## 3 Proposed Cross-Working Condition RUL Prediction Method
### JTMN-Based RUL Prediction
#### 3.2.1 Joint Domain Adaptation Loss  
The results demonstrated by the MMD in simulating the bearing degradation process are poor. Therefore, researchers have combined MMD with mutual information-based clustering to mitigate distribution discrepancies. [PERSON] et al. [27] investigated the working mechanism of MMD, aiming at the risk of poor distribution alignment due to the poor discrepancy representation of the mean statistic in reproducing kernel Hilbert space (RKHS) space. A new distribution alignment metric with a new Student kernel function called variance discrepancy representation (VDR) is explored to guarantee the robustness and generalization ability of VDR [28]. Let \(\mathcal{R}\) be an RKHS space. Given two data samples \(X:\{x_{k}\}_{k=1}^{m}\) and \(Y:\{y_{k}\}_{k=1}^{n}\) from source domain and target domain, the biased VDR statistic is the sum of two V-statistics and a sample average. \[\mathrm{VDR}_{b}^{2}\Big{[}\mathcal{R}_{s}\bigotimes\mathcal{R}_{t},X,Y\Big{]} =\frac{1}{m^{2}}\underset{i=1}{\overset{m}{\sum}}\underset{j=1}{ \overset{m}{\sum}}\tau\big{(}x_{i},x_{j}\big{)}+\frac{1}{n^{2}}\underset{i=1} {\overset{m}{\sum}}\underset{j=1}{\overset{m}{\sum}}\tau\big{(}y_{i},y_{j} \big{)}-\frac{2}{mn}\underset{i=1}{\overset{m}{\sum}}\underset{j=1}{ \overset{m}{\sum}}\tau\big{(}x_{i},y_{j}\big{)} \tag{11}\]  
where \(\mathcal{R}_{s}\) is the RKHS space from source domain and \(\mathcal{R}_{t}\) is the RKHS space from target domain. \(x_{k}\) is the \(k\)th sample from source domain, and \(y_{k}\) is the \(k\)th sample from target domain. \(m\) is the number of \(X\), and \(n\) is the number of \(Y\). According to \(\mu_{x}=\frac{1}{m}\underset{k=1}{\overset{m}{\sum}}\kappa\big{(}x_{k}, \cdot\big{)}\),  
\[\tau\big{(}x_{i},x_{j}\big{)} =\Big{\langle}(\kappa(x_{i},\cdot)-\mu_{x})^{\otimes 2},\big{(} \kappa\big{(}x_{j},\cdot\big{)}-\mu_{x}\big{)}^{\otimes 2}\Big{\rangle}_{ \varphi_{1}\otimes\varphi_{2}}=\big{\langle}\kappa(x_{i},\cdot)-\mu_{x},\kappa \big{(}x_{j},\cdot\big{)}-\mu_{x}\big{\rangle}_{\varphi_{1}\otimes\varphi_{2} }^{2}= \tag{12}\] \[\bigg{\{}\kappa\big{(}x_{i},x_{j}\big{)}-\frac{1}{m}\underset{l=1 }{\overset{m}{\sum}}\kappa(x_{i},x_{l})-\frac{1}{m}\underset{k=1}{\overset{m}{ \sum}}\underset{k=1}{\overset{m}{\sum}}\kappa\big{(}x_{j},x_{k}\big{)}\bigg{\}} ^{2}\]Similar to \(\tau\big{(}x_{i},x_{j}\big{)}\), \(\tau\big{(}y_{i},y_{j}\big{)}\), and \(\tau\big{(}x_{i},y_{j}\big{)}\) can be respectively represented with \(\mu_{y}=\frac{1}{n}\sum_{l=1}^{n}\kappa(y_{k}\cdot)\):  
\[\tau\big{(}y_{i},y_{j}\big{)}=\Bigg{\{}\kappa\big{(}y_{i},y_{j}\big{)}-\frac{1}{ n}\sum_{l=1}^{n}\kappa(y_{i},y_{l})-\frac{1}{n}\sum_{k=1}^{m}\kappa\big{(}x_{j},x_{ k}\big{)}-\frac{1}{n^{2}}\sum_{l=1}^{n}\sum_{k=1}^{n}\kappa(y_{l},y_{k})\Bigg{\}}^{2} \tag{13}\]  
\[\tau\big{(}x_{i},y_{j}\big{)}=\Bigg{\{}\kappa\big{(}x_{i},y_{j}\big{)}-\frac{1 }{n}\sum_{l=1}^{n}\kappa(x_{i},y_{l})-\frac{1}{m}\sum_{k=1}^{m}\kappa\big{(}x_ {k},y_{j}\big{)}-\frac{1}{nm}\sum_{l=1}^{n}\sum_{k=1}^{n}\kappa(x_{k},y_{l}) \Bigg{\}}^{2} \tag{14}\]  
It can be seen that the kernel function \(\kappa(x,y)\) is crucial for the discrepancy representation of VDR. A student kernel function \(\kappa(x,y)\) is designed and introduced to solve the poor robustness for the abnormal samples and longtail distribution. \[\kappa(x,y)=\frac{\Gamma\Big{(}\frac{d+1}{2}\Big{)}}{\sqrt{d}\Gamma\Big{(} \frac{d}{2}\Big{)}}\Bigg{(}1+\frac{\|x-y\|^{2}}{d}\Bigg{)}^{-\frac{d+1}{2}} \tag{15}\]  
where \(d>0\) is the degree of freedom and \(\Gamma(\cdot)\) is the gamma function. \[\Gamma(x)=\int_{0}^{+\infty}t^{x-1}e^{-t}\mathrm{d}t,x>0 \tag{16}\]  
Although VDR is a novel distributional alignment metric that expresses variance information to enhance difference representation, it is still a metric-based approach to distributional differences. Conventional DA metrics, while effective in extracting domain-invariant representations, encounter scale misalignment issues in deep feature distributions that deteriorate regression robustness during cross-domain adaptation.