## 2 Lifetime Prediction Model for Lithium-Ion Batteries  
### Least Squares Support Vector Machine(LSSVM) Principle  
Because of the good generalization ability of support vector machine (SVM), SVM has been widely used for forecasting [38]. However, the training process of SVM is a constrained quadratic programming problem. The sample size equals the number of constraints. When the sample is too large, the training process takes a long time. A linear system of equations needs to be solved in the training process of LSSVM, which improves the training efficiency of SVM to some extent. The principle of LSSVM model is as follows:  
\(Q=\big{\{}(u_{j},v_{j})\big{|}j=1,2,\ldots,n\big{\}}\) is the training sample, where \(u_{j}(u_{j}\in R^{m})\) is the input value, and \(v_{j}(v_{j}\in R)\) is the output value. The kernel function maps samples. And regression function \(\beta\), as indicated in Equation (1), is constructed in multidimensional space [39].  
\[\beta=\frac{1}{2}\mu^{T}\mu+\frac{1}{2}\delta\sum\limits_{i=1}^{n}e_{i}^{2} \tag{1}\]  
Note: \(\mu\) is the weight vector; \(\delta\) is the penalty coefficient; and \(e_{i}\) is the prediction error.  
According to the principle of structural risk minimization, the constrained optimization problem is shown in Equation (4) [40]:  
\[\left\{\begin{array}{l}\min\beta(\mu,e)=\frac{1}{2}\mu^{T}\mu+\frac{1}{2} \delta\sum\limits_{i=1}^{n}e_{i}^{2}\\ s.t.\;v_{i}=\mu^{T}\Phi(u_{i})+o+e_{i},i=1,2,3\ldots n\end{array}\right. \tag{2}\]  
where \(o\) is the offset and \(\Phi(\bullet)\) represents the mapping method.  
Based on the theory of constrained optimization, the corresponding Lagrange form can be obtained by introducing Lagrange multiplier \(g_{i}(g_{i}\geq 0)\):  
\[\psi=\frac{1}{2}\|\mu\|^{2}+\frac{1}{2}\delta\sum\limits_{i=1}^{n}e_{i}^{2}- \sum\limits_{i=1}^{n}g_{i}\big{[}\mu^{T}\Phi(u_{i})+o+e_{i}-v_{i}\big{\}} \tag{3}\]Equation (4) is obtained by the Karush-Kuhn-Tucher condition:  
\[\left\{\begin{array}{l l}\frac{\partial\psi}{\partial\mu}=0\rightarrow&\mu= \sum\limits_{i=1}^{n}g_{i}\Phi(u_{i})\\ \frac{\partial\psi}{\partial\phi}=0\rightarrow&\sum\limits_{i=1}^{n}g_{i}=0 \\ \frac{\partial\psi}{\partial\psi_{i}}=0\rightarrow&g_{i}=\delta e_{i}\\ \frac{\partial\psi}{\partial g_{i}}=0\rightarrow&\mu\Phi(u_{i})+o+e_{i}-v_{i }=0\end{array}\right. \tag{4}\]  
Equation (5) is shown as follows:  
\[\left[\begin{array}{cc}0&U\\ U^{T}&H+\delta^{-1}E\\ \end{array}\right]\left[\begin{array}{c}o\\ g\\ \end{array}\right]=\left[\begin{array}{c}0\\ v\\ \end{array}\right] \tag{5}\]  
where \(v=[v_{1},v_{2},\ldots,v_{l}]\); \(U=[1;1;\ldots;1]\); \(g\) is the Lagrange multiplier; \(E\) is the unit matrix; and \(H\) is the kernel function.  
According to the Mercer condition, the relationship between the mapping function \(\Phi(\bullet)\) and the kernel function H can be obtained using Equation (6):  
\[H(u_{i},u_{j})=\exp(-\frac{\|u_{i}-u_{j}\|}{2\theta^{2}}) \tag{6}\]  
where \(\theta\) is the Gauss kernel parameter.  
Therefore, the regression function of LSSVM is represented by Equation (7).  
\[F(u)=\sum\limits_{j=1}^{n}\delta_{j}H(u,u_{i})+o \tag{7}\]  
In LSSVM, parameters \(g\) and \(\theta\) have a great influence on the generalization ability of LSSVM. The right \(g\) and \(\theta\) can improve the prediction performance.