系统科学与信息学报(英文)

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A Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation and Its Application to the Gross Regional Product in the Cheng-Yu Area

Wenqing WU, Xin MA, Bo ZENG, Yuanyuan ZHANG

系统科学与信息学报(英文), 2024, 12(2): 245-273. DOI: 10.21078/JSSI-2023-0119

Year	Data	GM$ (1, 1) $	CFGM$ (1, 1) $	NGBM$ (1, 1) $	CFNGBM$ (1, 1) $	CFNGBM$ (1, 1, \lambda) $
2013	9450.66	9450.6600	9450.6600	9450.6600	9450.6600	9450.6600
2014	10368.43	9802.0478	9697.7092	9451.2892	9988.7884	9872.8210
2015	10662.31	10988.5260	11136.1566	11011.4264	11035.3619	11440.4793
2016	11874.07	12318.6202	12510.5143	12459.4319	12416.0122	12869.0126
2017	13931.39	13809.7142	13886.5734	13881.3030	13862.1596	14248.9026
2018	15698.94	15481.2961	15297.2487	15317.4627	15336.5046	15622.0957
2019	17012.65	17355.2127	16763.0519	16791.7802	16834.8710	17012.6500
2020	17716.70	19455.9556	18298.9072	18320.7100	18360.1984	18436.4628
2021	19916.98	21810.9806	19916.9800	19916.9799	19916.9801	19905.2171
MAPE$ _{\rm simu} $(%)		2.9051	3.8307	3.9678	2.9060	4.6452
MAPE$ _{\rm pred} $(%)		7.1134	1.5844	1.5692	1.5590	1.3739
MAPE$ _{\rm total} $(%)		4.4832	2.9883	3.0683	2.4009	3.4185

Table 6 The results of the gross regional product of Chengdu by the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models

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Figure 1 The main contents of the study
Table 1 Abbreviations for various grey forecasting models
Table 2 Notations for different variables
Figure 2 The relationship among the CFNGBM$ (1, 1, \lambda) $, CFNGBM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, GVM(1, 1) and GM$ (1, 1) $ models
Figure 3 The general framework of the MFO algorithm
Table 3 Raw data of gross regional product in the Cheng-Yu area (million yuan)
Figure 4 The numerical design of the application in the gross regional product
Table 4 The minimum MAPE$ _{\rm pred} $ and corresponding nonlinear parameters of grey models
Figure 5 The convergence curve of MAPE$ _{\rm pred} $ and $ \alpha $ of the CFGM$ (1, 1) $ model in Chongqing case
Figure 6 The convergence curve of MAPE$ _{\rm pred} $ and $ \gamma $ of the NGBM$ (1, 1) $ model in Chongqing case
Figure 7 The convergence curve of MAPE$ _{\rm pred} $, $ \alpha $ and $ \gamma $ of the CFNGBM$ (1, 1) $ model in Chongqing case
Figure 8 The convergence curve of MAPE$ _{\rm pred} $, $ \alpha $, $ \gamma $ and $ \lambda $ of the CFNGBM$ (1, 1, \lambda) $ model in Chongqing case
Table 5 The results of the gross regional product of Chongqing by the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models
Table 7 The results of the gross regional product of Mianyang by the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models
Table 8 The results of the gross regional product of Leshan by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$ and CFNGBM$(1, 1)$ models
Table 9 The results of the gross regional product of Zigong by the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models
Table 10 The results of the gross regional product of Deyang by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models
Table 11 The results of the gross regional product of Meishan by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models
Table 12 The results of the gross regional product of Luzhou by the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models
Table 13 The results of the gross regional product of Suining by the GM$(1,1)$, CFGM$(1,1)$, NGBM$(1,1)$, CFNGBM$(1,1)$ and CFNGBM$(1,1,\lambda)$ models
Table 14 The results of the gross regional product of Neijiang by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models
Table 15 The results of the gross regional product of Nanchong by the GM$(1,1)$, CFGM$(1,1)$, NGBM$(1,1)$, CFNGBM$(1,1)$ and CFNGBM$(1,1,\lambda)$ models
Table 16 The results of the gross regional product of Guang'an by the GM$(1,1)$, CFGM$(1,1)$, NGBM$(1,1)$, CFNGBM$(1,1)$ and CFNGBM$(1,1,\lambda)$ models
Table 17 The results of the gross regional product of Yibin by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models.
Table 18 The results of the gross regional product of Ya'an by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models
Table 19 The results of the gross regional product of Dazhou by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models
Table 20 The results of the gross regional product of Ziyang by the GM$(1, 1)$, CFGM$(1, 1)$, NGBM$(1, 1)$, CFNGBM$(1, 1)$ and CFNGBM$(1, 1, \lambda)$ models
Table 21 Summary of ranks for MAPE$_{\rm simu} $ of the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models in all cities
Table 22 Summary of ranks for MAPE$ _{\rm pred} $ of the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models in all cities
Table 23 Summary of ranks for MAPE$ _{\rm total} $ of the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models in all cities
Figure 9 Summary of ranks for MAPE$ _{\rm simu} $ of the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models in all cities
Figure 10 Summary of ranks for MAPE$ _{\rm pred} $ of the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models in all cities
Figure 11 Summary of ranks for MAPE$ _{\rm total} $ of the GM$ (1, 1) $, CFGM$ (1, 1) $, NGBM$ (1, 1) $, CFNGBM$ (1, 1) $ and CFNGBM$ (1, 1, \lambda) $ models in all cities
Table 24 The gross regional product of all cities in Cheng-Yu area by the CFNGBM$(1, 1, \lambda)$ model in the next several years
Figure 12 The plots of the gross regional product of all cities in Cheng-Yu area by the CFNGBM$ (1, 1, \lambda) $ model in the next several years
Table 25 The annual increase rate of the gross regional product in the next several years (%)

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