# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
### Limitations and Opportunities for Future Research

Although careful and systematic efforts were taken in this study to investigate the performance of the proposed opportunity matrix algorithm in comparison with that of the standard random sampling and \(k\)-means clustering methods, there nevertheless remain several limitations to this work that require acknowledgement. First, the experiments described herein relied on one real-world dataset and four different mathematical optimization and statistical functions. Care was taken to select well-known functions and a publicly available dataset that relied on both continuous and discrete input parameters, but any number of other functions or real-world datasets could have been used instead of the those that were considered in the study. Future work in this area should therefore endeavor to assess the performance of the opportunity matrix algorithm with other real-world datasets and other mathematical or statistical functions. In addition to relying on just four different functions and one real-world dataset, the findings of the experiments reported herein are also limited insofar as each of the five testing scenarios considered in the study used a relatively small number of input parameters and involved just one output value each. While the opportunity matrix framework allows for any positive number of input parameters, and while there is no theoretical or mathematical reason to expect that the opportunity matrix algorithm would perform differently with functions that require a large number of input parameters or that have more than one output value, these conditions have not yet been assessed experimentally. Future work in this area should therefore endeavor to assess the performance of the opportunity matrix algorithm when using functions that require a wider variety of inputs and outputs. Next, the opportunity matrix algorithm includes several tunable parameters whose values must be chosen by the AI/ML practitioner. Examples of these parameters include the number of intervals into which the range of values for each input parameter is partitioned, the size of the sample data that are used to iteratively identify RBF node center locations, and the stopping criterion that is used to terminate the algorithm. These tunable parameters are suitable targets for hyperparameter optimization, and future work in this area may be able to identify appropriate optimization techniques that can further improve the performance of the opportunity matrix algorithm. Finally, the opportunity matrix algorithm requires substantially more wall-clock time than the random sampling or \(k\)-means clustering methods in order to choose the locations for an equivalent number of RBF nodes. It is reasonable to expect that this notable differential in wall-clock time could be reduced through the use of a hardware accelerator when running the opportunity matrix algorithm; however, this expectation has not yet been asserted experimentally. AI/ML practitioners may take some comfort in knowing that the opportunity matrix algorithm will almost always yield a better-performing RBF neural network, but are nevertheless advised to consider the comparatively large amount of wall-clock time required by the opportunity matrix approach before deciding to apply it to their particular use case. ### Concluding Remarks

This paper clearly demonstrated that the locations chosen for the RBF node centers in a radial basis function network are critically important to the network's ability to generate accurate predictions. While it is true that an AI/ML practitioner can increase the accuracy of an RBF neural network by simply and lazily adding more RBF nodes to the network architecture, this is, I think, an incorrect approach. Indeed, the law of parsimony, as elegantly expressed in Occam's razor, should serve as a guiding principle in the design of RBF neural network architectures.