Forecasting time series data is a critical area of research with applications spanning from stock prices to early epidemic prediction. While numerous statistical and machine learning methods have been proposed, real-life prediction problems often require hybrid solutions that bridge classical forecasting approaches and modern neural network models. In this study, we introduce the Probabilistic AutoRegressive Neural Networks (PARNN), capable of handling complex time series data exhibiting non-stationarity, nonlinearity, non-seasonality, long-range dependence, and chaotic patterns. PARNN is constructed by improving autoregressive neural networks (ARNN) using autoregressive integrated moving average (ARIMA) feedback error, combining the explainability, scalability, and "white-box-like" prediction behavior of both models. Notably, the PARNN model provides uncertainty quantification through prediction intervals, setting it apart from advanced deep learning tools. Through comprehensive computational experiments, we evaluate the performance of PARNN against standard statistical, machine learning, and deep learning models, including Transformers, NBeats, and DeepAR. Diverse real-world datasets from macroeconomics, tourism, epidemiology, and other domains are employed for short-term, medium-term, and long-term forecasting evaluations. Our results demonstrate the superiority of PARNN across various forecast horizons, surpassing the state-of-the-art forecasters. The proposed PARNN model offers a valuable hybrid solution for accurate long-range forecasting. By effectively capturing the complexities present in time series data, it outperforms existing methods in terms of accuracy and reliability. The ability to quantify uncertainty through prediction intervals further enhances the model's usefulness in decision-making processes.

翻译：时间序列预测是涵盖从股票价格到早期疫情预测等众多应用的关键研究领域。尽管已有大量统计与机器学习方法被提出，现实预测问题往往需要融合经典预测方法与现代神经网络模型的混合解决方案。本研究提出概率自回归神经网络（PARNN），该模型能处理呈现非平稳性、非线性性、非季节性、长期依赖及混沌模式等复杂特性的时间序列数据。PARNN通过利用自回归积分滑动平均模型（ARIMA）的反馈误差改进自回归神经网络（ARNN），融合了两类模型的可解释性、可扩展性与"白盒式"预测特性。值得注意的是，PARNN模型通过预测区间实现不确定性量化，使其区别于先进深度学习工具。通过全面的计算实验，我们评估了PARNN与标准统计模型、机器学习及深度学习模型（包括Transformer、NBeats和DeepAR）的性能差异。采用涵盖宏观经济、旅游、流行病等领域的多样化真实数据集进行短期、中期与长期预测评估。结果表明PARNN在不同预测跨度上均展现出优越性，显著超越现有最优预测器。所提出的PARNN模型为精确长期预测提供了宝贵的混合解决方案。通过有效捕捉时间序列数据的复杂特征，其在准确性与可靠性方面优于现有方法。通过预测区间实现不确定性量化的能力进一步增强了该模型在决策过程中的实用价值。