# Neural Multivariate Grey Model and Its Applications
## 4 Experiments
### China's per Capita Energy Consumption Prediction  
The experiment in this section was carried out to predict energy consumption per capita in China using publicly available historical data. As shown in Table 1, we used China's per capita energy consumption data from 2012 to 2021, which comes from the Energy Statistical Yearbook 2022 [37]. The total energy unit is kilos of standard coal, while electricity is measured in kilowatt-hours and coal and oil in kilograms. Before constructing our models, we divided the dataset into two distinct segments. We used the period from 2012 to 2018, shown in Table 1, as the test dataset. Subsequently, the latest data from 2019 to 2021 are used as the test dataset to evaluate and compare the prediction accuracy. When selecting predictive indicators, we prioritize total energy consumption per capita as the primary system characteristic sequence, categorizing the remaining indicators as influencing factors. Table 2 displays the forecast results of the GM, FDGM, FGM, NOGDM, and NMGM models for the data presented in Table 1. The most optimal outcomes are indicated in bold, while the second most optimal outcomes are underlined. In addition to that, it shows the MAPE of all models based on the data used for training and testing. As indicated in Table 2, the NMGM model outperforms all other grey models in the MAPE. The NMGM model achieves an MAPE of 0.2537% on the training data and 0.7381% on the test data, surpassing all other tested models. The NMGM model exhibits a superior generalization capacity in comparison with the other grey models. Simultaneously, we specifically noted \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Year** & **Total Energy** & **Electricity** & **Coal** & **Oil** \\ \hline
2012 & 2977 & 3684 & 3018 & 354 \\
2013 & 3058 & 3976 & 3113 & 367 \\
2014 & 3122 & 4215 & 3015 & 378 \\
2015 & 3146 & 4205 & 2998 & 406 \\
2016 & 3181 & 4410 & 2802 & 416 \\
2017 & 3285 & 4721 & 2803 & 433 \\
2018 & 3364 & 5098 & 2833 & 444 \\
2019 & 3463 & 5318 & 2855 & 458 \\
2020 & 3531 & 5501 & 2869 & 463 \\
2021 & 3724 & 6032 & 3042 & 484 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Per capita energy consumption in China (2012–2021). that the predictive accuracy of the multi-variable NMGM surpassed that of the uni-variable NODGM in both the training and test sets. The major reason for this is that the model takes into account the impact factor. In order to make a more accurate comparison of the discrepancies between these models, the MAPE findings are compared in Figure 3, while the APE results are displayed in Figure 4. When comparing MAPE and APE, it is evident that the NMGM model exhibits superior prediction accuracy in contrast to other grey models. As illustrated in Figure 5, the per capita energy consumption in China exhibits a clear upward trend. Along with societal development, energy consumption continues to persist, and the sustained growth in energy usage is expected to remain stable. The predictions in this study can help government departments make better decisions and policies by providing more accurate data. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Year** & **GM(I,N)** & **FDGM(I,N)** & **FGM(I,N)** & **NODGM** & **NMGM** \\ \hline
2012 & 2977 & 2977 & 2977 & 2977 & 2977 \\
2013 & 2910.35 & 2993.94 & 2993.62 & 3033.55 & 3054.94 \\
2014 & 3156.44 & 3072.23 & 3071.91 & 3093.55 & 3098.41 \\
2015 & 3136.18 & 3152.59 & 3149.20 & 3157.22 & 3137.33 \\
2016 & 3160.87 & 3234.98 & 3216.73 & 3224.75 & 3194.37 \\
2017 & 3280.37 & 3319.57 & 3321.35 & 3296.41 & 3271.18 \\
2018 & 3387.67 & 3406.40 & 3376.2 & 3372.41 & 3367.31 \\ \hline MAPE & 1.4503\% & 1.1589\% & 0.9082\% & 0.5298\% & **0.2537**\% \\ \hline
2019 & 3486.65 & 3495.43 & 3488.27 & 3453.02 & 3482.64 \\
2020 & 3537.06 & 3586.83 & 3589.80 & 3538.51 & 3617.58 \\
2021 & 3744.13 & 3680.61 & 3682.52 & 3629.16 & 3771.59 \\ \hline MAPE & 1.2702\% & 1.106\% & 0.9927\% & 1.794\% & **0.7381**\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Modeling result of per capita energy consumption in China.