Journal of Systems Science and Information

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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

Journal of Systems Science and Information, 2024, 12(2): 245-273. DOI: 10.21078/JSSI-2023-0119

Symbol	Meaning
CFA	The conformable fractional-order accumulation
CFD	The conformable fractional-order difference
$ \boldsymbol X ^{(0)} $	The non-negative original data sequence
$ \boldsymbol X ^{(\alpha)} $	The $ \alpha $th-order accumulated generating sequence
$ \nu $	The number of data used to build models
$ n $	The length of the total sample data
$ a, b $	The linear parameters of grey models
$ \gamma $	The exponent power in the nonlinear grey Bernoulli model
$ \lceil\ \cdot\ \rceil $	The ceil function
$ \mathcal{T} $	The transpose operator
MFO	The moth flame optimization algorithm

Table 2 Notations for different variables

Other figure/table from this article

Figure 1 The main contents of the study
Table 1 Abbreviations for various grey forecasting models
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
Table 25 The annual increase rate of the gross regional product in the next several years (%)

模态框（Modal）标题

Figure/Table detail