## 3 Planning Strategy and Basic Assumptions
### Basic Assumptions
In addition, each degradation increment between successive measurement times (\(\Delta t_{ijkl}=t_{ijkl}-t_{ijk,l-1}\)), \(\Delta y_{ijkl}\big{(}=y_{ijkl}-y_{ij,l-1}\big{)}\) also follows a normal distribution with mean \(\eta_{ij}\Delta t_{ijkl}\) and variance \(\sigma^{2}\Delta t_{ijkl}\)Thus, a probability density function, \(f\left(\Delta y_{ijkl}\right)\), and the log-likelihood function, \(\ln L\), of \(\Delta y_{ijkl}\) are respectively expressed as:  
\[f\left(\Delta y_{ijkl}\right)=\frac{1}{\sqrt{2\pi\sigma^{2}\Delta t_{ijkl}}} \exp\left[-\frac{\left\{\Delta y_{ijkl}-\eta_{ij}\Delta t_{ijkl}\right\}^{2} \right]}{2\sigma^{2}\Delta t_{ijkl}}\Bigg{]},-\infty<\Delta y_{ijkl}<\infty\]  
\[\ln L=\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\sum_{k=1}^{n_{ij}}\sum_{l=1}^{m} \left[-\frac{1}{2}\ln\left(2\pi\cdot\Delta t_{ijkl}\right)-\ln\sigma-\frac{ \left(\Delta y_{ijkl}-\eta_{ij}\Delta t_{ijkl}\right)^{2}}{2\sigma^{2}\Delta t _{ijkl}}\right] \tag{1}\]  
The stress levels of each variable, \(T_{i}^{\prime}\) and \(V_{j}^{\prime}\), were standardized as \(T_{i}\) and \(V_{j}\) respectively, for simplicity and without loss of generality. In terms of \(T_{i}\) and \(V_{j}\), the drift \(\eta^{\prime}\left(T_{i}^{\prime},V_{j}^{\prime}\right)\) can be rewritten as  
\[T_{i}=\frac{1/T_{1}^{\prime}-1/T_{i}^{\prime}}{1/T_{1}^{\prime}-1/T_{M}^{ \prime}},\ V_{j}=\frac{V_{j}^{\prime}-V_{1}^{\prime}}{V_{M}^{\prime}-V_{1}^{ \prime}}\]  
\[\eta\left(T_{i},V_{j}\right)=\exp\left(\delta_{1}+\delta_{2}T_{i}+\delta_{3}V_ {j}+\delta_{4}T_{i}V_{j}\right)=\eta_{ij} \tag{2}\]  
where \(\delta_{1}=\ln\delta_{1}^{\prime}+\frac{\delta_{2}^{\prime}}{T_{1}^{\prime}}+ \delta_{3}^{\prime}V_{1}^{\prime}+\delta_{4}^{\prime}\frac{V_{1}^{\prime}}{T_ {1}^{\prime}}\), \(\delta_{2}=\left(\delta_{2}^{\prime}+\delta_{4}^{\prime}V_{1}^{\prime}\right) \left(\frac{1}{T_{M}}-\frac{1}{T_{1}^{\prime}}\right)\), \(\delta_{3}=\left(\delta_{3}^{\prime}+\frac{\delta_{4}^{\prime}}{T_{1}^{\prime}} \right)\left(V_{M}^{\prime}-V_{1}^{\prime}\right)\), \(\delta_{4}=\delta_{4}^{\prime}\left(\frac{1}{T_{M}}-\frac{1}{T_{1}^{\prime}} \right)\left(V_{M}^{\prime}-V_{1}^{\prime}\right)\). Equation (2) is a modified drift of the WP that organizes \(\eta^{\prime}\left(T_{i}^{\prime},\ V_{j}^{\prime}\right)\) by \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\) and \(\delta_{4}\). Note that \(\delta_{2}\), \(\delta_{3}\), \(\delta_{2}+\delta_{4}\) and \(\delta_{3}+\delta_{4}\) are always positive since \(\delta_{2}^{\prime}+\delta_{4}^{\prime}V_{j}^{\prime}<0\), \(\delta_{3}^{\prime}+\delta_{4}^{\prime}/T_{i}^{\prime}>0\), \(T_{1}^{\prime}<T_{M}^{\prime}\) and \(V_{1}^{\prime}<V_{M}^{\prime}\). For example, the relationship between the drift parameter and the stress variables, temperature, and voltage is assumed to follow the generalized Eyring model. The use and maximum stress levels (\(r_{1}=r_{2}=3\)) of each stress variable were assumed to be (\(45\ \left(=318\text{ K}\right)\), and \(130\ \left(=403\text{ K}\right)\)) and (\(3.8\text{ V}\), \(4.4\text{ V}\)), respectively. The stress levels standardized based on the above standardization equation are given below. If the middle-stress levels are given as \(100\ \left(T_{2}^{\prime}\right)\) and \(4.1\text{ V}\left(V_{2}^{\prime}\right)\), they also are standardized as follows.