# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
Comments: Any reasonable stopping criterion may be used to determine if the current set of RBF nodes is satisfactory. Examples include stopping once the neural network contains a specified number of RBF nodes or stopping once the value of the loss function crosses a specified threshold. If the latter strategy is used, the algorithm has the additional benefit of being able to select the optimal number of RBF nodes to use in the neural network. 8 For each element in the opportunity matrix that does not already have an associated RBF node in the neural network: 1. Compute the mean prediction error for all of the input values that fall within the current element's bounded region. 2. If the mean prediction error for the current element is the largest thus far observed, then fetch and store the element's corresponding centroid (which was computed in Step #3). 9 Add a new RBF node to the neural network structure using the centroid identified in Step #8 as the new node's center. 10 Compute and apply the new global RBF width value using Equation (2). 11 Go to Step #4. ```

**Algorithm 1**Using an opportunity matrix to select centers for an RBF neural network. The proposed opportunity matrix algorithm is summarized graphically in Figure 6 below. ### Evaluative Experiments

Having presented the proposed opportunity matrix algorithm for selecting RBF node centers, it is now possible to describe the series of experiments that were conducted in order to compare the performance of the opportunity matrix approach against that of the standard random sampling and \(k\)-means clustering techniques. In total, 25 experiments were conducted in order to evaluate the performance characteristics of the competing RBF node selection methods under a variety of different conditions. The first 20 of these experiments involved the task of using RBF neural networks to approximate four different functions, two of which were mathematical optimization functions ([PERSON]'s function [24] and the Hosaki function [25]) and two of which were statistical functions (the Poisson distribution probability mass function [26] and the \(t\)-distribution cumulative distribution function [27]). These specific functions were chosen because of their common use in optimization research and statistics, as well as to enhance the generalizability of the experiments by including functions from different domains that involve both continuous and discrete parameters. Visualizations of these four functions are provided in Figures 7-10 below. The color spectrum in these figures ranges from blue to red, signaling comparatively small to large output values, respectively. For the remaining five experiments, the performance of the proposed opportunity matrix algorithm was compared with that of the standard random sampling and \(k\)-means clustering techniques using a real-world energy consumption dataset from the steel industry [28]. This dataset, which is publicly available from the UC Irvine Machine Learning Repository [29], contains several continuous and categorical features that were used to predict the electrical power consumption of a steel manufacturing facility. Figure 6: Using an opportunity matrix to select centers for an RBF neural network. Figure 7: [PERSON]’s function. Figure 8: Hosaki function. Figure 10: \(t\)-Distribution cumulative distribution function. Figure 9: Poisson distribution probability mass function. the \(t\)-distribution cumulative distribution function were computed using the Python SciPy library [30]. [PERSON]'s function:

\[f(x,y)=\left(x^{2}+y-11\right)^{2}+\left(x+y^{2}-7\right)^{2} \tag{3}\]

[PERSON] function:

\[f(x_{1},x_{2})=\left(1-8x_{1}+7x_{1}^{2}-\frac{7}{3}x_{1}^{3}+\frac{1}{4}x_{1 }^{4}\right)x_{2}^{2}e^{-x_{2}} \tag{4}\]

Poisson distribution probability mass function (PMF):

\[pmf(\eta,\lambda)=\frac{\lambda^{\eta}e^{-\lambda}}{\eta!} \tag{5}\]

where \(\eta\) is the number of occurrences and \(\lambda\) is the expected value.