# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
\(t\)-Distribution cumulative distribution function (CDF):

\[cdf(t,df)=1-\frac{1}{2}I_{x(t)}\left(\frac{df}{2},\frac{1}{2}\right) \tag{6}\]

where \(t\) is the observed \(t\)-value, \(df\) is the degrees of freedom, \(I\) is the regularized incomplete beta function [31], and \(x(t)=\frac{df}{2}\). In addition to the steel industry power consumption dataset, the functions defined in Equations (3)-(6), and the three different RBF node selection methods, five possible values for the number of RBF nodes (\(k\)) were used in the experiments, with \(k\in\{16\), 32, 64, 128, 256\(\}\). Each experimental condition thus involved (1) using three RBF networks, each of which sought to approximate one of the four functions defined above or predict the power consumption of a steel manufacturing facility; (2) using one of the possible values of \(k\) for the number of RBF nodes in those networks; and (3) using the three competing RBF node selection methods to choose the locations of the RBF node centers, with one of the different methods (random sampling, \(k\)-means clustering, or the proposed opportunity matrix algorithm) being used for each of the experiment's three RBF networks. Given the five different function approximation tasks and the five possible values for the number of RBF nodes, a total of \(5\times 5=25\) experiments were carried out in the study, each of which compared the performances of the three competing RBF node selection methods. Each experiment was also repeated 30 times in order to ensure that the distributions of the resulting performance metrics would be approximately Gaussian, per the central limit theorem [32]. In each experiment, the same data were used to train all three competing RBF neural networks. Each of these neural networks also had precisely the same structure (including the number of RBF nodes) and relied on the same Adam stochastic optimization algorithm for learning the network's weights [33]. The only difference among the competing neural networks was therefore the locations of their RBF node centers, which, as noted above, were chosen by a different RBF node selection method for each network. In this way, any statistically significant differences in the function approximation performance among the three competing neural networks could be attributed solely to the method that was used to select the networks' RBF node centers. With specific respect to the data, the real-world steel industry dataset contained 35,040 cases, while 1,000,000 cases were used in each experimental trial involving one of the four mathematical optimization and statistical functions. With respect to the latter, the data were generated randomly within the experimental boundaries of each function's input parameters. Either a uniform continuous or uniform discrete sampling distribution was used when randomly generating these data, depending on the nature of each input parameter. After computing the input parameters' associated output values for each mathematical or statistical function, each dataset was then randomly split into training and testing sets, with the training set consisting of 80% of the cases and the testing set containing the remaining 20% of the cases. For the experiments involving [PERSON]'s function, the \(x\) and \(y\) input parameters were both bounded by the interval \([-5.0,\,5.0]\), while for the experiments involving the Hosaki function, the \(x_{1}\) and \(x_{2}\) input parameters were both bounded by the interval \([0.0,\,5.0]\). For the experiments involving the Poisson distribution probability mass function, the \(\eta\) parameter used values in the set \(\{0,\,1,\dots,\,30\}\), while the \(\lambda\) parameter was bounded by the interval \([0.0,\,30.0]\). Finally, for the experiments involving the \(t\)-distribution cumulative distribution function, values for the \(t\) parameter were bounded by the interval \([-5.0,\,5.0]\), while the \(df\) parameter was constrained to values in the set \(\{1,\,2,\dots,\,30\}\).