# Fuzzy Clustering Approaches Based on Numerical Optimizations of Modified Objective Functions
## 4 Experimental Studies
Table 3 given below summarizes the results of the experimental procedures, where for each dataset are applied the five modified versions of the objective function, respectively L2 REG (L\({}_{2}\) Regularization Term explained in Section 2.1), ENTR-T (Entropy term explained in Section 2.2), SP-IN (A Sparsity-Inducing Term explained in Section 2.3), PEN-LSC (A Penalty For Large Sized Clusters explained in Section 2.4), and SPAT-REG (Spatial Regularization Term explained in Section 2.5). For each of the modified versions, firstly, three numerical methods are applied: gradient descent (GD), root mean square propagation (RMSprop), and adaptive moment estimation (Adam). In order to provide a robust evaluation, each method is repeated 20 times, and the values shown in the table are the average scores obtained after these executions. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{2}{c}{**Classical FCM**} \\ \cline{2-3}  & **F-ARI** & **F-NMI** \\ \hline Banknotes & 0.73 & 0.74 \\ Liver & 0.68 & 0.72 \\ Yeast & 0.65 & 0.69 \\ Dermatology & 0.70 & 0.73 \\ Vehicle Silhouettes & 0.68 & 0.74 \\ LANDSAT Satellite & 0.69 & 0.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuzzy ARI and Fuzzy NMI scores for the classical FCM algorithm. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{
\begin{tabular}{c} **Objective Function** \\ **Modification** \\ \end{tabular} } & \multicolumn{2}{c}{**GD**} & \multicolumn{2}{c}{**RMSprop**} & \multicolumn{2}{c}{**Adam**} \\ \cline{3-8}  & & **F-ARI** & **F-NMI** & **F-ARI** & **F-NMI** & **F-ARI** & **F-NMI** \\ \hline \multirow{8}{*}{Banknotes} & L2 REG & 0.73 & 0.76 & 0.74 & 0.77 & 0.75 & 0.78 \\ \cline{2-8}  & ENTR-T & 0.76 & 0.79 & 0.77 & 0.8 & 0.78 & 0.81 \\ \cline{2-8}  & SP-IN & 0.75 & 0.8 & 0.76 & 0.81 & 0.77 & 0.82 \\ \cline{2-8}  & PEN-LSC & 0.78 & 0.81 & 0.79 & 0.82 & **0.80** & 0.83 \\ \cline{2-8}  & SPAT-REG & 0.77 & 0.82 & 0.78 & 0.83 & 0.79 & **0.84** \\ \hline \multirow{8}{*}{Liver} & L2 REG & 0.68 & 0.72 & 0.69 & 0.73 & 0.7 & 0.74 \\ \cline{2-8}  & ENTR-T & 0.71 & 0.75 & 0.72 & 0.76 & 0.73 & 0.77 \\ \cline{2-8}  & SP-IN & 0.72 & 0.77 & 0.73 & 0.78 & 0.74 & **0.81** \\ \cline{2-8}  & PEN-LSC & 0.74 & 0.78 & 0.75 & 0.79 & **0.76** & 0.8 \\ \cline{2-8}  & SPAT-REG & 0.73 & 0.79 & 0.74 & 0.8 & 0.75 & 0.79 \\ \hline \multirow{8}{*}{Yeast} & L2 REG & 0.66 & 0.7 & 0.67 & 0.71 & 0.68 & 0.72 \\ \cline{2-8}  & ENTR-T & 0.69 & 0.73 & 0.70 & 0.74 & 0.71 & 0.75 \\ \cline{2-8}  & SP-IN & 0.70 & 0.74 & 0.71 & 0.75 & 0.72 & 0.76 \\ \cline{2-8}  & PEN-LSC & 0.72 & 0.76 & 0.73 & 0.77 & **0.74** & 0.78 \\ \cline{2-8}  & SPAT-REG & 0.71 & 0.77 & 0.72 & 0.78 & 0.73 & **0.79** \\ \hline \multirow{8}{*}{Dermatology} & L2 REG & 0.70 & 0.75 & 0.71 & 0.76 & 0.72 & 0.77 \\ \cline{2-8}  & ENTR-T & 0.73 & 0.78 & 0.74 & 0.79 & 0.75 & 0.8 \\ \cline{1-1} \cline{2-8}  & SP-IN & 0.74 & 0.79 & 0.75 & 0.8 & 0.76 & 0.81 \\ \cline{1-1} \cline{2-8}  & PEN-LSC & 0.75 & 0.8 & 0.76 & 0.81 & **0.77** & 0.82 \\ \cline{1-1} \cline{2-8}  & SPAT-REG & 0.75 & 0.81 & 0.76 & 0.82 & **0.77** & **0.83** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fuzzy ARI and Fuzzy NMI scores across modifications and optimization methods. The Fuzzy ARI and Fuzzy NMI scores are visualized in two different line charts, as shown in Figures 1 and 2 below. The abbreviations for each line given in the chart legend are formed by concatenating the objective function modification abbreviations, such as L2 REG (L2 Regularization Term explained in Section 2.1), ENTR-T (Entropy term explained in Section 2.2), SP-IN (A Sparsity-Inducing Term explained in Section 2.3), PENLSC (A Penalty For Large Sized Clusters explained in Section 2.4), and SPAT-REG (Spatial Regularization Term explained in Section 2.5) with the abbreviation of the three employed numerical methods for optimization, gradient descent (GD), root mean square propagation (RMSprop), and adaptive moment estimation (Adam).