
Q = Q1 + Q2;
% Variable read, I use a xlsread instead
p1a = [-5, -5, -5, -5, -5, -5, -5, -5, -5, -5];
p2a = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9];
qa = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1];
p1b = [-6, -6, -6, -6, -6, -6, -6, -6, -6, -6];
p2b = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9];
qb = [12, 11, 10, 9, 8, 7, 6, 5, 4, 3];
% Variable initialization
coef = [50, 1, 1, 10, 1, 1];
% Function calculation
q1a = flow(p1a,p2a,coef(1),coef(2),coef(3),coef(4),coef(5),coef(6));
q1b = flow(p1b,p2b,coef(1),coef(2),coef(3),coef(4),coef(5),coef(6));
% Least squares
LQa = (qa-q1a).^2;
LQb = (qb-q1b).^2;
Sa = sum(LQa);
Sb = sum(LQb);
St = Sa+Sb;
% Optimization (minimize the least squares sum)
func = #(coef)(St);
init = coef;
opt = optimoptions('fminunc', 'Algorithm', 'quasi-newton', 'Display', 'iter','TolX', 1e-35, 'TolFun', 1e-30);
[coefmin, Stmin] = fminunc(func, init, opt);
If you run this, you should get a result of 15546 for Stmin, but if you change the coefficients, you'll get another result, and it will also be considered as a local minimum.
What am I doing wrong?

Answer 1

The problem is that your func is just a constant. It simply returns a pre-calculated value, St, which is constant no matter what input you pass to func. Try calling func with various different inputs to test this.
Your objective function needs to contain all the calculations that got you to St. So I suggest you replace your func with a function saved in an m-file looking something like this:
function St = objectiveFunction(coef, p1a, p2a, p1b, p2b, qa, qb, q1a, q1b)
% Function calculation
q1a = flow(p1a,p2a,coef(1),coef(2),coef(3),coef(4),coef(5),coef(6));
q1b = flow(p1b,p2b,coef(1),coef(2),coef(3),coef(4),coef(5),coef(6));
% Least squares
LQa = (qa-q1a).^2;
LQb = (qb-q1b).^2;
Sa = sum(LQa);
Sb = sum(LQb);
St = Sa+Sb;
end
And then in your script call objectiveFunction using an anonymous function like this:
[coefmin, Stmin] = fminunc(#(coef)(objectiveFunction(coef, p1a, p2a, p1b, p2b, qa, qb, q1a, q1b)), init, opt);
The idea is to create an anonymous function that only takes a single parameter, coef, which is the variable that fminunc will peturb and pass back to your objective function. The other parameters that your objectiveFunction needs (i.e. p1a, p2a, p1b,...) are now considered to be pre-calculated by your anonymous function and thus by fminunc.
The rest of your code can stay the same.