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.

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.

In this paper, we investigate the stability of following max-type difference equation , where , with , , and , the initial values are positive. By constructing a system of equations and binary function, we show the equation has a unique positive equilibrium solution, and the positive equilibrium solution is globally asymptotically stable. The conclusion of this paper extends and supplements the existing results.