# Optimization Algorithms for Scalable Stream Batch Clustering with k Estimation
## 5 Experiments
### Quantitative Analysis of Quality  
The quantitative analysis presented here compares the scores of the algorithms among themselves. In terms of ARI, the algorithm's score is the number of micro-batches (out of 100) for which the algorithm has the best or statistically equivalent result. For each data set, these results are tested over the repetitions of the algorithms. High scores indicate that the algorithm obtained better-quality results more often. For example, the algorithm SGMS achieved the best-quality result (or statistically equivalent) for 88 out of 100 micro-batches over the data set 100k5-2. Table 4 shows the score of each algorithm over each data set. The bold cells highlight the highest sum of points for each data set. For fairness, Total 1 sums the scores for the data sets with moderate dimensionality, which could be processed by SPGMS (100k5-2, 1M5-2, 1M10-5, 10M10-5, KC99). Total 2 summarizes the score for all the data sets. ISESC and ISESC-CD produced similar-quality results, mainly on artificial data sets, as shown in Table 4. Moreover, ISESC-CD was even better than ISESC in some 1M10-5, 10M10-5, KC99, and PPA micro-batches. This result evidences that the change-detection component of ISESC-CD did not affect the algorithm's quality in most scenarios. A similar result is presented in [13]. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**\begin{tabular}{} \end{tabular}** & **\begin{tabular}{} \end{tabular}** & **\begin{tabular}{} \end{tabular}** & **\begin{tabular}{} \end{tabular}** & **\
\begin{tabular}{} \end{tabular}** & **
\begin{tabular}{} \end{tabular}** \\ \hline
100k5-2 & 42 & 83 & 83 & **88** & 37 & 86 \\
1M5-2 & 40 & 65 & 65 & 68 & 29 & **78** \\
1M10-5 & 25 & 51 & 52 & 49 & 47 & **73** \\
10M10-5 & 30 & 64 & **65** & 22 & 19 & **65** \\ KC99 & 34 & 74 & **78** & 1 & 11 & 62 \\ YTG & **74** & 53 & 44 & 0 & 0 & 62 \\ PPA & 70 & 38 & **71** & 0 & 0 & 46 \\ \hline Total 1 & 171 & 337 & 343 & 228 & 143 & **364** \\ Total 2 & 315 & 428 & 458 & 228 & 143 & **472** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ARI rank scores. The table shows in how many micro-batches the algorithm was the best, in terms of ARI, or statistically equivalent to the best. The bold cells highlight the best algorithm for each data set. Figure 8: SS comparison—PPA data set. The vertical axis represents the SS score and the horizontal axis represents the micro-batch position. SESC performed worse than ISESC and ISESC-CD for most data sets, except for PPA and YTG. It is interesting to note that SESC performed better than other algorithms in specific situations, such as in the interval between micro-batch 43 and 76 of YTG or 21 and 33 of PPA (as shown in Figure 7d and 7e, respectively). Reflecting on the ARI values, such intervals contain overlapping clusters, and SESC could detect and learn them better. Summarizing data on micro-clusters smooths the overlapping of clusters for the SS evaluation compared to raw data. Sometimes, this smoothness improved the SS evaluation, resulting in higher ARI values. Such an effect rarely happens for algorithms that analyze raw data. It is important to note that the systematic definition of the number of micro-clusters, described at the beginning of Section 5, improved the quality results for YTG and PPA concerning [13]. SGMS produced high-quality results for 1M5-2 and 1M10-5, the best algorithm for 100K5-2. Such results reside in the well-behaved Gaussian clusters of these data sets, the best scenario for SGMS. However, low-quality results were obtained for 10M10-5, mainly due to the increased number of clusters overlapping in the last 30 micro-batches. On real data sets, SGMS was the worst algorithm (also shown in Figure 7). It did not even obtain a score on the PPA and YTG data sets. This occurred because SGMS inherited from G-means the drawback of overestimating \(k\) in non-Gaussian-distributed clusters.