# Forecasting per Capita Energy Consumption in China Using a Spatial Discrete Grey Prediction Model

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions by 18% compared to the respective level in 2020. In this context, a reasonable forecast of China’s future energy consumption is of high practical significance for ensuring energy security and formulating energy conservation policies.

- (1)
- In this paper, spatial characteristics are incorporated into DGM(1,1), which enhances spatial data predictions. SDGM(1,1,m) is proposed and is used to analyze spatial spillover effects in the selected modeling interval, and a grey model is used to process panel data.
- (2)
- SDGM(1,1,m) is compared with DGM(1,n), and the differences between them are analyzed in terms of modeling purposes and requirements.
- (3)
- In this paper, considering the time lag effect that often accompanies spatial interaction processes, L1-SDGM(1,1,m) is proposed, thus providing a conceptual approach for establishing other time-lag-based spatial discrete grey models.
- (4)
- Using the PCEC data from 30 provinces in China, SDGM(1,1,m) and L1-SDGM(1,1,m) are compared with DGM(1,1), DGM(1,n), NDGM(1,1), and BP neural network models to verify the effectiveness and superiority of SDGM(1,1,m) and L1-SDGM(1,1,m) for predicting the PCEC of China.
- (5)
- Based on a metabolic concept, we use SDGM(1,1,m) to predict the PCECs of 30 provinces in China from 2020–2025.

## 2. Construction of SDGM(1,1,m) and L1-SDGM(1,1,m)

#### 2.1. Introduction of DGM(1,1)

#### 2.2. Definition of SDGM(1,1,m)

**Definition**

**1.**

#### 2.3. Definition of L1-SDGM(1,1,m)

**Definition**

**2.**

#### 2.4. Error Evaluation Index

#### 2.5. Flow Chart of SDGM(1,1,m)

## 3. Applications in Forecasting PCEC in China

#### 3.1. Data Collection

#### 3.2. Establishment of the Spatial Weight Matrix

#### 3.3. Model Comparison

#### 3.4. Projections of PCEC of China

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Name | Formulation |
---|---|

The Root Mean Square Percentage Error (RMSPE) | $RMSPE=\sqrt{\frac{1}{n\times m}{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{k=1}^{n}{\left(\frac{{x}_{j}^{(0)}\left(k\right)-{\widehat{x}}_{j}^{(0)}\left(k\right)}{{x}_{j}^{(0)}\left(k\right)}\right)}^{2}}}}\times 100\%$ |

The Root Mean Square Error (RMSE) | $RMSE=\sqrt{\frac{1}{n\times m}{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{k=1}^{n}{\left({x}_{j}^{(0)}\left(k\right)-{\widehat{x}}_{j}^{(0)}\left(k\right)\right)}^{2}}}}$ |

The Mean Absolute Error (MAE) | $MAE=\frac{1}{n\times m}{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{k=1}^{n}\left|{x}_{j}^{(0)}\left(k\right)-{\widehat{x}}_{j}^{(0)}\left(k\right)\right|}}$ |

The Index of Agreement (IA) | $IA=\frac{1}{m}{\displaystyle \sum _{j=1}^{m}(1-\frac{{{\displaystyle \sum}}_{k=1}^{n}{({x}_{j}^{(0)}\left(k\right)-{\widehat{x}}_{j}^{(0)}\left(k\right))}^{2}}{{{\displaystyle \sum}}_{k=1}^{n}{\left(\left|{x}_{j}^{(0)}\left(k\right)-{\overline{x}}_{j}\left|+\right|{\widehat{x}}_{j}^{(0)}\left(k\right)-{\overline{x}}_{j}\right|\right)}^{2}}})$ |

The Correlation Coefficient (R) | $R=\frac{1}{m}{\displaystyle \sum _{j=1}^{m}\frac{\mathrm{cov}\left({x}_{j}^{(0)}\left(k\right),{\widehat{x}}_{j}^{(0)}\left(k\right)\right)}{\sqrt{\mathrm{Var}\left. [{\widehat{x}}_{j}^{(0)}\left(k\right)\left]\mathrm{Var}\right. [{x}_{j}^{(0)}\left(k\right)\right]}}}$ |

Year | Moran’s I | Z-Score | p-Value |
---|---|---|---|

2010 | 0.225 | 2.813 | 0.002 |

2011 | 0.197 | 2.517 | 0.006 |

2012 | 0.191 | 2.453 | 0.007 |

2013 | 0.193 | 2.446 | 0.007 |

2014 | 0.181 | 2.328 | 0.010 |

2015 | 0.181 | 2.343 | 0.010 |

2016 | 0.174 | 2.274 | 0.011 |

2017 | 0.170 | 2.251 | 0.012 |

2018 | 0.167 | 2.250 | 0.012 |

2019 | 0.156 | 2.150 | 0.016 |

Error Metrics | MDGM | DGM | SDGM | L1-SDGM | NDGM | BP |
---|---|---|---|---|---|---|

MRSPE (%) | 1.1887 × 10^{3} | 1.8251 | 1.6163 | 1.6999 | 1.5425 | 3.7021 |

MRFPE (%) | 1.0076 × 10^{4} | 5.8939 | 3.5159 | 5.2328 | 4.7667 | 7.1758 |

CMRPE (%) | 2.9661 × 10^{3} | 2.6388 | 1.9962 | 2.4065 | 2.1873 | 4.3968 |

RMSPE | 6.4272 × 10^{3} | 4.1807 | 3.1759 | 4.7898 | 3.5840 | 7.6856 |

RMSE | 34,167.3452 | 24.4039 | 18.8417 | 24.2914 | 22.6363 | 35.8206 |

MAE | 129.7515 | 0.1143 | 0.0847 | 0.1012 | 0.0911 | 0.1711 |

IA | 0.0049 | 0.8264 | 0.9072 | 0.8807 | 0.8527 | 0.7236 |

R | 0.0172 | 0.7514 | 0.8644 | 0.8429 | 0.7892 | 0.5647 |

Province | Spatial Correlation Coefficient-b | Province | Spatial Correlation Coefficient-b | ||
---|---|---|---|---|---|

SDGM | L1-SDGM | SDGM | L1-SDGM | ||

Beijing | 0.3845 | 0.3267 | Henan | 0.1176 | 0.0709 |

Tianjin | 0.8436 | 0.6084 | Hubei | 0.3368 | 0.2932 |

Hebei | 0.1196 | −0.1231 | Hunan | 0.3292 | 0.2901 |

Shanxi | 0.4113 | 0.8088 | Guangdong | 0.6824 | 0.5985 |

Inner Mongolia | 2.1688 | 1.2967 | Guangxi | 0.1482 | 0.0030 |

Liaoning | 1.4674 | −0.1139 | Hainan | −0.0448 | −0.0584 |

Jilin | 0.0669 | 0.0186 | Chongqing | 0.3304 | 0.2542 |

Heilongjiang | 0.2612 | −0.0964 | Sichuan | 0.2506 | 0.1302 |

Shanghai | 0.4931 | 0.5391 | Guizhou | 0.6152 | 0.6303 |

Jiangsu | 0.5556 | 0.1995 | Yunnan | 0.4339 | 0.1157 |

Zhejiang | −0.7091 | −0.3981 | Shaanxi | 0.1336 | 0.0887 |

Anhui | 0.1713 | 0.1522 | Gansu | 0.2695 | −0.3935 |

Fujian | 0.6435 | 0.5127 | Qinghai | 1.1506 | 1.4352 |

Jiangxi | 0.0582 | 0.0570 | Ningxia | −0.6829 | −0.8404 |

Shandong | 0.7867 | 0.3192 | Xinjiang | 0.6587 | 0.6787 |

Province | 2020 | 2021 | 2022 | 2023 | 2024 | 2025 |
---|---|---|---|---|---|---|

Beijing | 3.4015 | 3.4225 | 3.4171 | 3.3649 | 3.2611 | 3.0888 |

Tianjin | 5.9607 | 6.1118 | 6.2208 | 6.2472 | 6.2527 | 6.2041 |

Hebei | 4.4683 | 4.5062 | 4.5709 | 4.6455 | 4.7347 | 4.8103 |

Shanxi | 6.0551 | 6.3351 | 6.5418 | 6.7544 | 6.9954 | 7.2678 |

Inner Mongolia | 10.3993 | 11.2928 | 11.8828 | 12.3648 | 12.7920 | 13.4055 |

Liaoning | 5.4333 | 5.7046 | 5.8732 | 6.0661 | 6.2458 | 6.4316 |

Jilin | 2.8981 | 3.0800 | 3.1546 | 3.2611 | 3.3602 | 3.4692 |

Heilongjiang | 3.5072 | 3.7621 | 3.8624 | 3.9873 | 4.1115 | 4.2505 |

Shanghai | 4.8301 | 4.9320 | 5.0331 | 5.1702 | 5.3241 | 5.4703 |

Jiangsu | 3.8777 | 3.9323 | 3.9998 | 4.0874 | 4.1772 | 4.2650 |

Zhejiang | 3.5749 | 3.6706 | 3.7472 | 3.8415 | 3.9517 | 4.0679 |

Anhui | 2.3283 | 2.3957 | 2.4640 | 2.5334 | 2.6019 | 2.6726 |

Fujian | 3.3444 | 3.4510 | 3.5465 | 3.6656 | 3.7879 | 3.9184 |

Jiangxi | 2.1837 | 2.2514 | 2.3182 | 2.3878 | 2.4714 | 2.5623 |

Shandong | 4.2566 | 4.3510 | 4.4373 | 4.5540 | 4.6636 | 4.7802 |

Henan | 2.2553 | 2.2854 | 2.2958 | 2.3177 | 2.3462 | 2.4001 |

Hubei | 2.9657 | 3.0748 | 3.1673 | 3.2718 | 3.3901 | 3.5205 |

Hunan | 2.4674 | 2.5592 | 2.6294 | 2.7188 | 2.8234 | 2.9354 |

Guangdong | 2.7815 | 2.8356 | 2.8822 | 2.9400 | 3.0042 | 3.0731 |

Guangxi | 2.3030 | 2.3915 | 2.4663 | 2.5548 | 2.6558 | 2.7686 |

Hainan | 2.3343 | 2.4036 | 2.4748 | 2.5577 | 2.6469 | 2.7466 |

Chongqing | 2.8665 | 2.9959 | 3.0987 | 3.2294 | 3.3844 | 3.5581 |

Sichuan | 2.5476 | 2.6653 | 2.7668 | 2.8900 | 3.0316 | 3.1954 |

Guizhou | 2.7783 | 2.8838 | 2.9696 | 3.0880 | 3.2208 | 3.3631 |

Yunnan | 2.6431 | 2.7154 | 2.8217 | 2.9346 | 3.0577 | 3.1948 |

Shaanxi | 3.5389 | 3.6699 | 3.8253 | 4.0066 | 4.2078 | 4.4288 |

Gansu | 3.2307 | 3.3675 | 3.5202 | 3.6727 | 3.8754 | 4.1317 |

Qinghai | 7.3175 | 7.3303 | 7.3759 | 7.3487 | 7.3797 | 7.4859 |

Ningxia | 12.0485 | 13.3855 | 15.0643 | 16.8951 | 19.3048 | 22.1072 |

Xinjiang | 7.5810 | 7.7859 | 8.0738 | 8.4171 | 8.8585 | 9.2682 |

Modeling Interval | MRSPE (%) |
---|---|

2012–2019 | 1.1395 |

2013–2020 | 0.6917 |

2014–2021 | 0.5570 |

2015–2022 | 0.4549 |

2016–2023 | 0.3510 |

2017–2024 | 0.3232 |

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**MDPI and ACS Style**

Wang, H.; Zhang, Z.
Forecasting per Capita Energy Consumption in China Using a Spatial Discrete Grey Prediction Model. *Systems* **2023**, *11*, 285.
https://doi.org/10.3390/systems11060285

**AMA Style**

Wang H, Zhang Z.
Forecasting per Capita Energy Consumption in China Using a Spatial Discrete Grey Prediction Model. *Systems*. 2023; 11(6):285.
https://doi.org/10.3390/systems11060285

**Chicago/Turabian Style**

Wang, Huiping, and Zhun Zhang.
2023. "Forecasting per Capita Energy Consumption in China Using a Spatial Discrete Grey Prediction Model" *Systems* 11, no. 6: 285.
https://doi.org/10.3390/systems11060285