## 3 Proposed Method
### Classification and Severity Diagnosis  
Finally, in the third stage, the automatic fault detection and severity diagnosis by means of a classical NN-based structure are performed. It should be highlighted that an interesting advantage of the proposed hybrid-based feature calculation in conjunction with the feature extraction and fusion domain is that it leads to a reduction in the complexity related to the configuration of the required classification algorithm; that is, a high-performance pattern characterization of the considered conditions is achieved facilitating the classification task.  
\begin{table}
\begin{tabular}{c c} \hline
**Statistical Feature** & **Mathematical Equation** \\ \hline C Factor & \(F_{7}=\sqrt{\frac{\sum_{i=1}^{M}f_{i}^{2}s(i)}{\sum_{i=1}^{M}s(i)}}\) \\ D Factor & \(F_{8}=\sqrt{\frac{\sum_{i=1}^{M}f_{i}^{4}s(i)}{\sum_{i=1}^{M}f_{i}^{2}s(i)}}\) \\ E Factor & \(F_{9}=\frac{\sum_{i=1}^{M}f_{i}^{4}s(i)}{\sqrt{\sum_{i=1}^{M}s(i)}\sum_{i=1}^ {M}s(i)}\) \\ G Factor & \(F_{10}=\frac{F_{8}}{F_{5}}\) \\ Third moment 1 & \(F_{11}=\frac{\sum_{i=1}^{M}(f_{i}-F_{5})^{3}s(i)}{M_{F}^{2}}\) \\ Fourth moment 1 & \(F_{12}=\frac{\sum_{i=1}^{M}(f_{i}-F_{5})^{4}s(i)}{M_{F}^{2}}\) \\ H Factor & \(F_{13}=\frac{\sum_{i=1}^{M}(f_{i}-F_{5})^{1/2}s(i)}{M_{V}f_{5}}\) \\ J Factor & \(F_{14}=\frac{(F_{7}+F_{8})}{F_{1}}\) \\ \hline \end{tabular}
\end{table}
Table 2: _Cont._In this regard, the following parameters are taken into account for the proposed NN-based classifier: (i) the structure consists of a simple multi-layer NN that comprises three-layers: the input, hidden, and output layers; (ii) the number if neurons for each layer are defined as two, ten, and six neurons for the input, hidden, and output layer, respectively; (iii) the backpropagation is used as the training algorithm; (iv) the sigmoid function is used as the activation function; (v) 50 epochs are the number of iterations. Specifically, the number of neurons considered in the input layer is equal to the number of features extracted by means of the LDA technique, and for this proposed work, the features extracted by the LDA are represented in a 2D space. For the hidden layer, the number of neurons is fixed to ten neurons, which have been defined by following the literature suggestions [43,44]. Such literature suggestions are supported by the results obtained in practices of performance analysis based on trial and error for the selection of the number of neurons in the hidden layer [45]. In the output layer, a number of neurons equal to six are considered, where each neuron considered in the output layer represents each one of the evaluated conditions that are associated to a particular fault severity in the outer race of bearings. Moreover, it should be mentioned that the proposed NN-based classifier is trained and tested under a \(k\)-fold cross-validation scheme to obtain statistical and significant results. During the training and testing of the NN-based classifier, the sigmoid function is used as the activation function, and the backpropagation is used as the training algorithm during 50 epochs. Additionally, even though optimization algorithms can be optionally used to improve the performance of some NN-structures during the training procedure. The proposed multi-layered NN-based classifier is not supported by any optimizer algorithm since it is a simple multi-layer construction. In Figure 3, the structure of the proposed multi-layer NN-based classifier is shown.