The prediction of a time series such as climate indices and the sunspot number (SSN) with long-term oscillatory behaviors has been a challenging task due to the complex combination of oscillations. Frequency extraction algorithms have been developed to separate a time series into different oscillation components according to frequency, such as empirical model decomposition (EMD) and wavelet analysis. In the current study, the deep learning long short-term memory (LSTM) model was employed to predict the oscillation components extracted using EMD. The SSN series was modeled with the hybrid EMD-LSTM model. The simulation study results indicate that the LSTM model reproduces the smooth cyclic pattern of the sine function, and only a few hidden units are needed to model it. The EMD-LSTM model achieves better performance than does the LSTM model for mid-range SSN predictions while the LSTM achieves better performance within the first few time lags. However, the cyclic prediction of the SSN requires mid-range lags; thus, the superior performance of the EMD-LSTM model for these lags cannot be ignored. Furthermore, the remaining components from the significant EMD signals can be modeled to reveal the variability (or uncertainty) in the prediction. The summed residual is fitted by k-nearest neighbor resampling. The final SSN prediction results show that the EMD-LSTM model predicts a later and larger SSN for Solar Cycle 25 than does the LSTM model. Overall, the results lead to the conclusion that the EMD-LSTM model might be a suitable alternative for modeling complex sunspot time series with cyclic patterns.

中文翻译：

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用于预测具有循环模式的太阳黑子数时间序列的 EMD 和 LSTM 混合深度学习模型

由于振荡的复杂组合，预测具有长期振荡行为的气候指数和太阳黑子数 (SSN) 等时间序列一直是一项具有挑战性的任务。已经开发了频率提取算法来根据频率将时间序列分离为不同的振荡分量，例如经验模型分解 (EMD) 和小波分析。在当前的研究中，采用深度学习长短期记忆 (LSTM) 模型来预测使用 EMD 提取的振荡分量。SSN 系列采用混合 EMD-LSTM 模型建模。仿真研究结果表明，LSTM 模型再现了正弦函数的平滑循环模式，只需要很少的隐藏单元对其进行建模。EMD-LSTM 模型在中程 SSN 预测方面比 LSTM 模型实现了更好的性能，而 LSTM 在前几个时间延迟内实现了更好的性能。然而，SSN 的循环预测需要中程滞后；因此，EMD-LSTM 模型对于这些滞后的卓越性能不容忽视。此外，可以对来自重要 EMD 信号的其余分量进行建模，以揭示预测中的可变性（或不确定性）。求和残差通过 k 最近邻重采样拟合。最终的 SSN 预测结果表明，与 LSTM 模型相比，EMD-LSTM 模型预测的太阳周期 25 的 SSN 晚且更大。总体而言，结果得出的结论是，EMD-LSTM 模型可能是对具有循环模式的复杂太阳黑子时间序列进行建模的合适替代方案。