# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
All of these experimental boundaries are included in the depictions of the four test functions provided in Figures 7-10. Since using the opportunity matrix algorithm to obtain the locations of RBF node centers requires an AI/ML practitioner to make several choices, it is necessary and appropriate to provide specific details about how the algorithm was used in the current study in order to ensure that the study's results can be reproduced. To begin, Step #2 of Algorithm 1 requires a decision about the number of intervals into which the range of values for each input parameter should be partitioned. In the current study, the range of values for all of the mathematical and statistical functions' continuous input features was subdivided into 100 equidistantly spaced intervals, while for discrete input features, each unique item in the sets defined immediately above was used as the basis for its own partition in the feature's corresponding dimension of the opportunity matrix. For the real-world steel industry dataset, each of the continuous input features was partitioned using a quartile split, with the values for each discrete input feature again being used as the basis for its own partition in the opportunity matrix. Step #4 in Algorithm 1 requires a reasonably sized sample of the overall training data. For the experiments involving the mathematical and statistical functions, 5% (or 40,000) of the available cases were randomly sampled from the overall training data for use in the opportunity matrix algorithm, while a random sample of 10,000 cases was used for the experiments involving the steel industry power consumption dataset. For Step #7 in Algorithm 1, the value of \(k\) that was being considered in the current experimental trial was used as the stopping criterion, thus ensuring that a sufficient number of RBF node centers would be available to support the needs of the current trial. Finally, the RBF networks used in the opportunity matrix algorithm were trained using the Adam stochastic optimizer [33] with a batch size of 128 cases across 256 training epochs. As noted previously, the overall goal of the experiments was to compare the RBF node selection performance of the proposed opportunity matrix algorithm against that of the standard random sampling and \(k\)-means clustering methods under a wide variety of different conditions. Each of these conditions was evaluated in a distinct experiment involving 30 consecutive trials. For each trial, three RBF neural networks were used that relied on the same data, structure, and training algorithm, with the only difference among the networks being the method that was used to select the locations of their RBF node centers. Each final neural network in the trials was trained using a batch size of 128 training cases across 1024 epochs, after which training was considered complete. The experiments themselves were run sequentially using a fixed hardware configuration on the Google Colaboratory platform [34]. When the three RBF neural networks that were considered in each experimental trial had been fully trained, their performance was assessed by measuring the mean absolute error (MAE) of the networks' associated predictions on the data in the testing set. As with the training data, exactly the same testing data were used to evaluate all of the neural networks in each trial. With 30 trials per experiment, every experiment resulted in 30 MAE values for each of the three competing methods of selecting RBF node centers that corresponded to the experiment's unique combination of test function and number of RBF nodes within the study's overall experimental framework. Finally, the MAE valuesobtained from the opportunity matrix algorithm, the random sampling method, and the \(k\)-means clustering method were statistically compared against each other using [PERSON]'s \(t\)-tests [35]. Unlike most other \(t\)-tests, [PERSON]'s \(t\)-tests allow for the comparison of independent samples that have unequal variances.