# Clustering Algorithm with a Greedy Agglomerative Heuristic and Special Distance Measures
## 2 Kohonen Neural Networks for Clustering Problem
### Proposed Algorithms
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**BVQ**} & \multirow{2}{*}{**SOM**} & \multicolumn{2}{c}{**BVQ-**} & \multicolumn{2}{c}{**BVQ-**} & \multicolumn{2}{c}{**SOM-**} & \multicolumn{2}{c}{**SOM-**} \\  & & & **GREEDY(2)** & **GREEDY(3)** & **GREEDY(2)** & **GREEDY(3)** \\ \hline \hline \multicolumn{7}{c}{Two-batch mixed lot} \\  & \multicolumn{2}{c}{\(KI=2\)} & \multicolumn{2}{c}{\(KI=4\)} & \multicolumn{2}{c}{\(KI=6\)} & \multicolumn{2}{c}{\(KI=4\)} & \multicolumn{2}{c}{\(KI=6\)} \\ Min & 186.160 & 185.934 & 186.346 & 186.372 & 186.346 & 186.372 \\ Max & 186.160 & 186.214 & 335.401 & 337.849 & 335.401 & 337.849 \\ Mean & 186.160 & 186.071 & 226.094 & 297.455 & 236.031 & 266.997 \\ \(\sigma\) & 0.000 & 0.064 & 68.228 & 69.337 & 72.732 & 78.067 \\ V & 0.000 & 0.035 & 30.177 & 23.310 & 30.814 & 29.239 \\ R & 0.000 & 0.281 & 149.055 & 151.477 & 149.055 & 151.477 \\ \multicolumn{7}{c}{Three-batch mixed lot} \\  & \multicolumn{2}{c}{\(KI=3\)} & \multicolumn{2}{c}{\(KI=6\)} & \multicolumn{2}{c}{\(KI=9\)} & \multicolumn{2}{c}{\(KI=6\)} & \multicolumn{2}{c}{\(KI=9\)} \\ Min & 314.924 & 241.851 & 241.887 & 241.991 & 314.924 & 310.298 \\ Max & 700.891 & 555.231 & 706.183 & 705.449 & 711.926 & 892.642 \\ Mean & 692.693 & 508.407 & 645.549 & 633.408 & 705.483 & 714.682 \\ \(\sigma\) & 4.243 & 25.699 & 72.850 & 81.067 & 5.039 & 49.382 \\ V & 0.613 & 5.055 & 11.285 & 12.799 & 0.714 & 6.910 \\ R & 10.247 & 62.187 & 154.257 & 212.213 & 15.941 & 196.311 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Objective function value summarized after 30 attempts (Manhattan distance, initialization by k-means). ## References

* (1) [PERSON]; [PERSON]; [PERSON] A comparison study on similarity and dissimilarity measures in clustering continuous data. _PLoS ONE_**2015**, _10_, e0144059. [CrossRef] [PubMed]
* (2) [PERSON]; [PERSON] A clustering method based on k-means algorithm. _Phys. Procedia_**2012**, _25_, 1104-1109. * (3) [PERSON] Sur la divisiondes corps materiels en parties. _Bull. Acad. Polon. Sci._**1956**, \(4\), 801-804. * (4) [PERSON]; [PERSON] On the point for which the sum of the distances to n given points is minimum. _Ann. Oper. Res._**2009**, _167_, 7-41. [CrossRef]
* (5) [PERSON] A sequential method for discrete optimization problems and its application to the assignment, traveling salesman and tree scheduling problems. _J. Inst. Math. Appl._**1965**, _13_, 362-375. * (6) [PERSON] Least squares quantization in PCM. _IEEE Trans. Inf. Theory_**1982**, _28_, 129-137. [CrossRef]
* (7) [PERSON]; [PERSON] K-means++: The advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, USA, 7-9 January 2007; pp. 1027-1035. * (8) [PERSON]; [PERSON], U.M. Refining initial points for k-means clustering. In Proceedings of the Fifteenth International Conference on Machine Learning (ICML 1998), Madison, WI, USA, 24-27 July 1998; Volume 98, pp. 91-99. * (9) [PERSON]; [PERSON]; [PERSON] Comparison of k-means clustering initialization approaches with brute-force initialization. In _Advanced Computing and Systems for Security. Advances in Intelligent Systems and Computing_; Springer: Singapore, 2017; Volume 567, pp. 103-114. * (10) [PERSON]; [PERSON]; [PERSON] Less is more: Simple algorithms for the minimum sum of squares clustering problem. _IMA J. Manag. Math._**2021**, dpab031. [CrossRef]
* (11) [PERSON]; [PERSON] A hybrid approach using genetic algorithm and the differential evolution heuristic for enhanced initialization of the k-means algorithm with applications in text clustering. _Soft Comput._**2019**, _23_, 6361-6378. [CrossRef]
* (12) [PERSON] Data clustering: 50 years beyond k-means. _Pattern Recogn. Lett._**2010**, _31_, 651-666. [CrossRef]
* (13) [PERSON]; [PERSON]; [PERSON]. The k-means algorithm: A comprehensive survey and performance evaluation. _Electronics_**2020**, \(9\), 1295. [CrossRef]* [PERSON] et al.