Article Contents
Article Contents

Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application

• * Corresponding author: Maolin Cheng

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.

Mathematics Subject Classification: Primary: 62F99; Secondary: 93A30.

 Citation:

• 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

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

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

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

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
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