# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix RBF Center Selection Algorithm
The proposed opportunity matrix algorithm can, of course, be expected to require substantially more wall-clock time than either of the random sampling or \(k\)-means clustering methods, since it relies on an iterative approach in which one RBF node is identified per iteration, with the additional requirement that the weights of an RBF neural network be learned during each of those iterations. The expectations related above notwithstanding, a direct evaluation of the memory and wall-clock time required by each method of selecting RBF center locations was undertaken in order to provide a measure of empirical insight into this issue, with the approximation of the Hosaki function serving as the basis of the evaluation. To ensure equitable comparisons, the memory and wall-clock time requirements for all three RBF node selection methods were measured using a single thread on a single CPU. When conducting the comparisons, the sample size and RBF neural network training parameters that were used to select the RBF center locations with the opportunity matrix technique matched those described previously in Section 2. The results of these measurements are provided in Table 6 below. Note that the peak memory requirements shown in the table include overhead items that were needed to run the Google Colaboratory environment, such as external libraries and the memory profiler itself. Nevertheless, these values provide insights into the comparative memory demands of the three different methods of selecting RBF center locations. As shown in Table 6 above, the random sampling method required both the least memory and the least wall-clock time to select RBF node centers for the Hosaki function. Although the wall-clock time required by the \(k\)-means clustering method fell between the time required by the random sampling and opportunity matrix methods, the \(k\)-means clustering method consistently required the largest amount of memory to complete the RBF center selection task. As expected, the opportunity matrix algorithm required much more wall-clock time than either of the other two competing methods, with its memory requirements falling between those of the random sampling and \(k\)-means clustering methods. It is important to note, however, that the measurements reported in Table 6 were obtained using a single thread on a single CPU. Since the opportunity matrix algorithm involves the iterative training of a series of RBF neural networks, one might reasonably expect that substantially less wall-clock time would be required by the opportunity matrix method if a hardware accelerator, such as a GPU or a tensor processing unit, were used to facilitate the RBF node selection process. ## 4 Summary, Limitations, and Concluding Remarks

### Summary of Contributions and Findings

This paper proposed a new algorithm based on the concept of an opportunity matrix for selecting the locations of the centers in a radial basis function (RBF) neural network. The opportunity matrix approach involves first partitioning the input space into several regions,