# A Novel Hybrid Model for the Prediction and Classification of Rolling Bearing Condition
## 4 Experimental Analysis
### Experimental Results and Analysis
#### 4.2.2 Prediction Based on Hybrid Prediction Model  
Based on the aforementioned VMD, the original signal was decomposed into several IMF time series, using the hybrid prediction model to obtain the trend of subseries. First, the obtained six IMF series by VMD were subjected to ARMA. We used the AR model of order 12 (AR(12)) and MA model of order 16 (MA(16)) for IMF1. Then the three-layered \(1\times 4\times 2\) ANN architecture was used for the residuals. Figures 4-6 present the expectations and predictions of IF IMF1 series for ARMA, ANN and ARMA-ANN, respectively. The predictions of the IMF1 for NS, BF, OF, using ARMA-ANN are presented in Figures 7-9, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameter** & **Setting** \\ \hline \multirow{6}{*}{VMD} & \(K\) & 6 \\  & \(\alpha\) & 2891 \\ \cline{1-1}  & DC & 0 \\ \cline{1-1}  & tau & 0 \\ \cline{1-1}  & init & 1 \\ \cline{1-1}  & tol & \(10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings for VMD. Figure 3: Time-domain waveform and frequency spectrum of original signal and IMFs for inner race signal by VMD (\(K=6\), \(\alpha=2891\)). Figure 4: Prediction result of ARMA model for IF IMF1 time series. Figure 5: Prediction result of ANN model for IF IMF1 time series. Figure 6: Prediction result of ARMA-ANN model for IF IMF1 time series. Figure 7: Prediction result of ARMA-ANN model for NS IMF1 time series. To further quantify the performance of the prediction models, the comparison of quantitative criteria is shown in Table 3, including the root-mean-square error (RMSE): \(\sqrt{\frac{1}{n}\sum_{t=1}^{n}\epsilon_{t}^{2}}\), the mean average percentage error (MAPE): \(\frac{100\%}{n}\sum_{t=1}^{n}|\frac{\epsilon_{t}}{Y_{t}}|\), and the mean absolute error (MAE): \(\frac{1}{n}\sum_{t=1}^{n}|e_{t}|\), which are used to quantify the prediction accuracy, where \(e_{t}\) denotes the error between expectation and the prediction. We further compared the prediction of subseries with the original data series. The prediction of the IF original data series for the ARMA-ANN obtained by VMD is shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Condition** & **Model** & **RMSE** & **MAPE** & **MAE** \\ \hline \multirow{3}{*}{IF} & ARMA & \(1.0279\times 10^{-1}\) & \(1.1276\times 10^{0}\) & \(8.2175\times 10^{-2}\) \\  & ANN & \(9.6119\times 10^{-2}\) & \(2.9847\times 10^{0}\) & \(7.5156\times 10^{-2}\) \\  & ARMA-ANN & \(3.6904\times 10^{-4}\) & \(2.2367\times 10^{-2}\) & \(2.6025\times 10^{-4}\) \\ \hline \multirow{3}{*}{NS} & ARMA & \(1.4100\times 10^{-2}\) & \(1.8500\times 10^{0}\) & \(1.1707\times 10^{-2}\) \\  & ANN & \(1.1744\times 10^{-2}\) & \(1.8433\times 10^{0}\) & \(9.4530\times 10^{-3}\) \\  & ARMA-ANN & \(4.6082\times 10^{-4}\) & \(3.0578\times 10^{-2}\) & \(2.9941\times 10^{-4}\) \\ \hline \multirow{3}{*}{BF} & ARMA & \(4.2184\times 10^{-2}\) & \(1.3759\times 10^{0}\) & \(3.4340\times 10^{-2}\) \\  & ANN & \(3.9973\times 10^{-2}\) & \(4.0775\times 10^{0}\) & \(3.2100\times 10^{-2}\) \\  & ARMA-ANN & \(8.1888\times 10^{-4}\) & \(4.3402\times 10^{-2}\) & \(2.6100\times 10^{-4}\) \\ \hline \multirow{3}{*}{OF} & ARMA & \(1.2997\times 10^{-1}\) & \(1.0736\times 10^{0}\) & \(1.0533\times 10^{-1}\) \\  & ANN & \(1.2400\times 10^{-1}\) & \(2.1900\times 10^{0}\) & \(9.9000\times 10^{-2}\) \\ \cline{1-1}  & ARMA-ANN & \(2.4100\times 10^{-3}\) & \(2.9969\times 10^{-2}\) & \(1.5100\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of performance criteria of three models for all dataset series. Figure 8: Prediction result of ARMA-ANN model for BF IMF1 time series. Figure 9: Prediction result of ARMA-ANN model for OF IMF1 time series. The quantitative evaluation criteria shown in Table 4 demonstrate that the prediction of subseries obtained by VMD achieves better performance than the original data series. This result motivated us to make original data series into subseries to improve the accuracy of results.