The return series of cryptocurrencies, which are emerging digital assets, exhibit nonstationarity, nonlinearity, and volatility clustering compared to other traditional financial markets, making them exceptionally difficult to forecast. Therefore, accurate cryptocurrency price forecasting is essential for market participants and regulators. It has been demonstrated that improved data forecasting accuracy can be achieved through decomposition, but few researchers have performed information extraction on the residual series generated by data decomposition. Based on the construction of a “decomposition-optimization-integration” hybrid model framework, in this paper, we propose a multi-scale hybrid forecasting model that combines the residual components after primary decomposition for secondary decomposition and integration. This model uses the variational modal decomposition (VMD) method to decompose the original return series into a finite number of components and residual terms. Then, the residual terms are decomposed, and the features are extracted using the completed ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method. The components are predicted by an extreme learning machine optimized by the sparrow search algorithm, and the final predictions are summed to obtain the final results. Forecasts for the returns of Bitcoin and Ethereum, which are significant cryptocurrency assets, are compared with other benchmark models constructed based on different ideas. We find that the proposed quadratic decomposition VMD-Res.-CEEMDAN-SSA-ELM hybrid model demonstrates the optimal and most stable forecasting performance in both one-step and multi-step ahead prediction of the cryptocurrency return series.

Research on the price prediction of natural gas is of great significance to market participants of all kinds. In order to predict natural gas prices more reliably, this paper introduces a quadratic decomposition technology based on the combination of variational modal decomposition (VMD) and ensemble empirical modal decomposition (EEMD), which decomposes the residual term (Res) after VMD by EEMD; then, a new hybrid model called VMD-EEMD-Res.-LSTM is constructed in combination with the long short-term memory (LSTM) prediction model. The contribution of this new hybrid model is that, unlike existing application research that combines existing decomposition technology with the LSTM model, it does not ignore the important information contained in the residual after the VMD. In order to verify the predictive performance of the proposed new model, this paper uses the data of the spot price of natural gas in the United States to conduct a multistep-ahead empirical comparative analysis. The results show that the new hybrid model constructed in this paper has significant predictive advantages.