# An Automatic Partition Time-Varying Markov Model for Reliability Evaluation
## 2 Materials and Methods
### The Automatic Partition Time-Varying Markov Model for Reliability Evaluation
#### 2.2.1 The Automatic Partition Method for State Number Determination of the Markov Model  
The automatic partition method based on fuzzy time series segmentation is used to automatically obtain the state number of the Markov model. It is assumed that \(x_{l}\) is an \(M\) dimensional multivariate sample, and \(S\) is the sample sequence with length, \(L\), where \(S=\left\{d_{l}=[t_{l},x_{l}^{T}]^{T}|1\leq l\leq L\right\}\), \(x_{l}=\left\{x_{lm}|m=1,2,\ldots,M\right\}\). According to the trend of data change, that is based on the equipment performance degradation, sample sequence S can be partition into several segments. In this article, sample sequence \(S\) is assumed to be divided into \(P\) segments, noted as \(Sub_{S}^{p}=\left\{Sub_{S}^{p}|p=1,2,\ldots,P\right\}\), and the boundaries are \(s_{1}<s_{2}<\cdots s_{P}\), where \(s_{1},s_{2},\ldots,s_{P}\in S\) and \(s_{1}=d_{1},s_{P}=d_{L}\). That is, \(Sub_{S}^{p}=(s_{p-1}+1,s_{p})\), \(p=1,2,\ldots,P\). Define the sample sequence segmentation objective function as \(C(Sub_{S}^{p})\). \[C(Sub_{S}^{p})=\sum_{p=1}^{P}C(Sub_{S}^{p})=\sum_{p=1}^{P}C(s_{p-1}+1,s_{p}) \tag{2}\]  
The objective function is generally represented as the distances between the monitoring data in each segment and their corresponding fitting function. \[C(Sub_{S}^{p})=\sum_{p=1}^{P}\sum_{l=s_{p-1}+1}^{s_{p}}\left\|x_{l}-v_{p}^{x} \right\|^{2}=\sum_{p=1}^{P}\sum_{l=1}^{L}\beta_{p}(t_{l})D^{2}(x_{l},v_{p}^{x}) \tag{3}\]  
where \(v_{p}^{x}\) is the cluster center of the \(p\)-th segment, \(D^{2}(x_{l},v_{p}^{x})\) is the distance between \(x_{l}\) and \(v_{p}^{x}\), \(\beta_{p}(t_{l})\) is the membership function of \(x_{l}\) subordinate to the \(p\)-th segment. The objective function mentioned above is minimized to obtain the segment boundaries with optimal segment positions \(s_{p-1}+1,s_{p},p=1,2,\ldots,P\). It is generally conducted by various heuristic algorithms. \(\beta_{p}(t_{l})\) is defined as a (0, 1) binary function in references [35; 36], which is not appropriate in practice. In this article, the multivariate mixture Gaussian function is introduced into a cluster prototype as the sequence fitting function to describe the fuzzy characteristic of  
Figure 1: Basic steps of the proposed method. boundaries between segments. The optimal segment positions are obtained by minimizing the weighted sum of squared distances between the monitoring data in each segment and the center of the cluster prototype. \[C(Sub_{S}^{p})=\sum\limits_{p=1}^{P}\sum\limits_{l=1}^{L}\left(\mu_{p,l}\right)^ {m}D^{2}\left(d_{l},\eta_{p}\right) \tag{4}\]  
where \(\mu_{p,l}\) is the membership of \(d_{l}=\left[t_{l},x_{l}^{T}\right]^{T}\) belonging to the \(p\)-th segment, \(p=1,2,\ldots,P\); \(m=\left[1,\infty\right)\) is the weighted index of fuzzy clustering, generally taking \(m=2\)[37]; \(D^{2}\left(d_{l},\eta_{p}\right)\) represents the distance between the monitoring data in each segment and the center of the cluster prototype; \(\eta_{p}\) is the clustering prototype function of \(p\)-th segment. It is assumed that the expectation and the covariance matrix of the used multivariate Gaussian distribution for sample sequence are \(v_{p}\) and \(F_{p}\), respectively. The probability density function \(f(\mathbf{d}_{l}|\eta\,)\) denoting the adsorption of monitoring data to the \(p\)-th segment is  
\[f(\mathbf{d}_{l}|\eta\,)=\sum\limits_{p=1}^{P}f(\mathbf{d}_{l}|\eta\,_{p})f(\eta_{p}) \tag{5}\]  
where \(f(\mathbf{d}|\eta\,_{p})=\frac{1}{(2\pi)^{(l+1)/2}\sqrt{\det(F_{p})}}\exp(-\frac{ 1}{2}(d-\mathrm{v}_{p})^{T}(F_{p})^{-1}(d-\mathrm{v}_{p}))\); \(f(\eta_{p})\) is the unconditional clustering probability, \(\sum\limits_{p=1}^{P}f(\eta_{p})=1\); \(\eta_{p}\) is the clustering prototype function of the \(p\)-th segment, \(\eta_{p}=\left\{f(\eta_{p}),\mathrm{v}_{p},F_{p}|p=1,2,\ldots,P\right\}\). According to probability theory, the distance function and the membership in the \(p\)-th segment are inversely proportional based on the Gath-Geva clustering algorithm [38].