## 1 Introduction  
A wide range of practical problems can be considered as classification or regression problems. Such problems occur in areas such as physics [1, 2], astronomy [3, 4], chemistry [5, 6], medicine [7, 8], economics [9, 10], etc. A machine learning tool commonly used to tackle such problems is the radial basis function (RBF) network, expressed as the following function:  
\[R\big{(}\overrightarrow{x}\big{)}=\sum_{i=1}^{k}w_{i}\phi\big{(}\big{\|} \overrightarrow{x}-\overrightarrow{c_{i}}\big{\|}\big{)} \tag{1}\]  
In this equation, the following definitions are used for the symbols:  
1. The vector \(\overrightarrow{x}\) represents the input pattern with \(d\) number of features. 2. The parameter \(k\) stands for the number of weights of the network. These weights are represented by the values \(w_{i}\), \(i=1,\ldots,k\). 3. The vectors \(\overrightarrow{c_{i}^{\ast}}\), \(i=1,\ldots,k\) represent the so-called centers of the network. The function \(\phi(x)\) commonly is a Gaussian function with the following definition:  
\[\phi(x)=\exp\left(-\frac{\left(x-c\right)^{2}}{\sigma^{2}}\right) \tag{2}\]  
A typical plot of the Gaussian function for \(c=0,\ \sigma=1\) is depicted in Figure 1. From this graph it is deduced that the Gaussian RBF decreases regarding the distance from \(c\). The parameters \(c\) and \(\sigma\) can be estimated using the K-Means algorithm, introduced by [PERSON] [11]. The training error of an RBF network is defined as  
\[E\big{(}R\left(\overrightarrow{x}\right)\big{)}=\sum_{i=1}^{M}\big{(}R\left( \overrightarrow{x_{i}}\right)-y_{i}\big{)}^{2} \tag{3}\]  
where \(\left(\overrightarrow{x_{i}},y_{i}\right),\ i=1,\ldots,M\) represents the training set of the objective problem and the values \(y_{i}\) are the expected outputs for every pattern \(\overrightarrow{x_{i}}\). RBF networks have been incorporated in various cases, such as face recognition [12], solutions of differential equations [13; 14], stock prediction [15], robotics [16; 17], network security [18; 19], etc. Due to the widespread use of these networks, several papers have been presented in recent years that study their basic characteristics. For example, [PERSON] et al. [20] presented a discussion on kernel widths in RBF networks. Similarly, [PERSON] et al. [21] presented a novel method for the estimation of the kernel density. Moreover, [PERSON] et al. [22] proposed an automatic method for the initialization of the parameters of RBF networks. Furthermore, a variety of pruning techniques have also been proposed [23; 24], used to efficiently reduce the number of processing units of RBF networks in order to avoid the overfitting problem. Also, for the effective training of RBF networks a variety of methods has been proposed in the recent literature, such as the incorporation of genetic algorithms [25; 26], usage of the Particle Swarm Optimization (PSO) method [27; 28], the usage of methods based on Differential Evolution [29], etc. Furthermore, since there has been an extremely large spread of parallel programming techniques in recent decades, publications have also appeared that exploit such techniques for the efficient and fast training of the above networks [30; 31]. In most cases, RBF networks are trained using a two-stage technique. In the first stage, the centers and variances are estimated using the K-means technique. In the second stage,  
Figure 1: A typical plot of the Gaussian function. a linear system is solved to find the weights of the Gaussian units. Although the above procedure is extremely fast, it cannot effectively explore the value space for the network parameters and in many cases numerical stability problems occur when solving the linear system. This paper proposes a three-stage method for the efficient training of RBF networks. In the first stage, an initial estimate of the centers and fluctuations of the RBF network is made using the K-means technique.