# An Automatic Partition Time-Varying Markov Model for Reliability Evaluation
## 2 Materials and Methods
### The Automatic Partition Time-Varying Markov Model for Reliability Evaluation
#### 2.2.2 The Time-Varying Markov Model
}^{N-1}\alpha_{ij}(st_{t}^{i})\beta_{t+1}(j,1)+\beta_{t+1}(i,st_{t}^{i}+1),1<i<N \tag{31}\]  
\[\beta_{t}(1,st_{1}^{i})=\sum\limits_{j=2}^{N-1}a_{1j}(st_{1}^{i})\beta_{t}(j,1) \tag{32}\]Based on the forward probability formula and the backward probability formula, the conditional probability formula \(S=(s_{1},s_{2},\ldots,s_{T})\), \(s_{1},s_{2},\ldots,s_{T}\in\{i,\,1\leq i\leq N\}\) of the state vector can be obtained as  
\[P_{S}=P(S|M)=\sum\limits_{i=1}^{N}\sum\limits_{s_{t}^{i}}^{T^{i^{*}}}\alpha_{i }(i,st_{t}^{i})\beta_{i}(i,st_{t}^{i}) \tag{33}\]  
(3) The re-estimated parameters of the time-varying Markov model can be obtained by the formula  
\[ \begin{array}{ll}\zeta_{t}(i,j,st)=p(s_{t}=i,s_{t^{\prime}}=j,st_{t}^{i}=st| S,M)\\ =\left\{\begin{array}{ll}\frac{1}{P_{S}}\alpha_{t}(1,st_{t}^{i})a_{1j}(s_{t} ^{i})\beta_{t}(j,1),&i=1\\ \frac{1}{P_{S}}\alpha_{t}(1,st)p(st_{t}^{i})a_{ij}(st_{t}^{i})\beta_{t+1}(j,1),&2\leq i,j<N\\ \frac{1}{P_{S}}\alpha_{t}(i,st)p(st)a_{iN}(st)\beta_{t+1}(j,st),&2\leq i<N,j=N \end{array}\right.\end{array}  \tag{34}\]  
\(\zeta_{t}(i,j,st)\) is the probability with a given model, \(M\), and state sequence of \(S=(s_{1},s_{2},\ldots,s_{T})\) and \(s_{1},s_{2},\ldots,s_{T}\in\{i,\,1\leq i\leq N\}\) show that the performance state of equipment will transit to state \(j\), after a sojourn time \(st\) in state \(i\). \(\gamma_{t}(i,st)\) is the probability that when the device stays in state \(i\) for a sojourn time, \(st\), at time \(t\), with a given model of \(M\) and state sequence of \(S\), \(\gamma_{t}(i,st)=p(s_{t}=i,st_{t}^{i}=st|S,M)=\frac{1}{P_{S}}\alpha_{t}(i,st) \beta_{t}(i,st)\). \(\eta_{t}(i,st)\) is the probability when the device stays in state \(i\) for a sojourn time of \(st\) at time of \(t\),  
\[\eta_{t}(i,st)=\frac{1}{P_{S}}\alpha_{t}(i,st)p(st_{t}^{i}=st)[\begin{array}[ ]{ll}\sum\limits_{ \begin{array}{ll}j=2\\ j\
eq i\end{array} }^{N-1}a_{ij}(st)\beta_{t+1}(j,1)+a_{iN}(st)\beta_{t}(N,st_{ t}^{N})] \tag{35}\]  
It can be seen from \(a_{ij}(st)=p(s_{t+1}=j|s_{t}=i,st_{t}^{i}=st)\) that the revaluation formula of \(a_{ij}(st)\) is  
\[\overline{a}_{ij}(st)=\frac{\sum\limits_{t=1}^{T}\zeta_{t}(i,j,st)}{\sum \limits_{t=1}^{T}\eta_{t}(i,st)} \tag{36}\]  
where \(\sum\limits_{t=1}^{T}\zeta_{t}(i,j,st)\) is noted as the expected transition number from state \(i\) to state \(j\), when it stays in state \(i\) for a sojourn time, \(st\). \(\sum\limits_{t=1}^{T}\eta_{t}(i,st)\) is the sum of expected transition number out of state \(i\), when it stays in state \(i\) for a sojourn time, \(st\). Assume that the state sojourn time probability obeys a Gaussian distribution \((\mu_{i},\sigma_{i}^{2})\),  
\[\overline{\mu}_{i}=\frac{\sum\limits_{t=1}^{T}\sum\limits_{s=1}^{T^{i^{*}}} \eta_{t}(i,st)\cdot st}{\sum\limits_{t=1}^{T}\sum\limits_{st=1}^{T^{i^{*}}}\eta _{t}(i,st)} \tag{37}\]  
\[\overline{\sigma}_{i}^{2}=\frac{\sum\limits_{t=1}^{T}\sum\limits_{st=1}^{T^{i^{* }}}\eta_{t}(i,st)\cdot st^{2}}{\sum\limits_{t=1}^{T}\sum\limits_{st=1}^{T^{i^{* }}}\eta_{t}(i,st)}-\overline{\mu}_{i}^{2} \tag{38}\]  
(4) Reliability and remaining useful life prediction The device in the completely failed segment means that it cannot complete the specified task as required. Therefore, it can be considered that the probability \(\zeta_{t}(i,N,st)\) of the device transferring to the completely failed segment, \(N\), at time, \(t\), with a sojourn time, \(st\), is its unreliability. The sum probabilities for device from the current state transferring to every other states is its reliability. The reliability of a device at different times can be defined as  
\[R(t)=1-\zeta_{t}(i,N,st) \tag{39}\]  
The failure rate function is noted as \(\lambda(t)\), and its life distribution function is as \(F(t)\); density function is \(f(t)\),\(f(t)=F^{\prime}(t)\), the reliability function is \(R(t)\), \(F(t)+R(t)=1\), and \(\lambda(t)=f(t)/R(t)\). For \(\forall t\), there is \(R(t)>0\). Therefore, it can be considered that \(\lambda(t)\) is an estimate of the interval fault conditional probability.