# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
Since there was no ex ante reason to expect the distributional variances of the performance metrics obtained from each competing RBF center selection method to be equal, [PERSON]'s \(t\)-tests provided an appropriate statistical foundation for comparing the experimental results. ## 3 Results and Discussion

As described in the previous section, the experiments carried out in this study were designed to compare the performance of the proposed opportunity matrix algorithm against that of the standard random sampling and \(k\)-means clustering methods of selecting RBF node centers under a variety of different conditions. Each of these experimental conditions involved assessing the function approximation performance of an RBF neural network with 16, 32, 64, 1 \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Method of Selecting** \\ **Centers for RBF Nodes** \\ \end{tabular} & \begin{tabular}{c} **Average Observed Mean Absolute Error Across 30 Trials** \\ **16 RBF Nodes** \\ \end{tabular} & \begin{tabular}{c} **16 RBF Nodes** \\ \end{tabular} & \begin{tabular}{c} **Average Observed Mean Absolute Error across 30 Trials** \\ **64 RBF Nodes** \\ \end{tabular} & \
\begin{tabular}{c} **128 RBF Nodes** \\ \end{tabular} &
\begin{tabular}{c} **256 RBF Nodes** \\ \end{tabular} \\ \hline Random Sampling & 0.00349 & 0.00194 & 0.00194 & 0.00132 & 0.00154 \\ \hline \(k\)-Means Clustering & 0.00297 & 0.00217 & 0.00183 & 0.00129 & 0.00175 \\ \hline Opportunity Matrix & 0.00184 & 0.00135 & 0.00091 & 0.00063 & 0.00037 \\ \hline \multirow{2}{*}{**Method Comparison**} & \multicolumn{4}{c}{\(-\)**\(t\)-Values and Significances for Welch’s \(t\)-Tests**} \\  & **16 RBF Nodes** & **32 RBF Nodes** & **64 RBF Nodes** & **128 RBF Nodes** & **256 RBF Nodes** \\ \hline Opportunity Matrix vs. & \(-\)4.313 *** & \(-\)2.917 *** & \(-\)4.600 *** & \(-\)5.576 *** & \(-\)12.497 *** \\ \hline \multirow{2}{*}{\(k\)-Means Clustering} & \multicolumn{4}{c}{\(-\)3.406 ***} & \(-\)2.849 *** & \(-\)4.785 *** & \(-\)5.552 *** & \(-\)15.383 *** \\ \hline \hline \multicolumn{1}{c}{} & \multicolumn{4}{c}{\(*\)\(p\) < 0.01, *** \(p\) < 0.001.} \\ \end{tabular}
\end{table} 
Table 4: Experimental results for the approximation of the \(t\)-distribution CDF. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Method of Selecting** \\ **Centers for RBF Nodes** \\ \end{tabular} & \begin{tabular}{c} **16 RBF Nodes** \\ \end{tabular} & \begin{tabular}{c} **Average Observed Mean Absolute Error across 30 Trials** \\ **32 RBF Nodes** \\ \end{tabular} & \begin{tabular}{c} **1** \\ **64 RBF Nodes** \\ \end{tabular} & \
\begin{tabular}{c} **128 RBF Nodes** \\ \end{tabular} &
\begin{tabular}{c} **256 RBF Nodes** \\ \end{tabular} \\ \hline Random Sampling & 0.00423 & 0.00068 & 0.00029 & 0.00022 & 0.00025 \\ \hline \(k\)-Means Clustering & 0.00379 & 0.00062 & 0.00025 & 0.00021 & 0.00022 \\ \hline Opportunity Matrix & 0.00176 & 0.00032 & 0.00021 & 0.00019 & 0.00020 \\ \hline \multirow{2}{*}{**Method Comparison**} & \multicolumn{4}{c}{\(-\)**\(t\)-Values and Significances for Welch’s \(t\)-Tests**} \\  & **16 RBF Nodes** & **32 RBF Nodes** & **64 RBF Nodes** & **128 RBF Nodes** & **256 RBF Nodes** \\ \hline \multirow{2}{*}{\(k\)-Means Clustering} & \multicolumn{4}{c}{\(-\)3.079 ***} & \(-\)3.778 *** & \(-\)7.162 *** & \(-\)15.374 *** & \(-\)25.258 *** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results for the approximation of the Poisson distribution PMF.