Table 1.
Calculation Results of Grey Modelling for China's GDP
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May
2022, 18(3): 2017-2032.
doi: 10.3934/jimo.2021054
Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application
| 1. | School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, China |
| 2. | School of Business, Suzhou University of Science and Technology, Suzhou, 215009, China |
In the grey prediction, the nonlinear Grey Bernoulli model NGBM (1, 1) is an important type. The NGBM (1, 1) has good adaptability to data fitting and then small prediction errors, and thus has been applied widely. However, if we improve the modelling method, the prediction precision shall be improved to some extent. The important factors of prediction error are the approximation of background value and the approximation of power exponent. Therefore, the paper tries to combine the optimisation of background value with the optimisation of the power exponent of NGBM (1, 1) model and then improves the model from parameter estimation. The paper gives three methods for the following three cases respectively: the background value in the form of exponential curve, the background value in the form of the polynomial curve and the background value in the form of interpolation function, to combine background value optimisation with power exponent optimisation for parameter optimisation. The final section of the paper builds the NGBM (1, 1) models of China's GDP and energy consumption with three improvement methods. The simulation and prediction results show the three improvement methods all have high precision. The methods given offer good approaches for the in-depth study on nonlinear grey Bernoulli model, enrich the method system of grey modelling and can be applied to the studies on other grey models to promote the study and wide application of the grey model.
Keywords:
NGBM (1, 1) model,
parameter estimation,
background value,
power exponent,
prediction precision.
Mathematics Subject Classification:
Primary: 62F99; Secondary: 93A30.
Citation:
Maolin Cheng, Yun Liu, Jianuo Li, Bin Liu. Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application. Journal of Industrial and Management Optimization,
2022, 18
(3)
: 2017-2032.
doi: 10.3934/jimo.2021054
![]()
References:
| [1] |
C. I. Chen, H. L. Chen and S. P. Chen,
Forecasting of foreign exchange rates of Taiwan's major trading partners by novel nonlinear Grey Bernoulli model NGBM(1, 1), Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 1194-1204.
doi: 10.1016/j.cnsns.2006.08.008. |
| [2] |
Y. Y. Chen, G. W. Chen and A. H. Chiou,
Forecasting nonlinear time series using an adaptive nonlinear grey Bernoulli model: Cases of energy consumption, Journal of Grey System, 29 (2017), 75-93.
|
| [3] |
M. Cheng and G. Shi,
Modeling and application of grey model GM (2, 1) based on linear difference equation, Journal of Grey System, 31 (2019), 37-50.
|
| [4] |
M. Cheng and G. Shi,
Improved methods for parameter estimation of gray model GM (1, 1) based on new background value optimization and model application, Communications in Statistics - Simulation and Computation, 49 (2020), 1367-1384.
doi: 10.1080/03610918.2018.1498890. |
| [5] |
M. Cheng and M. Xiang,
Generalized GM(1, 1) model and its application, Journal of Grey System, 29 (2017), 110-122.
|
| [6] |
J. Cui, Y. G. Dang and S. F. Liu,
Novel grey forecasting model and its modeling mechanism, Control and Decision, 24 (2009), 1702-1706.
|
| [7] |
S. Ding, Y. G. Dang, N. Xu, J. J. Wang and S. S. Geng,
Construction and optimization of a multi-variables discrete grey power model, Systems Engineering and Electronics, 40 (2018), 1302-1309.
|
| [8] |
P. Hu,
GM(1, 1) power model for optimizing background values and its application, Mathematics in Practice and Theory, 47 (2017), 99-104.
|
| [9] |
Y. Huang, X. Chen and Y. Wang,
Application of GM(1, 1) power model based on sum of sine in port throughput prediction, Journal of Shanghai Maritime University, 40 (2019), 69-73.
|
| [10] |
J. Lan and Y. Zhou,
Death rate per million ton prediction of coal mine accidents based on improved gray Markov GM(1, 1) model, Mathematics in Practice and Theory, 44 (2014), 145-152.
|
| [11] |
J. L. Li, X. P. Xiao and R. Q. Liao,
Non-Equidistance GM(1, 1) power and its application, Systems Engineering-Theory & Practice, 30 (2010), 490-495.
|
| [12] |
L. Li, D. Zhang, J. Tang, J. Liu, C. Li, Z. Wang and Y. He,
Application of unequal interval GM (1, 1) power model in prediction of dissolved gases for power transformer failure, Power System Protection and Control, 45 (2017), 118-124.
|
| [13] |
S. F. Li and P. Y. Chen,
Unbiased GM(1, 1) power model and its application, Statistics & Information Forum, 25 (2010), 7-10.
|
| [14] |
J. S. Lu, W. D. Xie, H. B. Zhou and A. J. Zhang,
An optimized nonlinear grey Bernoulli model and its applications, Neurocomputing, 177 (2016), 206-214.
doi: 10.1016/j.neucom.2015.11.032. |
| [15] |
X. Ma, Z. B. Liu and Y. Wang,
Application of a novel nonlinear multivariate grey Bernoulli model to predict the tourist income of China, Journal of Computational and Applied Mathematics, 347 (2019), 84-94.
doi: 10.1016/j.cam.2018.07.044. |
| [16] |
Y. M. Ma and S. C. Wang,
Construction and application of improved GM(1, 1) power mode, Journal of Quantitative Economics, 36 (2019), 84-88.
|
| [17] |
L. Pei, W. Chen, J. Bai and Z. Wang,
The improved GM (1, N) models with optimal background values: A case study of Chinese high-tech industry, Journal of Grey System, 27 (2015), 223-233.
|
| [18] |
F. X. Wang, Improvement GM(1, 1) power model and its optimization, Pure and Applied Mathematics, 27 (2011), 148-150, 157. |
| [19] |
Q. Wang, S. Y. Li and R. R. Li,
Forecasting energy demand in China and India: Using single-linear, hybrid-linear, and non-linear time series forecast techniques, Energy, 161 (2018), 821-831.
doi: 10.1016/j.energy.2018.07.168. |
| [20] |
Q. Wang, S. Y. Li and R. R. Li,
Will Trump's coal revival plan work? - Comparison of results based on the optimal combined forecasting technique and an extended IPAT forecasting technique, Energy, 169 (2019), 762-775.
doi: 10.1016/j.energy.2018.12.045. |
| [21] |
Q. Wang, S. Y. Li, R. R. Li and M. L. Ma,
Forecasting U.S. shale gas monthly production using a hybrid ARIMA and metabolic nonlinear grey model, Energy, 160 (2018), 378-387.
doi: 10.1016/j.energy.2018.07.047. |
| [22] |
Y. H. Wang, Y. G. Dang, Y. Q. Li and S. F. Liu,
An approach to increase prediction precision of GM(1, 1) model based on optimization of the initial condition, Expert Systems with Applications, 37 (2010), 5610-5644.
doi: 10.1016/j.eswa.2010.02.048. |
| [23] |
Z. X. Wang,
GM(1, 1) power model with time-varying parameters and its application, Control and Decision, 29 (2014), 1828-1832.
|
| [24] |
Z. X. Wang, Y. G. Dang, S. F. Liu and Z. W. Lian,
Solution of GM(1, 1) power model and its properties, Systems Engineering and Electronics, 31 (2009), 2380-2383.
|
| [25] |
Z. Wu, J. Shuai and S. Wang,
Forecasting of Chinese copper demand based on improved gray model, Industrial Technology & Economy, 33 (2014), 9-14.
|
| [26] |
N. M. Xie and S. F. Liu,
Discrete grey forecasting model and its optimization, Applied Mathematical Modelling, 33 (2009), 1173-1186.
doi: 10.1016/j.apm.2008.01.011. |
| [27] |
B. H. Yang and J. S. Zhao,
Fractional order discrete grey GM(1, 1) power model and its application, Control and Decision, 30 (2015), 1264-1268.
|
| [28] |
Z. Yu, C. Yang, Z. Zhang and J. Jiao,
Error correction method based on data transformational GM(1, 1) and application on tax forecasting, Applied Soft Computing, 37 (2015), 554-560.
doi: 10.1016/j.asoc.2015.09.001. |
| [29] |
L. Zeng,
Grey GM(1, 1| sin) power model based on oscillation sequences and its application, Journal of Zhejiang University(Science Edition), 46 (2019), 697-704.
|
| [30] |
S. J. Zhang and S. Y. Chen,
Optimization of GM(1, 1) power model and its application, Systems Engineering, 34 (2016), 154-158.
|
| [31] |
J. Z. Zhou, R. C. Fang, Y. H. Li, Y. C. Zhang and B. Peng,
Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization, Applied Mathematics and Computation, 207 (2009), 292-299.
doi: 10.1016/j.amc.2008.10.045. |
show all references
References:
| [1] |
C. I. Chen, H. L. Chen and S. P. Chen,
Forecasting of foreign exchange rates of Taiwan's major trading partners by novel nonlinear Grey Bernoulli model NGBM(1, 1), Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 1194-1204.
doi: 10.1016/j.cnsns.2006.08.008. |
| [2] |
Y. Y. Chen, G. W. Chen and A. H. Chiou,
Forecasting nonlinear time series using an adaptive nonlinear grey Bernoulli model: Cases of energy consumption, Journal of Grey System, 29 (2017), 75-93.
|
| [3] |
M. Cheng and G. Shi,
Modeling and application of grey model GM (2, 1) based on linear difference equation, Journal of Grey System, 31 (2019), 37-50.
|
| [4] |
M. Cheng and G. Shi,
Improved methods for parameter estimation of gray model GM (1, 1) based on new background value optimization and model application, Communications in Statistics - Simulation and Computation, 49 (2020), 1367-1384.
doi: 10.1080/03610918.2018.1498890. |
| [5] |
M. Cheng and M. Xiang,
Generalized GM(1, 1) model and its application, Journal of Grey System, 29 (2017), 110-122.
|
| [6] |
J. Cui, Y. G. Dang and S. F. Liu,
Novel grey forecasting model and its modeling mechanism, Control and Decision, 24 (2009), 1702-1706.
|
| [7] |
S. Ding, Y. G. Dang, N. Xu, J. J. Wang and S. S. Geng,
Construction and optimization of a multi-variables discrete grey power model, Systems Engineering and Electronics, 40 (2018), 1302-1309.
|
| [8] |
P. Hu,
GM(1, 1) power model for optimizing background values and its application, Mathematics in Practice and Theory, 47 (2017), 99-104.
|
| [9] |
Y. Huang, X. Chen and Y. Wang,
Application of GM(1, 1) power model based on sum of sine in port throughput prediction, Journal of Shanghai Maritime University, 40 (2019), 69-73.
|
| [10] |
J. Lan and Y. Zhou,
Death rate per million ton prediction of coal mine accidents based on improved gray Markov GM(1, 1) model, Mathematics in Practice and Theory, 44 (2014), 145-152.
|
| [11] |
J. L. Li, X. P. Xiao and R. Q. Liao,
Non-Equidistance GM(1, 1) power and its application, Systems Engineering-Theory & Practice, 30 (2010), 490-495.
|
| [12] |
L. Li, D. Zhang, J. Tang, J. Liu, C. Li, Z. Wang and Y. He,
Application of unequal interval GM (1, 1) power model in prediction of dissolved gases for power transformer failure, Power System Protection and Control, 45 (2017), 118-124.
|
| [13] |
S. F. Li and P. Y. Chen,
Unbiased GM(1, 1) power model and its application, Statistics & Information Forum, 25 (2010), 7-10.
|
| [14] |
J. S. Lu, W. D. Xie, H. B. Zhou and A. J. Zhang,
An optimized nonlinear grey Bernoulli model and its applications, Neurocomputing, 177 (2016), 206-214.
doi: 10.1016/j.neucom.2015.11.032. |
| [15] |
X. Ma, Z. B. Liu and Y. Wang,
Application of a novel nonlinear multivariate grey Bernoulli model to predict the tourist income of China, Journal of Computational and Applied Mathematics, 347 (2019), 84-94.
doi: 10.1016/j.cam.2018.07.044. |
| [16] |
Y. M. Ma and S. C. Wang,
Construction and application of improved GM(1, 1) power mode, Journal of Quantitative Economics, 36 (2019), 84-88.
|
| [17] |
L. Pei, W. Chen, J. Bai and Z. Wang,
The improved GM (1, N) models with optimal background values: A case study of Chinese high-tech industry, Journal of Grey System, 27 (2015), 223-233.
|
| [18] |
F. X. Wang, Improvement GM(1, 1) power model and its optimization, Pure and Applied Mathematics, 27 (2011), 148-150, 157. |
| [19] |
Q. Wang, S. Y. Li and R. R. Li,
Forecasting energy demand in China and India: Using single-linear, hybrid-linear, and non-linear time series forecast techniques, Energy, 161 (2018), 821-831.
doi: 10.1016/j.energy.2018.07.168. |
| [20] |
Q. Wang, S. Y. Li and R. R. Li,
Will Trump's coal revival plan work? - Comparison of results based on the optimal combined forecasting technique and an extended IPAT forecasting technique, Energy, 169 (2019), 762-775.
doi: 10.1016/j.energy.2018.12.045. |
| [21] |
Q. Wang, S. Y. Li, R. R. Li and M. L. Ma,
Forecasting U.S. shale gas monthly production using a hybrid ARIMA and metabolic nonlinear grey model, Energy, 160 (2018), 378-387.
doi: 10.1016/j.energy.2018.07.047. |
| [22] |
Y. H. Wang, Y. G. Dang, Y. Q. Li and S. F. Liu,
An approach to increase prediction precision of GM(1, 1) model based on optimization of the initial condition, Expert Systems with Applications, 37 (2010), 5610-5644.
doi: 10.1016/j.eswa.2010.02.048. |
| [23] |
Z. X. Wang,
GM(1, 1) power model with time-varying parameters and its application, Control and Decision, 29 (2014), 1828-1832.
|
| [24] |
Z. X. Wang, Y. G. Dang, S. F. Liu and Z. W. Lian,
Solution of GM(1, 1) power model and its properties, Systems Engineering and Electronics, 31 (2009), 2380-2383.
|
| [25] |
Z. Wu, J. Shuai and S. Wang,
Forecasting of Chinese copper demand based on improved gray model, Industrial Technology & Economy, 33 (2014), 9-14.
|
| [26] |
N. M. Xie and S. F. Liu,
Discrete grey forecasting model and its optimization, Applied Mathematical Modelling, 33 (2009), 1173-1186.
doi: 10.1016/j.apm.2008.01.011. |
| [27] |
B. H. Yang and J. S. Zhao,
Fractional order discrete grey GM(1, 1) power model and its application, Control and Decision, 30 (2015), 1264-1268.
|
| [28] |
Z. Yu, C. Yang, Z. Zhang and J. Jiao,
Error correction method based on data transformational GM(1, 1) and application on tax forecasting, Applied Soft Computing, 37 (2015), 554-560.
doi: 10.1016/j.asoc.2015.09.001. |
| [29] |
L. Zeng,
Grey GM(1, 1| sin) power model based on oscillation sequences and its application, Journal of Zhejiang University(Science Edition), 46 (2019), 697-704.
|
| [30] |
S. J. Zhang and S. Y. Chen,
Optimization of GM(1, 1) power model and its application, Systems Engineering, 34 (2016), 154-158.
|
| [31] |
J. Z. Zhou, R. C. Fang, Y. H. Li, Y. C. Zhang and B. Peng,
Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization, Applied Mathematics and Computation, 207 (2009), 292-299.
doi: 10.1016/j.amc.2008.10.045. |
| Year | No. | Conventional Method of NGBM (1, 1) Model | Improvement Method 1 | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 187318. | - | - | - | - |
| 2006 | 2 | 219438.5 | 238832.77 | 8.84 | 217434.79 | 0.913 |
| 2007 | 3 | 270092.3 | 284734.09 | 5.42 | 270086.82 | 0.00203 |
| 2008 | 4 | 319244.6 | 328292.04 | 2.83 | 319272.6 | 0.00877 |
| 2009 | 5 | 348517.7 | 372350.46 | 6.84 | 367555.16 | 5.46 |
| 2010 | 6 | 412119.3 | 418204.34 | 1.48 | 416204.97 | 0.991 |
| 2011 | 7 | 487940.2 | 466656.87 | 4.36 | 465997.29 | 4.5 |
| 2012 | 8 | 538580.0 | 518315.16 | 3.76 | 517476.83 | 3.92 |
| 2013 | 9 | 592963.2 | 573702.33 | 3.25 | 571068.67 | 3.69 |
| 2014 | 10 | 641280.6 | 633309.02 | 1.24 | 627132.03 | 2.21 |
| 2015 | 11 | 685992.9 | 697620.56 | 1.7 | 685989.35 | 5.18e-4 |
| 2016 | 12 | 740060.8 | 767133.24 | 3.66 | 747943.29 | 1.07 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 820754.3 | 842365.18 | 2.63 | 813287.48 | 0.9098 |
| 2018 | 14 | 900309.5 | 923864.46 | 2.62 | 882313.59 | 1.999 |
| Average Simulation Relative Error (2005-2016) | - | 3.94 | - | 2.07 | ||
| Average Prediction Relative Error (2017-2018) | - | 2.62 | - | 1.45 | ||
| Average Relative Error (2005-2018) | - | 3.74 | - | 1.97 | ||
| Year | No. | Conventional Method of NGBM (1, 1) Model | Improvement Method 1 | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 187318. | - | - | - | - |
| 2006 | 2 | 219438.5 | 238832.77 | 8.84 | 217434.79 | 0.913 |
| 2007 | 3 | 270092.3 | 284734.09 | 5.42 | 270086.82 | 0.00203 |
| 2008 | 4 | 319244.6 | 328292.04 | 2.83 | 319272.6 | 0.00877 |
| 2009 | 5 | 348517.7 | 372350.46 | 6.84 | 367555.16 | 5.46 |
| 2010 | 6 | 412119.3 | 418204.34 | 1.48 | 416204.97 | 0.991 |
| 2011 | 7 | 487940.2 | 466656.87 | 4.36 | 465997.29 | 4.5 |
| 2012 | 8 | 538580.0 | 518315.16 | 3.76 | 517476.83 | 3.92 |
| 2013 | 9 | 592963.2 | 573702.33 | 3.25 | 571068.67 | 3.69 |
| 2014 | 10 | 641280.6 | 633309.02 | 1.24 | 627132.03 | 2.21 |
| 2015 | 11 | 685992.9 | 697620.56 | 1.7 | 685989.35 | 5.18e-4 |
| 2016 | 12 | 740060.8 | 767133.24 | 3.66 | 747943.29 | 1.07 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 820754.3 | 842365.18 | 2.63 | 813287.48 | 0.9098 |
| 2018 | 14 | 900309.5 | 923864.46 | 2.62 | 882313.59 | 1.999 |
| Average Simulation Relative Error (2005-2016) | - | 3.94 | - | 2.07 | ||
| Average Prediction Relative Error (2017-2018) | - | 2.62 | - | 1.45 | ||
| Average Relative Error (2005-2018) | - | 3.74 | - | 1.97 | ||
Table 3.
Coefficients of the Cubic Spline Interpolation Function
| 2 | -946.81174 | 26774.212 | 193611.1 | 187318.9 |
| 3 | 1839.4352 | 23933.777 | 244319.09 | 406757.4 |
| 4 | -7912.4291 | 29452.082 | 297704.95 | 676849.7 |
| 5 | 9931.0812 | 5714.7949 | 332871.82 | 996094.3 |
| 6 | 2516.6044 | 35508.038 | 374094.66 | 1344612.0 |
| 7 | -7778.1987 | 43057.852 | 452660.55 | 1756731.3 |
| 8 | 3415.0903 | 19723.256 | 515441.65 | 2244671.5 |
| 9 | -2138.7625 | 29968.526 | 565133.44 | 2783251.5 |
| 10 | -925.84013 | 23552.239 | 618654.2 | 3376214.7 |
| 11 | 2237.0231 | 20774.718 | 662981.16 | 4017495.3 |
| 12 | 1333.3479 | 27485.788 | 711241.66 | 4703488.2 |
| 2 | -946.81174 | 26774.212 | 193611.1 | 187318.9 |
| 3 | 1839.4352 | 23933.777 | 244319.09 | 406757.4 |
| 4 | -7912.4291 | 29452.082 | 297704.95 | 676849.7 |
| 5 | 9931.0812 | 5714.7949 | 332871.82 | 996094.3 |
| 6 | 2516.6044 | 35508.038 | 374094.66 | 1344612.0 |
| 7 | -7778.1987 | 43057.852 | 452660.55 | 1756731.3 |
| 8 | 3415.0903 | 19723.256 | 515441.65 | 2244671.5 |
| 9 | -2138.7625 | 29968.526 | 565133.44 | 2783251.5 |
| 10 | -925.84013 | 23552.239 | 618654.2 | 3376214.7 |
| 11 | 2237.0231 | 20774.718 | 662981.16 | 4017495.3 |
| 12 | 1333.3479 | 27485.788 | 711241.66 | 4703488.2 |
Table 2.
Calculation Results of Grey Modelling for China's GDP (Continued Table of Table 1)
| Year | No. | Improvement Method 2 | Improvement Method 3 | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 187318 | - | - | - | - |
| 2006 | 2 | 219438.5 | 217479.83 | 0.893 | 217472.44 | 0.896 |
| 2007 | 3 | 270092.3 | 270092.3 | 8.83e-8 | 270087.67 | 0.00171 |
| 2008 | 4 | 319244.6 | 319244.64 | 1.31e-5 | 319242.61 | 6.22e-4 |
| 2009 | 5 | 348517.7 | 367502.09 | 5.45 | 367502.19 | 5.45 |
| 2010 | 6 | 412119.3 | 416135.74 | 0.975 | 416137.41 | 0.975 |
| 2011 | 7 | 487940.2 | 465921.3 | 4.51 | 465923.92 | 4.51 |
| 2012 | 8 | 538580.0 | 517404.02 | 3.93 | 517406.92 | 3.93 |
| 2013 | 9 | 592963.2 | 571009.63 | 3.7 | 571012.09 | 3.7 |
| 2014 | 10 | 641280.6 | 627098.13 | 2.21 | 627099.39 | 2.21 |
| 2015 | 11 | 685992.9 | 685992.9 | 1.59e-7 | 685992.1 | 1.16e-4 |
| 2016 | 12 | 740060.8 | 747997.67 | 1.07 | 747993.9 | 1.07 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 820754.3 | 813407.26 | 0.895 | 813399.51 | 0.896 |
| 2018 | 14 | 900309.5 | 882514.71 | 1.98 | 882501.89 | 1.98 |
| Average Simulation Relative Error (2005-2016) | - | 2.07 | - | 2.07 | ||
| Average Prediction Relative Error (2017-2018) | - | 1.44 | - | 1.44 | ||
| Average Relative Error (2005-2018) | - | 1.97 | - | 1.97 | ||
| Year | No. | Improvement Method 2 | Improvement Method 3 | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 187318 | - | - | - | - |
| 2006 | 2 | 219438.5 | 217479.83 | 0.893 | 217472.44 | 0.896 |
| 2007 | 3 | 270092.3 | 270092.3 | 8.83e-8 | 270087.67 | 0.00171 |
| 2008 | 4 | 319244.6 | 319244.64 | 1.31e-5 | 319242.61 | 6.22e-4 |
| 2009 | 5 | 348517.7 | 367502.09 | 5.45 | 367502.19 | 5.45 |
| 2010 | 6 | 412119.3 | 416135.74 | 0.975 | 416137.41 | 0.975 |
| 2011 | 7 | 487940.2 | 465921.3 | 4.51 | 465923.92 | 4.51 |
| 2012 | 8 | 538580.0 | 517404.02 | 3.93 | 517406.92 | 3.93 |
| 2013 | 9 | 592963.2 | 571009.63 | 3.7 | 571012.09 | 3.7 |
| 2014 | 10 | 641280.6 | 627098.13 | 2.21 | 627099.39 | 2.21 |
| 2015 | 11 | 685992.9 | 685992.9 | 1.59e-7 | 685992.1 | 1.16e-4 |
| 2016 | 12 | 740060.8 | 747997.67 | 1.07 | 747993.9 | 1.07 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 820754.3 | 813407.26 | 0.895 | 813399.51 | 0.896 |
| 2018 | 14 | 900309.5 | 882514.71 | 1.98 | 882501.89 | 1.98 |
| Average Simulation Relative Error (2005-2016) | - | 2.07 | - | 2.07 | ||
| Average Prediction Relative Error (2017-2018) | - | 1.44 | - | 1.44 | ||
| Average Relative Error (2005-2018) | - | 1.97 | - | 1.97 | ||
Table 4.
Calculation Results of Grey Modelling for China's Energy Consumption
| Year | No. | GM (1, 1) Model | Improvement Method 3 | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 261369.0 | - | - | - | - |
| 2006 | 2 | 286467.0 | 301511.04 | 5.25 | 286467.0 | 2.49e-10 |
| 2007 | 3 | 311442.0 | 314255.12 | 0.903 | 311442.0 | 1.74e-8 |
| 2008 | 4 | 320611.0 | 327537.85 | 2.16 | 331239.51 | 3.32 |
| 2009 | 5 | 336126.0 | 341382.0 | 1.56 | 348511.62 | 3.69 |
| 2010 | 6 | 360648.0 | 355811.32 | 1.34 | 364321.66 | 1.02 |
| 2011 | 7 | 387043.0 | 370850.52 | 4.18 | 379210.19 | 2.02 |
| 2012 | 8 | 402138.0 | 386525.39 | 3.88 | 393491.8 | 2.15 |
| 2013 | 9 | 416913.0 | 402862.8 | 3.37 | 407366.8 | 2.29 |
| 2014 | 10 | 425806.0 | 419890.74 | 1.39 | 420971.29 | 1.14 |
| 2015 | 11 | 429905.0 | 437638.41 | 1.8 | 434402.48 | 1.05 |
| 2016 | 12 | 435819.0 | 456136.2 | 4.66 | 447732.61 | 2.73 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 448529.0 | 475415.9 | 5.99 | 461017.19 | 2.78 |
| 2018 | 14 | 464000.0 | 495510.47 | 6.79 | 474300.09 | 2.22 |
| 2019 | 15 | 479312.0 | 516454.39 | 7.75 | 487616.83 | 1.73 |
| Average Simulation Relative Error (2005-2016) | - | 2.77 | - | 1.76 | ||
| Average Prediction Relative Error (2017-2019) | - | 6.84 | - | 2.35 | ||
| Average Relative Error (2005-2019) | - | 3.64 | - | 1.88 | ||
| Year | No. | GM (1, 1) Model | Improvement Method 3 | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 261369.0 | - | - | - | - |
| 2006 | 2 | 286467.0 | 301511.04 | 5.25 | 286467.0 | 2.49e-10 |
| 2007 | 3 | 311442.0 | 314255.12 | 0.903 | 311442.0 | 1.74e-8 |
| 2008 | 4 | 320611.0 | 327537.85 | 2.16 | 331239.51 | 3.32 |
| 2009 | 5 | 336126.0 | 341382.0 | 1.56 | 348511.62 | 3.69 |
| 2010 | 6 | 360648.0 | 355811.32 | 1.34 | 364321.66 | 1.02 |
| 2011 | 7 | 387043.0 | 370850.52 | 4.18 | 379210.19 | 2.02 |
| 2012 | 8 | 402138.0 | 386525.39 | 3.88 | 393491.8 | 2.15 |
| 2013 | 9 | 416913.0 | 402862.8 | 3.37 | 407366.8 | 2.29 |
| 2014 | 10 | 425806.0 | 419890.74 | 1.39 | 420971.29 | 1.14 |
| 2015 | 11 | 429905.0 | 437638.41 | 1.8 | 434402.48 | 1.05 |
| 2016 | 12 | 435819.0 | 456136.2 | 4.66 | 447732.61 | 2.73 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 448529.0 | 475415.9 | 5.99 | 461017.19 | 2.78 |
| 2018 | 14 | 464000.0 | 495510.47 | 6.79 | 474300.09 | 2.22 |
| 2019 | 15 | 479312.0 | 516454.39 | 7.75 | 487616.83 | 1.73 |
| Average Simulation Relative Error (2005-2016) | - | 2.77 | - | 1.76 | ||
| Average Prediction Relative Error (2017-2019) | - | 6.84 | - | 2.35 | ||
| Average Relative Error (2005-2019) | - | 3.64 | - | 1.88 | ||
Table 5.
Calculation Results of Grey Modelling for China's Energy Consumption
| Year | No. | Improvement Method Proposed by Zhang and Chen [30] | Improvement Method Proposed by Ma and Wang [16] | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 261369.0 | - | - | - | - |
| 2006 | 2 | 286467.0 | 267260.08 | 6.7 | 264410.21 | 7.7 |
| 2007 | 3 | 311442.0 | 302546.19 | 2.86 | 301168.24 | 3.3 |
| 2008 | 4 | 320611.0 | 329138.11 | 2.66 | 328840.42 | 2.57 |
| 2009 | 5 | 336126.0 | 350612.31 | 4.31 | 351024.11 | 4.43 |
| 2010 | 6 | 360648.0 | 368636.25 | 2.21 | 369441.76 | 2.44 |
| 2011 | 7 | 387043.0 | 384142.61 | 0.749 | 385074.85 | 0.509 |
| 2012 | 8 | 402138.0 | 397713.35 | 1.1 | 398544.77 | 0.894 |
| 2013 | 9 | 416913.0 | 409739.41 | 1.72 | 410274.19 | 1.59 |
| 2014 | 10 | 425806.0 | 420497.92 | 1.25 | 420565.97 | 1.23 |
| 2015 | 11 | 429905.0 | 430193.48 | 0.0671 | 429645.9 | 0.0603 |
| 2016 | 12 | 435819.0 | 438982.1 | 0.726 | 437687.7 | 0.429 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 448529.0 | 446985.87 | 0.344 | 444828.45 | 0.825 |
| 2018 | 14 | 464000.0 | 454302.42 | 2.09 | 451178.59 | 2.76 |
| 2019 | 15 | 479312.0 | 461011.22 | 3.82 | 456828.7 | 4.69 |
| Average Simulation Relative Error (2005-2016) | - | 2.21 | - | 2.29 | ||
| Average Prediction Relative Error (2017-2019) | - | 2.08 | - | 2.76 | ||
| Average Relative Error (2005-2019) | - | 2.18 | - | 2.39 | ||
| Year | No. | Improvement Method Proposed by Zhang and Chen [30] | Improvement Method Proposed by Ma and Wang [16] | |||
| Simulation Value | Relative Error % | Simulation Value | Relative Error % | |||
| 2005 | 1 | 261369.0 | - | - | - | - |
| 2006 | 2 | 286467.0 | 267260.08 | 6.7 | 264410.21 | 7.7 |
| 2007 | 3 | 311442.0 | 302546.19 | 2.86 | 301168.24 | 3.3 |
| 2008 | 4 | 320611.0 | 329138.11 | 2.66 | 328840.42 | 2.57 |
| 2009 | 5 | 336126.0 | 350612.31 | 4.31 | 351024.11 | 4.43 |
| 2010 | 6 | 360648.0 | 368636.25 | 2.21 | 369441.76 | 2.44 |
| 2011 | 7 | 387043.0 | 384142.61 | 0.749 | 385074.85 | 0.509 |
| 2012 | 8 | 402138.0 | 397713.35 | 1.1 | 398544.77 | 0.894 |
| 2013 | 9 | 416913.0 | 409739.41 | 1.72 | 410274.19 | 1.59 |
| 2014 | 10 | 425806.0 | 420497.92 | 1.25 | 420565.97 | 1.23 |
| 2015 | 11 | 429905.0 | 430193.48 | 0.0671 | 429645.9 | 0.0603 |
| 2016 | 12 | 435819.0 | 438982.1 | 0.726 | 437687.7 | 0.429 |
| Prediction Value | Relative Error % | Prediction Value | Relative Error % | |||
| 2017 | 13 | 448529.0 | 446985.87 | 0.344 | 444828.45 | 0.825 |
| 2018 | 14 | 464000.0 | 454302.42 | 2.09 | 451178.59 | 2.76 |
| 2019 | 15 | 479312.0 | 461011.22 | 3.82 | 456828.7 | 4.69 |
| Average Simulation Relative Error (2005-2016) | - | 2.21 | - | 2.29 | ||
| Average Prediction Relative Error (2017-2019) | - | 2.08 | - | 2.76 | ||
| Average Relative Error (2005-2019) | - | 2.18 | - | 2.39 | ||
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