## 3 Results and Discussion
### Determination of the Neural Network Architecture  
The performance of the ANN was mainly affected by the format of the training data, size of the hidden layers and the training function. This section mainly explores the influence of the training functions and the size of the hidden layers on the performance of the ANN.  
There are many training functions for ANNs, such as Levenberg-Marquardt (LM), Bayesian regularization (BR), BFGS quasi-Newton (BFGS-QN), resilient backpropagation (RB) and scaled conjugate gradient (SCG). These training functions have distinctive characteristics. For instances, the LM algorithm is fast with high accuracy; the BR algorithm takes longer but may process complicated problems; and the SCG algorithm uses less memory,  
Figure 11: The prediction accuracy and computation time based on features extracted by VMD, EMD and raw vibration signal.  
Figure 10: Normalized energy of the IMFs corresponding to the fault types based on EMD and VMD: (**a**) normalized energy of the IMFs corresponding to the fault types based on EMD; (**b**) normalized energy of the IMFs corresponding to the fault types based on VMD.  
etc. The architecture of the ANN applied the parameters shown in Table 2. Each training function was simulated 10 times, and the average values of the prediction accuracy, iteration error and computation time corresponding to each training function were obtained, as shown in Figure 12.  
As shown in Figure 12, the BR training function had the highest prediction accuracy of 99.3% and the smallest training error. Although the computation time of the BR training function was 2.4 s, it is feasible in many practical engineering applications. Therefore, the BR training function was used as the training function of the ANN considering its high accuracy.  
In addition, the size of the hidden layers included the number of hidden layers and the number of neurons in each hidden layers. The prediction accuracy corresponding to the number of hidden layers, ranging from 1 to 5, and the number of neurons in each layer, ranging from 5 to 15, were studied. The average value of the prediction accuracy and computation time were obtained through 10 ANN training processes. The influence of the number of hidden layers and number of neurons on the ANN's performance is shown in Figure 13. The prediction accuracy was the highest when the number of hidden layers was three and the number of neurons in each layer was 13. Meanwhile, the computation time rose with the increase in the number of hidden layers and neurons. The computation time of the ANN with three hidden layers and 13 neurons was 2.4 s, which satisfies most of the requirements of actual engineering applications.  
Figure 12: Effect of the training function on the ANN’s performance: (**a**) prediction accuracy of the different training functions; (**b**) training error and computation time of the different training functions.  
Figure 13: The influence of the number of hidden layers and number of neurons on the ANN’s performance: (**a**) influence of the number of hidden layers and number of neurons on the prediction accuracy; (**b**) influence of the number of hidden layers and number of neurons on the computation time.