## 2 Data Analysis and Features Extraction
### Health Features Correlation Analysis and Processing
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Battery** & **HF1** & **HF2** & **HF3** & **HF4** & **HF5** & **HF6** & **HF7** & **HF8** \\ \hline CS2\_35 & \(\rho\) & \(-0.9331\) & \(-0.9567\) & \(1\) & \(0.9951\) & \(-0.8023\) & \(-0.7511\) & \(0.2074\) & \(0.2747\) \\  & \(\gamma\) & \(-0.9590\) & \(-0.9641\) & \(0.9910\) & \(0.9668\) & \(-0.7683\) & \(-0.7156\) & \(0.3648\) & \(0.5196\) \\ \hline CS2\_36 & \(\rho\) & \(-0.9274\) & \(-0.9382\) & \(1\) & \(0.9913\) & \(-0.9067\) & \(-0.7907\) & \(0.1980\) & \(0.2633\) \\  & \(\gamma\) & \(-0.9332\) & \(-0.9478\) & \(0.9950\) & \(0.9518\) & \(-0.7485\) & \(-0.7279\) & \(0.4333\) & \(0.5533\) \\ \hline CS2\_37 & \(\rho\) & \(-0.9349\) & \(-0.9468\) & \(1\) & \(0.9931\) & \(-0.7602\) & \(-0.7486\) & \(0.2358\) & \(0.2833\) \\  & \(\gamma\) & \(-0.9451\) & \(-0.9553\) & \(0.9970\) & \(0.9790\) & \(-0.7542\) & \(-0.7426\) & \(0.3349\) & \(0.4214\) \\ \hline CS2\_38 & \(\rho\) & \(-0.9278\) & \(-0.9484\) & \(1\) & \(0.9977\) & \(-0.7760\) & \(-0.7162\) & \(0.2298\) & \(0.2923\) \\  & \(\gamma\) & \(-0.9304\) & \(-0.9502\) & \(0.9946\) & \(0.9755\) & \(-0.7598\) & \(-0.7165\) & \(0.3929\) & \(0.4906\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The dual correlation coefficient of battery HFs and SOH. Figure 4: Dual correlation coefficients of SOH among four batteries. (**a**) Pearson correlation coefficient; (**b**) Spearman correlation coefficient. To remove noise and improve data smoothness, Gaussian filtering was applied to preprocess the battery health features. Additionally, the local outlier factor (LOF) algorithm was utilized to identify and correct potential outliers, thereby improving the reliability of the features [33]. For time-series data associated with the battery's health features, Gaussian filtering can be applied using the following equation:  
\[y(t)=\sum_{i=1}^{n}X(i)\cdot\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{{{ \left(t-i\right)}^{2}}}{{2\sigma^{2}}}\right) \tag{8}\]  
where \(X(i)\) represents the original health feature data, \(y(t)\) denotes the filtered data, and \(\sigma\) is the standard deviation of the Gaussian filter. Figure 6: Heatmap of the dual correlation of health features for four batteries. (**a**) CS2_35 HFs Pearson correlation; (**b**) CS2_35 HFs Spearman correlation; (**c**) CS2_36 HFs Pearson correlation; (**d**) CS2_36 HFs Spearman correlation; (**e**) CS2_37 HFs Pearson correlation; (**f**) CS2_37 HFs Spearman correlation; (**g**) CS2_38 HFs Pearson correlation; (**h**) CS2_38 HFs Spearman correlation. Despite the application of Gaussian filtering, extreme outliers may still remain in the battery state features, potentially due to measurement errors or other influencing factors. To address this issue, the LOF algorithm is employed to detect and manage anomalous data points. If LOF > 1, The LOF algorithm compares local data densities and calculates an outlier score for each point to identify those significantly deviating from the norm. Points with a higher outlier score are considered anomalies, which are effectively removed by this method. Then, point p is considered an outlier. For the outliers removed from the battery health features, Lagrange interpolation is used to fill in the missing values, mitigating their potential impact on battery health assessment. \[LOF(p)=\frac{\sum\limits_{q\in N_{k}(p)}\frac{lrd(q)}{lrd(p)}}{|N_{k}(p)|} \tag{9}\]  
where \(LOF(p)\) is the local reachability density of point p, \(N_{k}(p)\) is the k-nearest neighbor set of point p, and \(lrd\) is the local reachability density of data point.