## 4 Double-Criteria Drift Streaming Peaks over Threshold (DCDSPOT)
### Weighted Source Parameter  
Since causality exists universally in telemetry, when we aim to detect the anomaly of a certain target parameter, it is necessary to consider both the target parameter and its source parameters. In other words, we need to additionally take the values of source parameters of the target parameter into consideration when determining whether the target parameter is abnormal.  
When we apply causality to detect anomalies of a target parameter, whether it is anomalous or not cannot be inferred merely based on the value of a certain source parameter or part of the source parameters. There are two reasons. First, some variables that have non-negligible impact on the system exist but we cannot observe them (or it is difficult to be accurately measured), such as the intensity of sunlight in the spacecraft system. Therefore, the source parameters we select may be only part of all the source parameters, and the anomaly of one source parameter does not inevitably cause the anomaly of the target parameter. Second, although the interactions among some source parameters and the target parameter are not significant, their extreme anomalous mode may also lead to anomaly of the target parameter. Hence, we need to comprehensively consider the influence of all source parameters on target parameter.  
Suppose the target parameter is \(\mathbf{X}\), and all its source parameters are \(\mathbf{U}_{1},\mathbf{U}_{2},...,\mathbf{U}_{k}\), the time lags of all causalities are \(\mathbf{t}=[\tau_{1},\tau_{2},...,\tau_{k}]\). When we are detecting the value of the target parameter \(\mathbf{X}\) at time \(t\), we need to take the value of the source parameter \(\mathbf{U}_{j}\) at time \(t-\tau_{j}\) (\(j=1,2,...,k\)) into consideration, because if the anomalous mode appears in \(\mathbf{U}_{j}\) at time \(t\), it will not spread to target parameter \(\mathbf{X}\) until time \(t+\tau_{j}\). When \(t\) is less than \(\tau_{j}\), we take \(\mathbf{U}_{j}[0]\) as a substitute for \(\mathbf{U}_{j}[t-\tau_{j}]\).  
Then we define the Weighted Source Parameter (WSP), which is denoted by \(\mathbf{V}_{X}\). WSP is a \"virtual\" parameter weighted and summed according to the strength of the source parameters' influence (value of NETE in our method) on the target parameter (see Equation (10). Equation (9) shows the mathematical definition of WSP, and the negative sign in superscript means the timestep moves backward. WSP does not actually exist, but it represents the weighted synthesis of the numerical modes of all the source parameters. The extreme value distribution of WSP can also be regarded as approximately obeying GPD [29].  
\[\mathbf{V}_{X}=\mathbf{w}^{\mathrm{T}}\mathbf{U}^{-\mathbf{t}}=\mathbf{w}^{ \mathrm{T}}(\mathbf{U}_{1}^{-\tau_{1}},\mathbf{U}_{2}^{-\tau_{2}},..., \mathbf{U}_{k}^{-\tau_{k}}) \tag{9}\]  
\[\mathbf{w}^{\mathrm{T}}=\frac{1}{\sum\limits_{i=1}^{k}NETE_{U_{i}\to X}}[NETE_{ U_{1}\to X},NETE_{U_{2}\to X},...,NETE_{U_{k}\to X}]^{\mathrm{T}} \tag{10}\]  
\(\mathbf{U}_{j}^{-\tau_{j}}\) denotes shifting the sequence \(\mathbf{U}_{j}\) backward by \(\tau_{j}\), that is \(\mathbf{U}_{j}^{-\tau_{j}}[t]=\mathbf{U}_{j}[t-\tau_{j}]\).