## 2 Model Framework and Methodology
### Simulation of Degradation Data  
The wear of contact surfaces in mechanical components is measured in terms of wear debris concentration (WDC) in lubricating oil. The online monitoring of WDC provides information on the wear rate of the component. WDC and wear rate follows a relationship in the lubrication system. The lubrication system has a cyclic oil flow with continuous removal of wear debris due to oil filtration, oil loss, sedimentation, and debris sticking to the surface. Therefore, the WDC formulation assumes the wear debris production rate to be equal to the wear rate. The evolution of WDC comprises three stages with different wear rates, as described in detail by [PERSON] et al. [34]. Lubricating oil is assumed to be frequently added to sustain the oil level in the lubrication system. Therefore, WDC degradation data need to be simulated, accounting for the effect of frequent oil replenishment and oil change. Our study here considered a mechanical system comprising four basic components: tribological component, oil filter, oil pump, and oil tank. The assumptions that we made to carry out the WDC simulation are listed as follows:  
* Oil replenishment is an external event, and oil consumption is precisely equal to oil replenishment;
* Wear production rate is equal to wear rate;
* Wear debris are homogeneously distributed in the oil with negligible mixing time;
* The system is in the wear-out (abnormal) phase of its life cycle;Oil is changed every time the WDC reaches a threshold of 40 ppm. The quantity WDC is defined as \(C(t)\), a function of wear rate and is given by the following equation based on the model presented in Ref. [34]:  
\[C(t)=\frac{\frac{m_{s}}{(c+k)}\Big{(}e^{c(t-t_{1})}-e^{-k(t-t_{1})}\Big{)}+\int_ {t_{1}}^{t}e^{-k(t-u)}dW_{F}(u)}{V_{0}+V_{r}*t-V_{q}*t} \tag{1}\]  
where we define \(V_{r}=V_{q}\), and \(V_{r}\) is the oil replenishment quantity, \(V_{q}\) is the oil consumption quantity, \(V_{0}\) is the initial volume of oil in the tank, \(c\) is the wear mode, and (\(c>0\)) for wear-out phase, \(m_{s}\) is the steady-state wear rate, \(W_{F}(t)\) denotes the white Gaussian noise, and \(k\) is the attenuation coefficient. Therefore, the expression for WDC becomes  
\[C(t)=\frac{m_{s}}{(c+k)V_{0}}\Big{(}e^{c(t-t_{1})}\;e^{-k(t-t_{1})}\Big{)}\; \frac{1}{V_{0}}\int_{t_{1}}^{t}e^{-k(t-u)}dW_{F}(u) \tag{2}\]  
where  
\[k=\frac{\Big{[}\Big{(}1-\frac{1}{B_{x}}\Big{)}Q+\Psi\Big{]}}{V_{0}} \tag{3}\]  
Here, \(\beta_{x}\) is the beta ratio, i.e., filter efficiency, a ratio of WDC before filter to the WDC after the filter, \(\Psi\) is the wear debris loss factor, and \(Q\) is the oil flow through the filter. The oil replenishment and oil change cause abrupt negative jumps (spikes) in the degradation trajectory, and the WDC trend continuously evolves. The degradation pattern is simulated for the abnormal state, using experimental parameters for the lubrication system given by [PERSON] et al. [34]. The model parameters are \(m_{s}=25\) mg/min, \(V_{0}=40\)-L, \(m_{0}=0.4975\), \(W_{F}{}^{\prime}(t)\sim\) N(0,1.5), \(c=0.028\), \(k=0.0001\), \(V_{r}=V_{q}=0\), \(t_{1}=0\) sec, and the oil replenishment effects are added as negative ppm values in the model. The complete life cycle of the system comprises several oil life cycles (OLC). In this work, we considered four OLC for the wear-out phase of the oil and the system. As mentioned earlier, [PERSON] et al. [28] studied a healthy portion of the oil data, citing limitations for optimal data segregation. Figure 2 illustrates the complexity in the WDC data generated for the 4 OLCs with oil changes occurring every time WDC reaches the 40-ppm threshold criterion as well as random oil replenishments based on minimal oil quantity to be maintained in the operating system.