In this paper, a fractional opposite-direction accumulating nonlinear grey Bernoulli Markov model (FOANGBMKM) is established to forecast the annual GDP of Huizhou city from 2017 to 2021. The optimal fractional order number and nonlinear parameters of the model are determined by particle swarm optimization (PSO) algorithm. An experiment is provided to validate the high fitting accuracy of this model, and the effect of prediction is better than that of the other four competitive models such as autoregressive integrated moving average model (ARIMA), grey model (GM (1, 1)), fractional accumulating nonlinear grey Bernoulli model (FANGBM (1, 1)) and fractional opposite-direction accumulating nonlinear grey Bernoulli model (FOANGBM (1, 1)), which proves the robustness of the opposite-direction accumulating nonlinear Bernoulli Markov model. This research will provide a scientific basis and technical references for the economic planning industries.

In this paper, we propose an optimal fractional-order accumulative Grey Markov model with variable parameters (FOGMKM (1, 1)) to predict the annual total energy consumption in China and improve the accuracy of energy consumption forecasting. The new model is built upon the traditional Grey model and utilized matrix perturbation theory to study the natural and response characteristics of a system when the structural parameters change slightly. The particle swarm optimization algorithm (PSO) is used to determine the number of optimal fractional order and nonlinear parameters. An experiment is conducted to validate the high prediction accuracy of the FOGMKM (1, 1) model, with mean absolute percentage error (MAPE) and root mean square error (RMSE) values of 0.51% and 1886.6, respectively, and corresponding fitting values of 0.92% and 6108.8. These results demonstrate the superior fitting performance of the FOGMKM (1, 1) model when compared to other six competitive models, including GM (1, 1), ARIMA, Linear, FAONGBM (1, 1), FGM (1, 1) and FOGM (1, 1). Our study provides a scientific basis and technical references for further research in the finance as well as energy fields and can serve well for energy market benchmark research.