## 5 Numerical Results
### Anomaly Detection  
The overall anomaly detection scheme is verified by checking whether or not the learned neural network classifier provides a valid indication for the time series input from the real flight test data. Figures 9 and 10 represent the results for a representative test data case when the input time series is not included in the training set. The swarm flight test data on 5 June 2018, whose clustering results are shown in Figures 6 and 7, are considered. Figure 8(a) depicts the output value of the neural network corresponding to the data of drone ID \(\times\)70 that turns out to be in failure, when the input time series is continuously entered into the network. As the 1D-CNN takes an input of length 160 and the data rate is 10 Hz, the first output is obtained in response to the kinematic variables over the first time segment of 16 s. Then, the neural network continuously computes the output in response to the latest 160 time points. It can be found that the probability of being normal is oscillating for some period of time and then becomes zero constantly. Figure 8(b) compares the average value of the normal probability of the  
Figure 8: Effect of imbalance rate and sampling: (**a**) test accuracy with respect to original imbalance rate; (**b**,**c**) illustrative anomaly detection result for imbalance rate 0.1 (N: normal, A: abnormal). data of five drones flow in this flight test, when the average is taken over the whole flight time. It can be clearly seen that the abnormal probability of this problematic drone is much greater than the normal probability. Figure 10 depicts the time history of the kinematic input variables corresponding to \(\times 70\). Note that this drone does not effectively respond to the changes in the setpoints in \(x\)- and \(z\)-directions around the times 90 s and 75 s, respectively; as such, the anomaly detection scheme shown in Figure 9 indicates that the system may not be normal from the initial data point at time 85 s, which is obtained based on the behavior of the past 16 s; then, from time 100 s and onwards, it clearly says that the system is in the abnormal state with almost probability 1. It should be pointed out that investigation of the original flight log indicates that drone \(\times 70\) actually crashed during the flight test. From the perspective of on-line anomaly monitoring, the two plots in Figure 9 may be considered as two extreme ways of monitoring. Continuous monitoring of the normal or abnormal probability as in Figure 9a provides the assessment of system anomaly in a high-frequency manner. This approach allows for responsiveness in the anomaly detection as it continuously provides updated information on the normality of the system. However, as can be seen in the figure, the output signal is particularly noisy in the transient phase. The average probability over the entire time span, as shown in Figure 9b provides a smoothed/collective decision on anomaly in the system. This approach may be least sensitive to noise, but the decision frequency may be too low. Figure 10: Time history of kinematic variables of drone \(\times 70\). Figure 9: Anomaly classification result: (**a**) normal probability of \(\times 70\), (**b**) average probability of vehicles. To mitigate the noise in continuous monitoring at the same time to allow for responsive detection of anomaly, this work investigates moving average-based monitoring with finite time window. The normal/abnormal probability is averaged over some specified time horizon, and is updated with a new set of data. The time window of the adjacent averaging window can be overlapped. The four subplots of Figure 11 represent the change in the normal probability of different drones as the time proceeds. The average is taken over 30 s, and every 20 s the average value is updated, meaning that 1/3 of data are overlapped between the consecutive time windows.