## 3 ADDT Analysis
### ADDT Model
Along with a decreasing degradation path, failure-time \(T\) being less than \(t\) is equivalent to an observed degradation being less than \(\mathcal{D}_{f}\) at time \(t\); that is,  
\[F_{T}(t;x)\equiv\Pr(T\leq t)=\Pr\Big{(}Y\leq\mathcal{D}_{f}\Big{)}=\Phi\bigg{[} \frac{\mathcal{D}_{f}-\mathcal{D}(t;x,\theta)}{\sigma}\bigg{]},\ \ \ t\geq 0. \tag{5}\]  
Figure 9 shows the ADDT data of return-spring samples in bi-functional DC motors at 300 \({}^{\circ}\)C and 325 \({}^{\circ}\)C. Note that the degradation of the operating force for the return-spring is of nonlinear form and the force degradations at 325 \({}^{\circ}\)C are steeper than those at 300 \({}^{\circ}\)C. We seek a model for relative operating force by dividing each measurement of the operating force by the initial value to easily derive the failure-time of the return-spring, where the failure is defined at the time when the relative operating force falls below 1.23 kgF/1.42 kgF = 86.89%. The values of 1.23 kgF and 1.42 kgF were derived from the preliminary experiment of users' cognitive speed for return-spring resilience. We fitted the four models, including one linear model, to the ADDT data, but we re-parameterized the exponential model to fit the relative actuating forces and to present the decreasing degradation rate as  
\[y_{ij}(t)=\exp\Big{\{}-\exp(\beta)\cdot\exp\Big{(}E_{a}\times\big{(}\frac{11,605}{Temp_{u}+273.16}-\frac{11,605}{Temp_{i}+273.16}\ \big{)}\Big{)}\cdot t\Big{\}}+e_{ij}(t). \tag{6}\]  
In addition, we also modified the logarithmic model and the power model to fit the relative actuating forces as  
\[y_{ij}(t)=1-\beta\cdot\exp\bigg{(}E_{a}\times\bigg{(}\frac{11,605}{Temp_{u}+2 73.16}-\frac{11,605}{Temp_{i}+273.16}\ \bigg{)}\bigg{)}\cdot\ln(t+1)+e_{ij}(t), \tag{7}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Exponential** & **Logarithmic** & **Power** & **Logistic** \\ \hline Mean degradation & \(\mathcal{D}=\beta_{0}exp(\beta_{1}t)\) & \(\mathcal{D}=\beta_{0}\cdot t^{\beta_{1}}\) & \(\mathcal{D}=\beta_{0}\cdot t^{\beta_{1}}\) & \(\mathcal{D}=\frac{1}{1+\exp(\beta)\cdot t}\) \\ Degradation over time & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean degradation path models applied to ADDT of springs. \[y_{ij}(t)=(t+1)^{\beta\cdot\exp\left(E_{a}\times(\frac{11\,\mathrm{\AA 05}}{Temp_{1}+2\,\mathrm{\AA 16}}-\frac{11\,\mathrm{\AA 05}}{Temp_{1}+2\,\mathrm{\AA 16}}\ ^{)}\right)}+\epsilon_{ij}(t), \tag{8}\]  
respectively. Figure 10 presents the fitting results of the mean degradation path from four candidate models, along with the estimated degradation path at the normal use condition (230 \({}^{\circ}\)C). The horizontal dashed line represents the failure threshold. We computed the log-likelihood, Akaike information criterion (AIC) [16], and Bayesian information criterion (BIC) [17] to support the best model selection procedure. The AIC is a general-purpose criterion which is defined as: \(AIC=-2l(\hat{\mathbf{\theta}})+k\), where \(l(\hat{\mathbf{\theta}})\) is the log-likelihood for the ML estimates \(\mathbf{\theta}\) and \(k\) is the dimension of \(\mathbf{\theta}\), that is, the number of parameters to be estimated. The BIC is defined as: \(BIC=-2l(\hat{\mathbf{\theta}})+\mathrm{klog}(n)\), where \(n\) is the number of observations for the ADDT. The intention of these two information criteria is to help us find the simpler model by assigning the penalty for increasing the complexity of an assumed model. If we are using the log-likelihood, we prefer the model with the larger log-likelihood, while we prefer the model with the lower AIC, or lower BIC to compare several models for the same data. Table 2 summarizes the statistics from the best model selection procedure. All the results advocate the power model as the best model for the ADDT data of return-springs. The parameters of the power model were estimated as: \(\hat{\beta}=-0.0114\), \(\hat{E}_{a}=0.2571\), and \(\hat{\sigma}^{2}=0.0168^{2}\).