## 4 Experiment
### Experimental Results and Analysis  
In this section, as we employ our proposed model to conduct multistep, multidimensional predictions on inter-commodity futures price spread data, it is imperative to validate the forecasting effectiveness of different models. According to the study by [PERSON] et al. [32], multistep predictions can be realized through either iterative or independent methods. 1. The iterative method uses a single step-ahead model to iteratively generate forecasts. 2. The independent method uses a dedicated network to forecast each forecast horizon. For comparative experimentation purposes, we have selected the following models: iterative LSTM, iterative GRU, Transformer, and FEDformer. The first two models belong to the iterative category, while our employed ConvLSTM, Transformer, and FEDtransformer adopt the independent prediction approach. Specifically, recurrent neural networks (RNNs) have been appropriated in the realm of financial forecasting due to their distinct advantages in handling time series problems. However, they do present certain drawbacks, particularly the issue of gradient explosion. LSTM networks, capable of remedying the setbacks of RNNs, offer enhanced feature extraction and generalization abilities. The GRU model is another variant of the LSTM model. We have made minor modifications to these models enabling them to perform iterative predictions, thereby accomplishing multistep predictions. Additionally, the Transformer [33] is an encoder-decoder structure primarily encompassing position encoding, position embedding, and a self-attention module. The multihead attention feature in the Transformer allows for parallel computation, thereby reducing the training duration. Since its introduction in 2017, the Transformer has been widely applied to various data types, including text and image data. The FEDformer is a variant of the Transformer model [34] that combines the Transformer with the seasonal-trend decomposition method and uses randomly selected Fourier components to maintain a compact representation of time series. This not only overcomes the Transformer's high computation cost and inability to capture a global view of time series, but it also further enhances prediction accuracy. Thus, we have also selected these two models for comparison. By evaluating our proposed model against these models, we can ascertain its performance and effectiveness in predicting cross-commodity futures price spreads, supplying valuable references for further research and application. In the experiments, the batch size for all models was set to 128. For LSTM and GRU models, a two-layer structure was adopted with a learning rate of 0.0001 over 100 training epochs. The number of neurons was set to 100 and 20 for the first and second layers, respectively, while the Transformer and FEDformer models utilized the official default parameters. To comprehensively evaluate the performance of our proposed model and validate its superiority under different prediction horizons, we selected prediction settings of one, four, and eight steps. Specifically, the one-step prediction was primarily employed for benchmarking against LSTM-based models, highlighting the competitive edge of our proposed model in single-step forecasting. On the other hand, the four-step and eight-step predictions were utilized to assess the model's refinement capabilities in handling multistep forecasting tasks. For ease of reference, the models were assigned numerical identifiers: the GRU model was Model-1, the LSTM was Model-2, the Transformer was Model-3, FEDforemer-W was Model-4, FEDformer-F was Model-5, and our proposed PSO Deep-ConvLSTM was Model-6. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Search Range** & **Optimal Value** \\ \hline Learning rate & [0.00001, 0.0005] & 5.7399496072129 \(\times\) 10\({}^{5}\) \\ Epoch & [1, 400] & 365 \\ ConvLSTM layers numbers & [2; 7] & 6 \\ Kernel size & [1; 9] & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of hyperparametric optimization.