Fitting data to an equation with complex part

Hi,

I hope you are all well.

I have experimental data for a diaphragm displacement and want to fit it to the following equation. The experimental data is displacement (H in meters) and frequency (w in rad/s).

I have a working code but it is changes a lot with the initial guess. I need help to improve this.

x(1) is the alpha (numerator) and x(2) is the damping. omega_r is the resonant frequency and it is known. Gamma should be between 0.04 and 0.07. I am interested in gamma, the damping term in the denominator.

% non linear fitting
fun = @(x,omega)(x(1)./(omega_r^2 - omega.^2 + 1i*(2*x(2)*omega_r.*omega)));
x0 = [-35.9811 + 1i*23.8154,0.06];
opts = optimoptions(@lsqcurvefit,'Display','off','Algorithm','trust-region-reflective');
[vestimated,resnorm] = lsqcurvefit(fun,x0,omega,H,[],[],opts);

Looking forward to your suggestions.

Best regards,

Baris

Edit: x0 is changed.

[Alex Sha]

Post your data please

[Baris Gungordu]

Hi, thanks.

omega_r is 10920. The data is attached(data_v1.txt). First column is omega and second column is H.

[darova]

Can you please collaborate?

[Baris Gungordu]

omega_r is 10920 as per my previous post. Please let me know if you need something further. Equation that I want to fit is also given above.

[J. Alex Lee]

Before thinking about the optimization code, do you understand your problem well? For example you can analyze how the norm of your model-to-data residual changes with your parameters - luckily you only have 2 parameters so you can visualize it very easily in a contour plot, c.f.

produced by code below. It doesn't appear to me that gamma should be bounded by based on this plot...

data = readmatrix(datapath);

omega = data(:,1);
H     = data(:,2);

omega_r = 10920;
fun = @(x,omega)(x(1)./(omega_r^2 - omega.^2 + 1i*(2*x(2)*omega_r.*omega)));

x0 = [-1.5780e+04,0.05];

N = 100;
M = 100;

alph = x0(1) + 1e5*linspace(-1,1,N);
gamm = x0(2) + 1e-1*linspace(-1,1,M);

[X1,X2] = ndgrid(alph,gamm);
obj = nan(size(X1));
for k = 1:numel(X1)
	obj(k) = localobj([X1(k),X2(k)] , fun, H,omega);
end

figure(1); cla; hold on;
contourf(X1,X2,obj,logspace(log10(min(obj(:))),log10(max(obj(:))),10))
plot(x0(1),x0(2),'o','MarkerFaceColor','w','LineWidth',2)
set(gca,'ColorScale','log')
colorbar

% opts = optimoptions(@lsqcurvefit,'Display','off','Algorithm','trust-region-reflective');
% [vestimated,resnorm] = lsqcurvefit(fun,x0,omega,H,[],[],opts);

function obj = localobj(x , fun, H,omega)

res = localres(x , fun, H,omega);

obj = norm(res);

end

function res = localres(x , fun, H,omega)

Hmdl = fun(x,omega);

res = H - Hmdl;

end

[Baris Gungordu]

Hi J. Alex,

Thanks for this. This is the first time I am studying this equation. The damping of the diaphragm is definitely between 0.03 and 0.08. Damping does not have units but expressed in %. For example 0.06 means 6% of damping. I expect it to be a positive complex number.

I had a look at the initial conditions again:

If damping for instance is taken as 0.06 the alpha should be -35.9811 + i (23.8154) for omega = 1.0933E4 (1740 Hz)

Best regards

[J. Alex Lee]

Hi Baris,

Sorry, I missed that alpha is complex, somehow i thought only the model was complex. You could try the above exercise with the real and imaginary parts of alpha, if you have reason to believe gamma can be fixed to some value, or do it for several values of gamma within your bounds

[Alex Sha]

if parameter x1 and x2 are all complex-type values, then there will be reults below:

Root of Mean Square Error (RMSE): 1.17407781380902E-7
Sum of Squared Residual: 4.54891375249931E-12
Correlation Coef. (R): 0.999813998640161
R-Square: 0.999628031876828
Adjusted R-Square: 0.999625756842435
Determination Coef. (DC): 0.999625508902353
F-Statistic: 290981.048034672

Parameter          	Best Estimate
--------------------	-------------
x1.realpart	-386.462851740156
x1.imagpart	416.525052408265
x2.realpart	0.031303394360693
x2.imagpart	0.0114859438407334

[Baris Gungordu]

Hi Alex,

Thanks. The values makes sense. Can you send me the code so that I can test it further?

Best regards

[Alex Sha]

Hi, Baris, rather than Matlab, the results above are obtained by using a package called "1stOpt", the code looks like:

ComplexStr = i;
ComplexPar x1,x2;
Constant omega_r = 10920;
Variable omega, H[realPart];
Function H=(x1/(omega_r^2 - omega^2 + 1*i*(2*x2*omega_r*omega)));
Data;
10737	4.6598e-05
10738	4.6863e-05
10740	4.7127e-05
10741	4.739e-05
10743	4.7654e-05
10744	4.7916e-05
10746	4.8178e-05
....

[J. Alex Lee]

I am no longer clear on the problem or objective. Baris, what do you mean gamma is a "positive complex number"? If both alpha and gamma are complex, are there any constraints that you need to ensure that H_model remains real? Or as Alex has asserted, are you chopping off the imaginary part of the H_model? Actually, even if gamma is strictly real, the second part of the question remains.

Is the issue that your solution is too sensitive to initial guess?

[Baris Gungordu]

Hi J. Alex,

Both values (gamma & alpha) can be complex. Damping is usually a complex number with real positive part. Due to the structure of the equation this implies that alpha can be complex or real.

Right now, the my code with lsqcurve fit works but it produces the results same as x0 with 'trust-region-reflective' algorithm. With 'levenberg-marquardt' its very sensitive to X0.

[Baris Gungordu]

Hi Alex Sha,

Can we have that code transferred in MATLAB by any chance? What would you think about the use of lsqcurvefit or another algorithm to match that results in MATLAB?

Answer 1

[Alex Sha]

lsqcurvefit is seneitive to initial start-values since it use local optimization algorithm, same as cftool, there is a GA toolbox with global optimization algorithm in Matlab, unfortunately, the effects of GA are not so good as expected.

[Baris Gungordu]

Well maybe MATLAB support team knows how to deal with this.

[J. Alex Lee]

I don't know that it's helpful to try to characterize the methods' sensitivities...it's really sensitivity, i.e., the nature of the optimization problem that is important. If your optimization landscape is bumpy, as Alex Sha says, methods like Levenberg and trust-region may only find local bumps, depending on how much control you have over the size of the search space. On the other hand, if you have a very smooth optimization landscape, you might be fine. On the other hand, if your optimization landscape is smooth but flat, most algorithms will have trouble because no set of parameters will look any better than the next.

This is why I recommended characterizing your specific problem to the extent possible by studying norm(res) (and its gradient) as a function of your fit parameters.

It looks like you have a 4 parameter fit (2 real parts and 2 complex parts). But depending on how the model works (since you only care about real part of H), it's possible that you're over-specifying, in which case your sensitivity might have to do with multiple combinations of your 4 parameters result in the same real part of the model.

Answer 2

[Baris Gungordu]

Hi Alex Sha,

Would you mind studying the same equation with the attached data? Your 0.03 with 1stOpt was very accurate and I want to double check a model I found with a different data.

Best regards,

Baris

[Alex Sha]

Hi, if all are as previous except the data, then the result will be:

Root of Mean Square Error (RMSE): 0.478025304038979
Sum of Squared Residual: 133.220275528809
Correlation Coef. (R): 0.994221627528188
R-Square: 0.9884766446448

Parameter          	Best Estimate
--------------------	-------------
x1.realpart	828282884.340485
x1.imagpart	-136314338.186314
x2.realpart	0.681537210289169
x2.imagpart	9.3260946979766