## 5 Conclusions  
As the world's leading and most diverse derivatives marketplace, CME Group's wheat, corn and soybean futures prices are not only important reference prices of agricultural production and processing but also the authority of prices in the international trade of agricultural products, which can reflect the change trend of the corresponding agricultural products' spot prices in advance to some extent. Thus, forecasting their prices is expected to be an effective method for controlling market risks and making appropriate and sustainable food grain policy for the government. However, current research pays little attention to forecasting food grain futures prices and does not take their nonlinear and nonstationary characteristics into account while making predictions. Based on the above consideration, we propose four hybrid models which combine the PSO-BPNN model with the EMD, WPT, ITD and VMD methods respectively, to forecast wheat, corn and soybean futures prices, which can enrich empirical research on agricultural commodity futures prices forecasting, to some extent. According to our experimental results, three main conclusions are drawn as follows: (1) it has been proved that the VMD-PSO-BPNN model outperforms the EMD-PSO-BPNN model, WPT-PSO-BPNN model, ITD-PSO-BPNN model, PSO-BPNN model and ARIMA model in all study cases, in terms of three forecasting performance evaluation criteria, namely the MAE, RMSE and MAPE, which suggests that the proposed VMD-PSO-BPNN model has a high common adaptability and serviceability in forecasting the wheat, corn and soybean futures prices; (2) in all study cases, the forecasting performance of all four hybrid models with decomposition methods are superior to the performance of the PSO-BPNN model, which demonstrates that the EMD, WPT, ITD and VMD methods play an extremely significant role in improving the PSO-BPNN model's forecasting performance of these futures prices; (3) after comparing the results of different \"decomposition and ensemble\" hybrid models, we find out that the prediction ability of VMD-PSO-BPNN model and WPT-PSO-BPNN model is much better than the ability of EMD-PSO-BPNN model and ITD-PSO-BPNN model, meaning that the VMD and WPT methods, especially the VMD method, are more suitable to be applied to analyzing the prices data of wheat, corn and soybean futures than other two approaches, namely the EMD and ITD. Conclusively, based on the three evaluation criteria, four \"decomposition and ensemble\" hybrid models developed in this study perform better than the forecasting model without decomposition techniques, namely the PSO-BPNN model, with respect to the price forecasting of the wheat, corn and soybean futures, which provides a new promising research approach to forecasting prices of agricultural commodity futures. We would like to acknowledge that this paper was supported by the National Natural Science Foundation, China (No. 71301153); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China; the Science Foundation of Mineral Resource Strategy and Policy Research Center, China University of Geosciences (Grant No. H2017011B). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multicolumn{3}{c}{**Criteria**} \\ \cline{2-4}  & **MAE** & **RMSE** & **MAPE (\%)** \\ \hline PSO-BPNN & 11.71 & 18.98 & 1.20 \\ EMD-PSO-BPNN & 9.74 & 16.13 & 1.01 \\ WPT-PSO-BPNN & 6.80 & 12.03 & 0.70 \\ ITD-PSO-BPNN & 8.93 & 14.50 & 0.91 \\ VMD-PSO-BPNN & **5.45 \({}^{*}\)** & **7.22 \({}^{*}\)** & **0.57 \({}^{*}\)** \\ ARIMA & 10.87 & 18.49 & 1.11 \\ \hline \hline \end{tabular} \({}^{*}\) The smallest value of each column is marked in boldface and with an asterisk.  
\end{table} 
Table 5: Forecasting performance evaluation of soybean. **Author Contributions:**: [PERSON] designed the experiment for testing the proposed hybrid forecasting model. [PERSON] and [PERSON] made the program in MATLAB and analyzed the data. [PERSON] and [PERSON] wrote the manuscript.