Accurate and stable natural gas price forecasts are essential for effective management of energy systems. However, due to the mixed frequency of data and the inherent nonlinear fluctuation characteristics of natural gas price changes, it is difficult to achieve satisfactory forecasting performance. In order to effectively improve the forecasting results of mixed frequency data. In this study, the MIDAS regression model and the machine learning MIDAS models are successfully combined to form a novel combination forecasting model. Moreover, the extreme learning machine with the multi-objective grey-wolf algorithm is used to combine univariate MIDAS results and further improve the forecasting accuracy. In the empirical analysis, the weekly natural gas futures prices of the Intercontinental Exchange UK NBP are used to generate real-time forecasts to evaluate the forecasting performance of the proposed combination model. The experimental results show that the comprehensive forecasting accuracy of the novel combination MIDAS model is 26.35%, 8.82% and 12.91% higher than that of the benchmark MIDAS regression models, the combination MIDAS models and the multivariate MIDAS models, respectively. Based on the forecasting results, the non-linear, non-stationary and irregular natural gas futures prices can be effectively managed, which provides better investment and management tools.

Carbon trading prices are crucial for carbon emissions and transparent carbon market pricing. Previous studies mainly focused on data mining in the prediction direction to quantify carbon trading prices. Although the prospect of high-frequency data forecasting mechanisms is considerable, more mixed-frequency ensemble forecasting is needed for carbon trading prices. Therefore, this article designs a new type of ensemble prediction model to increase the scope of model research. The module is divided into three parts: data denoising, mixed frequency and machine learning, multi-objective optimization, and ensemble forecasting. Precisely, the data preprocessing technology enhanced by adopting a self-attention mechanism can better remove noise and extract effective features. Furthermore, mixed frequency technology is introduced into the machine learning model to achieve more comprehensive and efficient prediction, and a new evaluation criterion is proposed to measure the optimal submodel. Finally, the ensemble model based on deep learning strategy can effectively integrate the advantages of high-frequency and low-frequency data in complex datasets. At the same time, a new multi-objective optimization algorithm is proposed to optimize the parameters of the ensemble model, significantly improving the predictive ability of the integrated module. The results of four experiments and the Mean Absolute Percent Error index of the proposed model improved by 28.3526% compared to machine learning models, indicating that the ensemble model established can effectively address the time distribution characteristics and uncertainty issues predicted by carbon trading price models, which helps to mitigate climate change and develop a low-carbon economy.

Effective crude oil price forecasting is essential for energy supply stabilization, investment decisions, policy formulation, and economic impact assessment. However, previous studies of crude oil futures price forecasting either consider only a single variable without adequately accounting for the influence of various factors on crude oil prices, or they consider only the same frequency of influences and ignore the importance of mixed frequencies, resulting in forecasts that do not achieve the desired results. To overcome these problems, this paper proposes a new multivariate combinatorial mixed frequency forecasting system to predict the weekly closing price of crude oil futures. The system is divided into four modules: Data denoising, Feature selection, Combined forecasting, and Performance evaluation module. To obtain smooth data, ICEEMDAN is used to denoise the original data. Furthermore, to select appropriate variables and reduce model redundancy, recursive feature elimination is used to select appropriate variables with low frequency. Considering the importance of mixed-frequency data, the mutual information method was used to select appropriate high-frequency variables for modeling the crude oil price forecast model. To overcome the shortcomings of Back Propagation Neural Network, Gate Recurrent Unit, and Radial Basis Function Neural Network models, integrate their advantages, and obtain accurate and stable prediction results, a combined forecasting mechanism based on a Multi-Objective Sparrow Search algorithm was developed to obtain both point and interval forecasting results, and finally, two data sets were selected for empirical analysis. The results show that the mean absolute percentage errors of the point forecast of this model are 1.96% and 1.84%, respectively, about 31% and 15% higher than those of the competing models (mean absolute percentage errors 2.57% and 2.13%, respectively). For interval forecasting, the accumulated width deviation is 0.0037 and 0.002, respectively, about 35% and 25% higher than those of the competing models (accumulated width deviation 0.005 and 0.0025, respectively). Thus, the proposed forecasting framework outperforms all comparative models and can be used effectively for forecasting crude oil prices.

In this paper, we study the eventual periodicity of the fuzzy max-type difference equation \(x_{n} =\max \{C , \frac{x_{n-m-k}}{x_{n-m} }\}, n\in \{0,1,\ldots \} \), where m and k are positive integers, C and the initial values are positive fuzzy numbers. Let the support \(\operatorname{supp} C=\overline{\{t : C(t) > 0\}}=[C_{1},C_{2}]\) of C. We show that: (1) if \(C_{1}>1\), then every positive solution of this equation equals C eventually; (2) there exists a positive fuzzy number C with \(C_{1}=1\) such that this equation has a positive solution which is not eventually periodic; (3) if \(C_{2}\leq 1\), then this equation has a positive solution which is not eventually periodic; (4) if \(C_{1}<1<C_{2}\), then every positive solution of the above equation is not eventually periodic.