# Modeling of Distorted Degradation Data Based on Oil Analysis
## 5 Discussion and Conclusions  
As we all know, most of the actual degradation process is monotone. However, in some circumstances, such as the observation being disturbed by some factors, the observed degradation data lose their monotonicity and are distorted. In this paper, we propose a model to characterize this distorted case and come up with the corresponding approach to solving this model.  
Compared with the piecewise linear method [17], the simulation results demonstrate that our methodology can automatically identify disturbed points, thereby avoiding the subjective judgment required for the piecewise linear method, and can also more accurately recover the actual degradation curves. Importantly, our method is flexible enough to model both linear and non-linear degradation patterns, which expands its applicability and practical utility beyond that of the piecewise linear method. Case studies demonstrate that our methodology can describe the effect of disturbed factors, as well as recovering the actual degradation curve and automatically identifying the disturbed points. This is significantly meaningful for condition monitoring, because it avoids a false diagnosis of degradation to some degree.  
We remark that our proposed model can be generalized to a situation where the physical degradation process is monotone while the observed process is not monotone due to the disturbance of a point process. Specifically, without observed error, the observation is a perturbation of the actual physical degradation process, as follows:  
\[f(t)=h(t)+\int_{0}^{t}f(\tau)dN(\tau), \tag{6}\]  
where the actual physical degradation \(h(t)\) is a monotone process, such as a Gamma process or inverse Gaussian process, \(N(t)=\sum_{t_{i}\leq t}z_{i}^{\delta_{i}}-1\) is a marked point process, the random variable \(z_{i}\) indicates the influence of disturbance, and the random variable \(\delta_{i}\) takes one of two values, 0 or 1; it equals 1 when there is an existing disturbance at the \(i\)th observed time \(t_{i}\) and equals 0 otherwise. If the observed error cannot be neglected, then the observed degradation is as follows:  
\[y(t)=f(t)+\varepsilon(t), \tag{7}\]  
where \(\varepsilon(t)\) is the stochastic observed error with \(E(\varepsilon(t))=0\).  
Particle morphology influences wear mechanism identification. Therefore, these morphological parameters should be incorporated into future investigations. Furthermore, while the current validation is limited to the PSST, extending model verification to multiple machinery types would strengthen the generalizability. Additionally, integrating data-driven methodology with equipment-specific engineering knowledge represents a critical direction for subsequent research.  
**Author Contributions:** Conceptualization, Y.C.; methodology, Y.C.; supervision, J.S.; writing--review and editing, Y.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by Youth Academic Innovation Team Construction project of Capital University of Economics and Business, grant No.QNMTD202303.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.