## 4 Experimental Results and Discussions
### Macro SOStream  
Initially, there was validation of the create function of the macroclusters. For this, we used the Horseshoe dataset with \(\alpha=0.001\), \(\eta=2\), \(\gamma=0.01\), and \(p=1.5\). Figure 9 illustrates how clustering occurs. The first macrocluster is created with the data from the outer ring of the dataset, as shown in Figure 9a. Although the samples exhibit the behavior of data drift over time, the algorithm identifies that they belong to the same macrocluster. Only when the inner ring samples arrive at the algorithm is a new macrocluster created, as shown in Figure 9b. The Horseshoe dataset clustering result is illustrated in Figure 9c. In the second experiment, there was the validation of the merge function of the macroclusters. For that, we used the dataset named T with the settings of \(\alpha=0.001\), \(\eta=2\), \(\gamma=0.0001\), and \(p=2.5\). Figure 10 illustrates clustering behavior, where in Figure 10a shows that the algorithm detects the presence of two macroclusters initially. Over time, it realizes that it was just one macrocluster, as shown in Figure 10b. Figure 10c illustrates the microclusters identified of the macrocluster. Another dataset is the Moon with \(\alpha=0.001\), \(\eta=2\), \(\gamma=0.01\), and \(p=2.5\). We can see in Figure 11a that the algorithm detected four macroclusters initially because of the samples' order. After some macroclusters are merged, it results in only two macroclusters, as illustrated in Figure 11b. Figure 11c shows the microclusters. Figure 9: Validation of the Macro SOStream create function with the Horseshoe dataset. (**a**) Clustering of the first 800 samples. (**b**) Clustering of the first 1950 samples. (**c**) Final result, where different colors represent different macroclusters. The best hyper-parameterization of Macro SOStream to clustering the Three Spirals dataset is \(\alpha=0.65\), \(\eta=3\), \(\gamma=0.0001\), and \(p=2.0\). Figure 11(a) shows the clustering of the first 261 samples, with the two macroclusters identified. Figure 11(b) illustrates the final result of the clustering, and Figure 11(c) presents the microclusters found. Thus, the Macro SOStream clustered correctly, even with the concept drift from the Three Spirals dataset. We exhaustively searched for the optimal hyperparameters. We are used the graph Parallel Categories Diagram of the Plotly package to evaluate the performance. Figure 12(a) shows the variance of the Macro SOStream hyperparameters and ARI metric for the Horseshoe dataset. The graph columns represent the hyperparameters; the lines are the connections between the hyperparameters' values. Moreover, the colors of the lines illustrate the ARI metric value to the configuration determined. It is possible to see the effect of the \(\eta\) as the minimum number of clusters, where \(\eta\) equal to 2 performs better. Moreover, the hyperparameter \(p\) equal 1.5 may have been decisive in the ARI's best value. Figure 11: Clustering of the Moon dataset with Macro SOStream. (**a**) Clustering of the first 3500 samples forming four distinct macroclusters, represented by different colors. (**b**) Final result with two macroclusters. (**c**) Representation of the microclusters. Figure 12: Clustering of the Three Spirals dataset with Macro SOStream. (**a**) Clustering of the first 261 samples forming two distinct macroclusters, represented by different colors. (**b**) Final result with three macroclusters. (**c**) Representation of the microclusters. Figure 10: Validation of the Macro SOStream merge function with the T dataset. (**a**) Clustering of the first 1800 samples forming two distinct macroclusters, represented by different colors. (**b**) Final result. (**c**) Representation of the microclusters. In the clustering with the T dataset, as illustrated in Figure 13b, most hyperparameters result \(ARI=1.0\). This excellent performance could be because the T dataset has just one cluster, resulting from the merging function application.