## 4 Experimental Procedure and Analysis of Results
### Analysis of Results
#### 4.2.2 Results of Comparison with the Benchmark Models  
The results of the prediction performance comparison between the proposed optimal combined model LSTM-GP and the three sets of benchmark models are shown in Figure 9a-c. It can be seen from the figures that the prediction performance of LSTM-GP was better than that of the GRU-GARCH combined models, the CNN-GARCH-family combined models, and the LSTM model with an attention mechanism. Below we analyze the reasons in detail.  
First, we compare and analyze the performances of the models obtained by combining LSTM, GRU, and CNN with GARCH-family models. From Figure 9a,b, we found that all three types of deep-learning models combined with GARCH-family models improved the prediction performances more than single deep-learning models. The results of the three types of combined models, LSTM-GARCH, GRU-GARCH, and CNN-GARCH, proved that combining deep-learning models with GARCH-family models can improve the prediction performance. The reason is that the GARCH-family models expand the historical price information by calculating the price volatility at each time node and mining the price volatility features at each time node, which enables the deep-learning models to capture feature information other than price and effectively improve the prediction performances of the models. Among them, the addition of GARCH and PGARCH models contributed the most to improving performance.  
Second, we compare and analyze the extraction abilities of three deep-learning models, LSTM, GRU, and CNN, for garlic price sequence features. From Figure 9a,b, it can be seen that among the various types of single and combined models, the LSTM-GP model produced the lowest MAE, RMSE, and MAPE metrics, which proves that LSTM has the best extraction ability for garlic-price series features. We briefly analyzed the reasons in terms of the three types of model structures. Although the CNN can capture the spatial structure of data, it does not have a long-term memory function and lacks consideration of the correlation of time-series data. Hence, its prediction performance is inferior to those of LSTM and GRU. Compared with the LSTM model, although the GRU improves  
Figure 8: Comparison of MAE, RMSE, and MAPE between the combined models and LSTM model.  
computational efficiency by simplifying the gate structure, GRU has a slightly inferior performance to LSTM. The LSTM-GP model has the best performance and outperformed the GRU-GP and CNN-GP models.  
Finally, to verify the effectiveness of the LSTM-GP model, we compared it with the attention-LSTM model, which shows a better current temporal prediction effect. Figure 9c shows that the LSTM model with the attention mechanism has better prediction performance than the LSTM model alone. The reason is that by adding the attention mechanism, the LSTM model can take into account the importance of different hidden states and focus more attention on the relatively essential states. Therefore, the model can better capture the sequence's long-term dependencies. Compared to the LSTM model with the attention mechanism, the LSTM-GP model, by simultaneously capturing characteristic information such as volatility aggregation and temporal correlation of garlic-price series, captures more effective information at each time node. For this reason, our proposed combined LSTM-GP model produced lower values for the three evaluation metrics and better performance in garlic price prediction.