Technical Implementations

This document explains the methods and calculations used in impedance analysis.

Data Quality Assessment

Lin-KK Analysis

The Lin-KK validation uses the impedancepy package, implementing the method from Schönleber et al. [1]. This implementation:

• Uses a Kramers-Kronig circuit model with ohmic resistor and RC elements

• Finds the best number of RC elements automatically

• Analyzes residuals to check data quality

• Confirms if measurements follow physical principles

Equivalent Circuit Model (ECM) Fitting

Parameter Estimation Process

The fitting process has these steps:

1. Parameter Transformation

   For bounded optimization:

   \[ \begin{align}\begin{aligned}p_{\text{int}} = \log_{10}\left(\frac{p - lb}{1 - p/ub}\right)\\p_{\text{ext}} = \frac{lb + 10^p}{1 + 10^p/ub}\end{aligned}\end{align} \]

2. Objective Function

   Using weighted residuals:

   \[\text{WRSS} = \sum_{i=1}^N \frac{(Z_{\text{exp},i} - Z_{\text{model},i})^2}{\sigma_i^2}\]

3. Optimization

   Uses BFGS algorithm with weighted residuals and parameter bounds.

Weighting Schemes

We offer three weighting options:

\[\begin{split}\sigma_i = \begin{cases} 1 & \text{for unit weighting} \\ |Z_{\text{exp},i}| & \text{for proportional weighting} \\ \sqrt{(Re(Z_{\text{exp},i}))^2 + (Im(Z_{\text{exp},i}))^2} & \text{for modulus weighting} \end{cases}\end{split}\]

Parameter Uncertainties

Calculated using QR decomposition of the weighted Jacobian:

\[\sigma_j = \|R^{-1}_j\| \sqrt{\text{WRMS}}\]

where: - R comes from QR decomposition of the weighted Jacobian - WRMS is weighted root mean square error - Jacobian elements: \(J_{ij} = \frac{\partial Z_i}{\partial p_j}\)

Correlation Analysis

Using the Hessian of the objective function:

\[C_{ij} = \frac{H^{-1}_{ij}}{\sqrt{H^{-1}_{ii}H^{-1}_{jj}}}\]

Understanding the values: - \(|r| > 0.9\): Strong correlation - \(0.7 < |r| < 0.9\): Medium correlation - \(|r| < 0.7\): Weak correlation

Fit Quality Metrics

Vector Difference Analysis

Measures point-by-point agreement:

\[\text{VD} = \frac{1}{N}\sum_{i=1}^N \frac{|Z_{\text{fit},i} - Z_{\text{exp},i}|}{|Z_{\text{exp},i}|}\]

Quality guides: - Excellent: < 0.05 (5% average deviation) - Good: < 0.10 (10% average deviation) - Poor: > 0.10

Path Following Analysis

Checks if model follows data trajectory:

\[\text{PD} = \frac{1}{N-1}\sum_{i=1}^{N-1} \left|\frac{\Delta Z_{\text{fit},i}}{|\Delta Z_{\text{fit},i}|} - \frac{\Delta Z_{\text{exp},i}}{|\Delta Z_{\text{exp},i}|}\right|\]

Quality guides: - Excellent: < 0.05 (5% path deviation) - Good: < 0.10 (10% path deviation) - Poor: > 0.10 (shows model structure issues)

Model Selection Metrics

Akaike Information Criterion (AIC)

For different weighting schemes [2]:

Unit weighting:

\[\text{AIC} = 2N\ln(2\pi) - 2N\ln(2N) + 2N + 2N\ln(\text{WRSS}) + 2k\]

Modulus/proportional weighting:

\[\text{AIC} = 2N\ln(2\pi) - 2N\ln(2N) + 2N - \sum\ln(w_i) + 2N\ln(\text{WRSS}) + 2(k+1)\]

Sigma weighting:

\[\text{AIC} = 2N\ln(2\pi) + \sum\ln(\sigma_i^2) + \text{WRSS} + 2k\]

where: - N is number of data points - k is number of model parameters - WRSS is weighted residual sum of squares

Distribution of Relaxation Times (DRT)

We use Kulikovsky’s method [3] for DRT analysis, which:

• Combines Tikhonov regularization with projected gradient method

• Handles the ill-posed nature of DRT calculations

• Ensures physically meaningful results (non-negative distribution)

• Provides fast calculations

The objective function is:

\[\text{Objective Function} = \|Z_{\text{exp}} - Z_{\text{fit}}\|^2 + \lambda \|L \gamma\|^2\]

where: - λ is regularization parameter - L is regularization operator - γ is distribution of relaxation times

Implementation Details

Optimization Algorithm

• BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm

• Bounded optimization through parameter transformation

• Automatic differentiation for gradients

Numerical Stability

• SVD for correlation matrix calculation

• QR decomposition for uncertainty estimation

• DRT regularization

• Parameter scaling

LLM Integration

The analysis workflow integrates these metrics with the LLM to: - Evaluate model validity based on path following - Guide model structure modifications - Interpret parameter correlations - Provide physically meaningful recommendations

The LLM system is structured to prioritize analysis based on: 1. Path following assessment (primary metric for ECM fits) 2. Data quality assessment (primary focus for non-ECM analysis)

References

[1]

Schönleber, M., et al. (2014). A Method for Improving the Robustness of linear Kramers-Kronig Validity Tests. Electrochimica Acta, 131, 20-27.

[2]

Ingdal, M., Johnsen, R., & Harrington, D. A. (2019). The Akaike information criterion in weighted regression of immittance data. Electrochimica Acta, 317, 648-653.

[3]

Kulikovsky, A. (2020). PEM fuel cell distribution of relaxation times: A method for calculation and behavior of oxygen transport peak. Physical Chemistry Chemical Physics, 19, 19131.

Notes

• JAX helps with automatic differentiation and fast computation

• Error calculations assume normal distribution of residuals

• CPE and Warburg elements need special attention for correlations

• DRT needs careful selection of regularization parameter