# Machine Learning-Based Lithium Battery State of Health Prediction Research
## 2 Review of Related Algorithms
### Review of the SVR Algorithm  
SVR is an extension of the Support Vector Machine (SVM) and is specifically designed to address regression problems. It achieves data fitting by introducing the \(\epsilon\)-insensitive loss function and utilizes kernel functions to map nonlinear problems into a high-dimensional feature space, thereby capturing complex nonlinear relationships.  
The objective of SVR is to find a function \(f(x)=w\cdot x+b\) that fits the data within an error margin of \(\epsilon\) while maintaining a simple model with strong generalization capability. The key steps of SVR are as follows:  
(1) \(\epsilon\)-insensitive loss function: SVR introduces the \(\epsilon\)-insensitive loss function to define an allowable error margin \(\epsilon\). If the error between the predicted value and the actual value exceeds \(\epsilon\), a loss is incurred; otherwise, the error is considered negligible.  
(2) Optimization objective function: The goal of SVR is to balance the complexity of the model (achieved by regularizing the weights w) and the prediction error (measured by the loss function). The objective function is as follows:  
\[\min\left\{\frac{1}{2}\|w\|^{2}+c\cdot\sum_{i=1}^{s}\left(\xi_{i}+\xi_{i}^{*} \right)\right\} \tag{8}\]\[s.t\left\{\begin{array}{l}w\cdot x_{i}+b-y_{i}\leq\varepsilon+\xi_{i}\\ y_{i}-w\cdot x_{i}-b\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i}\geq 0,\xi_{i}^{*}\geq 0\end{array}\right. \tag{9}\]  
where \(\frac{1}{2}\left\|w\right\|^{2}\) is the regularity term, which controls the model complexity; \(c\) is the penalty factor, which controls the trade-off between the error term and the model complexity; and \(\xi_{i}\) and \(\xi_{i}^{*}\) are the relaxation factors, which are used to deal with sample points with errors exceeding \(\varepsilon\).  
(3) Convex optimization: SVR transforms the regression problem into a convex optimization problem. By leveraging the dual form and Lagrange multipliers, SVR enhances its ability to fit nonlinear data effectively.  
(4) Kernel function: To handle nonlinear problems, SVR employs kernel functions to map low-dimensional data into a high-dimensional feature space, where linear regression can be performed. Common kernel functions include the radial basis function (RBF), linear kernel, and polynomial kernel.  
SVR aims to control prediction errors within an acceptable range, making it particularly suitable for precise modeling of battery SOH. Additionally, SVR exhibits strong generalization performance even under small sample conditions, effectively addressing the limitation of limited experimental data for lithium-ion batteries.  
By summarizing relevant algorithms, it becomes evident that the LSTM model excels in handling time-series data, effectively capturing the complex temporal dynamics inherent in the data. Since SOH prediction for lithium-ion batteries is a typical time-series problem, LSTM leverages its memory capabilities to effectively track the long-term degradation of battery performance. The CNN model, on the other hand, is proficient in extracting local features efficiently, enabling it to identify critical characteristics from lithium battery parameter data while demonstrating strong noise tolerance. Lastly, the SVR model, through kernel functions (e.g., RBF kernel), effectively handles the complex nonlinear relationships in battery SOH and exhibits robust generalization capability with reduced risk of overfitting. Based on these considerations, this study selects LSTM, CNN, and SVR models to comprehensively address the challenges of lithium battery SOH prediction from three perspectives: long-term temporal dependency, local feature extraction, and nonlinear modeling.