# An Automatic Partition Time-Varying Markov Model for Reliability Evaluation
## 3 Results
### Reliability Evaluation  
In order to verify the validity and accuracy of the state division of mechanical devices of rail transit trains proposed in this article, Bearing1_1, Bearing1_2, Bearing2_1, Bearing2_4, and Bearing3_1 datasets are used for training, and other bearing data are used for testing. In the time-varying Markov process model, it is assumed that the state transition probability obeys the probability density function of mixed Gaussian distribution, and the probability distribution function of state sojourn time obeys the probability density function of a single Gaussian distribution. Based on the automatic partition method, the state number of the time-varying Markov model is 4, that is \(N=4\). The maximum number of iteration steps is set to be 1000, and the stop criteria of the error convergence is \(1\times 10^{-5}\). The initial matrix of mutual conversion between each state is shown in Table 3. The average sojourn time in each state can be obtained based on training datasets; its result is shown in Table 4. A 4-state, time-varying Markov process reliability and life prediction model is obtained. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bearing** & **State 1** & **State 2** & **State 3** & **State 4** \\
**Number** & **Running-in** & **Normal** & **Degradation** & **Complete** \\  & **Stage** & **Operation Stage** & **Stage** & **Failure Stage** \\ \hline Bearing1\_1 & [1, 360] & [361, 1800] & [1801, 2740] & [2741, 2803] \\ Bearing1\_2 & [1, 289] & [290, 392] & [393, 844] & [845, 871] \\ Bearing1\_4 & [1, 329] & [330, 1084] & [1085, 1376] & [1377, 1428] \\ Bearing1\_5 & [1, 612] & [613, 1020] & [1021, 2421] & [2422, 2463] \\ Bearing1\_7 & [1, 540] & [541, 1980] & [1981, 2210] & [2211, 2259] \\ Bearing2\_1 & [1, 660] & [661, 817] & [818, 874] & [875, 977] \\ Bearing2\_4 & [1, 402] & - & [403, 740] & [741, 751] \\ Bearing2\_5 & [1, 566] & [567, 2254] & - & [2255, 2311] \\ Bearing3\_1 & [1, 246] & [247, 490] & - & [491, 517] \\ Bearing3\_3 & [1, 134] & [135, 294] & [295, 423] & [424, 436] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Segmentations result of dataset. Figure 9: Result of DTW on Bearing2_4 data. The trained time-varying Markov model is updated when a new monitoring data sample is arrived. The bearing staying in state 1 for 10 h is taken as an example. After feature extraction of the monitoring data sample, the data is sent into the trained time-varying Markov model, and state transition matrix is updated at this time. At this time, the state transition matrix is shown in Table 5, while the mean and variance of the state sojourn time are shown in Table 6.  
The bearing reliability at current time is \(R(i=1,st_{1}=10)=1-0.0003=0.9997\). To demonstrate the adaptability of the proposed model to changes in time, we also show the model changing parameters when the bearing stays in state 2 for 40 h. At this time, the state transition matrix is shown in Table 7, while the mean and variance of the state sojourn time are shown in Table 8. \begin{table}
\begin{tabular}{c c c c c} \hline
**States** & **State 1** & **State 2** & **State 3** & **State 4** \\ \hline Expectation of sojourn time & 390.8451 & 739.5547 & 383.93594 & 44.3543 \\ Variance of sojourn time & 2.9458 & 1.5942 & 1.5026 & 1.4358 \\ \hline \end{tabular}
\end{table} 
Table 6: The mathematical expectation and variance of sojourn time when bearing stays in state 1 for 10 h. \begin{table}
\begin{tabular}{c c c c c} \hline
**States** & **State 1** & **State 2** & **State 3** & **State 4** \\ \hline State 1 & 0.8712 & 0.1112 & 0.0173 & 0.0003 \\ State 2 & 0.0000 & 0.7496 & 0.2511 & 0.0993 \\ State 3 & 0.0000 & 0.0000 & 0.7028 & 0.2972 \\ State 4 & 0.0000 & 0.0000 & 0.0000 & 1.0000 \\ \hline \end{tabular}
\end{table} 
Table 3: Initial transformation matrix.