# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 1 Introduction
### Selecting Widths and Centers for RBF Nodes  
Although it is feasible to assign individual width values to each of the RBF nodes in an RBF neural network, doing so is typically unnecessary. Indeed, research has shown that Gaussian RBF networks using the same width value (i.e., the same standard deviation) for every RBF node retain their universal function approximation capability [18]. Equation (2) below defines a simple method for identifying the global width value \(\sigma\) to use for all of the RBF nodes in an RBF network [10]. The parameter \(d_{max}\) in Equation (2) represents  
Figure 1: Standard RBF neural network architecture. the maximum (typically Euclidean) distance between any two RBF node centers, while \(k\) denotes the total number of RBF nodes in the neural network. \[\sigma=d_{max}\sqrt{2k} \tag{2}\]  
While calculating the global width value to use in an RBF neural network is trivial, choosing the most suitable locations for the RBF nodes' centers is much less straightforward. As noted previously, RBF neural networks are, in theory, universal function approximators; however, in practice, the ability of an RBF network to accurately approximate a function depends critically on the locations that have been chosen for the centers of its RBF nodes [10; 19]. The reason for this is that the influence of each RBF node is localized to a specific region within the input space, with the extent of the region of influence depending on the value of the node's width parameter [18]. A given RBF node's influence on the network's output will be quite strong when the distance between an input vector and the node's center is small, but the extent of the node's influence on the output will diminish rapidly as the distance between the node's center and the input vector increases (vide supra, Equation (1)). Well-chosen centers will therefore allow the RBF network to more accurately approximate a target function than if the locations of the centers had been chosen poorly. This concept is illustrated for a target function with a single input value in Figure 2 below. The width of each Gaussian curve in Figure 2 indicates the region of the input space over which the corresponding RBF node exerts influence, while the size of each RBF node center represents the magnitude of the corresponding node's weight in the RBF network (denoted \(w_{1},\ w_{2},\ \ldots\ w_{n}\) in Figure 1). As shown in Figure 2, the location of the RBF nodes' centers plays a very important role in the overall performance of the RBF neural network [19]. For this reason, the overall task of training an RBF neural network is typically subdivided into two distinct phases, with the first phase involving the selection of the RBF nodes' centers and widths, and the second phase involving the adjustment of the network's weights with a view to minimizing error [10]. By a wide margin, the two most common strategies for choosing the locations of an RBF network's centers are random sampling and \(k\)-means clustering [10; 13]. Random sampling is the simplest and fastest of these two common strategies. Using this approach, a subset of \(k\) input values is randomly chosen from the available training data, with these \(k\) input values then being used directly as the locations for the RBF network's centers. This approach is rational from the perspective of probability theory, since a sufficiently large random sample will approximate the distribution of the underlying population reasonably well [20]. As such, the resulting RBF nodes are likelier to be located near the regions of the input space that have the highest probability density than if the centers had been positioned equidistantly from one another within the input space. The rationale underlying the \(k\)-means clustering strategy is similar to that of the random sampling strategy; i.e., to locate the RBF centers in areas of the input space that best represent the distribution of the underlying training data.