## 4 Results
### Algorithms Comparison
\begin{table}
\begin{tabular}{c l} \hline
**Algorithm** & **Parameters for Determination** \\ \hline \multirow{2}{*}{AVQ} & 1. Maximum number of iterations \\  & 2. Constant parameter of the learning rate \\ \hline \multirow{4}{*}{SOM} & 1. Dimension (1D or 2D) \\  & 2. Map shape \\  & 3. Map size \\  & 4. Weights initialization \\  & 6.Learning method \\  & 6.Learning function (type, initial learning rate, training epochs) \\  & 7.Neighborhood function (type, initial neighbourhood radius) \\ \hline Hopfield & Maximum number of iterations \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters of the neural network-based algorithms. Criterion#3 refers to the comparison via the validity indicators. The comparison per algorithm category is shown in Figures 1-10. In the present paper, no information about the number of clusters is available, therefore this number should be determined by the validity indicator. The algorithms are executed for 2 to 30 clusters and for each number the score of the validity indicator is checked. Each algorithm is applied separately to the data set of the consumer. In the present paper, the maximum number of 30 is near the 10% of the patterns population, \(N\) = 365. The superiority of an algorithm over the others is indicated when it leads, depending on the indicator, to lower or higher values in most of the clusters if not all. In some cases, an algorithm is more robust for certain number of clusters but it is surpassed by another for other number of clusters. Therefore, the general behaviour of an algorithm over a validity indicator should be examined. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters for Determination** \\ \hline \multirow{3}{*}{FDL} & 1. Maximum number of iterations \\  & 2. Initial centroids (_optional_) \\  & 3. Parameter \(\rho\) \\ \hline \multirow{3}{*}{ISODATA} & 1. Maximum number of clusters \\  & 2. Maximum number of clusters for merging \\  & 3. Maximum number of iterations \\  & 4. Threshold of number of patterns in a cluster \\  & 5. Threshold of distance for cluster merging \\  & 6. Threshold of standard deviation for cluster split \\  & 7. Minimum distance between patterns and centroid \\ \hline BCEC1 & Merging stopping criterion \\ \hline BCEC2 & Merging stopping criterion \\ \hline CSC & Merging stopping criterion \\ \hline \multirow{3}{*}{SVC} & 1.Parameter that controls the number of outliers \\  & 2. Scale parameter of the Gaussian kernel \\  & 3. Minimum distance \\  & 4. Cluster formation threshold \\ \hline IRC & 1. Maximum number of iterations \\  & 2. Parameter \(\rho\) \\ \hline \multirow{3}{*}{CLA} & 1. Maximum number of iterations \\  & 2. Initial centroids (_optional_) \\  & 3. Constant term of learning rate (winner neuron) \\  & 4. Constant term of learning rate (rest neurons) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the rest algorithms. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameter** \\ \hline K-means & - \\ Modified K-means\#1 & - \\ Modified K-means\#2 & - \\ WFA K-means & - \\ IWFA K-means & - \\ Hopfield K-means & - \\ minCEntropy & - \\ K-means\_A & - \\ K-means\_B & - \\ K-medoids & - \\ SL & - \\ CL & - \\ UPGMA & - \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters updating requirements. The comparison of the partitional algorithms per validity indicator is illustrated in Figures 1 and 2. The number of pair of values of the coefficients \(\{a,b\}\) for the Modified K-means#1 and IWFA K-means are 1295. This means that 1295 clustering are generated from each algorithm. Only the one that leads to the lowest error is kept. In Figures 1 and 2 the term \"optimal\" refers to the pair of values with lower error and the term \"average\" refer to the average value of the 1295 clusterings. For each validity indicator, different optimal pair of values is obtained. It can be noticed that there is no algorithm that wins the competition in all indicators, a finding that confirms the conclusions of the literature in algorithms comparison.