# An Optimal and Stable Algorithm for Clustering Numerical Data
## 3 Proposed Clustering Algorithm with a Constant Seeding Selection
### Computational Complexity
\begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline \multirow{2}{*}{**FMI Score**} & \multirow{2}{*}{**Algorithm**} & \multicolumn{8}{c}{**Dataset**} \\ \cline{3-10}  & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Mean & k-means & 0.641 & 0.507 & 0.563 & 0.540 & 0.660 & **0.522** & 0.311 & 0.247 \\  & k-means++ & 0.694 & 0.514 & 0.597 & 0.569 & 0.699 & 0.414 & 0.312 & 0.248 \\  & Fuzzy c-means & 0.891 & 0.502 & 0.633 & 0.701 & **0.898** & 0.509 & 0.455 & 0.266 \\  & Fuzzy c-means++ & 0.823 & **0.516** & 0.602 & 0.589 & 0.757 & 0.407 & 0.447 & 0.251 \\  & k-AMH numeric & 0.901 & 0.488 & **0.661** & 0.713 & 0.885 & 0.446 & 0.498 & 0.261 \\  & k-AMH numeric++ & 0.900 & 0.487 & 0.645 & 0.689 & 0.883 & 0.455 & 0.486 & 0.262 \\  & Zk-AMH & **0.908** & 0.494 & 0.623 & **0.732** & 0.893 & 0.413 & **0.583** & **0.278** \\ \hline Min. & k-means & 0.373 & 0.421 & 0.307 & 0.425 & 0.332 & 0.333 & 0.185 & 0.162 \\  & k-means++ & 0.495 & 0.392 & 0.320 & 0.480 & 0.486 & 0.277 & 0.181 & 0.151 \\  & Fuzzy c-means & 0.536 & 0.490 & 0.623 & 0.455 & **0.898** & 0.362 & 0.326 & 0.209 \\  & Fuzzy c-means++ & 0.457 & 0.406 & 0.310 & 0.395 & 0.456 & 0.239 & 0.255 & 0.186 \\  & k-AMH numeric & 0.884 & 0.418 & 0.619 & 0.441 & 0.705 & 0.315 & 0.435 & 0.223 \\  & k-AMH numeric++ & 0.870 & 0.472 & 0.603 & 0.451 & 0.804 & 0.297 & 0.320 & 0.205 \\  & Zk-AMH & **0.908** & **0.494** & **0.623** & **0.732** & 0.893 & **0.413** & **0.583** & **0.278** \\ \hline Max. & k-means & 0.844 & 0.643 & **0.729** & 0.672 & 0.880 & **0.653** & 0.467 & **0.334** \\  & k-means++ & 0.829 & **0.675** & 0.714 & 0.725 & 0.885 & 0.594 & 0.458 & 0.319 \\  & Fuzzy c-means & 0.904 & 0.568 & 0.641 & 0.728 & 0.898 & 0.594 & 0.578 & 0.307 \\  & Fuzzy c-means++ & 0.960 & 0.659 & 0.723 & **0.765** & **0.911** & 0.590 & **0.659** & 0.304 \\  & k-AMH numeric & 0.928 & 0.569 & 0.696 & 0.743 & 0.904 & 0.624 & 0.595 & 0.310 \\  & k-AMH numeric++ & 0.925 & 0.512 & 0.696 & 0.752 & 0.901 & 0.624 & 0.590 & 0.306 \\  & Zk-AMH & **0.908** & 0.494 & 0.623 & 0.732 & 0.893 & 0.413 & 0.583 & 0.278 \\ \hline Std. Dev. & k-means & 0.118 & 0.040 & 0.136 & 0.063 & 0.125 & 0.062 & 0.057 & 0.034 \\  & k-means++ & 0.101 & 0.050 & 0.071 & 0.061 & 0.112 & 0.067 & 0.060 & 0.037 \\  & Fuzzy c-means & 0.051 & 0.009 & 0.005 & 0.046 & 0.000 & 0.089 & 0.070 & 0.022 \\  & Fuzzy c-means++ & 0.126 & 0.054 & 0.086 & 0.121 & 0.129 & 0.075 & 0.086 & 0.027 \\  & k-AMH numeric & 0.009 & 0.026 & 0.028 & 0.038 & 0.020 & 0.077 & 0.047 & 0.019 \\  & k-AMH numeric++ & 0.008 & 0.009 & 0.029 & 0.066 & 0.013 & 0.079 & 0.045 & 0.023 \\  & Zk-AMH & **0** & **0** & **0** & **0** & **0** & **0** & **0** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean, Maximum, Minimum, and Standard Deviation for Each Algorithm and Dataset. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**FM Index—Games–Howell** & & & & & **95\% CI** \\ \hline
**(I) Algo.** & **(J) Algo.** & **Mean Diff. (I-)** & **Std. Err.** & \(p\)**-Value** & **Lower Bound** & **Upper Bound** \\ \hline Zk-AMH & k-means & 0.12 & 0.01 & \textless{}0.01 & 0.09 & 0.14 \\  & k-means++ & 0.11 & 0.01 & \textless{}0.01 & 0.08 & 0.14 \\  & Fuzzy c-means & 0.01 & 0.01 & 0.98 & \(-0.02\) & 0.04 \\  & Fuzzy c-means++ & 0.07 & 0.01 & \textless{}0.01 & \(-0.04\) & 0.10 \\  & k-AMH numeric & 0.01 & 0.01 & 0.98 & \(-0.02\) & 0.04 \\  & k-AMH numeric++ & 0.01 & 0.01 & 0.81 & \(-0.02\) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Multiple Comparison of the Zk-AMH Algorithm with the other Six Algorithms for a Combined Dataset. 5; and marginally lower mean accuracies for datasets 2, 3 and 6. Furthermore, the Zk-AMH algorithm produced the highest minimum score for all datasets except dataset 5, which had a marginally lower score. The Zk-AMH algorithm also demonstrated its competitiveness compared with the other six algorithms with marginally lower maximum scores. ## 6 Discussion

The first consideration regarding the Zk-AMH algorithm is the cluster optimality.