# Optimization Algorithms for Scalable Stream Batch Clustering with k Estimation
## 5 Experiments
### Qualitative Analysis of Quality  
A qualitative quality analysis, measured by ARI, among the six algorithms and over the most representative data set results, is presented in the charts of Figure 7. The vertical axis represents the average ARI score values, and the horizontal axis measures the micro-batch position in the stream from 1 to 100. For each algorithm, the markers indicate the average of ARI, and the vertical grey bars are their standard deviations. Note that concept drift and concept evolution in the data streams cause variations in quality, reflected by the ARI scores. The charts show discrepancies in the algorithms' behaviors for different data sets. One of the most notable discrepancies resulted from SGMS in the real data sets (the right arrows in Figure 7c-e). SGMS overestimated the number of clusters up to six times more that of other algorithms for such data sets, which caused a decrease in its ARI values. Similar to [19], the split policy inherited from G-means overestimated the number of clusters here. SPGMS had better quality than SGMS in the KDD data set because the goodness-of-fit test inherited from PG-means was designed to deal with the G-means overestimation of \(k\). However, SPGMS was shown to be computationally prohibitive for high-dimensional data sets. Fortunately, SPKMS had good performance and we found \begin{table}
\begin{tabular}{l l r r r} \hline
**Data Set** & **Type** & **Attributes** & **Instances** & **\# of Clusters** \\ \hline
100k5-2 & Artificial & 10 & \(10^{5}\) & 3...7 \\
1M5-2 & Artificial & 10 & \(10^{6}\) & 3...7 \\
1M10-5 & Artificial & 10 & \(10^{6}\) & 5...15 \\
10M10-5 & Artificial & 10 & \(10^{7}\) & 5...15 \\ KDDCup99 (10\%) [30] & Real & 34 & 494,021 & 2...6 \\ PaperAbstracts [12] & Real & 5871 & \(6.65\times 10^{4}\) & 2...6 \\ YouTube Games [31] & Real & 1000 & \(8.79\times 10^{4}\) & 2...7 \\ \hline \end{tabular}
\end{table} 
Table 3: Characteristics of artificial and real data sets: the name and type of each data set, their numbers of attributes and instances, and the range of the number of clusters. that its goodness-of-fit test, inherited from PG-means, dealt with the \(k\) overestimation, even in scenarios with scalable batch streams. Over artificial Gaussian data sets, the SGMS algorithm performed as well as the other algorithms. It is notable that SESC presented a lower-quality result during the micro-batches from 71 to 80 of 100k5-2 (Figure 7a) compared to the other algorithms. In this scenario, the granularity and density of the micro-clusters were not sufficient to detect two close clusters with the SS index, which caused SESC to underestimate the number of clusters and, consequently, impacted the ARI. For most scenarios, ISESC and ISESC-CD have similar quality, which is expected since their only difference is the change-detection component. However, there are some punctual differences, as presented in the periods between the micro-batches 48 and 52, 67, and 74 of the YTG data set. In such periods, the ISESC can promptly detect changes in the data stream, while the ISESC-CD needs more time to discover (and react) to these changes. A different scenario occurred for the PPA data set, where the SS index that guides the F-EAC did not behave similarly to the ARI, as described in [12]. Such behavior caused significant ARI differences between ISESC and ISESC-CD, mainly between micro-batches 5 and 20 and between micro-batches 56 and 75. In both intervals, ISESC replaced the model with a  
Figure 7: ARI comparison among the presented algorithms over the five most relevant data set results. The markers represent the ARI score of each algorithm over each data set. The vertical grey bars represent the standard deviation. better-evaluated one (according to the SS index), while ISESC-CD incrementally updated its current model. As the SS and ARI indexes do not behave similarly to PPA, the increase in the SS value in ISESC caused a decrease in its ARI values. Figure 8 shows how ISESC was slightly better in terms of SS, when compared to ISESC-CD in the same intervals.