## 4 Performance Evaluation
### Constant Faults  
Constant faults are just to add the constant magnitude to the original values. Rather than injecting several constant values selected randomly, the constant magnitudes are generated by using the predefined values for intensity in order to give consistency to the constant values. The constant magnitude is determined by using the predefined intensity and the magnitude of several samples picked randomly from the original data. Hence, the magnitude of the constant fault is the magnitude of the picked sample multiplied by the intensity plus the magnitude of the picked sample. The values of the intensity used is \(\eta=\{1.5,2,5\}\)[33]. The magnitude for constant faults, denoted as \(\mu\), is calculated by  
\[\mu=\mu_{r}+\mu_{r}\times\eta_{0}, \tag{10}\]  
where the magnitude of the data that is randomly selected is denoted as \(\mu_{r}\), and \(\eta_{0}\) is one of the values of the intensity. From Equation (10), when \(\eta_{0}=1.5\) is applied, the constant faults with one value in the range of \([\mu_{r},\mu_{r}+m]\) are generated by about 100% of the total constant faults where \(m\) is the mean value of the training data at the certain velocity. When \(\eta_{0}=5\) is applied, constant faults in the range of \([\mu_{r}+m,\mu_{r}+m\times 2]\) account for about 100%. Figure 4 shows the performance results in terms of accuracy calculated by \(\{(\text{n(TPs)}+\text{n(TNs)})/\text{n(TPs+TNs+FPs+FNs)}\). A true positive (TP) means a transient fault that is correctly identified, a true negative (TN) is a normal data including a soft fault that is correctly \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Velocity \(\
u\)**} & **AT** & **KT** & **RT** \\ \cline{2-4}  & **(\(\rho^{\
u}\),\(\sigma^{\
u}\))** & **(\(\rho^{\
u}\),\(\sigma^{\
u}\))** & **Mean RMS** \\ \hline
0.7 m/s & (0.117679, 5) & (0.1, 8) & 0.078927 \\
1.0 m/s & (0.1249, 1.8) & (0.1, 6) & 0.080349 \\
1.3 m/s & (0.167154, 2.3) & (0.1, 13) & 0.092686 \\
1.6 m/s & (0.203168, 2) & (0.18, 6) & 0.142931 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The list of thresholds for all velocities when the size of the time window (\(w\)) is 100.  
identified, an FP is an actual normal data that is identified as a transient fault, and an FN means an actual faulty data that is identified as a normal data in fault detection. Overall, AT demonstrates better performance than RT and KT. For all velocities, it shows the trend that the performance of AT is not significantly affected by the magnitude of the intensity unlike KT and RT, and AT has also high performance level regardless of the duration as the velocity increases. Only when the duration is low (\(\delta=5\)) at the low velocity (0.7 m/s), the performance of AT is less than 2% (three samples) of that of KT. Under the small duration of the constant faults injected at low velocity, the acquired measurements are very small and the change (standard deviation) in the acquired measurements is not large. In this regard, the small difference between the given thresholds of \(\rho^{v}\) (i.e., 0.117679 of AT and 0.11 of KT) seems to have a great effect on the determination of the anomalous region. Except in the case above, the performance of AT overcomes those of RT and KT performance regardless of the duration and intensity as the speed increases. Thus, our AT shows good detection results because it is sensitive to the very small intensity of injected faults. In the case of KT, although the accuracy performance of KT shows the similar tendency to that of our AT, it notes that the accuracy of KT is highly affected by the length of duration rather than the magnitude of the intensity. For example, the performance at the small intensity (\(\eta_{0}=1.5\)) is about 4% lower than that at the largest intensity (\(\eta_{0}=5\)) at the lowest speed. In addition, at the short duration (\(\delta=5\) and 10) the performance of KT is not any the better even if the velocity increases.