# Modeling of Distorted Degradation Data Based on Oil Analysis
## 3 Numerical Simulations  
In order to compare the proposed methodology with the piecewise linear method [17], two kinds of degradation functions are considered in this subsection. Specifically, the observed degradations \(y_{i}\) is given by \(y_{i}=f_{i}+\varepsilon_{i}\), where \(f_{i}=h_{i}-h_{i-1}+z_{i}^{\delta_{i}}f_{i-1}\). Here, the actual degradation \(h_{i}\) equals \(h(t_{i})\). The observed points \(t_{i}(i=1,...,n)\) with \(n=30\) are equally spaced in \([0,1]\). The measure error \(\varepsilon_{i}\) follows a normal distribution with zero mean and standard deviation \(\sigma=0.5\). \(\delta_{i}\) follows Bernoulli distribution with the parameter \(p=0.3\). \(z_{i}\) follows the normal distribution with mean \(\alpha\) and variance \(0.01\), and we set \(\alpha=0.6\). We simulate 1000 datasets based on the above parameter configuration and calculate the estimations for each dataset. Since it is impossible to enumerate all degradation patterns, here, we will only choose certain representative patterns for illustration. We consider the following two degradation scenarios for all the three cases defined in Section 2:  
* **Scenario (I)**: The actual degradation \(h(t)\) is linear given by \(h(t)=a+bt\) where \(a=10,b=30\). * **Scenario (II)**: The actual degradation \(h(t)\) is non-linear given by \(h(t)=a+bt^{c}\) where \(a=12,b=28,c=0.5\). For the parameter estimations obtained by the parametric approach, we calculate the mean of these 1000 estimations, as well as the standard deviation (SD) and the mean square error (MSE). The results of **Scenario (I)** and those of **Scenario (II)** are shown in Table 2 and Table 3, respectively. We can see that the SD and MSE are very small for each parameter. Moreover, the mean values of estimations are very close to the true values. These results show that the proposed parametric approach indeed works well. The results obtained by the non-parametric approach in **Scenario (I)** and **Scenario (II)** are, respectively, plotted in Figures 3 and 4. The top, middle, and bottom rows display the non-parametric estimations for **Case I**, **Case II**, and **Case III**, respectively. The 95% point-wise confidence intervals are computed by taking the 2.5 th percentile and the 97.5 th percentile of 1000 estimations at each observed time point. It can be seen that the mean values of the non-parametric estimations are very close to the actual degradation curve and the confidence intervals are narrow. These results indicate that the proposed non-parametric approach performs well. Figure 4: _Cont._  
Figure 3: **Scenario (I)**: The results obtained by the non-parametric approach and the piecewise linear method. Moreover, the piecewise linear method [17] is also used in our simulation. The results obtained by the piecewise linear method are also shown in Figures 3 and 4. We can see that the following is true: (a) The mean values of estimations obtained by the piecewise linear method are farther away from the actual degradation curve than those obtained by our non-parametric approach. (b) The confident intervals obtained by the piecewise linear method are wider than those obtained by our non-parametric approach. These results imply that the performances of our non-parametric approach are better than those of the piecewise linear method. In addition, from Tables 2 and 3 and Figures 3 and 4, we can conclude that the estimations for **Case I** are more precise than those for **Case II**, and the estimates for **Case II** are better than those for **Case III**. This is reasonable, because the more complete the observed data, the more accurate the obtained estimations. Moreover, we computed the coverage probabilities for the 95% bootstrap confidence intervals, and we list them in Tables 4 and 5. We can observe that the coverage probabilities are all around 95%. This indicates that our model and the corresponding estimation approaches are effective.