## 2 Nonlinear Wiener Process with Covariates
### Deducing Relation of Parameters in the ADT for Nonlinear Wiener Model  
ADT accelerates the degradation process of products by increasing the stress, and the collected degradation data could be extrapolated with information by accelerating the model to evaluate the life or reliability of products under normal operating conditions. To ensure the extrapolation accuracy, the failure mechanism must remain unchanged under the accelerated and the normal stress based on the basic assumptions, originally proposed by [PERSON] [21] in the accelerated test. That is, the acceleration factor should be independent of test time and subject to a constant principle. The definition of widely used acceleration factors under [PERSON]'s assumption [22] is described below. Let \(F_{h}(t_{h})\), \(F_{k}(t_{k})\) be the CDF of the product under any two stress levels \(S_{k}\), \(S_{h}\), separately. If \(F_{h}(t_{h})=F_{k}(t_{k})\), the acceleration factor \(A_{k,h}\) from \(S_{k}\) to \(S_{h}\) is defined as [22; 23]. \[A_{k,h}=t_{h}/t_{k} \tag{6}\]  
The degradation mechanism of products should be consistent under two different levels \(S_{k}\) and \(S_{h}\) based on the acceleration factor constant principle. Then, the following equation should always satisfy for any \(t_{h},t_{k}>0\). \[F_{k}(t_{k})=A_{k,h}F_{h}(t_{h}) \tag{7}\]Considering an equivalent relation between CDF and PDF with Equations (4) and (5), the following PDF equation is deduced in Equation (8). \[f_{k}(t_{k})=\frac{dF_{k}(t_{k})}{dt_{k}}=\frac{dA_{k,k}F_{h}(t_{h})}{dt_{h}}=A_{k,h}f_{h}(t_{h}) \tag{8}\]  
Specify \(\Lambda(t)=t\) and substitute Equation (4) into Equation (8), it is obtained as follows. \[ \begin{split} A_{k,h}&=f_{k}(t_{k})/f_{h}(t_{h})\\ &=\frac{(\sigma_{8})_{h}A_{k,h}^{3/2}}{(\sigma_{8})_{k}}\cdot \exp\biggl{[}\left(\frac{D\lambda_{k}}{(\sigma_{8}^{2})_{h}}-\frac{D\lambda_{k }}{(\sigma_{8}^{2})_{k}}\right)+\frac{1}{t_{k}}\left(\frac{D^{2}}{2(\sigma_{8 }^{2})_{h}A_{k,h}}-\frac{D^{2}}{2(\sigma_{8}^{2})_{k}}\right)+t_{k}\left(\frac {\lambda_{k}^{2}A_{k,h}}{2(\sigma_{8}^{2})_{h}}-\frac{\lambda_{k}^{2}}{2( \sigma_{8}^{2})_{k}}\right)\biggr{]}\end{split}  \tag{9}\]  
To confirm that \(A_{k,h}\) is independent of test time \(t_{k}\), the coefficient of the term concerned with \(t_{k}\) in Equation (9) is required to be zero. \[\left\{ \begin{array}{c}\frac{D^{2}}{2(\sigma_{8}^{2})_{h}A_{k,h}}-\frac{D^ {2}}{2(\sigma_{8}^{2})_{k}}=0\\ \frac{\lambda_{k}^{2}A_{k,h}}{2\left(\sigma_{8}^{2}\right)_{h}}-\frac{\lambda_ {k}^{2}}{2(\sigma_{8}^{2})_{k}}=0\end{array} \right. \tag{10}\]  
The relationship of the following parameter is inferred from Equation (10). \[A_{k,h}=\frac{\lambda_{k}}{\lambda_{h}}=\frac{\left(\sigma_{B}^{2}\right)_{k} }{\left(\sigma_{B}^{2}\right)_{h}} \tag{11}\]  
It can be concluded that both the drift parameter \(\lambda\) and diffusion parameter \(\sigma_{B}^{2}\) relate to stress variables due to \(A_{k,h}\
eq 1(k\
eq h)\), rather than the assumption that the diffusion parameter is fixed and does not vary with the applied stresses. Let \(\Lambda(t)=t^{\theta}\), Wiener degradation model can effectively simulate concave, convex, and linear degenerate trajectories. If \(\Lambda(t)=t^{\theta}\), it can be deduced according to the above ideas:  
\[ \begin{split} A_{k,h}&=\frac{\lambda_{k}}{\lambda_{ h}}=\frac{\left(\sigma_{B}^{2}\right)_{k}}{\left(\sigma_{B}^{2}\right)_{h}}\\ \Lambda_{k}(t)&=\Lambda_{h}(t)=\Lambda(t)\end{split}  \tag{12}\]  
From the above analysis, \(\lambda\) and \(\sigma_{B}^{2}\) should keep the same ratio under any two different stress levels, \(\Lambda(t)\) has no relationship with stresses and should remain unchanged under various stresses.