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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

* Corresponding author: Maolin Cheng

Received  September 2020 Revised  December 2020 Published  March 2021

Fund Project: This work is supported in part by the National Natural Science Foundation of China(11401418)

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.

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 & Management Optimization, doi: 10.3934/jimo.2021054
References:
[1]

C. I. ChenH. 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.  Google Scholar

[2]

Y. Y. ChenG. 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[5]

M. Cheng and M. Xiang, Generalized GM(1, 1) model and its application, Journal of Grey System, 29 (2017), 110-122.   Google Scholar

[6]

J. CuiY. G. Dang and S. F. Liu, Novel grey forecasting model and its modeling mechanism, Control and Decision, 24 (2009), 1702-1706.   Google Scholar

[7]

S. DingY. G. DangN. XuJ. 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.   Google Scholar

[8]

P. Hu, GM(1, 1) power model for optimizing background values and its application, Mathematics in Practice and Theory, 47 (2017), 99-104.   Google Scholar

[9]

Y. HuangX. 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.   Google Scholar

[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.   Google Scholar

[11]

J. L. LiX. P. Xiao and R. Q. Liao, Non-Equidistance GM(1, 1) power and its application, Systems Engineering-Theory & Practice, 30 (2010), 490-495.   Google Scholar

[12]

L. LiD. ZhangJ. TangJ. LiuC. LiZ. 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.   Google Scholar

[13]

S. F. Li and P. Y. Chen, Unbiased GM(1, 1) power model and its application, Statistics & Information Forum, 25 (2010), 7-10.   Google Scholar

[14]

J. S. LuW. D. XieH. 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.  Google Scholar

[15]

X. MaZ. 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.  Google Scholar

[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.   Google Scholar

[17]

L. PeiW. ChenJ. 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.   Google Scholar

[18]

F. X. Wang, Improvement GM(1, 1) power model and its optimization, Pure and Applied Mathematics, 27 (2011), 148-150, 157.  Google Scholar

[19]

Q. WangS. 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.  Google Scholar

[20]

Q. WangS. 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.  Google Scholar

[21]

Q. WangS. Y. LiR. 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.  Google Scholar

[22]

Y. H. WangY. G. DangY. 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.  Google Scholar

[23]

Z. X. Wang, GM(1, 1) power model with time-varying parameters and its application, Control and Decision, 29 (2014), 1828-1832.   Google Scholar

[24]

Z. X. WangY. G. DangS. F. Liu and Z. W. Lian, Solution of GM(1, 1) power model and its properties, Systems Engineering and Electronics, 31 (2009), 2380-2383.   Google Scholar

[25]

Z. WuJ. Shuai and S. Wang, Forecasting of Chinese copper demand based on improved gray model, Industrial Technology & Economy, 33 (2014), 9-14.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

Z. YuC. YangZ. 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.  Google Scholar

[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.   Google Scholar

[30]

S. J. Zhang and S. Y. Chen, Optimization of GM(1, 1) power model and its application, Systems Engineering, 34 (2016), 154-158.   Google Scholar

[31]

J. Z. ZhouR. C. FangY. H. LiY. 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.  Google Scholar

show all references

References:
[1]

C. I. ChenH. 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.  Google Scholar

[2]

Y. Y. ChenG. 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[5]

M. Cheng and M. Xiang, Generalized GM(1, 1) model and its application, Journal of Grey System, 29 (2017), 110-122.   Google Scholar

[6]

J. CuiY. G. Dang and S. F. Liu, Novel grey forecasting model and its modeling mechanism, Control and Decision, 24 (2009), 1702-1706.   Google Scholar

[7]

S. DingY. G. DangN. XuJ. 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.   Google Scholar

[8]

P. Hu, GM(1, 1) power model for optimizing background values and its application, Mathematics in Practice and Theory, 47 (2017), 99-104.   Google Scholar

[9]

Y. HuangX. 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.   Google Scholar

[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.   Google Scholar

[11]

J. L. LiX. P. Xiao and R. Q. Liao, Non-Equidistance GM(1, 1) power and its application, Systems Engineering-Theory & Practice, 30 (2010), 490-495.   Google Scholar

[12]

L. LiD. ZhangJ. TangJ. LiuC. LiZ. 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.   Google Scholar

[13]

S. F. Li and P. Y. Chen, Unbiased GM(1, 1) power model and its application, Statistics & Information Forum, 25 (2010), 7-10.   Google Scholar

[14]

J. S. LuW. D. XieH. 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.  Google Scholar

[15]

X. MaZ. 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.  Google Scholar

[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.   Google Scholar

[17]

L. PeiW. ChenJ. 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.   Google Scholar

[18]

F. X. Wang, Improvement GM(1, 1) power model and its optimization, Pure and Applied Mathematics, 27 (2011), 148-150, 157.  Google Scholar

[19]

Q. WangS. 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.  Google Scholar

[20]

Q. WangS. 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.  Google Scholar

[21]

Q. WangS. Y. LiR. 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.  Google Scholar

[22]

Y. H. WangY. G. DangY. 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.  Google Scholar

[23]

Z. X. Wang, GM(1, 1) power model with time-varying parameters and its application, Control and Decision, 29 (2014), 1828-1832.   Google Scholar

[24]

Z. X. WangY. G. DangS. F. Liu and Z. W. Lian, Solution of GM(1, 1) power model and its properties, Systems Engineering and Electronics, 31 (2009), 2380-2383.   Google Scholar

[25]

Z. WuJ. Shuai and S. Wang, Forecasting of Chinese copper demand based on improved gray model, Industrial Technology & Economy, 33 (2014), 9-14.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

Z. YuC. YangZ. 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.  Google Scholar

[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.   Google Scholar

[30]

S. J. Zhang and S. Y. Chen, Optimization of GM(1, 1) power model and its application, Systems Engineering, 34 (2016), 154-158.   Google Scholar

[31]

J. Z. ZhouR. C. FangY. H. LiY. 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.  Google Scholar

Table 1.  Calculation Results of Grey Modelling for China's GDP
Year No. $ x^{(0)}(t) $} 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. $ x^{(0)}(t) $} 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
$ k $ $ a_{k} $ $ b_{k} $ $ c_{k} $ $ d_{k} $
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
$ k $ $ a_{k} $ $ b_{k} $ $ c_{k} $ $ d_{k} $
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. $ x^{(0)}(t) $} 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. $ x^{(0)}(t) $} 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. $ x^{(0)}(t) $ 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. $ x^{(0)}(t) $ 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. $ x^{(0)}(t) $ 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. $ x^{(0)}(t) $ 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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