## 3 Improved PIM under Dynamic Conditions
### Model Estimation  
The parameter estimation of the PIM with a Weibull process as a baseline intensity can refer to the solutions in the literature [17; 33]. Firstly, the covariate coefficients are estimated by using the partial likelihood estimation method [34]. Then the shape and scale parameters in the Weibull function can be estimated according to the maximum likelihood estimation method. In the estimation of covariate coefficients, the WC factors that have little impact on the FI can be eliminated by the likelihood ratio test [35]. The end-of-life data and censored data are distinguished in the partial likelihood estimation. Supposing that there are \(np\) units of observation, each unit is composed of an observation time and a covariate group. For example, the \(i\)-\(th\) individual unit of observation \((t_{i},\mathbf{Z}(\mathbf{i}))\) is composed of the observation time \(t_{i}\) and the covariate vector \(\mathbf{Z}(\mathbf{i})\). The \(np\) individual units of observation are arranged in descending observation times to obtain the \(mp\) of individual unit of observation groups. The \(i\)-\(th\) unit of observation group is denoted as \(S_{i}\). Since the observation times in different individual units of observation may be equal, there is \(mp\leq np\). Supposing that among all individual units of observation there are \(kp\) individual units of observation whose observation times are end-of-life data, the remaining \(np-kp\) individual units' observation times are censored. Arranging the \(kp\) individual units of observation with end-of-life observation times in ascending order obtains \(\tau p\) end-of-life groups with different observation times. Then, recording the \(i\)-\(th\) group as \(B_{j}\), the number individual unit of observation are denoted in \(B_{j}\) as \(b_{j}\). Arranging the remaining \(np-kp\) individual units of observation whose observation time belongs to the censored data in ascending order, obtains \(\mu p\) censored groups with different observation times. Then, the \(k\)-\(th\) group is recorded as \(C_{k}\). Since the observation time in different individual units of observation may be equal, there are \(\tau p\leq kp\), \(\mu p\leq np-kp\). For the observation time \(t\) of a specific individual unit of observation, all units of observation that have not expired and are not censored at that moment constitute the risk group, and the risk set corresponding to the observation time \(t_{i}\) can be denoted as \(R_{i}\). The specific definition results are shown in Figure 2.  
Comparing the FIs in \(e_{1}\) and \(e_{2}\), it can be seen that the FI in \(e_{1}\) is always greater than that in \(e_{2}\), which means that the combined effect of each factor in \(e_{1}\) is more likely to cause failure than that in \(e_{2}\). By using Equation (10), it can be discerned that the unit time damage in \(e_{1}\) is 1.13 times the unit time damage in \(e_{2}\). Based on this, the relationship of the cumulative damage between these two WCs throughout the same time span can be obtained. For example, the cumulative damage from 0 to 800 h in \(e_{1}\) is equal to the cumulative damage from 0 to 905.00800 h in \(e_{2}\). Equal cumulative damage does not mean the same FI, which is also related to the current WC. For example, the FI at 800 h is 0.00225 in \(e_{1}\),which is not equal to the FI at 905.00800 h, which is 0.00199. Though their cumulative damage is equal at this point, the ratio of their FIs is also equal to 1.13. Comparing the FIs in \(e_{2}\) and \(e_{3}\), it can be seen that the FI in \(e_{3}\) is always lower than in \(e_{2}\), which means that the combined effect of each factor in \(e_{3}\) is less likely to cause failure than that in \(e_{2}\). The unit time damage in \(e_{3}\) is 0.84 times the unit time damage in \(e_{2}\). So, the cumulative damage from 0 to 800 h in \(e_{3}\) is equal to the cumulative damage from 0 to 673.24800 h in \(e_{2}\).