The fractional non-equidistant grey opposite-direction model with time-varying characteristics

The grey opposite-direction model with fractional-order accumulation ($$ {\text{GOM}}^{\text{r}} (1,1) $$) has been appealed and interested in non-equidistant cases. However, there exists the drawback that it does not consider the effect of time-varying factor. In other words, the fixed grey control parameter defined as a certain constant limits the prediction performance of the model. By fully studying modelling procedure of the model, the optimized non-equidistant $$ {\text{GOM}}^{\text{r}} (1,1) $$ model with time-varying characteristics is proposed in this paper, which is abbreviated as $$ {\text{NTVGOM}}^{\text{r}} (1,1) $$ model. In the new model, a polynomial with time-varying characteristics is applied on grey control parameter, and the optimal fractional order could be automatically determined by minimizing the mean absolute percentage error. Then, the two empirical examples are employed to verify the effectiveness of the proposed model, and the numerical results show the proposed model has a better prediction performance.