## 4 Performance Evaluation
### Random Faults  
The magnitude of random faults injected in our experiments is determined by using one value selected randomly in the distribution of the magnitude of the original data at each velocity. The duration variation follows that of the constant fault. The intensity is \(\eta=\{1,\ 5,\ 10\}\) by which the selected magnitude is multiplied. The magnitude of random faults is given by  
\[\mu=\mu_{r}+|(m+\gamma\times d\times\eta_{0})|, \tag{11}\]  
where \(\gamma\) is a random number drawn from the standard normal distribution, \(\eta_{0}\) is one of the intensity values, and \(m\) and \(d\) are the mean value and the standard deviation of the training data at the certain velocity respectively. The accuracy of the random fault detection is presented in Figure 7. At all velocities except for 0.7 m/s of the velocity, AT achieves the best performance close to 100% when exceeding the intensity of a certain magnitude (\(\eta_{0}>1\)). As with constant faults, AT has lower performance than KT only in the shortest duration and velocity (\(\delta=5\) and \(\
u=0.7\)m/s) and it seems to be due to the slightly high threshold of AT at the lowest velocity. From Equation (11), when \(\eta_{0}=1\) is applied, the random faults with one value in the range of \([\mu_{r},\,\mu_{r}+m]\) are generated by about 47% of the total random faults and random faults in the range of \([\mu_{r},\,\mu_{r}+m\times 2]\) account for about 82%. Hence, when \(\eta_{0}=1\) is applied, these results are so meaningful in terms of sensitivity to small changes. The ability to identify them is superior at all velocities compared to other methods when the faults with the magnitude similar to that of the original data are injected. Obviously, the larger the intensity is, the closer to 100% the performance of AT is. Unlike AT, the accuracy of KT is significantly influenced by the intensity and duration irrespective of the velocity. Furthermore, when \(\eta_{0}=5\) and 10, although the magnitude that goes beyond the value of \((\mu_{r}+m\times 4)\) account for about 47% and about 71% of the total faults generated, respectively, the performance of KT is not as good as that of AT. RT tends to decrease in the performance as the duration and intensity increase. In the case of RT, when the intensity is small (\(\eta_{0}=1\)), the TPs do not increase regardless of the duration, but the actual normal data are identified well. When the intensity is medium and high, the TPs increase but the ability is insufficient for identifying the actual normal data as the duration increases. Therefore, these results mean that KT and RT are not adaptive and sensitive even though they are determined dynamically. Figure 6: Precision according to each intensity as a function of constant fault duration: (**a**) Cumulative sum of the precision at 0.7 m/s of Jackal UGV; (**b**) Cumulative sum of the precision at 1.0 m/s; (**c**) Cumulative sum of the precision at 1.3 m/s; (**d**) Cumulative sum of the precision at 1.6 m/s. Figure 8 shows the results of the recall of three methods. It is similar to the tendency of the accuracy of both AT and KT except RT. When the velocity is 0.7 m/s, the recall performance is low as the duration becomes shorter. This is due to the increase in the occurrence of FNs, that is, the faults are mistaken as normal data. The cause of the FN increase seems to be closely related to the size of the time window. However, from the results, we find that this tendency shows that AT adapts quickly as the intensity becomes stronger. At the velocity above 1.0 m/s, we also find that AT and KT produce many FNs only when \(\eta_{0}\) = 1. However, the frequency of FNs in AT is much smaller than that of KT even though the duration is very short. Furthermore, as the intensity and the velocity increase, AT does not generate any FN error, but KT still has many FN errors. In the random faults, RT is unable to detect faults with similar size to the original data, and classify almost all injected faults as normal data.