# Modeling of Distorted Degradation Data Based on Oil Analysis
## 2 Methodology
### Parameter Estimation
#### 2.2.1 Parametric Estimation Approach  
If the parametric representation of the physical degradation process can be derived based on available physical theories, then the parametric estimation approach is a good choice. Here, the physical degradation \(h\) is a function of the observed time \(t\) and the unknown parameter vector \(\mathbf{\theta}\), given by \(h(t,\mathbf{\theta})\). Through a simple derivation, Equation (3) for **Case II** can be reformulated as follows:  
\[f(t_{i},\mathbf{\theta},\alpha)= \begin{cases}h(t_{i},\mathbf{\theta}),&i=1,\\ h(t_{i},\mathbf{\theta})+\left(\alpha^{\delta_{i}}-1\right)h(t_{i-1},\mathbf{\theta})& i=2,\\ h(t_{i},\mathbf{\theta})+\left(\alpha^{\delta_{i}}-1\right)h(t_{i-1},\mathbf{\theta})+ \sum\limits_{k=1}^{i-2}\left[\left(\alpha^{\delta_{k+1}}-1\right)h(t_{k},\mathbf{ \theta})\prod\limits_{j=k+2}^{i}\alpha^{\delta_{i}}\right],&\text{otherwise,} \end{cases} \]  
where \(f(t_{i},\mathbf{\theta},\alpha)\equiv f_{i}\) and \(h(t_{i},\mathbf{\theta})\equiv h_{i}\). Under the assumption that the observed error follows the normal distribution, the likelihood function is given by  
\[L_{n}(\mathbf{\theta},\alpha,\sigma)=\frac{1}{(2\pi)^{2/n}\sigma^{n}}\exp{\left( -\frac{\sum\
olimits_{i=1}^{n}\left(f(t_{i},\mathbf{\theta},\alpha)-y_{i}\right)^ {2}}{2\sigma^{2}}\right)}. \tag{4}\]  
We use traditional optimization methods such as the Gauss-Newton iteration method to maximize this likelihood function. Then, the maximum likelihood estimations of the unknown parameters obtained are denoted by \((\widehat{\mathbf{\theta}},\hat{\mathbf{\alpha}},\hat{\sigma})\). Similarly, we can obtain the maximum likelihood estimations \((\widehat{\mathbf{\theta}},\hat{\sigma})\) for **Case I**. Compared with **Case I**, there is an additional parameter \(\alpha\) under **Case II**. Under **Case III**, the locations of oil top-up points are unknown; therefore, it is necessary to identify them. Inspired by the monotonic non-decreasing nature of the physical degradation, we propose an explosion search algorithm, which detects the oil top-up pointsfrom the decreased points. Note that the decreased point refers to the point where the observed degradation is lower than the previous point. More specifically, the indicator of whether the fresh oil is added may be 1 or 0 at the decreased point, while it is fixed to 0 at the increased point. Therefore, with \(d\) decreased points, there are \(2^{d}\) candidate degradation paths. For each candidate degradation path, we calculate its maximum likelihood estimation based on Equation (4). Then, we obtain \(2^{d}\) estimations and select one with the minimum residual sum of squares as the final estimation. The procedure for the parametric estimation approach for **Case III** is described by Algorithm 1 in detail. ```
Step 1:  Find the candidate oil top-up paths:  Find the locations of the decreased points;  Calculate the number of the decreased points, denoted by \(d\);  Define the candidate oil top-up paths \(\delta_{i},i=1,\ldots,2^{d}\). Step 2:  Calculate the maximum likelihood estimation of each candidate oil top-up path: for all\(i=1,\ldots,2^{d}\)do  Calculate maximum likelihood estimations \((\hat{\mathbf{\theta}}_{i}\hat{\mathbf{x}}_{i},\hat{\mathbf{\sigma}}_{i})\) by Equation (4), where \(\delta=\delta_{i}\);  Calculate the residual sum of squares, \(re_{i}=\sum_{j=1}^{n}\left(y_{j}-f(t_{j},\hat{\mathbf{\theta}}_{i},\hat{\mathbf{x}}_{i} )\right)^{2}\). endfor Step 3:  Obtain the final estimation: \[j=\arg\min_{i}re_{i},\quad\hat{\mathbf{\theta}}=\hat{\mathbf{\theta}}_{j},\quad\hat{ \mathbf{\kappa}}=\hat{\mathbf{\kappa}}_{j},\quad\hat{\mathbf{\
u}}=\hat{\mathbf{\
u}}_{j}.\]
```  
**Algorithm 1** The three-step explosion search algorithm  
In addition to point estimation, engineers may be also interested in interval estimation. We take **Case II** as an example to illustrate how to use the percentile bootstrap to construct confidence intervals. 1.