Error when fitting complex data with leasqr algorithm in Octave

Question

I have data as real and imaginary parts of complex number, and I want to fit them according to a complex function. More in detail, they are data from electrochemical impedance spectroscopy (EIS) experiments, and the function comes from an equivalent circuit. I am using Octave 7.2.0 in a Windows 10 computer. I need to use the leasqr algorithm, present in the Optim package. The leasqr used the Levenberg-Marquardt nonlinear regression, typical in EIS data fitting. Regarding the data, xdata are linear frequency, y data are ReZ + j*ImZ. If I try to fit the complex data with the complex fitting function, I get the following error:

error: weighted residuals are not real
error: called from
    __lm_svd__ at line 147 column 20
    leasqr at line 662 column 25
    Code_for_StackOverflow at line 47 column 73

I tried to fit the real part of the data with the real part of the fitting function, and the imaginary parts of the data, with the imaginary part of the function. The fits are successfully performed, but I have two sets of fitted parameters, while I need only one set. Here the code I wrote.

clear -a;
clf;
clc;
pkg load optim;
pkg load symbolic;

Linear_freq = [1051.432, 394.2871, 112.6535, 39.42871, 11.59668, 3.458659, 1.065641, 0.3258571, 0.1000221];
ReZ = [84.10412, 102.0962, 178.8031, 283.0663, 366.7088, 431.3653, 514.4105, 650.5853, 895.9588];
MinusImZ = [27.84804, 59.56786, 116.5972, 123.2293, 102.6806, 117.4836, 178.1147, 306.256, 551.2337];
Z = [88.5946744, 118.2030626, 213.4606653, 308.7264008, 380.8131426, 447.0776424, 544.3739605, 719.0646495, 1051.950932];
MinusPhase = [18.32042302, 30.26135402, 33.1083029, 23.52528583, 15.64255593, 15.23515301, 19.09841797, 25.2082044, 31.60167787];
ImZ = -MinusImZ;
Angular_freq = 2*pi*Linear_freq;
xdata = Angular_freq;
ydata = ReZ + j*ImZ; 

Fitting_Function = @(xdata, p) (p(1) + ((p(2) + (1./(p(3)*(j*xdata).^0.5))).^-1 + (1./(p(4)*(j*xdata).^p(5))).^-1).^-1);

p = [80, 300, 6.63E-3, 5E-5, 0.8]; # True parameters values, taken with a dedicated software: 76, 283, 1.63E-3, 1.5E-5, 0.876
options.fract_prec = [0.0005, 0.0005, 0.0005, 0.0005, 0.0005].';
niter=400;
tol=1E-12;
dFdp="dfdp";
dp=1E-9*ones(size(p));
wt = abs(sqrt(ydata).^-1);
#[Fitted_Parameters_ReZ pfit_real cvg_real iter_real corp_real covp_real covr_real stdresid_real z_real r2_real] = leasqr(xdata, ReZ, p, Fitting_Function_ReZ, tol, niter, wt, dp, dFdp, options);
#[Fitted_Parameters_ImZ pfit_imag cvg_imag iter_imag corp_imag covp_imag covr_imag stdresid_imag z_imag r2_imag] = leasqr(xdata, ImZ, p, Fitting_Function_ImZ, tol, niter, wt, dp, dFdp, options);
[Fitted_Parameters pfit cvg iter corp covp covr stdresid z r2] = leasqr(xdata, ydata, p, Fitting_Function, tol, niter, wt, dp, dFdp, options);

#########################################################################
# Calculate the fitted functions, with the fitted paramteres array
#########################################################################
Fitted_Function_Real = real(pfit_real(1) + ((pfit_real(2) + (1./(pfit_real(3)*(j*xdata).^0.5))).^-1 + (1./(pfit_real(4)*(j*xdata).^pfit_real(5))).^-1).^-1);
Fitted_Function_Imag = imag(pfit_imag(1) + ((pfit_imag(2) + (1./(pfit_imag(3)*(j*xdata).^0.5))).^-1 + (1./(pfit_imag(4)*(j*xdata).^pfit_imag(5))).^-1).^-1);
Fitted_Function = Fitted_Function_Real + j.*Fitted_Function_Imag;
Fitted_Function_Mod = abs(Fitted_Function);
Fitted_Function_Phase = (-(angle(Fitted_Function))*(180./pi));

################################################################################
# Calculate the residuals, from https://iopscience.iop.org/article/10.1149/1.2044210

# An optimum fit is obtained when the residuals are spread randomly around the log Omega axis.
# When the residuals show a systematic deviation from the horizontal axis, e.g., by forming
# a "trace" around, above, or below the log co axis, the complex nonlinear least squares (CNLS) fit is not adequate.
################################################################################
Residuals_Real = (ReZ-Fitted_Function_Real)./Fitted_Function_Mod;
Residuals_Imag = (ImZ-Fitted_Function_Imag)./Fitted_Function_Mod;

################################################################################
# Calculate the chi-squared - reduced value, with the fitted paramteres array NOVA manual page 452
################################################################################
chi_squared_ReZ = sum(((ReZ-Fitted_Function_Real).^2)./Z.^2)
chi_squared_ImZ = sum(((ImZ-Fitted_Function_Imag).^2)./Z.^2)
Pseudo_chi_squared = sum((((ReZ-Fitted_Function_Real).^2)+((ImZ-Fitted_Function_Imag).^2))./Z.^2)

disp('The values of the parameters after the fit of the real function are '), disp(pfit_real);
disp('The values of the parameters after the fit of the imaginary function are '), disp(pfit_imag);
disp("R^2, the coefficient of multiple determination, intercept form (not suitable for non-real residuals) is "), disp(r2_real), disp(r2_imag);

###################################################
## PLOT Data and the Function
###################################################
#Set plot parameters
set(0, "defaultlinelinewidth", 1);
set(0, "defaulttextfontname", "Verdana");
set(0, "defaulttextfontsize", 20);
set(0, "DefaultAxesFontName", "Verdana");
set(0, 'DefaultAxesFontSize', 12);

figure(1);
## Nyquist plot (Argand diagram)
subplot(1,2,1, "align");
plot((ReZ), (MinusImZ), "o", "markersize", 2, (Fitted_Function_Real), -(Fitted_Function_Imag), "-k");
axis ("square");
grid on;
daspect([1 1 2]);
title ('Nyquist Plot - Argand Diagram');
xlabel ('Z'' / \Omega' , 'interpreter', 'tex');
ylabel ('-Z'''' / \Omega', 'interpreter', 'tex');

## Bode Modulus
subplot (2, 2, 2);
loglog((Linear_freq), (Z), "o", "markersize", 2, (Linear_freq), (Fitted_Function_Mod), "-k");
grid on;
title ('Bode Plot - Modulus');
xlabel ('\nu (Hz)' , 'interpreter', 'tex');
ylabel ('|Z| / \Omega', 'interpreter', 'tex');

## Bode Phase
subplot (2, 2, 4);
semilogx((Linear_freq), (MinusPhase), "o", "markersize", 2, (Linear_freq), (Fitted_Function_Phase), "-k");
set(gca,'YTick',0:10:90);
grid on;
title ('Bode Plot - Phase');
xlabel ('\nu (Hz)' , 'interpreter', 'tex');
ylabel ('-\theta (°)', 'interpreter', 'tex');

figure(2)
## Bode Z'
subplot (2, 1, 1);
semilogx((Linear_freq), (ReZ), "o", "markersize", 2, (Linear_freq), (Fitted_Function_Real), "-k");
grid on;
title ('Bode Plot Z''');
xlabel ('\nu (Hz)' , 'interpreter', 'tex');
ylabel ('Z'' / \Omega', 'interpreter', 'tex');

## Bode -Z''
subplot (2, 1, 2);
#subplot (2, 2, 4);
semilogx((Linear_freq), (MinusImZ), "o", "markersize", 2, (Linear_freq), -(Fitted_Function_Imag), "-k");
grid on;
title ('Bode Plot -Z''''');
xlabel ('\nu (Hz)' , 'interpreter', 'tex');
ylabel ('-Z'''' / \Omega', 'interpreter', 'tex');

figure(3)
## Residuals Real
subplot (2, 1, 1);
semilogx((Angular_freq), (Residuals_Real), "-o", "markersize", 2);
grid on;
title ('Residuals Real');
xlabel ('\omega (Hz)' , 'interpreter', 'tex');
ylabel ('\Delta_{re} / \Omega', 'interpreter', 'tex');

## Residuals Imaginary
subplot (2, 1, 2);
#subplot (2, 2, 4);
semilogx((Angular_freq), (Residuals_Imag), "-o", "markersize", 2);
grid on;
title ('Residuals Imaginary');
xlabel ('\omega (Hz)' , 'interpreter', 'tex');
ylabel ('\Delta_{im} / \Omega', 'interpreter', 'tex');

Octave should be able to handle complex numbers. What do I do wrong? I was thinking to fit the real part of the data with the real part of the fitting function, and then using the Kramers-Kronig relations to get the imaginary part of the fitted function, but I would like to avoid this method, if possible.

Any help would be greatly appreciated, thanks in advance.

Answer 1

From your data drawing the complex impedances diagram makes appear a rather common shape that can be model with a lot of equivalent circuits :

Reference : https://fr.scribd.com/doc/71923015/The-Phasance-Concept

You chose the model n°2 probably according to some physical considerations. This is not the subject to be discussed here.

Also according to physical consideration and/or by graphical inspection you correctly assumed that one phasance is of Warbourg kind (Phi=-pi/4 ; nu=-1/2).

The problem is to fit an equation with five adjustable parameters. This is a difficult problem of non linear regression of a complex equation. The usual method consists in an iterative process starting from "guessed values" of the five parameters.

The "guessed values" have to be not far from the unknown correct values. One can find some approximates from graphical inspection of the impedances diagram. Often this is a cause of failure of convergence of the iterative process.

A more reliable method consists in using a combination of linear regression wrt most of the parameters and non-linear regression wrt only few parameters.

In the present case it is shown below that the nonlinear regression can be reduced to only one parameter while the other parameters can be handled by a simple linear regression. This is a big simplification.

A software for mixed linear and nonlinear regression (in cases involving several phasors) was developed in years 1980-1990. Infortunately I have no access to it presently.

Nevertheless in the present case of one phasor only we don't need a sledgehammer to crack a nut. The Newton-Raphson method is sufficient. Graphical inspection gives a rough approximate of (nu) between -0.7 and -0.8 The chosen initial value is nu=-0.75 giving the next first run :

Since all calculus are carried out in complex numbers the resulting values are complex instead of real as expected. They are noted ZR1, ZR2, ZP1, ZP2 to distinguish from real R1, R2, P1, P2. This is because the value of (nu) isn't optimal.

The more (nu) converges to the final value the more the imaginary parts vanishes. After a few runs of the Newton-Raphson process the imaginary parts become quite negligible. The final result is shown below.

Publications :

"Contribution à l'interprétation de certaines mesures d'impédances". 2-ième Forum sur les Imédances Electrochimiques, 28-29 octobre 1987.

"Calcul de réseau électriques équivalents à partir de mesures d'impédances". 3-ième Forum sur les Imédances Electrochimiques, 24 novembre 1988.

"Synthèse de circuits électiques équivalents à partir de mesures d'impédances complexes". 5-ième Forum sur les Imédances Electrochimiques, 28 novembre 1991.