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Sulfur dioxide is an important source of atmospheric pollution. Many countries are developing policies to reduce sulfur dioxide emissions. In this paper, a novel prediction model is proposed, which could be used to forecast sulfur dioxide emissions. To improve the modeling procedure, fractional order accumulating generation operator and fractional order reducing generation operator are introduced. Based on fractional order operators, a discrete grey model with fractional operators is developed, which also makes use of genetic algorithms to optimize the modeling parameter . The improved performance of the model is demonstrated via comparison studies with other grey models. The model is then used to predict China's sulfur dioxide emissions. The forecast result shows that the amount of sulfur dioxide emissions is steadily decreasing and the policies of sulfur dioxide reduction in China are effective. According to the current trend, by 2020, the value of China's sulfur dioxide emissions will be only 86.843% of emissions in 2015. Fractional order generation operators can be used to develop other fractional order system models.

Gross domestic product (GDP) is the most widely-used tool for measuring the overall situation of a country’s economic activity within a specified period of time. A more accurate forecasting of GDP based on standardized procedures with known samples available is conducive to guide decision making of government, enterprises and individuals. This study devotes to enhance the accuracy regarding GDP forecasting with given sample of historical data. To achieve this purpose, the study incorporates artificial neural network (ANN) into grey Markov chain model to modify the residual error, thus develops a novel hybrid model called grey Markov chain with ANN error correction (abbreviated as GMCM_ANN), which assembles the advantages of three components to fit nonlinear forecasting with limited sample sizes. The new model has been tested by adopting the historical data, which includes the original GDP data of the United States, Japan, China and India from 2000 to 2019, and also provides predications on four countries’ GDP up to 2022. Four models including autoregressive integrated moving average model, back-propagation neural network, the traditional GM(1,1) and grey Markov chain model are as benchmarks for comparison of the predicted accuracy and application scope. The obtained results are satisfactory and indicate superior forecasting performance of the proposed approach in terms of accuracy and universality.

The aim of this work was to improve the forecasting performance of business failure prediction with all sample sizes by constructing a novel nonlinear integrated forecasting model (ANIFM) of individual linear forecasting models and individual nonlinear forecasting models. First, a new variable set including internal variables and external variables was proposed. Using scatter diagrams, all variables were placed in either the linear group or the nonlinear group. We considered logistic regression (LR) as the individual linear forecasting method to deal with each linear variable, the support vector machine (SVM) as the individual nonlinear forecasting method to deal with each nonlinear variable, and the residual SVM as the integration method to integrate the forecasts of LRs and SVMs. The proposed procedure was applied to real datasets from China. For performance comparison, single LR, SVM methods, integration forecasting models based on equal weights and on neural networks, and one based on rough set and Dempster-Shafer evidence theory (D-S theory) were also included in the empirical experiment as benchmarks. The experimental results demonstrate the superior forecasting performance of the proposed ANIFM in terms of forecasting accuracy and forecasting stability, especially with small sample sizes.