# State of Health Estimation for Lithium-Ion Batteries Based on TCN-RVM
## 2 Materials and Methods
### Extraction and Optimization of Health Indicators
#### 2.1.2 Extraction of Health Factors
The Gaussian filter is used in this study to smooth the original capacity-voltage data, and its one-dimensional discrete form is  
\[\widetilde{Q}(V_{i})=\sum_{j=-k}^{k}\frac{1}{\sqrt{2\pi\sigma}}\text{exp}( -\frac{j^{2}}{2\sigma^{2}})\cdot Q\big{(}V_{i+j}\big{)} \tag{4}\]  
After filtering, the central difference method is used to calculate the incremental capacity curve, thereby effectively suppressing the interference caused by voltage noise during the differentiation process and improving the smoothness and stability of the characteristic curve. Figure 2 shows the comparison of the IC curve before and after \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Battery** & **HF1** & **HF2** & **HF3** & **HF4** & **HF5** \\ \hline CS2 36 & 4.7475 & 4.5690 & 3.9298 & 4.7200 & 4.5343 \\ CS2\_37 & 4.6151 & 4.2351 & 4.2102 & 4.6628 & 4.4311 \\ CS2\_38 & 4.5351 & 4.1414 & 4.0741 & 4.7226 & 4.4662 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mutual Information. filtering. It can be observed that the original IC curve fluctuates sharply due to noise interference, making it difficult to identify key features. In contrast, after Gaussian filter (GF) processing, the curve becomes significantly smoother, with high-frequency noise interference effectively suppressed, and the feature identifiability is significantly improved, which verifies the effectiveness of this method in denoising IC curves. Figure 3 shows the IC curves of the CS2-35 battery for different cycles. It can be observed that the IC curve changes significantly when the voltage is between 3.8 V and 4.1 V. Due to the loss of active materials in the lithium-ion battery and the increase in internal resistance caused by electrolyte decomposition, the peak height decreases with the increase in the number of cycles, and the peak position shifts. Therefore, the peak value and peak position of the IC curve reflect the aging process of the battery and can be used as features for the SOH evaluation. Figure 3: IC curves of the CS2-35 cell at different cycle numbers. Figure 2: Comparison of the CS2-35 battery before and after Gaussian filtering processing. To further verify the degree of correlation between the extracted health factors and SOH, Pearson and Spearman correlation analysis methods are adopted as evaluation methods. The specific calculation formulas are as follows:  
\[Pearson=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^{2})-E^{2}(X)}\sqrt{E(Y^{2})-E^{2}(Y)}} \tag{5}\]  
\[Spearman=\frac{\sum_{i=1}^{n}{(x_{i}-\overline{x})\left(y_{i}-\overline{y} \right)}}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_ {i}-\overline{y})^{2}}} \tag{6}\]  
In the formula, the correlation coefficient range is \(\pm 1\), and the closer its absolute value is to 1, the stronger the correlation. Figure 4 shows the normalization results for the five health indicators for battery CS2_36, where HF1 is the peak height, HF2 is the peak position, HF3 is the area to the left of the peak, HF4 is the area to the right of the peak, and HF5 is the total peak area. Table 2 presents the Pearson correlation coefficients and Spearman correlation coefficients between the five health indicators and SOH. It can be seen from the data in Table 2 that the absolute values of the Pearson coefficients and Spearman coefficients of the health factors extracted in this paper are relatively high, indicating strong correlations with the SOH of lithium-ion batteries.