fminsearch gives weird zick zack solution

So, I wanna implement a parameter fit to fit meassurement data to an equation (specifically, I wanna fit data to the "improved generalised steinmetz equation", which you can see here. The average over time period T from eq 1.11 is ignored, since I want the current value).

For a reason I don't understand, the solution fminsearch (and also lsqcurvefit) delivers is just completely off and goes zick-zack through my meassured data. Can someone of you maybe spot the flaw? In my mind it has to be somewhere in the implementation of the 2 equations, which are linked above too, since lsqcurvefit and fminsearch both deliver the same bs.

EDIT: Here is a screenshot of the plot:

Here is my code: At the beginning, I import data (B, P) from a file (which is not included below):

B_pp = max(B) - min(B); % peak-to-peak flux density

T = 1./f;

dt = T ./ 1000;

t = 0:dt:T;

k1 = @(x) x(1)./( (2*pi).^(x(2)-1) .* ...

integral(@(theta) abs(cos(theta)).^x(2) .* 2.^(x(3)-x(2)), 0, 2*pi) );

pv = @(x) k1(x) .* abs(gradient(B, t(2)-t(1))).^x(2) .* B_pp.^(x(3)-x(2));

x = fminsearch(@(x) vecnorm(pv(x) - P), [1,1,1]);

%[x, resnorm, residual] = lsqcurvefit(pv, [0,0,0], B, P, [-Inf, -Inf, -Inf],[Inf, Inf, Inf],opts);

figure; grid on;

plot(B, P, 'bo', B, pv(x), 'r-');

xlabel('Magnetic field B [T]'); ylabel('Power loss p [W/m^3]');

legend('Experimental data', 'Fitted function'); title('Experimental vs. fitted');

[Torsten]

Could you include your fitting function in a mathematical notation ?

[sbr]

Those 2 equations are implemented, but without the in eq. 1.11:

[Torsten]

And you think you can just take the value of the integrand

pv = @(x) k1(x) .* abs(gradient(B, t(2)-t(1))).^x(2) .* B_pp.^(x(3)-x(2));

instead of the time-averaged integrand as done in (1.11) ?

[sbr]

Honestly, I am pretty sure this should work. The code above is from the "improved generalised steinmetz equation"-model, as the name says it is an improvement to the "generalised steinmetz equation", which has the following equations:

I implemented 1.8 and 1.9 and the fitting algorithm works perfectly fine. The main difference to the problematic equations is, that in eq. (1.11) the peak-to-peak value is used, instead of the absolute value in eq. (1.8). (1.10) is again the mean value, like in (1.11).

[Torsten]

My guess is that you are overfitting with the model you use.

k1(x)*B_pp^(x(3)-x(2)) is (almost) independ of your measurement data. Thus you can use only one constant to reflect this expression.

A second constant (in your code: x(2)) can be used for the exponent in the expression abs(gradient(B, t(2)-t(1))).^x(2).

But you use three free constants instead of two in your model.

I suggest you make a test with a model of the form

pv = @(x) x(1) * abs(gradient(B, t(2)-t(1))).^x(2)

and see if it works better.

[sbr]

Funny enough, it gives exactly the same curve:

[Torsten]

Could you plot abs(gradient(B,t(2)-t(1))) over t ?

[sbr]

Sure. It is more or less exactly the fitted function without a weight (same zick-zack, different amplitudes).

[Torsten]

And by multiplication by a constant and an exponentiation you expect a smooth curve like the blue one ? No chance. I think you have to reconsider if delta B in your integral really means max(B) - min(B). I doubt it. If it wouldn't depend on t as your setting implies , it could have been taken out of the integral.

Answer 1

[Matt J]

It seems unlikely to me that x0=[1,1,1] would be an accurate initial guess. Surely, you chose that arbitrarily. Since you only have 3 unknown parameters, it shouldn't be too computationally expensive to do a discrete grid search for a better initial point.

[sbr]

Normal values for or in this case x(1), x(2), x(3) are [10, 1.5, 3], but setting the initial guess to that doesn't improve the result sadly.

[Matt J]

Then your model (as coded currently) may be inconsistent with your data. But that is an additional reason a grid search could help you. If you get poor agreement even after a grid search, you know you have bug in your model function code.