# An Optimal and Stable Algorithm for Clustering Numerical Data
## 3 Proposed Clustering Algorithm with a Constant Seeding Selection
### Computational Complexity
It can be used to discover inherent data structures in the clustering results [1] and measures the degree of correspondence between the clusters and the priori classes assigned to them. The Fowlkes-Mallows (FM) Index [33] was used to measure the performance of clustering algorithms as Equation (12)

\[FM=\sqrt{\bigg{(}\frac{TP}{TP+FP}\bigg{)}\bigg{(}\frac{TP}{TP+FN}\bigg{)}} \tag{23}\]

where \(TP\) is the number of true positives, \(FP\) is the number of false positives, and \(FN\) is the number of false negatives. ## 5 Results

This section presents the results obtained from 100 experimental runs for each of the seven algorithms and eight datasets. The experimental results provide evidence that the new Zk-AMH seeding algorithm performs better than the existing algorithms. Based on the experimental results, the results of the analysis focus on two main factors according to the objectives stated earlier. First, to provide evidence that the new Zk-AMH seeding algorithm is optimal (Cluster optimality). This is the first goal that is primarily required for any new clustering algorithm. Second, to provide evidence that the Zk-AMH is stable (Cluster stability). Hence, FMI scores such as the mean, maximum, minimum, and standard deviation were chosen to compare the clustering performance of the Zk-AMH approach with respect to the six algorithms as listed in Table 2. Please note that for fuzzy clustering algorithms such as the fuzzy c-means, fuzzy c-means++, k-AMH, k-AMH++, and Zk-AMH algorithms, the weighting exponent of \(\psi\) was set to 1.1. ### Cluster Optimality

Figure 3 shows an initial comparison of the seven algorithms for the combination of the eight datasets considered. These results demonstrate that the Zk-AMH algorithm is certainly competitive and comparable to its counterparts, the k-means, k-means++, fuzzy c-means, fuzzy c-means++, k-AMH numeric, and k-AMH numeric++ algorithms. Overall, the Zk-AMH approach produced the highest minimum scores and marginally lower maximum scores than the other algorithms. Moreover, the box plot for the Zk-AMH algorithm is larger than those of the other algorithms, particularly in the third quartile. It indicates that the Zk-AMH algorithm attained scores that were greater than the median score and close to the maximum score as compared to the other algorithms. Since the above results indicate that the proposed approach is very competitive, a one-way ANOVA was conducted to support these results. The test indicated that the assumption of homogeneity of variance was violated; therefore, the Welch F-ratio was reported. There was also significant variance in the clustering scores among the seven algorithms, in which \(F(6,2483)=59,p<0.01,\omega 2=0.05\). Thus, the Games-Howell post hoc test was used to compare the seven algorithms. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Algorithm** & **Seeding Method** & **Cluster Approach** & **Cluster Center** \\ \hline
1. k-means [10] & Random & Hard & Mean \\
2. k-means++ [18] & Probability & Hard & Mean \\
3. Fuzzy c-means [11] & Random & Soft & Mean \\
4. Fuzzy c-means++ [20] & Probability & Soft & Mean \\
5. k-AMH Numeric [27] & Random & Soft & Object \\
6. k-AMH Numeric++ & & & \\ (Using k-means++ seeding) & Probability & Soft & Object \\
7. Zk-AMH & & & \\ (The proposed algorithm) & Zero-Seeding & Soft & Object \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of algorithms used for comparison. Table 3 compares the results of the Games-Howell post hoc test for the Zk-AMH algorithm with those for the other six algorithms. At the 5% significance level and with \(p<0.01\), the mean score of the Zk-AMH algorithm (\(mean=0.62,95\%\) confidence interval \([0.60,0.63]\)) differs from those of the k-means (\(mean=0.50,95\%\) confidence interval \([0.49,0.51]\)) and k-means++ (\(mean=0.51,95\%\) confidence interval \([0.49,0.52]\)) algorithms. Thus, the k-AMH algorithm outperformed the k-means and k-means++ algorithms in terms of clustering performance.