In the process of grey prediction modeling, the accumulation of original non-negative sequences can enhance the inherent regularity of data and the smoothness of sequences. The estimated method of background values and derivative values greatly affects the prediction accuracy and adaptability of the model. On the basis of the traditional GM(2,1) model, the GM(2,1) model with the fractional order accumulation is obtained by introducing a fractional order operator, which can be written as GM$ ^r $(2,1) with r-AGO. In this work, we estimated the background values and derivative values based on a $ C^1 $ convexity-preserving rational quadratic interpolation spline, and thereby established a new GM$ ^2 $(2,1) model. Numerical examples showed that the new GM$ ^2 $(2,1) model had better prediction quality of data than the classical GM(2,1) and GM$ ^2 $(2,1) model and improved the precision of prediction in practice.
Citation: Fengyi Chen. A new GM$ ^2(2,1) $ model based on a $ C^1 $ convexity-preserving rational quadratic interpolation spline[J]. AIMS Mathematics, 2024, 9(7): 17917-17931. doi: 10.3934/math.2024872
In the process of grey prediction modeling, the accumulation of original non-negative sequences can enhance the inherent regularity of data and the smoothness of sequences. The estimated method of background values and derivative values greatly affects the prediction accuracy and adaptability of the model. On the basis of the traditional GM(2,1) model, the GM(2,1) model with the fractional order accumulation is obtained by introducing a fractional order operator, which can be written as GM$ ^r $(2,1) with r-AGO. In this work, we estimated the background values and derivative values based on a $ C^1 $ convexity-preserving rational quadratic interpolation spline, and thereby established a new GM$ ^2 $(2,1) model. Numerical examples showed that the new GM$ ^2 $(2,1) model had better prediction quality of data than the classical GM(2,1) and GM$ ^2 $(2,1) model and improved the precision of prediction in practice.
| [1] |
J. Deng, Control problems of grey systems, Syst. Control Lett., 5 (1982), 288–294. https://doi.org/10.1016/S0167-6911(82)80025-X doi: 10.1016/S0167-6911(82)80025-X
|
| [2] | J. Deng, The basis of grey theory, Wuhan: Huazhong University of Science and Technology Press, 2002. |
| [3] | J. Shen, X. Zhao, Improvement of GM(2,1) model by minimum squares, Journal of Harbin Engineering University, 22 (2001), 64–66. |
| [4] |
X. Zeng, X. Xiao, Research on morbidity problem of accumulating method GM(2,1) model, J. Syst. Eng. Electron., 28 (2006), 542–544. https://doi.org/10.1360/jos172537 doi: 10.1360/jos172537
|
| [5] | L. Liu, X. Peng, J. Zhou, Two improved model of GM(2,1) and its application in ship rolling forecast, Journal of Xiamen University (Natural Science), 50 (2011), 515–519. |
| [6] | L. Li, R. Shan, H. Cui, The application of the imporved GM(2,1) grey model using Matlab, Math. Pract. Theor., 41 (2011), 179–183. |
| [7] |
N. Xu, Y. Dang, An optimized grey GM(2,1) model and forecasting of highway subgrade settlement, Math. Probl. Eng., 2015 (2015), 606707. https://doi.org/10.1155/2015/606707 doi: 10.1155/2015/606707
|
| [8] | Z. Zhou, Y. Zhang, The improved GM(2,1) model based on the Laplace transform and pattern search method, Science & Technology Vision, 2016. |
| [9] | Y. Zhou, Improved GM(2,1) model and experimental simulation, Comput. Knowl. Technol., 13 (2017), 171–173. |
| [10] |
L. Wu, S. Liu, L. Yao, S. Yan, D. Liu, Grey system model with the fractional order accumulation, Commun. Nonlenear Sci., 18 (2013), 1775–1785. https://doi.org/10.1016/j.cnsns.2012.11.017 doi: 10.1016/j.cnsns.2012.11.017
|
| [11] | L. Wu, S. Liu, L. Yao, Discrete model based on fractional order accumulate, Syst. Eng. Theory Pract., 34 (2014), 1822–1827. |
| [12] |
L. Wu, S. Liu, L. Yao, R. Xu, X. Lei, Using fractional order accumulation to reduce errors from inverse accumulated generating operator of grey model, Soft Comput., 19 (2015), 483–488. https://doi.org/10.1007/s00500-014-1268-y doi: 10.1007/s00500-014-1268-y
|
| [13] |
S. Mao, M. Gao, X. Xiao, Fractional order accumulation time-lag GM(1, N, $\tau$) model and its application, Syst. Eng. Theory Pract., 35 (2015), 430–436. https://doi.org/10.12011/1000-6788(2015)2-430 doi: 10.12011/1000-6788(2015)2-430
|
| [14] | J. Liu, S. Liu, Z. Fang, Fractional-order reverse accumulation generation GM(1,1) model and its applications, J. Grey Syst., 27 (2015), 52–62. |
| [15] | L. Zeng, Research on GM(2,1) model based on fractional order accumulation and its improvement, Fuzzy Syst. Math., 32 (2018), 69–79. |
| [16] |
Y. Zhu, Z. Jian, Y. Du, W. Chen, J. Fang, A new GM(1,1) model based on cubic monotonicity-preserving interpolation spline, Symmetry, 11 (2019), 420. https://doi.org/10.3390/sym11030420 doi: 10.3390/sym11030420
|
| [17] | F. Chen, Y. Zhu, A new GM(1,1) based on piecewise rational linear/linear monotonicity-preserving interpolation spline, Eng. Lett., 29 (2021), 849–855. |
| [18] | W. Chen, Y. Zhu, A new GM(1,1) model based on piecewise rational quadratic monotonicity-preserving interpolation spline, IAENG Int. J. Comput. Sci., 47 (2020), 130–135. |
| [19] |
R. Delbourgo, Shape preserving interpolation to convex data by rational functions with quadratic numerator and linear denominator, IMA J. Numer. Anal., 9 (1989), 123–136. https://doi.org/10.1093/imanum/9.1.123 doi: 10.1093/imanum/9.1.123
|
| [20] |
C. Li, C. Xiang, Evaluation and forecast of sunstainable utilization of water resources in Beijing, Pearl River, 43 (2022), 23–30. https://doi.org/10.3969/j.issn.1001-9235.2022.04.004 doi: 10.3969/j.issn.1001-9235.2022.04.004
|
| [21] | J. Pu, L. Kang, Application of grey GM(2,1) model to temperature and precipitation prediction, Journal of Sichuan Meteorology, 23 (2003), 10–12. |
| [22] |
L. Wu, S. Liu, L. Yao, S. Yan, The effect of sample size on the grey system model, Appl. Math. Model., 37 (2013), 6577–6583. https://doi.org/10.1016/j.apm.2013.01.018 doi: 10.1016/j.apm.2013.01.018
|
| [23] |
A. Saxena, Optimized fractional overhead power term polynomial grey model (OFOPGM) for market clearing price prediction, Electr. Pow. Syst. Res., 214 (2023), 108800. https://doi.org/10.1016/j.epsr.2022.108800 doi: 10.1016/j.epsr.2022.108800
|
Fengyi Chen. A new GM$ ^2(2,1) $ model based on a $ C^1 $ convexity-preserving rational quadratic interpolation spline[J]. AIMS Mathematics, 2024, 9(7): 17917-17931. doi: 10.3934/math.2024872
DownLoad: