## 4 Vibration Fault Diagnosis
### Back-Propagation Neural Network Based Fault Diagnosis  
The purpose of this subsection is to classify the bearings in the categories (Healthy bearing, Inner Ring Fault, Ball Fault, Outer Ring Fault) but with a better efficiency. The same database as above was used. It consists of 1044 lines corresponding to the number of signals and 16 columns corresponding to the descriptor extracted from the signal analysis. The data were subdivided as follows: 60% for Training (626 lines), 20% for validation (210 lines), and 20% for the test (210 lines). We used a neural network with one hidden layer. The number of neurons in the input layer is 16 corresponding to the number of the descriptors. The Back-propagation algorithm has been tested to train the neural network. The particularity of this network is that it contains 2 neurons in the output layer instead of 4. This choice allows decreasing the number of weights and consequently the calculation will be faster. The neurons in the output layer will take values 1 or \(-1\) depending on the desired classification decision. The set of possibilities is summarized in Table 3. We used the neural transfer function \"tansig\" to calculate the outputs of each layer. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Predicted Class**} \\ \cline{3-6}  & & **Healthy Bearing** & **Inner Ring Fault** & **Ball Fault** & **Outer Ring Fault** & **Recall** \\ \hline \multirow{4}{*}{**Actual Class**} & Healthy Bearing & 119 & 0 & 0 & 1 & 99.17\% \\  & Inner Ring Fault & 0 & 196 & 2 & 2 & 98.00\% \\  & Ball Fault & 0 & 2 & 200 & 3 & 97.56\% \\  & Outer Ring Fault & 1 & 2 & 2 & 201 & 97.57\% \\  & Precision & 99.17\% & 98.00\% & 204\% & 97.10\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classification rate of LDA model (Training data). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Predicted Class**} \\ \cline{3-6}  & & **Healthy Bearing** & **Inner Ring Fault** & **Ball Fault** & **Outer Ring Fault** & **Recall** \\ \hline \multirow{4}{*}{**Actual Class**} & Healthy Bearing & 57 & 0 & 1 & 2 & 95.00\% \\  & Inner Ring Fault & 0 & 85 & 2 & 1 & 96.59\% \\  & Ball Fault & 1 & 3 & 75 & 4 & 90.36\% \\  & Outer Ring Fault & 2 & 3 & 5 & 72 & 87.80\% \\  & Precision & 95.00\% & 93.41\% & 90.36\% & 91.14\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Classification rate of LDA model (Test data). Several tests were performed by changing the number of neurons in the hidden layer each time. The optimal neural network is composed of 16 neurons in its hidden layer (Figure 7a). This network converged after 24 epochs with Mean Squared Error MSE = 0.0124 (Figure 7b). As noted above, the data were subdivided to 3 parts. For the training data the F1-Score obtained is 99.04%, the validation data the score is 97.62% and for the test data the score is 96.67% (Tables 4-6). It can be seen from the test data that among the 56 cases of ball faults bearing, 3 were misclassified with a rate of false discovery is equal to 5.36%. Regarding the 59 defects of the outer ring, 3 of them were not well predicted. The rate of false discovery is equal to 5.08%. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \multicolumn{6}{c}{**Predicted Class**} \\ \cline{3-8}  & & **Healthy Bearing** & **Inner Ring Fault** & **Ball Fault** & **Outer Ring Fault** & **Recall** \\ \hline \multirow{4}{*}{**Actual Class**} & Healthy Bearing & 98 & 0 & 0 & 0 & 100\% \\  & Inner Ring Fault & 0 & 177 & 0 & 1 & 99.44\% \\ \cline{1-1}  & Ball Fault & 0 & 0 & 171 & 2 & 98.84\% \\ \cline{1-1}  & Outer Ring Fault & 0 & 0 & 3 & 167 & 98.31\% \\ \cline{1-1}  & Precision & 100\% & 100\% & 98.28\% & 98.31\% & \\ \hline \end{tabular}
\end{table} 
Table 4: Classification rate for Back-Propagation neural network (Training data).