## Materials and Methods
### Risk Decision Based on RUL Prediction
#### 2.2.3 System Portfolio Risk Determination Based on RUL  
Assume that \(Z=\left\{Z_{1},\,Z_{2},\,\cdots,\,Z_{l_{1}}\right\}\) represents the degradation reference track and \(Y=\left\{Y_{1},\,Y_{2},\,\cdots,Y_{l_{2}}\right\}\), \(l_{1}>l_{2}\), is the current degradation track. Then, the normalized cross correlation (NCC) is introduced to measure the similarity between tracks. The specific definition of NCC is given as follows:  
\[s_{\text{YZ}}(i)=\frac{(Y-\overline{Y})\times(Z_{i}-\overline{Z}_{i})^{\prime }}{\|Y-\overline{Y}\|_{2}\|Z_{i}-\overline{Z}_{i}\|_{2}}, \tag{10}\]  
where \(Z_{i}=\left\{Z_{i},\,Z_{i+1},\,\cdots,\,Z_{i+l_{2}-1}\right\}\) and \(Y\) are of the same length. \(\overline{Y}\) and \(\overline{Z}_{i}\) are the mean vectors of \(Y\) and \(Z_{i}\) respectively. Taking the \(j\)th time series \(x^{j}=\left\{x^{j}_{1},\,x^{j}_{2},\,\cdots,\,x^{j}_{N_{j}}\right\}\) as an example, the time points of the time series are denoted as \(t^{j}=\left\{t^{j}_{1},\,t^{j}_{2},\,\cdots,\,t^{j}_{N_{j}}\right\}\). Apparently, \(t^{j}_{N_{j}}\) is the running life of the \(j\) th degradation process. Taking time delay \(\tau\) as 1 and considering embedded dimension as \(d\) for all degradation track phase spaces (\(d\) is no less than the maximal embedded dimension of all phase spaces). Then the original time series of degradation process is transformed into the track of phase space. Denoting the reference track for the \(j\)th process as \(Z_{j}=\left[X^{j}_{d^{\prime}},\,X^{j}_{d+1},\,\cdots,\,X^{j}_{N_{j}}\right]^{T}\), the points in phase space are expressed as \(X^{j}_{i}=\left[x^{j}_{i-(d-1)},\,x^{j}_{i-(d-1)},\cdots x^{j}_{i}\right]\). Assuming the current degradation time series is \(y=\left\{y_{1},\,y_{2},\,\cdots,\,y_{c}\right\}\). Similarly, it is transformed into an incomplete degradation track form denoted as \(Y=\left[Y_{d},\,Y_{1+d},\,\cdots,\,Y_{c}\right]\), \(Y_{i}=\left[y_{i-(d-1)},\,y_{i-(d-2)},\,\cdots,\,y_{i-1},\,y_{i}\right]\). When performing track matching, a track subset \(\gamma^{k}=\left[Y_{k-l+1},\,\cdots,\,Y_{k}\right]\) is selected with a time window length of 1 in the current phase space. Then the NCC between the current track \(\gamma^{k}\) and the reference track \(Z_{j}\) should be modified as follows. \[s^{j}_{k}=s_{\text{YZ}}+(1-\frac{\left|t_{k}-t^{j}_{k}\right|}{t_{k}}), \tag{11}\]  
If the \(s^{j}_{k}\) reaches the maximum at the time point \(T^{j}_{k^{\prime}}\) then sub-tracks at time \(T^{j}_{k}\) are regarded as the best matching results in terms of degradation reference track \(Z_{j}\). This result also reflects the influence of the track shape to the degradation stage. Therefore, the RUL of the most similar part to the \(j\)th degradation reference track can be estimated as \(L^{j}_{k}=t^{j}_{N_{j}}-T^{j}_{k}\). On the basis of the above result of the best track matching, the weight of the remaining life calculated according to the \(j\)th degradation reference track is given as \(w^{j}_{k}=s^{j}_{k}(T^{j}_{k})/\left(\sum_{j=1}^{M}s^{j}_{k}(T^{j}_{k})\right)\), where \(M\) is the total number of the degradation processes. Then the remaining life of the system \(k\) is estimated as Equation (12). \[RUL_{k}=\sum_{j=1}^{M}\omega^{j}_{k}L^{j}_{k^{\prime}}, \tag{12}\]Then, the RUL should be transformed into risk according to the joint probability. It is identified as a risk event when any system in a system portfolio breaks. Therefore, for a system portfolio \(x_{i}=(s_{1},s_{2},\ldots,s_{n})\), the risk is calculated as Equation (13) shows. \[R(x_{i})=1-\prod\limits_{j=1}^{n}\left[\left(1-\frac{1}{RUL_{j}}\right)^{x_{ij }}\right], \tag{13}\]  
In conclusion, the RUL is an important criterion for the weapon system research because longer RUL can reduce the failure risk of systems in combat. The availability of each component system is the basis to guarantee the normal operation of systems portfolios. An operation activity will be broken or even unsuccessful if any system of a system portfolio malfunctions in the operation process.