linKK not working as intended[BUG]

I tried replicating results from the Schonleber et al. paper this module is based on, but found some key differences to their results. The focus, for now, will be on Figure 3 from the paper.

To Reproduce First, I simulated a circuit for their IS1 data using parameters listed in Table 1 of the paper.

import numpy as np
import matplotlib.pyplot as plt

from impedance.circuits import CustomCircuit
from impedance.plotting import plot_nyquist
from impedance.validation import linKK

circ = 'R1-p(R2,C3)-p(R4,E5)'

vals = [100, 200, 0.8e-6, 500, 0.4e-3, 0.5]

f = np.logspace(7,-6, num=131)

IS1 = CustomCircuit(circ, initial_guess=vals)

Z_IS1 = IS1.predict(f)

fig, ax = plt.subplots()

plot_nyquist(ax, f, Z_IS1)

ax.set_ylim(0, 400)
ax.set_xlim(0, 900)
plt.show()

Compare this result: To Figure 2 in the paper:

It looks pretty close, though a quantitative comparison is difficult without their data.

Now, add noise to IS1 to get IS1a:

sigma_a = 1000
Z_IS1a = (Z_IS1.real + np.abs(Z_IS1)/sigma_a) + 1j *(Z_IS1.imag + np.abs(Z_IS1)/sigma_a)

And use manual mode of linKK to fit with M=6 (“Underfitting”) and M=104 (“Overfitting”)

IS1a_uf = linKK(f, Z_IS1a, c=None, max_M=6)
IS1a_of = linKK(f, Z_IS1a, c=None, max_M=104)

fig, ax =plt.subplots()

plot_nyquist(ax, f, Z_IS1a, fmt='.')
plot_nyquist(ax, f, IS1a_uf[2], fmt='-')
plot_nyquist(ax, f, IS1a_of[2], fmt='--')

ax.set_ylim(0, 400)
ax.set_xlim(0, 900)
ax.legend(labels=('IS1a', 'Underfitting', 'Overfitting'))
plt.show()

fig, (ax0, ax1) =plt.subplots(nrows=2, figsize=(6,6))

ax0.plot(np.log10(f), IS1a_uf[3], '-', label=r'$\Delta_{Re}$')
ax0.plot(np.log10(f), IS1a_uf[4], '--', label=r'$\Delta_{Im}$')

ax1.plot(np.log10(f), IS1a_of[3], '-', label=r'$\Delta_{Re}$')
ax1.plot(np.log10(f), IS1a_of[4], '--', label=r'$\Delta_{Im}$')

ax0.set_ylim(-.12, .12)
ax1.set_ylim(-.12, .12)
ax0.set_xlim(-6, 6)
ax1.set_xlim(-6, 6)
ax1.set_xlabel('log(f) / Hz')
ax0.locator_params(axis='y', nbins=3, tight=True)
ax1.locator_params(axis='y', nbins=3, tight=True)
ax0.locator_params(axis='x', nbins=3, tight=True)
ax1.locator_params(axis='x', nbins=3, tight=True)
ax0.legend()
ax1.legend()
ax0.grid(True)
ax1.grid(True)

plt.show()

Compared to their result:

Notably, the first visible arc in the “underfitting” case from impedance.py linKK is much larger than their figure. Further, the residuals lines have the same general shape but don’t quantitatively agree – most easily seen by comparing the residuals lines at log(f) = 0.

I believe this behavior arises from the way impedanc.py linKK calculates the distribution of time constants using linear frequencies (vs radial frequencies as used in the paper).

[mdmurbach]

Wow @BGerwe this is a fantastic write up of your exploration 🚀 thanks for working on getting this implemented correctly!!!

[mdmurbach]

sounds good! 😎