## 4 Results and Analysis
### Case of Wheat Futures
#### 4.1.2 Comparison and Analysis  
Based on the decomposition results of these four decomposition methods shown in Section 4.1.1, we apply the PSO-BPNN model to implement the forecasting part. As for the decomposition results of each decomposition algorithm, the PSO-BPNN model is utilized to predict the last 20 percent of data, namely 300 forecasting values totally, of each sub-series. Then, 300 predictions of each sub-series are summed up in order to obtain the ultimate 300 forecasting values of the wheat futures prices. For a better comparison of the effectiveness of four hybrid models combined with four different decomposition techniques, we select the PSO-BPNN model without decomposition methods as a benchmark model, aiming to testify the effectiveness of decomposition techniques. Furthermore, the ARIMA model is treated as a comparative model in order to compare the superiority in predictive capability between the ANN-based models and the traditional statistic models. The comparison of one-day-ahead forecasting results among all these models is provided in Figure 11. And the forecasting performance is evaluated through three criteria, namely the MAE, RMSE and MAPE, which are given in Table 3 and Figure 12.  
According to the Figure 11, there exists a preliminary judgment that the forecasting accuracy of the VMD-PSO-BPNN model is higher than any other model proposed in this study, since, during the  
Figure 10: Decomposition results of the variational mode decomposition (VMD) method (wheat). Figure 9: Decomposition results of the intrinsic time-scale decomposition (ITD) method (wheat). Furthermore, it can be proved from Table 3 that, among all these forecasting models mentioned above, the VMD-PSO-BPNN model performs much better than any other model; namely the EMD-PSO-BPNN model, WPT-PSO-BPNN model, ITD-PSO-BPNN model, PSO-BPNN model and ARIMA model, in terms of the values of MAE, RMSE and MAPE, which respectively are 2.68, 3.41 and 0.55% for the VMD-PSO-BPNN model. For the comparison among ANN-based models, more specifically, on the one hand, the EMD-PSO-BPNN model, WPT-PSO-BPNN model, ITD-PSO-BPNN model and VMD-PSO-BPNN model have a more satisfactory performance than the PSO-BPNN model in terms of these three evaluation criteria, which means that the decomposition methods mentioned in this study can further make the PSO-BPNN model improve its forecasting accuracy of wheat futures; on the other hand, the VMD-PSO-BPNN model shows considerable superiority compared with other hybrid models combined with EMD, WPT and ITD methods respectively, signifying that the VMD method is proven to be more effective to conduct data pretreatment in contrast to the other three decomposition approaches. From a pure data perspective, the values of MAE, RMSE and MAPE of the VMD-PSO-BPNN model have all been reduced by about 68% compared with the PSO-BPNN model and approximately 63% compared with the ITD-PSO-BPNN model, and have decreased by 63.39%, 59.11% and 62.34% respectively compared with the EMD-PSO-BPNN model, and 35.58%, 57.09% and 36.05% respectively compared with the WPT-PSO-BPNN model. For the comparison between ANN-based models and the ARIMA model, in terms of the MAPE, the value of ARIMA model is a little smaller than the PSO-BPNN model, EMD-PSO-BPNN model and ITD-PSO-BPNN model, while its value is obviously much larger than the WPT-PSO-BPNN model and VMD-PSO-BPNN model. On the whole, it can be concluded that the WPT-PSO-BPNN model and VMD-PSO-BPNN model, especially the latter one, are more suitable to be used in forecasting the wheat futures prices because they raise the forecasting precision, namely the MAPE, by an order of magnitude, in contrast with the ARIMA model, PSO-BPNN model, EMD-PSO-BPNN model and ITD-PSO-BPNN model.