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Traditional discrete grey forecasting model can effectively predict the development trend of the stabilizing system. However, when the system has disturbance information, the prediction result will have larger error, and there will appear significant downward trend in the stability of the model. In the presence of disturbance information, this paper presents a fractional-order linear time-varying parameters discrete grey forecasting model to deal with the system that contains both linear trend and nonlinear trend. The modeling process of the model and calculation method are given. The perturbation bounds of the new model are analyzed by using the least-squares method of perturbation theory. And it is compared with that of the first-order linear time-varying parameters discrete grey forecasting model. Finally, two real cases are given to verify the effectiveness and practicality of the proposed method.

The grey system theory was proposed by Deng in 1982. It only needs a small amount of data to establish the model for analysis [

Dang gave the new method to select initial value [

Xie proposed the discrete grey model (DGM(1,1) model). The relationship between DGM(1,1) and GM(1,1) was studied deeply, and the reason for the instability of GM(1,1) was found. It only needs to use difference equation to solve the equation and does not need to convert the difference equation to differential equation. Therefore, the modeling accuracy is effectively improved [

The research mentioned above has positive significance for improving the accuracy of grey prediction model. However, for complex systems with disturbance, the model’s robustness is insufficient; how to deal with the system disturbance information is particularly important. To solve this problem, this paper presents a fractional-order accumulation linear time-varying parameters discrete grey prediction model (FTDGM(1,1)model). The modeling process and parametric calculation method of the model are given. It is proven that the model has good stability using the theory of matrix perturbation analysis. Finally, two real cases are given. And the calculation results show that FTDGM(1,1) model can effectively reduce the disturbance caused by disturbance information. The model's robustness and prediction accuracy are improved, and the validity and practicality of the model are further verified.

Assume that the nonnegative sequence is

Among them,

The equation

is called discrete grey prediction model (DGM(1,1) model).

The parameters of the DGM(1,1) model can be solved by using the following least-squares estimation:

and, among them,

Assume that the nonnegative sequence is

Among them,

The equation

is called linear time-varying parameters discrete grey model (TDGM).

The parameters of the TDGM(1,1) model can be solved by using the following least-squares estimation:

and, among them,

Assume that the nonnegative sequence is

Among them,

Assume the nonnegative sequence

is called fractional-order accumulative linear time-varying parameters discrete grey model (FTDGM).

The parameters of the FTDGM(1,1) model can be solved by using the following least-squares estimation:

and, among them,

The predicted value of FTDGM(1,1)model is as follows:

According to the calculation formula of fractional-order accumulation, it is not difficult to calculate the reduced value of the predicted sequence as follows:

Suppose that

Suppose that

Among them,

The perturbation bounds of TDGM(1,1) model will be analyzed in the following section.

The solution of TDGM model can be given as the following function:

Then

Assume that the solution of the new model

Since

we have

Then, the following result can be obtained according to Theorem

Assume that the conditions of Theorem

Then the perturbation bound of the solution is as follows:

Assume that the solution of the new model

Since

we have

Then, the following result can be obtained according to Theorem

The solution of TDGM(1,1) model can be given as the following function:

Then

Assume that the solution of the new model

Since

we have

Then, the following result can be obtained according to Theorem

Since

we have

It is not hard to get

The solution of TDGM(1,1) model can be given as the following function:

Then

We have

Then, the following result can be obtained according to Theorem

We have

Then

It can be seen from the conclusion of Theorems

In order to test the modeling effect of the model, two real cases will be given in the following section.

The GDP of Guangdong province in 2001-2009 is used to build different grey prediction model. The data is shown in Table

The highest prediction accuracy in [

Gross domestic product in Guangdong province in 2001-2014 (one hundred million yuan).

Year | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 |

| |||||||

GDP | 12039.25 | 13502.42 | 15844.64 | 18864.62 | 22557.37 | 26587.76 | 31777.01 |

| |||||||

Year | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |

| |||||||

GDP | 39482.56 | 36796.71 | 46013 | 53210 | 57068 | 62475 | 67792 |

Source: Guangdong Statistical Yearbook.

Predicted values and predicted errors of different grey prediction models.

Year | Original value | DGM(1,1) model | TVGM(1,1) model | FTDGM(1,1) model | |||
---|---|---|---|---|---|---|---|

Predicted value | Relative error (%) | Predicted value | Relative error (%) | Predicted value | Relative error (%) | ||

2010 | 46013 | 48326.89 | 5.03 | 44920.35 | 2.37 | 44989.65 | 2.22 |

2011 | 53210 | 56295.98 | 5.80 | 49471.34 | 7.03 | 50379.20 | 5.32 |

2012 | 57068 | 65579.18 | 14.91 | 54030.11 | 5.32 | 56267.53 | 1.40 |

2013 | 62475 | 76393.18 | 22.28 | 58593.05 | 6.21 | 62739.40 | 0.42 |

2014 | 67792 | 88990.4 | 31.27 | 63158.22 | 6.83 | 69893.15 | 3.10 |

MAPE | 15.86 | 5.55 | 2.49 |

MAPE (mean absolute percentage error) =

Calculation results of different models of Case

As one of the important symbols of China's transportation modernization, highway reflects the degree and level of a country's modernization. Compared with railway and air or water transportation, highway transportation is the more used mode in passenger and cargo transportation. Highway transportation is point-to-point direct, flexible, and convenient. It is very important to accurately predict the length of highway transportation route. The length of Chinese highway transportation route in 2010-2017 is used to build different grey prediction model. The unit of highway mileage is ten thousand kilometers. The calculation results of different models are shown in Table

The calculation results of different grey models.

Year | Original value | DGM(1,1) model | TDGM(1, 1) model | FTDGM(1,1) model, r=0.85 | |||
---|---|---|---|---|---|---|---|

Simulation value | Relative error(%) | Simulation value | Relative error (%) | Simulation value | Relative error (%) | ||

2010 | 7.41 | 7.41 | 0 | 7.41 | 0 | 7.41 | 0 |

2011 | 8.49 | 8.65 | 1.91 | 8.49 | 0.05 | 8.49 | 0.00 |

2012 | 9.62 | 9.46 | 1.69 | 9.65 | 0.30 | 9.64 | 0.23 |

2013 | 10.44 | 10.34 | 0.98 | 10.37 | 0.71 | 10.33 | 1.07 |

2014 | 11.19 | 11.30 | 0.97 | 11.28 | 0.80 | 11.45 | 2.29 |

2015 | 12.35 | 12.35 | 0.00 | 12.30 | 0.42 | 11.97 | 3.06 |

MAPE | 1.11 | 0.46 | 1.33 | ||||

2016 | 13.1 | 13.50 | 3.05 | 13.46 | 2.73 | 13.22 | 0.92 |

2017 | 13.64 | 14.76 | 8.18 | 14.78 | 8.36 | 13.69 | 0.37 |

MAPE | 5.61 | 5.54 | 0.64 |

Source: China Statistical Yearbook.

Calculation results of different models of Case

It can be seen from the calculation results that the TDGM(1,1) model is better than the DGM(1,1) model in describing the internal evolution. However, neither the DGM(1,1) model nor the TDGM(1,1) model can accurately describe the development trend of the system. In particular, the prediction error of the second step is relatively large, indicating that the memory of the integer-order model is insufficient for the FTDGM(1,1) model. The prediction error of the FTDGM(1,1) model is only 0.64%, which shows that the model has strong extrapolation ability and good memory.

On the basis of the traditional discrete grey prediction model, this paper proposed the FTDGM(1,1) model. The parametric solution method of the model was given. By using the matrix disturbance theory, the disturbance boundary of the model was analyzed, and it was proved that the FTDGM(1,1) model had better robustness than the TDGM(1,1) model.

Two real cases were used to test the effect of the proposed FTDGM(1,1) model. It was found that the prediction accuracy is higher than that of the existing models, which further verifies the superiority of the proposed FTDGM(1,1) model. At the same time, the FTDGM(1,1) model had better memory than other discrete grey prediction models. Research results of this paper further expand the application scope of the grey prediction model. And the reasons for the existence of short-term memory model deserve further discussion.

The data are from China Statistical Yearbook, website of China’s national bureau of statistics, http://www.stats.gov.cn/, Guangdong Statistical Yearbook, and Guangdong Statistical Information Network, http://www.gdstats.gov.cn/tjsj/gmjjhs/.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (71871084), 2016 Shanghai Maritime University National Key Project Development Project (A20201161107X), and 2015 Shanghai Maritime University Research Start-Up Funding Project (A15101154505Z). At the same time, the authors would like to acknowledge the support of the project of Chinese postdoctoral science foundation (2018M632777) and the training program of youth backbone teacher in institutions and universities in Henan province (2018GGJS115).