# State of Health Estimation for Lithium-Ion Batteries Based on TCN-RVM
## 2 Materials and Methods
### Causal Convolution
#### 2.2.2 Relevance Vector Machine  
The Relevance Vector Machine (RVM) is a probabilistic machine learning model based on Bayesian principles. It employs a Bayesian sparse prior and evidence maximization to automatically identify and retain only those training samples that make significant contributions to the prediction. Suppose the training dataset is given by \(\{x_{n,t_{n}}\}_{n-1}^{N}\), where \(x_{n}\) denotes the input vector with \(x_{i}\in R^{n},\ t_{n}\) is the corresponding output with \(t_{n}\in R\), and \(N\) is the total number of training samples. The relationship between \(t_{n}\) and \(x_{n}\) is defined as follows:  
\[t_{n}=y(x_{n};\omega)+\xi_{n} \tag{9}\]  
\[y(x_{n};\omega)=\sum_{n=1}^{N}w_{n}K(x,x_{n})+\omega_{0} \tag{10}\]  
In the above equation, \(\omega\) represents the weight vector, \(\omega=(\omega_{0},\omega_{1},\cdot\cdot\cdot,\omega_{N})^{T}\); \(\xi_{n}\) denotes the error term, which follows a Gaussian distribution with a mean of 0 and variance \(\sigma\); and \(K\) is the kernel function.  
According to Bayesian inference, assuming that \(t_{n}\) is independently distributed, the likelihood function can be expressed as follows:  
\[p\Big{(}t|\omega,\sigma^{2}\Big{)}=\Big{(}2\pi\sigma^{2}\Big{)}^{-N/2}\mathrm{ exp}\Big{\{}-\|t-\Phi\omega\|^{2}/\Big{(}2\sigma^{2}\Big{)}\Big{\}} \tag{11}\]  
When directly optimizing the model weights \(\omega\) using the maximum likelihood method, the absence of regularization constraints often leads to severe overfitting. To address this issue, the weights are assigned hyperparameters. [PERSON] modeled the weight parameters \(\omega\) as Gaussian random variables controlled by hyperparameters and assigned them a zero-mean prior probability distribution:  
\[p\Big{(}t|\omega,\sigma^{2}\Big{)}=\prod_{0}^{N}N\Big{(}\omega_{i}\Big{|}0, \alpha_{i}^{-1}\Big{)}=\prod_{0}^{N}\frac{\alpha_{i}}{\sqrt{2\pi}}\mathrm{ exp}\Bigg{(}\frac{\omega_{i}^{2}a_{i}}{2}\Bigg{)} \tag{12}\]  
Figure 6: Dilated convolution.  
Each weight \(\omega\) is associated with an independent hyperparameter \(\alpha\), resulting in a total of N + 1 hyperparameters. During the computation process, these hyperparameters essentially function as regulatory gates for the weights--their magnitudes directly determine the strength of the Gaussian prior's constraint on the corresponding weight parameters. This is the primary reason for the sparsity of the probabilistic model. Therefore, determining appropriate hyperparameters through computation and thereby obtaining the corresponding weight distribution and regression function are critical aspects of ensuring the effectiveness of the RVM.  
[PERSON] proposed a Gamma distribution for both the hyperparameters \(\alpha\) and the noise variance \(\sigma^{2}\):  
\[p(\alpha)=\prod_{i=0}^{N}Gamma(\alpha_{i}|a,b) \tag{13}\]  
\[p\Big{(}\sigma^{2}\Big{)}=\prod_{i=0}^{N}Gamma(\beta|c,d) \tag{14}\]  
Specifically, the Gamma distribution is given by \(Gamma(\alpha_{i}|a,b)=\Gamma(\alpha)^{-1}b^{\alpha}\alpha^{a-1}e^{-ab}\). To ensure that this prior distribution is non-informative, it is common practice to set the parameters to very small values, such as \(a=b=c=d=10^{-4}\).