## 2 Data Analysis and Features Extraction
### Health Features Extraction
#### 2.2.3 Similarity Health Features Extraction  
To accurately describe the voltage curve degradation trend and its extent, the voltage data during the battery charging phase are analyzed using both time-series analysis and probability distribution dispersion. Correlation analysis techniques are used to assess the similarity between the charging voltage curve of the initial cycle and that of the current cycle, facilitating the extraction of meaningful similarity features. In this study, multiple lithium-ion batteries of the same specifications were preliminarily measured in the laboratory to obtain the initial cycle charging voltage curves. The average of these curves was then calculated to serve as the standard initial cycle charging voltage curve for this specification  
Figure 2: Voltage curves under different cycles.  
of the battery. Dynamic time warping (DTW) is a method that adjusts the alignment of data points in two-time series to obtain an optimal matching strategy; a smaller DTW value indicates a higher similarity [29]. Thereby calculating the similarity between the two datasets. The time-series data \(\text{S}_{1}=\{\text{a}_{1},\text{a}_{2},\text{a}_{3},\ldots,\text{a}_{\text{m}}\}\) and \(\text{S}_{2}=\{\text{b}_{1},\text{b}_{2},\text{b}_{3},\ldots,\text{b}_{\text{n}}\}\) are given. The goal of DTW is to construct all possible warping paths based on the two time series, forming an m \(\times\) n cost matrix R, where the element at the i-th row and j-th column, \(R_{ij}\) is defined as:  
\[R_{ij}=\sqrt{\left(a_{i}-b_{j}\right)^{2}}1\leqslant i\leqslant m,1\leqslant j \leqslant n \tag{3}\]  
The DTW distance between time-series data \(a_{i}\) and \(b_{j}\) is represented by the accumulated distance r (\(i,j\)) along the optimal warping path, which is calculated as follows:  
\[r(i,j)=R_{ij}+\min\{r(i-1,j-1),r(i,j-1),r(i-1,j)\} \tag{4}\]  
where \(r\) (\(i-1,j-1\)) represents the subsequence distance when \(\text{a}_{i-1}\) and \(\text{b}_{j-1}\) are matched; r (\(i-1,j\)) represents the subsequence distance when \(\text{a}_{i-1}\) and bj are matched; \(r\) (\(i,j-1\)) represents the subsequence distance when ai and \(\text{b}_{j-1}\) are matched.  
The DTW algorithm selects the shortest distance from the three possible split options. By iteratively applying Equation (3), the optimal alignment between the variable sequences a and b can be obtained, resulting in the DTW distance \(\text{V}_{\text{DTW}}\), \(\text{V}_{\text{DTW}}=r(m,n)\). The DTW distance can be employed to evaluate the similarity between the charging voltage curves of the initial and current cycles from a time-domain perspective. This similarity can be utilized as a health feature 5 (HF5) to represent the battery's SOH.  
To more intuitively reflect the reduction in the time required for the same voltage change as the battery's SOH decreases, we introduce the coefficient of variation, a common statistical metric, as a similarity feature for characterizing the battery SOH, denoted as health feature 6 (HF6). The coefficient of variation is a normalized metric used to assess the dispersion of a probability distribution [30]. It quantifies the extent of variability relative to the mean across different sample data, with its mathematical expression provided in Equation (5).  
\[C_{\text{var}}=\frac{\sigma}{\mu}=\frac{1}{\mu}\sqrt{\frac{\sum\limits_{i}^{n} \left(x_{i}-\mu\right)^{2}}{n}} \tag{5}\]  
where \(C_{\text{var}}\) represents the coefficient of variation, \(\sigma\) denotes the standard deviation of the sample, and \(n\) indicates the sample size.