## 2 Artificial Neural Network Models Background, and Fundamental Capabilities
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ANN Model** & **Creator** & **Year** & **Utilization** & **Reference** \\ \hline Perceptron Networks & [PERSON] & 1958 & Prediction & [10] \\ Additive Yadaline & [PERSON] & 1960 & Prediction & [11] \\ Spatio-Temporal-Pattern Recognition (SPR) & [PERSON] & 1960–1970 & Association & [12] \\ Adaptative Resonance Theory Networks (ART) & [PERSON] & 1960–1986 & Conceptualization & [13] \\ Directed Random Search (DRS) Networks & [PERSON] y Solis & 1965–1981 & Classification & [14] \\ Brain State in a Box & [PERSON] & 1970–1986 & Association & [15] \\ Self-organizing Maps (SOM) & [PERSON] & 1979–1982 & Conceptualization & [16] \\ Hopfield Networks & [PERSON] & 1982 & Optimization & [17] \\ Back-Propagation & [PERSON] & 1985 & Prediction & [18] \\ The Boltzmann Machine & [PERSON] & 1985 & Association & [19] \\ Bi-Directional Associative Memory (BAM) Networks & [PERSON] & 1987 & Association & [20] \\ Counter-Propagation & [PERSON] & 1987 & Association & [21] \\ Hamming Networks & [PERSON] & 1987 & Association & [22] \\ Delta Bar Delta (DBD) Networks & [PERSON] & 1988 & Classification & [23] \\ Learning Vector Quantization (LVQ) Networks & [PERSON] & 1988 & Classification & [24] \\ Probabilistic Neural Network (PNN) & [PERSON] & 1988 & Association & [25] \\ Recirculation Networks & [PERSON] & 1988 & Filtering & [26] \\ Functional-link Networks (FLN) & [PERSON] & 1989 & Classification & [27] \\ Cascade-Correlation Networks & Fahhman y Lebiere & 1990 & Association & [28] \\ Digital Neural Networks Architecture (DNNA) & Neural Semiconductor Inc. & 1990 & Prediction & [29] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Catalogue of initial Artificial Neural Network (ANN) contributions. layer, is composed of M neurons, arranged as a M-dimensional vector; and the output layer has one neuron, Y. The output of a neuron (i) of the input layer is equal to \(X_{i}\), whereas the components of the output of the hidden layer, \(Z_{m}\) (Equation (1)) are equal to a function called the activation function, resembling the mechanism of a physiological neuron, of the linear combination of the input neurons \(X_{i}\), adjusted by a threshold \(\alpha_{0}^{m}\). Most usual activation functions are the sigmoid and the hyperbolic tangent. \[Z_{m}=\sigma(M_{m})=\sigma\left(\sum_{i=0}^{P}\alpha_{i}^{m}\cdot X_{i} \right);\ X_{0}=1 \tag{1}\]  
The output of the neural network, Y in Equation (2), is a function (usually the identity in most common regression problems) of the linear combination of the output of the hidden layer neurons, adjusted by another threshold \(\beta_{0}\), and considering \(Z_{0}=1\). \[Y=f(x)=g\left(\sum_{j=0}^{M}\beta_{j}\cdot Z_{j}\right) \tag{2}\]  
All the aforementioned parameters of the model (weights of linear combinations and thresholds) are calculated through the application of a learning algorithm using observed data (experience). The objective of the learning process is to find the parameter set that minimizes the so-called loss function over the training data set, whereas the validation set is used to find the hyperparameters of the learning algorithm by minimizing the loss function over the validation data set. On the other hand, the test data set is employed to evaluate the accuracy of the model. Furthermore, it is common to apply a k-fold cross-validation process to iteratively apply the algorithm to k different partition sets. Back-propagation with gradient descent is the most usual algorithm used to train the neural network when the activation functions are differentiable. [PERSON] and [PERSON] (1987) [42] have demonstrated that a type of ANN model, the Backpropagation Neural Network, exceeds in an order of magnitude the results of conventional linear and polynomial methods. The Backpropagation Network is a very popular learning mechanism for prediction problems solving in multiple-layer perceptron  
Figure 1: ANN structure. networks [9; 43].