# 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
**Method of Selecting** & \(\mathsf{r}\) & \multicolumn{3}{c}{**Average Peak Memory Requirement (MB)**} & \multicolumn{1}{c}{\({}^{1}\)} \\
**Centers for RBF Nodes** & **16 RBF Nodes** & **32 RBF Nodes** & **64 RBF Nodes** & **128 RBF Nodes** & **256 RBF Nodes** \\ \hline Random Sampling & 604.41 & 604.61 & 604.61 & 604.63 & 604.66 \\ \hline \(k\)-Means Clustering & 683.20 & 695.75 & 714.12 & 714.52 & 727.37 \\ \hline Opportunity Matrix & 601.33 & 610.98 & 616.45 & 621.48 & 628.59 \\ \hline
**Method of Selecting** & \(\mathsf{r}\) & \multicolumn{3}{c}{**Average Wall-Clock Time (Seconds)**} & \multicolumn{1}{c}{\({}^{1}\)} \\
**Centers for RBF Nodes** & **16 RBF Nodes** & **32 RBF Nodes** & **64 RBF Nodes** & **128 RBF Nodes** & **256 RBF Nodes** \\ \hline Random Sampling & \(<\)0.01 & \(<\)0.01 & \(<\)0.01 & \(<\)0.01 & \(<\)0.01 \\ \hline \(k\)-Means Clustering & 2.53 & 5.83 & 11.27 & 23.31 & 55.74 \\ \hline Opportunity Matrix & 534.05 & 1100.89 & 2236.67 & 4665.81 & 9720.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average peak memory and wall-clock time required to select RBF node centers for the Hosaki function. and then iteratively adding one RBF node at a time to the structure of the neural network. In each iteration, a small sample of the available training data is used as the basis for estimating the weights for the current RBF neural network architecture. These weights are subsequently utilized to generate predictions for the sample inputs, with the corresponding prediction errors being recorded in the appropriate regions of the opportunity matrix. Finally, the opportunity matrix is examined to identify the region of the input space that currently exhibits the largest average error, and the centroid of that region is used as the center for the next RBF node that is added to the neural network structure. New RBF nodes are iteratively added to the neural network in this manner until a stopping criterion is met. A large set of experiments was conducted in order to evaluate the performance of the proposed opportunity matrix algorithm in comparison with the two most widely used methods of selecting centers for RBF neural networks, namely, random sampling of the training data and \(k\)-means clustering. The experiments involved the task of using RBF neural networks to approximate four different well-known mathematical optimization and statistical functions or to predict the energy consumption for a steel manufacturing facility. To ensure generalizability, two of these functions used only continuous input parameters, while the remaining two functions and the real-world steel industry dataset included both continuous and discrete input parameters. A variety of neural network architectures were also considered in the experiments, with each network containing 16, 32, 64, 128, or 256 RBF nodes. The results of the experiments revealed the general superiority of the proposed opportunity matrix algorithm. Specifically, in head-to-head comparisons with the standard random sampling and \(k\)-means clustering techniques, RBF neural networks whose center locations were chosen using the opportunity matrix approach were found to generate statistically superior predictions 90% of the time. Among the remaining experimental conditions, the opportunity matrix algorithm yielded statistically identical outcomes in 8% of the cases, and was found to perform more poorly than one of the standard methods in only 2% of the cases. It can therefore be concluded that using the proposed opportunity matrix algorithm to select the locations of RBF node centers will, on average, result in RBF neural networks with superior function approximation capabilities.