# A Hybrid Degradation Modeling and Prognostic Method for the Multi-Modal System
## 2 Degradation Modeling and Prognostic Hybrid
### Long-Term Cumulative Degradation Assessment with Cdd-Hi
#### 2.1.2 Composite Health Indicator Construction
Samples of multi-modal systems with identical structures usually have different lengths of life cycle. Therefore, it is necessary to calculate the similarities between different lengths of degradation sequences. In this paper, the dynamic time warping (DTW) method is used to measure the similarities of different health indicator series. The similarity between \(H_{\alpha}(t_{\alpha})\), \(t_{\alpha}\in[0,q_{\alpha}]\) and \(H_{\beta}(t_{\beta})\), \(t_{\beta}\in\left[0,q_{\beta}\right]\) is calculated with the dynamic programming method based on finding the min-distance warp path. \(D=D_{1},D_{2},..D_{d},D_{N_{path}},N_{path}\in\left[\max(q_{\alpha},q_{\beta }),q_{\alpha}+q_{\beta}\right]\). The obtained warp path \(D\) indicates the correspondence between \(H_{\alpha}(t_{\alpha})\) and \(H_{\beta}(t_{\beta})\), permissible point-to-point, and many-to-one point correspondences. The obtained warp path \(D\) should satisfy three constraints: (1) The constraint of endpoint ensures that the starting point and the endpoint are consistent when the sequences are aligned, as \(D_{1}=(1,1),D_{N_{path}}=(q_{\alpha},q_{\beta})\). (2) The continuity constraint is used to limit the excessive expansion of the dynamic time warping and limit the jump correspondence, as \(t_{\beta_{d+1}}-t_{\beta_{d}}\leq\zeta,t_{\alpha_{d+1}}-t_{\alpha_{d}}\leq\zeta\), where \(\zeta\) is a positive integer. (3) The monotonicity constraint requires \(t_{\beta_{d+1}}-t_{\beta_{d}}\geq 0,t_{\alpha_{d+1}}-t_{\alpha_{d}}\geq 0\). Then, the minimum distance of dynamic time warping is revised as Equation (5),  
\[DTW(H_{\alpha},H_{\beta})=\min_{p}\sum_{d=1}^{N_{path}}d(H_{\alpha}(t_{\alpha _{d}}),H_{\beta}(t_{\beta_{d}})), \tag{5}\]  
where \(d(H_{\alpha}(t_{\alpha_{d}}),H_{\beta}(t_{\beta_{d}}))\) is the euclidean distance between two points. To reduce the amount of calculation, the upper triangular matrix of the DTW matrix which consists of \(DTW(H_{\alpha},H_{\beta})\), \(1\leq\alpha\leq p\), \(1\leq\beta\leq p\) is used to calculate the distance sum of samples. Then, the Property 3 could be formulated as Equation (6),  
\[ \begin{array}{l}\min_{w}\sum\limits_{\alpha=1}^{p-1}\sum\limits_{\beta= \alpha}^{p}\frac{2 DTW(H_{\alpha},H_{\beta})}{(p-1)^{2}}\\ s.t.w^{\prime}M^{\prime}1=1,Mw\geq 0.\end{array}  \tag{6}\]  
Above all, the optimization problem of composite health indicator construction model could be formulated as Equation (7),  
\[ \begin{array}{l}\min_{w}\lambda_{1}\sum\limits_{i=1}^{p}\sum\limits_{t=1}^{ q_{i}-1}\epsilon_{i,t}+\lambda_{2}w^{\prime}A^{\prime}Baw\\ \quad+\lambda_{3}\sum\limits_{\alpha=1}^{p-1}\sum\limits_{\beta=\alpha}^{p} \frac{2 DTW(H_{\alpha},H_{\beta})}{(p-1)^{2}}\\ s.t.w^{\prime}M^{\prime}1=1,Mw\geq 0,\epsilon_{ij}\geq 0,\lambda^{\prime}1=1,\\ \quad-1<w<1,\end{array}  \tag{7}\]where \(\lambda\) is the parameter adjusting the relative importance of the three properties. Based on solving the multi-objective programming equation to calculate \((w_{1},w_{2},...w_{o})\), the composite health indicator \(H(t)\) with good monotonicity and generalization will be calculated from cumulative dynamic physical difference feature with Equation (1).