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

	Chongqing	Chengdu	Mianyang	Leshan	Zigong	Deyang	Meishan	Luzhou
2022	8.4367	7.6514	8.2031	8.4250	6.5700	6.8074	6.5420	7.2650
2023	8.3019	7.3980	7.8842	8.2245	6.4129	6.5805	6.3809	6.8679
2024	8.1900	7.1899	7.6217	8.0485	6.2843	6.3943	6.2488	6.5302
2025	8.0955	7.0160	7.4019	7.8928	6.1770	6.2388	6.1386	6.2400
2026	8.0147	6.8686	7.2152	7.7540	6.0863	6.1071	6.0453	5.9879
mean	8.2078	7.2248	7.6652	8.0689	6.3061	6.4256	6.2711	6.5782

	Suining	Neijiang	Nanchong	Guang'an	Yibin	Ya'an	Dazhou	Ziyang
2022	7.2139	6.2011	6.9112	6.3806	10.3131	7.1325	6.8621	6.9146
2023	6.8057	5.8994	6.6369	6.0269	9.8986	6.7906	6.6186	6.8526
2024	6.4593	5.6403	6.4119	5.7244	9.5423	6.4956	6.4167	6.7900
2025	6.1620	5.4157	6.2243	5.4632	9.2325	6.2392	6.2467	6.7282
2026	5.9041	5.2195	6.0653	5.2356	8.9603	6.0144	6.1015	6.6678
mean	6.5090	5.6752	6.4499	5.7661	9.5893	6.5345	6.4491	6.7906

Table 25 The annual increase rate of the gross regional product in the next several years (%)

本文的其它图/表

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 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
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

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