## 4 Experiment
### Experimental Results and Analysis
Table 7 provides a detailed characterization of each model's performance indicators for one-step, four-step, and eight-step predictions across the \"open\", \"high\", \"low\", and \"close\" dimensions. To clearly observe that the PSO Deep-ConvLSTM model can approximately predict the trend, we have plotted a comparison between the model's eight-step predictions for the \"close\" dimension and the actual values (as shown in Figure 6). Additionally, Figure 7 illustrates the prediction errors of the PSO Deep-ConvLSTM model compared with the actual values, providing further insight into its predictive accuracy. From these data and figures, we can obtain an approximate understanding of the relative merits of each model. To enable a more intuitive and precise comparison of predictive effectiveness among the models and to underscore the superior performance of the PSO Deep-ConvLSTM model, we compiled the average of these indicators across all dimensions. Figure 6 clearly illustrates that, for one-step-ahead predictions, the LSTM model has a significant advantage over other models, such as the Transformer. The PSO Deep-ConvLSTM model yields RMSE, MAE, and MAPE values of 1.508, 1.184, and 0.884, respectively. Compared with the traditional GRU and LSTM models, our proposed model has achieved a reduction of 43.8% and 39.1% in RMSE, 47.9% and 41.6% in MAE, and 56.8% and 46.6% in MAPE, demonstrating its superior prediction performance in single-step forecasting. From Figure 7, we observe that in four-step-ahead predictions, the LSTM and GRU models start to falter, while the Transformer-type models begin to show their strengths. The performance of LSTM and GRU deteriorates significantly, whereas the proposed model attains RMSE, MAE, and MAPE values of 2.579, 1.800, and 1.286, still delivering satisfactory results. When compared with advanced Transformer and FEDformer models, the PSO Deep-ConvLSTM model exhibits a decrease in RMSE by 39.1%, 43.9%, and 33.1%; in MAE by 43.0%, 47.8%; and 32.3%, and in MAPE by 45.8%, 47.8%, and 31.0%. This implies that the discrepancy between predicted and actual values for the four-step-ahead forecast is smaller with higher prediction accuracy. Furthermore, the R2 value is closer to 1, indicating a strong fitting capability. In Figure 8, in the case of eight-step-ahead predictions, the performance of LSTM and GRU models is quite poor, which also justifies their advantage in short-term forecasting. As the number of prediction steps increases, the error metrics exhibit substantial deterioration. However, the proposed model, with RMSE, MAE, and MAPE values of 4.520, 3.293, and 2.369, can alleviate this issue to some extent. In comparison with the long-term forecasting-focused Transformer model, our model reduces the various metrics by 0.4%, 0.1%, and 3.1%, respectively. Even when compared with the FEDformer model, an expert in long-term forecasting, our model surpasses it under the wavelet exchange mode and remains fairly competitive under the Fourier exchange mode. Figures 9-11 display the average evaluation indicators for single-step and multiple-step predictions of the six models. In summary, the PSO Deep-ConvLSTM model achieves satisfactory prediction results in both single-step and multistep forecasting. Figure 8: Prediction errors.