## 5 Comparison of Forecasting Results by Different Forecasting Models
Without rolling mechanism, the further into the future, the more the forecasted annual electricity consumption deviates from the actual annual electricity consumption. However, a forecast with rolling mechanism can pull it back at each \"roll\". The primary reason is that a rolling forecast can use more up-to-date information, which takes not only the latest development of policy, the economy and society into account but also the features of forecasting objective into consideration. Therefore, the accuracy of forecasting results with rolling mechanism can be remarkably improved. This finding has also been verified in other practical issues, such as the prediction of GDP, interest rate, and energy consumption [42; 58]. Additionally, the forecasting performance of MFO-GM (1,1) is much better to that of LSR, GM (1,1) and FOA-GM (1,1), which implies using MFO to optimize the parameters of GM (1,1) is effective and promising. Figure 9: Forecasting results of Inner Mongolia’s electricity consumption from 2010 to 2014 by different models. To further evaluate the accuracy of each forecasting model, several evaluation indicators are selected to analyze the accuracy of forecasting results, namely Percentage Error (_PE_), Mean Absolute Percentage Error (_MAPE_), and Root Mean Square Error (_RMSE_), which can be calculated as Equations (24)-(26), respectively. \[PE=\frac{x(k)-\hat{\hat{x}}(k)}{x(k)}\times 100\% \tag{24}\]  
\[MAPE=\frac{1}{n}\sum_{k=1}^{n}\left|\frac{x(k)-\hat{\hat{x}}(k)}{x(k)}\right| \times 100\% \tag{25}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{k=1}^{n}{(x(k)-\hat{\hat{x}}(k))^{2}}} \tag{26}\]  
where \(x(k)\) is the actual value at time \(k\), and \(\hat{\hat{x}}(k)\) is the forecasting value at time \(k\). The _PE_ results of different forecasting models show the superior performance of Rolling-MFO-GM (1,1) when compared with that of other seven models, as shown in Table 4.  
The results of _MAPE_ and _RMSE_ shown in Table 5 verify again that the forecasting models with rolling mechanism perform better than that without rolling mechanism. Furthermore, as can be seen in Table 5, the _MAPE_ value of LSR is smaller than GM (1,1) but much larger than MFO-GM (1,1), which indicates that the MFO algorithm is effective. Meanwhile, according to the evaluating criterion of _MAPE_ listed in Table 6 [33], as the _MAPE_ results of the forecasting models with rolling mechanism are smaller than 10%, the forecasting power of these models is excellent, while the _MAPE_ results of the models without rolling mechanism are a little more than 20%, which indicate the forecasting power of these models is reasonable. What is more, the Rolling-MFO-GM (1,1) is the best forecasting model for annual electricity consumption due to its obtained smallest _MAPE_ (6.29%). We can draw the same conclusions from the results of _RMSE_ as that of _MAPE_. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**LSR**} & \multirow{2}{*}{**GM (1,1)**} & \multicolumn{2}{c}{**FOA-GM**} & \multicolumn{2}{c}{**MFO-GM**} & \multirow{2}{*}{**Rolling-LSR**} & \multicolumn{2}{c}{**Rolling-GM**} & \multicolumn{2}{c}{**Rolling-FOA-**} & \multicolumn{2}{c}{**Rolling-MFO-**} \\  & & & **(1,1)** & & **(1,1)** & & **(1,1)** & **GM (1,1)** & **GM (1,1)** \\ \hline _MAPE_ & 21.39\% & 25.01\% & 23.98\% & 21.09\% & 12.42\% & 9.55\% & 8.30\% & 6.29\% \\ _RMSE_ & 441.16 & 662.34 & 643.2 & 642.52 & 235.98 & 217.18 & 195.6 & 164.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Criterion of _MAPE_[33].