# Fault Prediction of Mechanical Equipment Based on Hilbert-Full-Vector Spectrum and TCDAN
## 4 Experiment Verification
### Performance Comparison of Prediction Models  
First, the amplitudes corresponding to different characteristic frequencies in the life cycle of Bearing 3_1 are extracted based on the Hilbert-full-vector spectrum. According to the failure mechanism of the bearing [38], the amplitudes corresponding to the first and second times of the five characteristic frequencies of the bearing, which are the rotation frequency, the outer-ring fault frequency, the rolling element fault frequency, the cage fault frequency, and the inner-ring fault frequency, are taken as the input of the model, and the output variable is the amplitude corresponding to the characteristic frequency of concern after 1 min. To illustrate the effectiveness of the prediction method proposed in this paper, the data of the past 2000 min (containing 2000 samples) is selected as the training set, and the data of the next 100 min (containing 100 samples) is selected as the test set. Based on the same dataset, the TCDAN proposed in this paper is compared with the TCN, GRU, and LSTM models. The computer processor used in this experiment is Intel Core i5-4200H, and the graphics card is an NVIDIA GTX950M standalone graphics card with 8 GB memory. The model is based on Python language. Both the TCDAN and TCN models use two residual modules, the size of the convolution kernel is 3, the number of convolution kernels is 64, and the time step is 8; The LSTM and GRU models have eight hidden layers and 128 neurons. The input dimension of each model is 10, and the output dimension is 1. All  
Figure 11: Spectral changes in Bearing 3_1 in different periods. In this figure, fr stands for the rotational frequency of Bearing 3_1, fo stands for the outer-ring fault frequency of Bearing 3_1, 2 \(\times\) fo, 3 \(\times\) fo \(\ldots\) 7 \(\times\) fo represents the doubling frequency of Bearing 3-1 outer-ring fault frequency; S1, S2, and S3 represent the spectrum of the Bearing 3_1 health state, S4 and S5 represent the spectrum at the initial stage of the fault, S6, S7 and S8 represent the spectrum at the middle stage of the fault, and S9 and S10 represent the spectrum at the late stage of the fault. models are developed based on the TensorFlow deep-learning framework and use the GPU to train the models. To speed up the convergence of the model, each variable is input into each model after normalization. Taking the prediction of the corresponding amplitude of the outer-ring fault frequency of Bearing 3_1 as an example, the prediction results of the corresponding amplitude of 1 and 2 times the outer-ring fault frequency of different models are shown in Figures 12 and 13, respectively. Through the comparison of the prediction effects of different models in Figures 12 and 13, it can be seen that the proposed TCDAN has higher prediction accuracy compared with the TCN without attention mechanism. At the same time, the prediction performance of TCDAN is also significantly better than that of the current mainstream prediction models such as BiLSTM, LSTM, and GRU. Combined with the evaluation indexes of the prediction results of each model in Table 2, it can be seen that the MAE, RMSE, and MAPE prediction indexes of the amplitude corresponding to 1 times the outer-ring fault frequency predicted by TCDAN are reduced to 0.0240, 0.0297, and 6.04%, respectively. The MAE, RMSE, and MAPE prediction indexes of the amplitude corresponding to 2 times the outer-ring fault frequency are reduced to 0.0317, 0.0382, and 6.34%, respectively. Compared with TCN, GRU, BiLSTM, and LSTM models, TCDAN has greatly improved the prediction performance. Figure 12: Prediction comparison of the amplitude corresponding to 1 \(\times\) fo. Figure 13: Prediction comparison of the amplitude corresponding to 2 \(\times\) fo. Through the comparison of the prediction effects of different models in Figures 14 and 15, the prediction performance of TCDAN is significantly better than that of TCN, BiLSTM, LSTM, and GRU.