## 4 Results and Discussion
### Balanced and Imbalanced Data Stream with Gradual Drift  
The gradual drift is the change of one concept to another gradually. In this study, this change is completed in 10 timestamps (5000 instances). The performance of the base classifier on balanced and imbalanced multi-class data streams is presented in Figure 6. Compared to the data stream with sudden drift, where the base classifier was able to produce good results without using the drift detection methods, and its performance was decreased with most of the drift detection methods. In the case of the data stream with gradual drift, some of the drift detection methods have contributed to improving the performance of the base classifier.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**True Detection**} & \multicolumn{2}{c}{**Drifts Missed**} & \multicolumn{2}{c}{**Distance (T)**} & \multicolumn{2}{c}{**Distance (I)**} & \multicolumn{2}{c}{**False Alarm**} \\ \cline{2-10}  & **Bal.** & **Imb.** & **Bal.** & **Imb.** & **Bal.** & **Imb.** & **Bal.** & **Imb.** & **Bal.** & **Imb.** \\ \hline DDM & 3 & 3 & 0 & 0 & 0.33 & 0 & 165 & 0 & 0 & 1 \\ EDDM & 1 & 1 & 2 & 2 & 1.67 & 1.333 & 835 & 666.65 & 1 & 3 \\ Cusum & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 3 & 3 \\ HDDM\_A & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ HDDM\_W & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 35 & 32 \\ GMA & 0 & 0 & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ PH & 3 & 3 & 0 & 0 & 2.33 & 2 & 1165 & 1000 & 0 & 0 \\ RDDM & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 5 \\ STEPD & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 27 & 32 \\ EWMAC & 3 & 2 & 0 & 1 & 0 & 2 & 0 & 1000 & 83 & 38 \\ \hline \hline \end{tabular} Bal. = Balanced (data stream); Imb. = Imbalanced (data stream).  
\end{table}
Table 4: Performance comparison on other metrics.  
Figure 7 shows the actual impact on the classifier's performance in percentage. On a balanced data stream, the performance of the base classifier is improved using DDM, EDDM, CuSUM, PH, RDDM, and EWMAC by 4.35%, 5.22%, 4.28%, 5.06%, 4.3%, 5.22%, and 6.22%, respectively, whereas HDDM_W and STEPD cause a decrease in the performance of the base classifier by 3.63% and 4.31%. The GMA again showed no effect on the base classifier's performance on both streams. On an imbalanced data stream with gradual drift, using DDM, EDDM, CUSUM, PH, and RDDM contributed to improving the base classifier's performance, whereas EWMAC badly affected the classifier's performance when the data stream had a class imbalance issue.  
Again, we looked into the false alarm factor, as we did for streams with sudden drift. Figure 8 shows the number of false alarms generated by each drift detection method on the balanced and imbalanced data stream. It can be observed that again HDDM_W, STEPD, and EWMAC generated a high number of false alarms for both streams so as they affected the base classifier's performance. Even though EWMAC generated 63 false alarms but still produced on the balanced data stream, on the imbalanced data stream, the false alarm rate was increased to 87 and which causes the decrease in the base classifier's  
Figure 6: F1 score on balanced and imbalanced data streams with gradual drift.  
Figure 7: Effect of drift detection methods on base classifier’s performance in case of gradual drift. (**a**) Gradual drift without class imbalance; (**b**) Gradual drift with class imbalance.  
performance. From Figure 8, it can also be observed that DDM, EDDM, CUSUM, PH, and RDDM generated either 0 or very few false alarms; hence, they contributed to improving the base classifier's performance.