# Clustering Algorithm with a Greedy Agglomerative Heuristic and Special Distance Measures
## 2 Kohonen Neural Networks for Clustering Problem
### Proposed Algorithms
Results were tested with Wilcoxon rank sum test. Statistically significant superiority in objective function value between BVQ algorithm and its greedy version, SOM algorithm and its greedy version at \(p\leq 0.05\) highlighted in bold. It turned out that, in the vast majority of cases, minimal objective function value was demonstrated by SOM algorithm without influence of initialization method or distance measure. Minimal value of coefficient of variation and span factor were achieved with Euclidean, squared Euclidean and Manhattan distance and initialization by average. From Figures 3-5, it can be seen that initialization by k-means increases the coefficient of variation value for any type of mixed lot composition. \begin{table}
\begin{tabular}{c l c c c c c} \hline \hline
**Algorithm** &
\begin{tabular}{c} **Initialization** \\ **Method** \\ \end{tabular} & **ChD** & **EuD** & **SEuD** & **MahD** & **ManD** \\ \hline \multirow{3}{*}{BVQ} & average & 0.39 & 0.60 & 0.60 & 0.65 & 0.63 \\  & k-means & 0.76 & 0.60 & 0.60 & 0.49 & 0.48 \\  & random & 0.71 & 0.59 & 0.59 & 0.52 & 0.59 \\ \hline \multirow{3}{*}{BVQ-GREEDY(2)} & average & 0.99 & 0.99 & 0.64 & 0.50 & 0.64 \\  & k-means & 0.92 & 0.92 & 0.92 & 0.50 & 0.99 \\  & random & 0.99 & 0.59 & 0.59 & 0.58 & 0.59 \\ \hline \multirow{3}{*}{BVQ-GREEDY(3)} & average & 0.99 & 0.99 & 0.99 & 0.50 & 0.99 \\  & k-means & 0.74 & 0.74 & 0.74 & 0.57 & 0.99 \\  & random & 0.99 & 0.59 & 0.74 & 0.56 & 0.59 \\ \hline \multirow{3}{*}{SOM} & average & 0.99 & 0.99 & 0.99 & 0.97 & 0.98 \\  & k-means & 0.38 & 0.98 & 0.65 & 0.98 & 0.98 \\  & random & 0.99 & 0.65 & 0.68 & 0.59 & 0.49 \\ \hline \multirow{3}{*}{SOM-GREEDY(2)} & average & 0.99 & 0.75 & 0.75 & 0.47 & 0.47 \\  & k-means & 0.72 & 0.74 & 0.74 & 0.52 & 0.59 \\  & random & 0.64 & 0.58 & 0.58 & 0.56 & 0.59 \\ \hline \multirow{3}{*}{SOM-GREEDY(3)} & average & 0.64 & 0.62 & 0.74 & 0.47 & 0.48 \\  & k-means & 0.58 & 0.74 & 0.74 & 0.57 & 0.59 \\ \cline{1-1}  & random & 0.66 & 0.59 & 0.59 & 0.56 & 0.37 \\ \hline \multirow{3}{*}{k-means} & random & 0.39 & 0.65 & 0.65 & 0.65 & 0.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Accuracy of device clustering for BVQ, SOM, BVQ-GREEDY(2), BVQ-GREEDY(3), SOM-GREEDY(2) and SOM-GREEDY(3) algorithms (four-batch mixed lot). Figure 4: Coefficient of variation of the objective function value for the three-batch mixed lot. Figure 3: Coefficient of variation of the objective function value for the two-batch mixed lot. ## 4 Discussion

It can be summarized that the use of the greedy agglomerative heuristic procedure with the simple competitive learning algorithm in online mode improves the objective function value in the majority of the cases. In the other cases, the difference between our new algorithms and the known algorithms is insignificant (see two exceptions in our computational experiments: the four batches with Chebyshev distance and two batches with Mahalanobis distance). It can be noted that, in the vast majority of cases, the use of a triple number of neurons in the greedy agglomerative heuristic procedure provided a smaller value of the objective function than double. Regarding the stability of the results (coefficient of variation and span factor values), none of the algorithms have demonstrated advantages over the others. The minimal values of the objective function were achieved with the squared Euclidean distance measure for two batches and with the Euclidean distance measure for three batches and with the Chebyshev distance measure for four batches. In almost all the cases, the clustering accuracy of the SCL-GREEDY version was the same as or better than the SCL algorithm. The only exception was the three-batch mixed lot with Chebyshev distance. Concerning batch mode algorithms, the situation was different. The greedy version of the BVQ algorithm demonstrated a better objective function value than the BVQ in 42% of the cases and a worse objective function value than the BVQ in 37% of the cases. In 23% of the cases, the difference was statistically insignificant.