# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
\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 & 2.56237 & 2.49915 & 2.43682 & 2.26433 & 1.94755 \\ \hline \(k\)-Means Clustering & 2.62083 & 2.52551 & 2.47267 & 2.42276 & 2.03874 \\ \hline Opportunity Matrix & 2.38783 & 2.35383 & 2.27311 & 1.87854 & 1.34394 \\ \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. & \(-\)2.264 * & \(-\)2.685 ** & \(-\)4.980 *** & \(-\)7.237 *** & \(-\)20.783 *** \\ \hline Opportunity Matrix vs. & \(-\)3.079 ** & \(-\)3.778 *** & \(-\)7.162 *** & \(-\)15.374 *** & \(-\)25.258 *** \\ \hline \hline \end{tabular}

* \(p\) < 0.05, *
* \(p\) < 0.01, **
* \(p\) < 0.001.
* Indicates that opportunity matrix algorithm outperformed competing method.

\end{table} 
Table 5: Experimental results for the real-world steel industry energy consumption dataset. and it is evident that this function approximation capability depends in part on the number of RBF nodes that are included in the network architecture. What is also evident from the average MAEs reported in Tables 1-5 above is that the ability of an RBF neural network to accurately approximate a function depends critically on the locations of the network's RBF node centers. Put simply, an RBF neural network with well-chosen centers will deliver more accurate predictions than an otherwise identical RBF neural network with poorly chosen centers. With this in mind, the primary goal of the current research was to propose a new algorithm for selecting RBF centers based on the concept of an opportunity matrix, and then experimentally investigate the proposed algorithm's performance in comparison with the two most widely used methods of selecting centers. Generally speaking, [PERSON]'s \(t\)-test results reported in Tables 1-5 reveal that the proposed opportunity matrix algorithm is typically able to statistically outperform the standard random sampling and \(k\)-means clustering methods of selecting centers for RBF neural networks, often by a very large margin. Indeed, among the 25 head-to-head comparisons with the random sampling method, the performance of the opportunity matrix algorithm was statistically superior in 24 of the experimental conditions and was statistically indistinguishable in the remaining condition. Among the 25 head-to-head comparisons with the \(k\)-means clustering method, the performance of the opportunity matrix algorithm was statistically superior in 21 of the experimental conditions, statistically indistinguishable in 3 experimental conditions, and statistically inferior in just 1 of the 25 conditions. Collectively, then, the proposed opportunity matrix algorithm was able to select RBF node centers that resulted in statistically better-performing RBF neural networks in 90% of the head-to-head comparisons with alternative methods that were considered in the experiments. Among the remaining head-to-head comparisons, the performance of the proposed opportunity matrix algorithm was statistically identical to an alternative method in 8% of the cases and was statistically inferior in only 2% of the cases. The results of the experiments, therefore, indicate that the proposed opportunity matrix algorithm is generally superior to the standard random sampling and \(k\)-means clustering methods in terms of its ability to select high-quality locations for RBF node centers.