Multi-step stock index forecasting is vital in finance for informed decision-making. Current forecasting methods on this task frequently produce unsatisfactory results due to the inherent data randomness and instability, thereby underscoring the demand for advanced forecasting models. Given the superiority of capsule network (CapsNet) over CNN in various forecasting and classification tasks, this study investigates the potential of integrating a 1D CapsNet with an LSTM network for multi-step stock index forecasting. To this end, a hybrid 1D-CapsNet-LSTM model is introduced, which utilizes a 1D CapsNet to generate high-level capsules from sequential data and a LSTM network to capture temporal dependencies. To maintain stochastic dependencies over different forecasting horizons, a multi-input multi-output (MIMO) strategy is employed. The model's performance is evaluated on real-world stock market indices, including S&P 500, DJIA, IXIC, and NYSE, and compared to baseline models, including LSTM, RNN, and CNN-LSTM, using metrics such as RMSE, MAE, MAPE, and TIC. The proposed 1D-CapsNet-LSTM model consistently outperforms baseline models in two key aspects. It exhibits significant reductions in forecasting errors compared to baseline models. Furthermore, it displays a slower rate of error increase with lengthening forecast horizons, indicating increased robustness for multi-step forecasting tasks.

翻译：多步股票指数预测对于金融领域的明智决策至关重要。由于数据固有的随机性和不稳定性，当前针对该任务的预测方法常常产生不理想的结果，从而凸显了对先进预测模型的需求。鉴于胶囊网络在各类预测和分类任务中优于CNN，本研究探讨了将一维胶囊网络与LSTM网络相结合用于多步股票指数预测的潜力。为此，提出了一种混合1D-CapsNet-LSTM模型，该模型利用一维胶囊网络从序列数据中生成高层胶囊，并利用LSTM网络捕捉时间依赖关系。为保持不同预测范围上的随机依赖关系，采用了多输入多输出策略。该模型在真实世界股票市场指数（包括S&P 500、DJIA、IXIC和NYSE）上进行了性能评估，并与LSTM、RNN和CNN-LSTM等基线模型通过RMSE、MAE、MAPE和TIC等指标进行了比较。所提出的1D-CapsNet-LSTM模型在两个关键方面持续优于基线模型：与基线模型相比，其预测误差显著降低；此外，随着预测范围的延长，其误差增加速度较慢，表明该模型在多步预测任务中具有更强的鲁棒性。