## 4 Discussion
While we implemented MERLIN purely in Python, MDI is implemented in C++ with a Python interface. All methods except RRCF were able to detect the \"steep increase\" anomaly shown in Figure 6(a), with varying UCR scores ranging from 0.5 for GANF and TranAD to 1.0 for MDI and MERLIN. The low F1-Score for MDI indicates a high number of false-positive results, which in this case depends on the length of the detected subsequence, as shown in Figure 0(a). In contrast, the low F1 score for TranAD is caused by a poor threshold, leading to an increased number of false-positive results, as shown in Figure 0(b). RRCF is unique in its ability to detect the \"smoothed increase\" anomaly shown in Figure 6(b), but not the \"steep increase\" anomaly detected by the other five methods. This behavior can be explained by the working principle of RRCF to isolate single points. The values in the smoothed subsequence occur only once in the time series and can therefore be isolated from all other values. That RRCF does not find the outlier anomalies seems contradicting, but this is due to the time series containing other extreme values, e.g., with an inverted sign, covering the true anomaly in the anomaly score. The comparison of different strategies for choosing the range of subsequence lengths for MDI and MERLIN presented in Section 3.3 reveals that both methods can utilize additional information about the anomalies, with a stronger effect for MERLIN. Providing additional information in terms of the subsequence length of the true anomaly increased the F1 score for MDI but decreased the UCR score by 0.02, which indicates that MDI utilized the additional information to reduce false-positive results. For MERLIN, the F1 score and UCR score increased, indicating that the information on the true anomaly length helped MERLIN to identify anomalies it missed before. In the final experiment, we used subsequence-based statistics instead of point-wise features for RRCF, which increased the UCR score and the AUC ROC. However, the macro-averaged F1 score slightly decreased due to its inability to detect the \"smoothed increase\" anomaly. Instead, RRCF@sequences was able to detect the \"steep increase\" anomaly like the other five methods, indicating that this anomaly can only be detected on the subsequence level. The low F1 scores for RRCF@sequences on those time series with a UCR score of one are mostly caused by the anomaly being detected slightly before or after the ground truth label but within the 100 time steps tolerance for short anomalies, or by the true anomaly being much shorter or longer than the subsequence length used for RRCF@sequences. Figure 12 illustrates these two cases. We conclude this section by summarizing the strengths and weaknesses of the methods analyzed in this study. MDI and MERLIN have the notable advantage of not requiring any hyperparameter tuning. The only parameters that need to be set are the minimal and maximal subsequence lengths, which practitioners select based on the specific application or domain. Despite the arbitrary choice of \(L_{min}=75\) and \(L_{max}=125\) time steps, MDI and MERLIN still outperform all other methods in this study. Additionally, these methods detect a wide range of anomaly types. However, a disadvantage of MDI and MERLIN is that they are not immediately applicable in an online setting. Although discord discovery can be performed online using a different algorithm such as DAMP [49], MERLIN cannot be directly applied to data streams. Similarly, while it may be possible to adapt MDI to consider only subsequences up to a given timestamp when estimating the density of \(\Omega(S)\), the current version of MDI does not support this. The isolation forest approach used in RRCF is intuitive and can be applied to data streams; these factors are advantages of RRCF. However, RRCF shows poor results in this study and may be more suitable for applications in which outliers have distinct values from normal data. All three classical methods have the advantage of being easily interpretable.