# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 2 Materials and Methods
### The Opportunity Matrix  
The core concept underlying the proposed algorithm is that of an \"opportunity matrix\". In brief, an opportunity matrix is a fixed-size, \(d\)-dimensional container whose dimensionality matches that of the number of input features in the RBF network's training data. If  
Figure 3: Overview of the current study. each training case contains a single input feature, the opportunity matrix will be a vector. Similarly, if each training case consists of two input features, the opportunity matrix will be two-dimensional, and so on. Each dimension in the opportunity matrix is mapped to one of the input features in the training data, with the number of items in each dimension being determined by partitioning the range of values in the dimension's associated input feature into \(n_{d}\) equidistantly spaced intervals. Alternatively, a quantile-based split rather than an equidistant split may be used if the AI/ML practitioner prefers to take the distributional properties of the data into account. Note that these methods of partitioning can be implemented for both continuous and discrete input features, and can be readily adapted for use with categorical features. As an illustrative example, imagine a scenario in which an AI/ML practitioner wishes to develop an RBF neural network. The available training data contain two continuous input features, \(a\) and \(b\), with the range of feature \(a\) being \(0\leq a\leq 10\) and the range of feature \(b\) being \(0\leq b\leq 5\). Further, assume that the AI/ML practitioner wishes to partition both of the input features into five equidistantly spaced intervals, such that \(n_{a}=5\) and \(n_{b}=5\). In this scenario, the resulting opportunity matrix would be structured as shown in Figure 4 below. In the scenario illustrated above, the opportunity matrix contains \(\prod\limits_{i=1}^{d}n_{i}=25\) unique elements, each of which represents a bounded region of the overall input space. The first foundational idea of the opportunity matrix approach is that the centroid of each of these bounded regions (which are signified by dots in Figure 3) represents a potential opportunity for the location of an RBF node in a radial basis function neural network. It naturally follows that the quantity \(\prod\limits_{i=1}^{d}n_{i}\) defines the upper limit for the number of RBF nodes that could feasibly appear in the network. An AI/ML practitioner must therefore be judicious when choosing the number of intervals into which each input feature should be partitioned. Note also that this framework can be easily extended into higher-dimensional spaces to accommodate any possible number of input features. The second foundational idea of the opportunity matrix approach is that the RBF network is constructed incrementally by adding one RBF node at a time. To accomplish this, the AI/ML practitioner begins by first selecting a reasonably sized random sample of the training data. An initial RBF node is then added to the neural network structure, with the location of the initial node's center being chosen either randomly or by relying on the centroid of all of the sample's input features. The single weight for this simplest of all possible RBF neural networks is then learned using the sample of the training data. When training is complete, each element in the opportunity matrix is populated with the average prediction error for all of the input values that lie within the element's boundaries. The opportunity matrix is then examined to identify the element with the largest average  
Figure 4: Example of an opportunity matrix structure for an RBF network with two input features. error that does not already have an associated RBF node, and that element's centroid is then used as the location for the next RBF node that is added to the neural network. The opportunity matrix is then emptied and the process is repeated, with one new RBF node being added during each iteration until a chosen stopping criterion has been met.