# A Two-Phase Evolutionary Method to Train RBF Networks
## 1 Introduction  
In machine learning, many practical problems appear such as classification and regression problems. A good programming tool that can be used to tackle this problem is radial basis function (RBF) networks [1]. These networks typically are expressed as a function:  
\[y(x)=\sum_{i=1}^{k}w_{i}\phi(\|x-c_{i}\|) \tag{1}\]  
where \(\overrightarrow{x}\) is the input pattern, the vector \(\overrightarrow{w}\) is called the weight vector and \(y(x)\) is the predicted value of the network. RBF networks are feedforward neural networks [2] with three computational layers:  
1. The input layer, where the problem is presented in the form of patterns to the neural network.
2. The processing layer, where a computation is performed using the Gaussian processing units \(\phi(x)\). These units can have many forms in the relevant literature but the most used form is the Gaussian function expressed as: \[\phi(x)=\exp(-\frac{(x-c)^{2}}{\sigma^{2}})\] (2) The value \(\phi(x)\) depends only on the distance of vector \(\overrightarrow{x}\) from some other vector \(\overrightarrow{c}\), which typically is called centroid.
3. The output layer where the output of every function \(\phi(x)\) is multiplied by a corresponding weight value \(w_{i}\).  
RBF networks have been used in many classification and regression problems from the areas of physics [3; 4; 5; 6], medicine [7; 8; 9], solution of differential equations [10; 11], chemistry [12; 13; 14], economics [15; 16; 17], digital communications [18; 19], etc. Furthermore, recently, RBF networks have been used in more difficult problems such as the authentication assurance of meat products [20], trajectory tracking for electrohydraulic servo systems, identification of geographical origin for foods [21], prediction of solution gas-oil ratio of crude oil systems [22], prediction of occurrences of haloketones in tap water [23], healthmonitoring [24], etc. Because of the extensive use of RBF networks, many methods have been proposed in the recent literature to enhance them. There are methods that parallelize the RBF networks [25; 26], methods that improve the initialization of the RBF parameters [27; 28; 29], methods that alter the architecture of the network [30; 31; 32], methods aimed to locate the best set of RBF parameters with global optimization techniques [33; 34; 35], etc. This article transforms the problem of RBF training into an optimization problem and applies a modified genetic algorithm technique to solve it. The global optimization problem is defined as:  
\[\min(E(y))=\sum_{i=1}^{m}(y(x_{i})-t_{i})^{2} \tag{3}\]  
where \(m\) is the total number of input patterns and \(t_{i}\) is the output for pattern \(x_{i}\). The suggested approach has two phases: firstly, reasonable bounds for the RBF parameters are estimated using the k-means [36] algorithm and in the second phase, the modified algorithm is used to solve the problem of Equation (3) inside the bounds located in the first phase.  
The rest of this paper is organized as follows: in Section 2 the proposed method is described, in Section 3 the conducted experiments are listed and the proposed method is compared against the traditional training of RBF networks and finally, in Section 4 some conclusions are derived.