## 3 A Wiener Process-Based Approach of RUL Prediction with Unequal Interval Data
### Restoration of Model Parameter and Wear Loss  
The estimated parameters in Section 3.2 need to be reversed to the original value at unequal interval mileage points. In this paper, the UIWGLRCM in [24] is applied to restore the estimation of the parameters. The estimated optimal parameters are \(\hat{\Theta}_{k}^{(i+1)}=[(\hat{\mu}_{\theta})_{k}^{(i+1)},(\hat{\sigma}_{ \theta}^{2})_{k}^{(i+1)},(\hat{\sigma})_{k}^{(i+1)},(\hat{\sigma}^{2})_{k}^{( i+1)}]^{\mathsf{T}}\) through the EM algorithm. The wear loss estimates \(\hat{X}_{1\pm}\) are obtained by one-step prediction in the KF process. \(\big{(}\hat{\mu}_{\theta}\big{)}_{k}^{(i+1)}\) is used to describe the proposed model in detail, let \(x_{1:k}=(\hat{\mu}_{\theta})_{1:k}^{(i+1)}\). The original parameter series of equal mileage intervals is given as:  
\[x^{(0)}(k)=\left\{x^{(0)}(1),x^{(0)}(2),\ldots,x^{(0)}(k)\right\},\;(k=1,2, \ldots,n) \tag{18}\]  
where \(x^{(0)}(1:k)=x_{1:k}\), \((k=1,2,\ldots,n)\). The original parameter series of equal mileage intervals is accumulated once, the series is written as:  
\[x^{(1)}(k)=\left\{x^{(1)}(1),x^{(1)}(2),\ldots,x^{(1)}(k)\right\},\;(k=1,2, \ldots,n) \tag{19}\]  
where \(x^{(1)}(k)=\sum\limits_{j=1}^{k}x^{(0)}(j)\), \((k=1,2,\ldots,n)\). According to the principle of the classical grey model (GM) [25], the response equation of equal interval GM (1,1) is formulated by:  
\[\hat{x}^{(1)}(k+1)=(x^{(0)}(1)-b/a)\exp(-ak)+b/a \tag{20}\]  
where \(a\) denotes development coefficient, \(b\) is grey action, they are calculated according to the principle of the least square method. The form of the Equation (20) can be written as:  
\[\hat{x}^{(1)}(k+1)=C_{1}\exp(vk)+C_{2} \tag{21}\]  
The linear regression equation \(Y=aX+b\) and exponential equation \(Y=ae^{X}\) are used to fit the accumulation series. The series is formulated by:  
\[\hat{x}^{(1)}(k+1)=C_{1}\exp(vk)+C_{2}k+C_{3} \tag{22}\]  
where \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(v\) are all unknown, they need to be estimated. \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(v\) are assumed to have a relationship of the Equation (23):  
\[ \begin{array}{l}L(k)=\hat{x}^{(1)}(k+1)-\hat{x}^{(1)}(k)\\ =C_{1}\exp(vk)[\exp(v)-1]+C_{2}\end{array}  \tag{23}\]  
\(Y_{\rm m}(k)=L(k+m)-L(k)\) is assumed to calculated \(v\), \(v\) is written as:  
\[v=\ln[Y_{\rm m}(k+1)/Y_{\rm m}(k)] \tag{24}\]  
The approximations of \(v\) in the Equation (24) are obtained by replacing \(\hat{x}^{(1)}(k)\) as \(x^{(1)}(k)\), the average approximations are taken as the estimation of \(v\), which are obtained by:  
\[\hat{v}=2\sum\limits_{m=1}^{n-3}\sum\limits_{k=1}^{n-2-m}v_{m}(k)/\left[(n-2) (n-3)\right] \tag{25}\]  
The value of different mileage points in the original data series should be assigned a weight to represent its reliability, and the weight increases exponentially with time. The weight matrix is formulated by:  
\[p=diag(R^{n-1}) \tag{26}\]  
where \(R\) is weight increasing factor, generally the value range of w is from 1 to 2. In this article, \(R=1.5\). According to the Equation (26), the least squares is used to calculate \(C\):  
\[C=(A^{\rm T}p\,A)^{-1}A^{\rm T}p\hat{X}^{(1)} \tag{27}\]where \(C=\left[ \begin{array}{c}C_{1}\\ C_{2}\\ C_{3}\end{array} \right]\), \(\hat{X}^{(1)}=\left[ \begin{array}{c}\hat{X}^{(1)}(1)\\ \hat{X}^{(1)}(2)\\ \vdots\\ \hat{X}^{(1)}(n)\end{array} \right]\), \(A=\left[ \begin{array}{ccc}e^{p}&1&1\\ e^{2v}&2&1\\ \vdots&\vdots&\vdots\\ e^{3v}&n&1\end{array} \right]\). The UIWGLRCM is written as:  
\[\hat{x}^{(1)}(\tau)=C_{1}\exp[v(\tau/\Delta\tau_{0}+1)]+C_{2}(\tau/\Delta\tau_ {0}+1)+C_{3} \tag{28}\]  
where \(\tau\) is the mileage interval which distances from the first cycle, \(\tau=\tau_{k}\)-\(\Delta\tau_{0}\). Through once inverse accumulated, the unequal interval wear series is formulated by:  
\[\left\{ \begin{array}{l}\hat{x}^{(0)}(\tau)=\hat{x}^{(0)}(\tau),\;\tau=0\\ \hat{x}^{(0)}(\tau)=\hat{x}^{(1)}(\tau)-\hat{x}^{(1)}(\tau-\Delta\tau_{0}),\; \tau>0\end{array} \right.