## 4 Experiments
### Experimental Results and Analysis  
To rigorously assess the predictive performance of the proposed IGWO-MALSTM model, a series of comparative experiments were conducted utilizing a proprietary dataset comprising the price spreads between rebar and hot-rolled coil futures. The experimental framework incorporated four recent deep learning models--CS-LSTM, PSO-LSTM, Transformer, LSTM-Transformer--as benchmark baselines. All models were trained and evaluated under identical experimental conditions. The performance of each model was quantitatively measured using three widely recognized evaluation metrics: MSE, MAE, MAPE, and R\({}^{2}\). As evidenced in Table 6, the IGWO-MALSTM model consistently outperforms all baseline models across various evaluation metrics. Quantitatively, it achieved an MSE of 2.63, an MAE of 1.091, a MAPE of 2.79%, and an R\({}^{2}\) of 0.997 on the rebar and HRC futures dataset. Relative to the standard LSTM (MSE: 4.52, MAE: 1.587, MAPE: 4.47%), the model yielded a 41.8% reduction in MSE, a 31.3% reduction in MAE, and a 37.6% reduction in MAPE. Compared to GRU (MSE: 5.60, MAE: 1.785, MAPE: 5.52%), IGWO-MALSTM reduced the MSE by 53.0%, the MAE by 38.9%, and the MAPE by 49.5%. In addition, it demonstrated superior convergence behavior and predictive accuracy compared to CS-LSTM (MSE: 2.91, MAE: 1.163, MAPE: 2.85%), underscoring the enhanced optimization capability of the IGWO algorithm over conventional swarm-based techniques. While IGWO-MALSTM requires a slightly longer training time (65.15 min), this is acceptable in financial applications, where training is typically conducted offline and inference time remains minimal. When benchmarked against state-of-the-art architectures such as Transformer and LSTM-Transformer, the proposed approach achieved comparable or better performance--particularly in reducing cumulative forecast error over longer horizons, where traditional models often struggle with gradient degradation. These results high \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Search Range** & **Optimal Value** \\ \hline look\_back & [1, 10] & 1 \\ neuron1 & [30, 100] & 98 \\ neuron2 & [30, 100] & 88 \\ neuron3 & [30, 100] & 66 \\ dropout & [0.0002, 0.5] & 0.133489 \\ batch\_size & [9000, 10,000] & 9583 \\ epochs & [50, 100] & 98 \\ \hline \end{tabular}
\end{table} 
Table 5: Results of hyperparametric optimization. light the robustness and practical utility of the IGWO-MALSTM framework for complex financial time series forecasting tasks. On the volatile, high-frequency CSI 300 1 min dataset, IGWO-MALSTM again delivered the best overall performance, with an MSE of 4.17, MAE of 3.466, MAPE of 6.92%, and R\({}^{2}\) of 0.996. Compared to the LSTM baseline, it reduced MSE by 59.7%, MAE by 67.1%, and MAPE by 42.4%. Against the Transformer model, it achieved improvements of 51.0%, 62.2%, and 29.7%, respectively. These results highlight IGWO-MALSTM's strong generalization across varying frequencies and data scales, making it highly suitable for both low- and high-frequency financial forecasting. Figure 9 further illustrates the superior performance of the IGWO-MALSTM model through visual analysis. The predicted curves indicate that the model is capable of closely tracking both the short-term fluctuations and long-term trends of the futures price spread series. Notably, even as the prediction window increases--a condition that typically leads to performance degradation--IGWO-MALSTM maintains a relatively stable error profile. This robustness can be attributed to the synergy between the IGWO algorithm and the multi-head self-attention mechanism: the former effectively optimizes the hyperparameter selection of LSTM, while the latter dynamically emphasizes critical time steps and reduces sensitivity to irrelevant or noisy inputs. In addition to the architectural advantages of the model, its optimization strategy plays a critical role in enhancing performance. The integration of IGWO enables adaptive and efficient hyperparameter tuning through the use of the Mersenne Twister.