# Modeling of Distorted Degradation Data Based on Oil Analysis
## 2 Methodology  
### Data and Model  
The actual physical degradation serves as a \"health indicator\" for system wear; however, the observation of the physical degradation process is affected by oil top-ups. To be specific, once the fresh oil is added, the observed degradation data, such as the concentration of metal elements, will be diluted. Therefore, it is necessary to correct the observed degradation data. Before establishing the model, we have the following assumptions: If the mass of the oil added is unknown, then the observed degradation data, namely, the concentrations of metal elements, are diluted in the same ratio each time fresh oil is added. This indicates that the oil top-up effect can be described by a constant.  
The definitions of all variables and mathematical symbols are listed in Table 1. If the observed error cannot be neglected, then the observed degradation is random, namely,  
\[y_{i}=f_{i}+\epsilon_{i}, \tag{1}\]  
where \(\epsilon_{i}\) is the random error and follows a normal distribution with zero mean and standard deviation \(\sigma\).  
We established a model to describe the relationship between observed degradation data without measurement errors and the actual physical degradation process. Typically, we consider three cases: complete observation data, partially missing data, and completely missing data.  
**Case I**: The observed data consist of \(\left(t_{i},y_{i},z_{i},\delta_{i}\right)_{i=1}^{n}\), representing complete observations. If fresh oil is added after the \((i-1)\)th observation, then the concentration becomes  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Symbol** \\ \hline The \(i\)th observed time & \(t_{i}\) \\ Actual physical degradation at \(i\)th observed time & \(h_{i}\) \\ Observed degradation with observed error at \(i\)th observed time & \(y_{i}\) \\ Observed degradation without observed error at \(i\)th observed time & \(f_{i}\) \\ Observed error at \(i\)th observed time & \(\epsilon_{i}\) \\ Indicator of whether the fresh oil is added & \(\delta_{i}\) \\ Dilution effect of fresh oil at \(i\)th observed time & \(z_{i}\) \\ Mean of the dilution effect & \(\alpha\) \\ Number of observation & \(n\) \\ Standard deviation of observed errors & \(\sigma\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nomenclature: summary of all variables and mathematical symbols.  
Figure 1: The concentration of Cu in the power shift steering transmission of Vehicle 1.  
\(z_{i}f_{i-1}\); otherwise, it remains \(f_{i-1}\). Therefore, neglecting measurement errors, the observed degradation \(f_{i}\) consists of two components: the cumulative degradation \(z_{i}^{\delta_{i}}f_{i-1}\) and the increment in the degradation process \(h_{i}-h_{i-1}\). Formally, the correction model can be expressed as follows:  
\[f_{i}=\begin{cases}h_{i},&i=1;\\ h_{i}-h_{i-1}+z_{i}^{\delta_{i}}f_{i-1},&i=2,\ldots,n.\end{cases} \tag{2}\]  
**Case II**: The observed data consist of \(\left(t_{i},y_{i},\delta_{i}\right)_{i=1}^{n}\), which are partial missing observations. Since the effects of oil top-ups are unknown, their mean value \(\alpha\) is adopted. Therefore, in this case, the model can be represented as follows:  
\[f_{i}=\begin{cases}h_{i},&i=1;\\ h_{i}-h_{i-1}+\alpha^{\delta_{i}}f_{i-1},&i=2,\ldots,n.\end{cases} \tag{3}\]  
**Case III**: The observed data consist of \(\left(t_{i},y_{i}\right)_{i=1}^{n}\), which are complete missing observations. The formula of the model is also given by Equation (3). However, we note that \(\delta_{i}\) is unknown in this case.