## 2 Background and Methodological Framework
A fully interconnected neural network occurs when the nodes in each layer are connected to the nodes in the next layer. The sole mission of the input layer is to distribute the information presented to the neural network for processing in the next layer. The nodes in the hidden layers and in the output layer process the signals by applying processing factors, known as synaptic weights. Each layer has an additional node referred to as bias, which adds an additional term to the output from all the nodes found in the layer. All inputs in a node are weighted, combined, and processed through a function called a transfer function or activation function, which controls the output flow from that node to enable connection with all the nodes in the next layer. The transfer function serves to normalize the output. The connections between processing elements are linked to a connection weight or force W, which determines the quantitative effect certain elements have on others. Figure 1: The general structure of a feedforward network. Figure 2: The general structure of a feedback network. Specifically, the transformation process for the inputs and outputs in a feedforward artificial neural network with \(r\) inputs, a sole hidden layer, composed of q processing elements and an output unit can be summarized in the following formulation of the network output function for the following model (1) and in Figure 3:  
\[Y=\hat{f}(x,W)=F\left(\beta_{0}+\sum_{j=1}^{q}\beta_{j}G(x^{\prime}\gamma_{j})\right) \tag{1}\]  
where:  
* \(x=(1,x_{1},x_{2},\ldots,x_{r})^{\prime}\) are the network inputs (independent variables), where 1 corresponds to the bias of a traditional model. * \(\gamma_{j}=(\gamma_{j0},\gamma_{j1},\ldots,\gamma_{jii},\ldots,\gamma_{jr})^{ \prime}\in\Re^{r+1}\) are the weights of the inputs layer neurons to those of the intermediate or hidden layer. * \(\beta_{j},j=0,\ldots,q\), represents the connection force of the hidden units to those of pertaining to output (\(j=0\) indexes the bias unit) and \(q\) is the number of intermediate units, that is, the number of hidden layer nodes. * \(W\) is a vector which includes all the synaptic weights of the network, \(\gamma_{j}\) and \(\beta_{j}\), or connections pattern. * \(Y=\hat{f}(x,W)\) is the network output (in our case, it refers to fraud probability)
* \(F\colon\Re\to\Re\) is the unit activation function and output while \(G\colon\Re\to\Re\) corresponds to the intermediate neurons activation function. Selection of both was considered optimum, in accordance with the software utilized (It is normal to use the sigmoid or logistic function \(G(a)=\hat{I}/(1+exp(-a))\), which produces a smooth sigmoid response. Notwithstanding, it is possible to use the hyperbolic tangent function. In the expression \(\hat{f}(x,W)\) if we consider that \(a=x^{\prime}\gamma_{j}\), we find that \(G(x^{\prime}\gamma_{j})\) tallies with the binary logit model). As to the creation and application of a neural network to a specific problem, the following steps are shown in Figure 4:  
Figure 4: Steps in the empirical application of the model. Figure 3: General representation of the network execution process. The neural network model we are going to apply in this study is the supervised learning model based on the multilayer perceptron. There are other neural network models such as the Radial Basis Function (RBF), which is also interesting for this kind of analysis. Accordingly, it was utilized to compare its efficiency against that of the model presented, confirming that the multilayer perceptron provides the best results. The results obtained with the Radial Basis Function model are available upon request), given that it presents an output pattern or dependent variable which allows for contrasting and correcting data. Due to this, it is a technique used for classification as well as for prediction, for market segmentation, the positioning of products, forecasting demand, evaluation of credit files or analysis of stock exchange value, in addition to a countless number of other applications.