# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
As shown in the experiments, these superior center locations directly result in better-performing RBF neural networks. Given the observed superiority of the proposed opportunity matrix algorithm, it may be instructive to consider the locations of the RBF node centers that were chosen by the algorithm in comparison with the standard random sampling and \(k\)-means techniques. Figure 11 below, therefore, provides an illustrative example of the RBF center locations that were chosen by the opportunity matrix, random sampling, and \(k\)-means clustering techniques for a neural network with 64 RBF nodes that was seeking to approximate Himmelblau's function. Figure 11: Comparison of RBF node center locations chosen by the opportunity matrix, random sampling, and \(k\)-means clustering techniques for a neural network seeking to approximate Himmelblau’s function using 64 RBF nodes. As shown in Figure 11 above, excepting for one centrally located RBF node center, all of the other RBF centers chosen by the opportunity matrix algorithm were peripherally located when attempting to approximate Himmelblau's function. By contrast, the RBF center locations chosen by the \(k\)-means clustering technique were approximately equidistantly distributed throughout the bounded region of the input space, while no obvious distributional pattern was present for the RBF center locations chosen by the random sampling technique. Recalling from Section 2 that the values for the input parameters in the experiments involving Himmelblau's function were randomly generated from a uniform distribution, the theoretical rationale underlying the use of random sampling and \(k\)-means clustering is evident in Figure 11; i.e., to locate the RBF node centers in areas of the input space that best represent the distribution of the input features in the training data. Rather than focusing exclusively on the distribution of the inputs, the theoretical rationale underlying the opportunity matrix technique is to locate the RBF node centers in areas of the input space that are, on average, the largest sources of prediction error. When considering the RBF center locations chosen by the opportunity matrix algorithm in Figure 11 in conjunction with the visualization of [PERSON]'s function provided in Figure 7, it becomes immediately clear why the opportunity matrix algorithm chose the RBF center locations that it did--to target the areas of the input space that otherwise would likely yield large prediction errors. These visual elements are reproduced and presented side by side as Figure 12 below for the convenience of the reader. The core insight to be learned from this discussion--and the reason for the opportunity matrix algorithm's superior performance--is that identifying and taking steps to mitigate the most significant sources of prediction error appears to be a very effective strategy for selecting the locations of the centers in an RBF neural network. Finally, some consideration of the computational resources and wall-clock time required by each competing method of selecting RBF node center locations is merited. Assuming that the available training cases have been randomly shuffled, then the random sampling technique can reasonably be expected to require minimal memory and wall-clock time, since one may simply use the input values for the first \(k\) cases in the training data as the center locations for a network with \(k\) RBF nodes. By contrast, the \(k\)-means clustering technique can be expected to require much more memory and wall-clock time than random sampling, since the \(k\)-means clustering algorithm is known to be NP-hard [36] and requires the repeated calculation of the distance between each of the \(k\) centroids and every available

Figure 12: [PERSON]’s function and the corresponding RBF center locations that were chosen by the opportunity matrix algorithm. case in the training data.