## 4 Prediction of Residual Life  
As described in detail above, the degradation process is defined by HDP-HMM with an estimated state space consisting of \(Z=\) {1, 2, 3, 4}. The state sequence can be inferred from Figures 6b and 7a, and the corresponding state transition matrix is obtained from Figure 7b. This procedure of state estimation and representation is rare in LCM DP since most of the focus in the past has been towards HMM, as mentioned earlier in Section 1.  
Figure 6: (**a**) Observed “true” state sequences corresponding to the simulated WDC evolution trend in Figure 2. The scatter plot shows observed data; (**b**) the observation distribution scatter plot, and the estimated states sampled at the 100 th iteration.  
Figure 7: (**a**) Inferred hidden states for lubricating oil degradation with irregular oil replenishments and oil changes; (**b**) state transition probability matrix for estimated state space.  
Following the state diagnosis above, for prognosis, we used the posterior probability to define a conditional reliability function (CRF) and the mean residual life (MRL) functions. Since degradation data filtration is required in most cases to remove the random noise in the data, the refined samples were then considered for DP. The present work used DRP and a stick-breaking process for sampling. After state estimation, we considered the data points as sampling epochs. The samples were collected at time intervals of \(\Delta=1\) h, and \(n=300\) samples were collected, and \(Y_{n\Delta}\) denotes the observation process. Assume that the lubrication oil does not fail at a decision epoch, \(n\), i.e., \(\xi>\Delta n\), \(\xi\) is the observation time. The posterior probability statistic, \(\pi_{n}\), is then used to express CRF and MRL, representing the likelihood that the oil is in a warning state given all the information until time \(n\Delta\) and defined as [23]  
\[\pi_{n}=P(Z_{n\Delta}=1\xi>n\Delta,\ Y_{\Delta},Y_{2\Delta},\ldots\ldots Y_{n \Delta}) \tag{23}\]  
The CRF denoting the probability of non-failure by \(n\Delta+t\), is then defined as  
\[R\left(\frac{t}{\pi_{n}}\right)\ =\ P(\xi>n\Delta+t\xi>n\Delta,Y_{\Delta},\ldots \ldots,\ \ Y_{n\Delta},\ \pi_{n}) \tag{24}\]  
Using the CRF expression, the MRL(\(\mu\)) at the nth epoch can then be extracted as  
\[\mu_{n\Delta}\ =\ \int_{0}^{\infty}R(t|\pi_{n})dt \tag{25}\]  
We can predict the RUL making use of the CRF function and the MRL value, as suggested by [PERSON] and [PERSON] in Ref. [26], estimated by applying Equations (24) and (25) to the degradation data set. Figure 8a plots the conditional reliability estimated at all the sampling epochs for the degradation data. The reliability drops significantly around the 120 th, 180 th, and 240 th sampling epochs when the oil is changed. The reliability drops flicker between 0.965 to 0.865 since the degradation is constantly affected by irregular oil replenishments and oil changes. The effect of oil replenishment and oil change is evident from the more minor reliability transitions in the plot. The reliability may decline drastically in the absence of oil replenishments.  
Figure 8b plots the posterior probability of lubricating oil degradation data presented in Figure 2. The posterior probability is estimated using Equation (23), taken from Ref. [23]. It may be inferred from the plot that the lubricating oil healthy state continues for a prolonged time between the 10 th and 250 th sampling epoch. Moreover, the oil has transitioned to the warning states several times between the 100 th to 300 th sampl  
Figure 8: (**a**) Trend of the conditional reliability as a function of the sampling epoch estimated at the 100 th iteration; (**b**) the posterior probability trend of transition to warning state for the oil degradation data with frequent oil replenishments and oil changes.  
changes. This shows a high probability that the lubricating oil state can change and that oil replenishments need to be initiated following such inspections.