## 4 Experimental Procedure and Analysis of Results
### Analysis of Results
#### 4.2.1 Results of LSTM-GARCH-Family Models  
The results of the LSTM-GARCH family combined models are shown in Table 6. From the experimental results, it can be seen that the prediction performance of the LSTM model is significantly better than those of the GARCH-family models alone. The values of MAE, RMSE, and MAPE for the LSTM model were 12.87%, 16.73%, and 5.21%, which are significantly lower than those of the GARCH, EGARCH, and PGARCH models. Compared with the GARCH-family models, the LSTM model can learn the temporal patterns and nonlinear characteristics of time series data more effectively due to its long memory. Figure 7 shows the prediction results of the LSTM model in the test set. In the combined models, the prediction performance was significantly improved over the LSTM model. Due to the addition of a GARCH-family model, the single combined models captured the information of economic characteristics in the garlic-price series. The experimental results show that LSTM-G had the best prediction performance among the single combined models. By adding GARCH coefficients to the input layer of LSTM, the LSTM-G model captured the information of economic characteristics, such as volatility aggregation and volatility persistence of garlic-price series. Thus, the model prediction \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **MODEL** & **MAE** & **RMSE** & **MAPE** \\ \hline Single model & LSTM & 0.1287 & 0.1673 & 5.21\% \\  & GARCH & 0.1690 & 0.2561 & 7.31\% \\  & EGARCH & 0.1718 & 0.2592 & 7.82\% \\  & PGARCH & 0.1691 & 0.2561 & 7.31\% \\ Single-combination model & LSTM-G & 0.1188 & 0.1600 & 4.87\% \\  & LSTM-E & 0.1247 & 0.1620 & 5.21\% \\  & LSTM-P & 0.1277 & 0.1683 & 5.34\% \\ Dual-combination model & LSTM-GE & 0.1155 & 0.1575 & 4.61\% \\  & LSTM-GP & 0.1144 & 0.1566 & 4.56\% \\  & LSTM-EP & 0.1148 & 0.1607 & 4.60\% \\ Triple-combination & LSTM-GEP & 0.1235 & 0.1619 & 5.12\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance comparison among different models. performance was significantly improved; and its values of MAE, RMSE, and MAPE were reduced to 11.88%, 16.00%, and 4.87%. Similarly, LSTM-E and LSTM-P, by the addition EGARCH coefficients and PGARCH coefficients to the input layer of LSTM, respectively, enable the models to capture the leverage effect of garlic-price series and enhance the flexibility of asymmetric fluctuations of the time series. GARCH coefficients had the highest contribution to the improvement of model performance among the single combined models. Figure 7b shows the prediction results of the LSTM-G combination model in the test set. Compared to the single models and single-combination models, it was found that the dual-combination models have better prediction performances. The LSTM-GP combined model had the best prediction performance. The combination of GARCH coefficients and PGARCH coefficients had the highest contribution to the improvement of model performance in the dual-combination models. The LSTM-GP model had the smallest MAE, RMSE and MAPE values of 11.44%, 15.66%, and 4.56%, respectively. Figure 7c shows the prediction results of the LSTM-GP model in the test set. The prediction performance of the LSTM-GEP combined model merged with three GARCH-family models did not further improve on the dual-combination models. Further additions did not provvide better prediction performance. Figure 7d shows the prediction results of the LSTM-GEP combination model in the test set. From the overall prediction results, it can be seen that the deep-learning model outperformed the econometric models and the combined models generally performed better than the single models. The combined models use the GARCH-family models to capture the fluctuation characteristic information of the garlic-price series, which effectively expands the price series information, thereby enabling the LSTM model to learn more effective  
Figure 7: The prediction results of the models on the test set. (**a**) LSTM, (**b**) LSTM-G, (**c**) LSTM-GP, (**d**) LSTM-GEP. information.