# An Optimal and Stable Algorithm for Clustering Numerical Data
## 3 Proposed Clustering Algorithm with a Constant Seeding Selection
### Computational Complexity
Based on the overall average scores for the combination of all eight datasets, the results provide evidence that the Zk-AMH algorithm is optimal and comparable to the other clustering algorithms. The box plot demonstrated the first piece of evidence in this regard, where the Zk-AMH approach yielded approximately similar median value with the k-AMH, k-AMH++, and fuzzy c-means scores (see Figure 3). Furthermore, the Zk-AMH scored the highest minimum score in 800 experimental runs (1 algorithm \(\times\) 100 runs \(\times\) 8 datasets). Finally, the one-way ANOVA test proved that the Zk-AMH algorithm is optimal and comparable to the fuzzy c-means and fuzzy c-means++ approaches. In addition, for the dataset comparison, the Zk-AMH algorithm clearly showed its advantages by producing the highest scores for four datasets, an approximately score for one dataset, and marginally lower values for the other three datasets (see Table 4). The second consideration regarding the Zk-AMH algorithm is the cluster stability. Most impressively, the Zk-AMH algorithm exhibited its strength by producing identical mean, minimum, and maximum scores, which led its standard deviation to decline to zero for all datasets (see Table 4). It is clearly evident that in the 100-run experiment for each dataset, the Zk-AMH managed to produce constant results. This scenario did not happen to the other algorithms, except for the fuzzy c-mean algorithm with dataset 5 alone. In fact, out of eight datasets, four produced the optimal values, whereas the other four datasets produced marginally lower scores. Further investigation of why these two datasets were less optimal is urgently required. Hence, the new seeding method seems to be exceptionally well suited and to work exclusively for the k-AMH clustering framework to achieve cluster optimality and stability. The new seeding might not work for the k-means and fuzzy c-means frameworks due to their center using the mean. The cluster optimality is actually inherited from the previous performance of k-AMH framework. Furthermore, the new seeding mechanism contributed to cluster stability. With the fixed seeding technique and using multidimensional zero points (the origin), the Zk-AMH algorithm converges in a globally optimal fashion toward better clustering performance. ## 7 Conclusions

Based on the experimental results above, the Zk-AMH algorithm is stable while seeding with zero-point multidimensional Cartesian spaces. In fact, the algorithm also maintains its optimal solution for numerical clustering and is comparable not only to its original algorithm, the k-AMH numeric algorithm and its extension k-AMH numeric++, but also its counterparts, the k-means, k-means++, fuzzy c-means, and fuzzy c-means++ algorithms. The most promising finding is that the Zk-AMH algorithm obtained identical results for all datasets in every run, due to its cluster stability. Thus, the seeding selection would no longer be an obstacle to producing optimal clustering results for every run. With its optimality and stability, the proposed Zk-AMH has the potential to be used in the future development of clustering tools, particularly for numerical clustering. This research received no external funding. **Institutional Review Board Statement:** Not Applicable. **Informed Consent Statement:** Not Applicable. Data Availability Statement: The datasets used in this study are taken from UCI Machine Learning Repository. We would like to thank the Dean, [PERSON] for supporting us in completing this study. The authors declare no conflict of interest. ## References

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