MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2019/6343298 6343298 Research Article Fractional-Order Accumulative Linear Time-Varying Parameters Discrete Grey Forecasting Model http://orcid.org/0000-0003-0820-1053 Gao Pumei 1 Zhan Jun 1 2 http://orcid.org/0000-0002-5213-0128 Liu Jiefang 3 Balsera Ines Tejado 1 School of Economics & Management Shanghai Maritime University Shanghai 201306 China shmtu.edu.cn 2 School of Business Management Shanghai Lixin University of Accounting and Finance Shanghai 201209 China lixin.edu.cn 3 School of Mathematical Science Henan Institute of Science and Technology Xinxiang 453003 China hist.edu.cn 2019 252019 2019 27 02 2019 04 04 2019 18 04 2019 252019 2019 Copyright © 2019 Pumei Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China 71871084 Shanghai Maritime University A20201161107X A15101154505Z China Postdoctoral Science Foundation 2018M632777 training program of youth backbone teacher in institutions and universities in Henan province 2018GGJS115
1. Introduction

The grey system theory was proposed by Deng in 1982. It only needs a small amount of data to establish the model for analysis [1, 2]. Grey prediction model is the main part of grey system theory, and its theoretical basis is the grey of accumulation generation. The prediction model is established by using the characteristic of grey exponential. Since the grey prediction model was proposed, it has been widely concerned by scholars. The existing studies mainly focus on the theoretical development from the aspects of background value optimization [3, 4], parameter optimization , model expansion , and the application of natural gas [11, 12], electric power , environment , economy [21, 22], and transportation [23, 24].

Dang gave the new method to select initial value . Cui proposed the NGM(1,1,k) model and described the modeling mechanism and modeling process of the model . Zhou proposed the generalized GM(1,1) model and used the new model to simulate and predict China's fuel output from 2003 to 2010 . Combined with the concept of Bernoulli differential equation, Chen proposed the NGBM(1,1) model on the basis of GM(1,1) model . Li proposed 3spGM(1, 1) model and applied the new model to the failure data sets of electric product manufacturing systems . These methods further improve the modeling effect of grey prediction model. However, the transformation of difference equation and differential equation is required in the solving process. There is still error in the grey prediction model for the sequences that conform to the exponential features.

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 . Wu proposed the discrete grey prediction model based on fractional-order accumulation, discussed the properties of the model, and gave the calculation method of the model . Liu proposed the fractional-order reverse accumulation discrete grey prediction model and discussed the properties of the model .

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.

2. The Fractional-Order Accumulative Linear Time-Varying Parameters Discrete Grey Model Definition 1 (see [<xref ref-type="bibr" rid="B31">30</xref>]).

Assume that the nonnegative sequence is X(0)=x(0)1,x(0)2,,x(0)n. X(1)=x(1)1,x(1)2,,x(1)n is the first-order accumulative sequence of X(0).

Among them, (1)x1k=i=1kx0i,k=1,2,,n.

The equation (2)x1k+1=β1x1k+β2,k=1,2,,n-1

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

Theorem 2 (see [<xref ref-type="bibr" rid="B31">30</xref>]).

The parameters of the DGM(1,1) model can be solved by using the following least-squares estimation:(3)β1β2=BTB-1BTY,

and, among them, (4)B=x11x12x1n-2x1n-11111,Y=x12x13x1n-1x1n.

Definition 3 (see [<xref ref-type="bibr" rid="B39">35</xref>]).

Assume that the nonnegative sequence is X(0)=x(0)1,x(0)2,,x(0)n. X(1)=x(1)1,x(1)2,,x(1)n is the first-order accumulative sequence of X(0).

Among them,(5)x1k=i=1kx0i,k=1,2,,n.

The equation(6)x1k+1=β1k+β2x1k+β3k+β4,k=1,2,,n-1

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

Theorem 4.

The parameters of the TDGM(1,1) model can be solved by using the following least-squares estimation:(7)β1β2β3β4=CTC-1CTY,

and, among them, (8)C=x11x11112x12x1221n-2x1n-2x1n-2n-21n-1x1n-1x1n-1n-11,Y=x12x13x1n-1x1n.

Definition 5 (see [<xref ref-type="bibr" rid="B19">18</xref>]).

Assume that the nonnegative sequence is X(0)=x(0)1,x(0)2,,x(0)n.

X ( r ) = x ( r ) 1 , x ( r ) 2 , , x ( r ) n is called the fractional-order accumulative sequence of X(0).

Among them, (9)xrk=i=1kCk-i+r-1k-ix0i,Cr-10=1,Ck-1k=0,k=1,2,,n.

Definition 6.

Assume the nonnegative sequence X(0); X(r)is defined as Definition 5. The equation(10)xrk+1=β1k+β2xrk+β3k+β4,k=1,2,,n-1

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

Theorem 7.

The parameters of the FTDGM(1,1) model can be solved by using the following least-squares estimation:(11)β1β2β3β4=DTD-1DTW,

and, among them,(12)D=xr1xr1112xr2xr221n-2xrn-2xrn-2n-21n-1xrn-1xrn-1n-11,W=xr2xr3xrn-1xrn.

The predicted value of FTDGM(1,1)model is as follows:(13)x^rk+1=β1k+β2x^rk+β3k+β4

According to the calculation formula of fractional-order accumulation, it is not difficult to calculate the reduced value of the predicted sequence as follows:(14)x^0k=x^rk-i=1kCk-i+r-1k-ix^ri,k=1,2,,n,

3. Disturbance Analysis of TDGM(1,1) Model and FTDGM(1,1) Model Theorem 8 (see [<xref ref-type="bibr" rid="B36">36</xref>, <xref ref-type="bibr" rid="B37">37</xref>]).

Suppose that ACm×n,bCm,A is the generalized inverse matrix of A. When the column vector of A has linear independence, the function Ax-b2=min has a unique solution.

Theorem 9 (see [<xref ref-type="bibr" rid="B36">36</xref>, <xref ref-type="bibr" rid="B37">37</xref>]).

Suppose that ACm×n,bCm,A is the generalized inverse matrix of A. B=A+E,c=b+kCn. Suppose that the solutions of function Bx-c2=min and Ax-b2=min are x+h and x, respectively. When rank(A)=rank(B)=n and A2E2<1, we have the following result:(15)h2stE2Ax+kA+stE2ArxA.

Among them,(16)s=A2A,t=1-A2E2,rx=b-Ax.

3.1. Disturbance Analysis of TDGM(1,1) Model

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

Theorem 10.

The solution of TDGM model can be given as the following function: Y-Cx2=min. Suppose that the solution of the TDGM(1,1) model is x, and x^(0)(1)=x(0)(1)+ε. Among them, ε is the disturbance information.

Then(17)h2εstxi=1n-1i2B+n-1B+sti=1n-1i2BrxB.

Proof.

(18) C ^ = C + Δ C = x 1 1 + ε x 1 1 + ε 1 1 2 x 1 2 + ε x 1 2 + ε 2 1 n - 2 x 1 n - 2 + ε x 1 n - 2 + ε n - 2 1 n - 1 x 1 n - 1 + ε x 1 n - 1 + ε n - 1 1 = x 1 1 x 1 1 1 1 2 x 1 2 x 1 2 2 1 n - 2 x 1 n - 2 x 1 n - 2 n - 2 1 n - 1 x 1 n - 1 x 1 n - 1 n - 1 1 + ε ε 0 0 2 ε ε 0 0 n - 2 ε ε 0 0 n - 1 ε ε 0 0 Y ^ = Y + Δ Y = x 1 2 x 1 3 x 1 n - 1 x 1 n + ε ε ε ε .

Assume that the solution of the new model Y^-C^x2=min is x^, and the disturbance is h. The equation Y-Cx2=min has a unique solution x=YC due to the linear independence of column vectors of C.

Since(19)ΔY=εεεε,ΔCTΔC=i=1n-1i2ε2i=1n-1iε200i=1n-1iε2n-1ε20000000000.ΔY2=εn-1,ΔC2=λmaxΔCTΔC,

we have(20)ΔC2=i=1n-1i2ε2=εi=1n-1i2

Then, the following result can be obtained according to Theorem 9:(21)h2stΔC2Bx+ΔYB+stΔC2BrxB=εstxi=1n-1i2B+n-1B+sti=1n-1i2BrxB=Qx01.

Theorem 11.

Assume that the conditions of Theorem 9 remain unchanged, and x^(0)(t)=x(0)(t)+ε.

Then the perturbation bound of the solution is as follows: (22)h2εstxi=tn-1i2B+n-t+1B+sti=tn-1i2BrxB.

Proof.

(23) C ^ = C + Δ C = x 1 1 x 1 1 1 1 t x 1 t + ε x 1 t + ε t 1 n - 2 x 1 n - 2 + ε x 1 n - 2 + ε n - 2 1 n - 1 x 1 n - 1 + ε x 1 n - 1 + ε n - 1 1 = x 1 1 x 1 1 1 1 t x 1 t x 1 t t 1 n - 2 x 1 n - 2 x 1 n - 2 n - 2 1 n - 1 x 1 n - 1 x 1 n - 1 n - 1 1 + 0 0 0 0 t ε ε 0 0 n - 2 ε ε 0 0 n - 1 ε ε 0 0 Y ^ = Y + Δ Y = x 1 2 x 1 t x 1 n - 1 x 1 n + 0 ε ε ε .

Assume that the solution of the new model Y^-C^x2=min is x^, and the disturbance is h. The equation Y-Cx2=min has a unique solution x=YC due to the linear independence of column vectors of C.

Since(24)ΔY=0εεε,ΔCTΔC=i=tn-1i2ε2i=tn-1iε200i=tn-1iε2n-t+1ε20000000000.ΔY2=εn-t+1,ΔC2=λmaxΔCTΔC,

we have(25)ΔC2=i=tn-1i2ε2=εi=tn-1i2

Then, the following result can be obtained according to Theorem 9:(26)h2stΔC2Bx+ΔYB+stΔC2BrxB=εstxi=tn-1i2B+n-t+1B+sti=tn-1i2BrxB=Qx1t.

3.2. Disturbance Analysis of TDGM(1,1) Model and FTDGM(1,1) Model Theorem 12.

The solution of TDGM(1,1) model can be given as the following function: W-Dx2=min. Suppose that the solution of the TDGM(1,1) model is x, and x^(0)(1)=x(0)(1)+ε. Among them, ε is the disturbance information.

Then(27)h2εsti=1n-1iCi+r-2i-12Bx+i=1n-1Ci+r-2i-12B+sti=1n-1iCi+r-2i-12BrxB.

Proof.

(28) D ^ = D + Δ D = x r 1 x r 1 1 1 2 x r 2 x r 2 2 1 n - 2 x r n - 2 x r n - 2 n - 2 1 n - 1 x r n - 1 x r n - 1 n - 1 1 + ε ε 0 0 2 r ε r ε 0 0 n - 2 C n - 4 + r n - 3 ε C n - 4 + r n - 3 ε 0 0 n - 1 C n - 3 + r n - 2 ε C n - 3 + r n - 2 ε 0 0 . W ^ = W + Δ W = x r 2 x r 3 x r n - 1 x r n + r ε C 1 + r 2 ε C n - 3 + r n - 2 ε C n - 2 + r n - 1 ε .

Assume that the solution of the new model Y^-B^x2=min is x^, and the disturbance is h. The equation Y-Bx2=min has a unique solution x=YB due to the linear independence of column vectors of B.

Since(29)ΔW=rεC1+r2εCn-3+rn-2εCn-2+rn-1ε,ΔDTΔD=i=1n-1iCi+r-2i-1ε2i=1n-1iCi+r-2i-1ε200i=1n-1iCi+r-2i-1ε2i=1n-1Ci+r-2i-1ε20000000000.ΔW2=εi=2nCi+r-2i-12,ΔD2=λmaxΔDTΔD

we have(30)ΔW2=εi=2nCi+r-2i-12,ΔD2=i=1n-1iCi+r-2i-1ε2=εi=1n-1i2Ci+r-2i-12

Then, the following result can be obtained according to Theorem 9:(31)h2stΔD2Bx+ΔYB+stΔD2DrxB=εsti=1n-1iCi+r-2i-12Bx+i=2nii+r-2i-12B+sti=1n-1iCi+r-2i-12BrxB=Lx01Qx01=εstxi=1n-1i2B+n-1B+sti=1n-1i2BrxB

Since(32)Ci+r-2i-1=Ci-1+r-1i-1<1

we have(33)i=1n-1i2Ci+r-2i-1ε2<i=1n-1i2,i=2nCi+r-2i-12<n-1.

It is not hard to get L(x(0)(1))<Q(x(0)(1)).

Theorem 13.

The solution of TDGM(1,1) model can be given as the following function: W-Dx2=min. Suppose that the solution of the TDGM(1,1) model is x, and x^(0)(t)=x(0)(t)+ε. Among them, ε is the disturbance information.

Then(34)h2εsti=tn-1iCi+r-t-1i-t2Bx+i=tnCi-t+r-1i-t2B+sti=tn-1iCi+r-t-1i-t2BrxB=Lx0t.

Proof.

(35) D ^ = D + Δ D = x r 1 x r 1 1 1 t x r t x r t t 1 n - 2 x r n - 2 x r n - 2 n - 2 1 n - 1 x r n - 1 x r n - 1 n - 1 1 + 0 0 1 1 t ε ε 0 0 n - 2 C n - t + r - 3 n - t - 2 ε C n - t + r - 3 n - t - 2 ε 0 0 n - 1 C n - t + r - 2 n - t - 1 ε C n - t + r - 2 n - t - 1 ε 0 0 W ^ = W + Δ W = x r 2 x r t x r n - 1 x r n + 0 ε C n - t + r - 3 n - t - 2 ε C n - t + r - 2 n - t - 1 ε

We have(36)ΔW=0εCn-t+r-3n-t-2εCn-t+r-2n-t-1ε,ΔDTΔD=i=tn-1iCi-t+r-1i-tε2i=tn-1iCi-t+r-1i-tε200i=tn-1iCi-t+r-1i-tε2i=tn-1Ci-t+r-1i-tε20000000000.ΔW2=εi=tnCi-t+r-1i-t2,ΔD2=λmaxΔDTΔD=εi=tn-1iCi-t+r-1i-t2

Then, the following result can be obtained according to Theorem 9.(37)h2stΔD2Bx+ΔWB+stΔD2DrxB=εsti=tn-1iCi-t+r-1i-t2Bx+i=tnCi-t+r-1i-t2B+sti=tn-1iCi-t+r-1i-t2BrxB=Lx0t

We have(38)Qx0t=εstxi=tn-1i2B+n-t+1B+sti=tn-1i2BrxBi=1n-tiCk+r-2i-12<i=1n-ti2<i=tn-1i2,i=1n-t+1Ck+r-2i-12<n-t+1.

Then L(x(0)(t))<Q(x(0)(t)).

It can be seen from the conclusion of Theorems 12 and 13 that when r<1, the disturbance bound of FTDGM(1,1) model is smaller than that of TDGM(1,1) model. Generally speaking, compared with the TDGM(1,1) model, the FTDGM(1,1) model has better robustness. The proposed FTDGM(1,1) model can effectively reduce the prediction error caused by system disturbance and improve the prediction accuracy of the grey forecasting model. And the solution of optimal parameter r can be given by genetic algorithms.

4. Numerical Illustrations

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

Case 1.

The GDP of Guangdong province in 2001-2009 is used to build different grey prediction model. The data is shown in Table 1. Different grey prediction models were established based on the given data. And the advantages of the model were tested by comparing the prediction accuracies of different models.

The highest prediction accuracy in  is time-varying parameters grey model (TVGM(1,1) model). This Paper established DGM(1,1) model, TVGM(1,1) model, and FTDGM(1,1) model, respectively. The calculation results are shown in Table 2 and Figure 1. The parameters of FTDGM(1,1) model are as follows: β1=0.0185,β2=0.5166,β3=3530.48,β4=7817.70,r=0.41. It can be seen from Table 2 that, because of the nature of exponential function, large errors will appear after the second prediction of the DGM(1,1) model. TVGM(1,1) model can effectively increase the prediction accuracy along with adjusting and optimizing the time-varying parameters. However, the prediction error is still relatively large. The proposed FTDGM(1,1) model in this paper shows good robustness, and the prediction error does not increase with the passage of time. It indicated that the FTDGM(1,1) model has better anti-interference and long-term memory, which can be used to predict medium-term goals.

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) = 100%1/nk=1nx(k)-x^(k)/x(k).

Calculation results of different models of Case 1.

Case 2.

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 3 and Figure 2. The parameters of FTDGM(1,1) model are as follows: β1=0.06, β2=-1.46,β3=17.44,β4=7.76,r=0.85.

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 2.

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.

5. Concluding Remarks

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.

Data Availability

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/.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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).

Deng J. L. Introduction to grey system theory The Journal of Grey System 1989 1 1 1 24 MR1025897 Zbl0701.90057 Deng J. L. Grey Prediction and Grey Decision 2002 Wuhan, China Huazhong University of Science and Technology Press Zeng B. Li C. Improved multi-variable grey forecasting model with a dynamic background-value coefficient and its application Computers & Industrial Engineering 2018 118 278 290 2-s2.0-85042859741 10.1016/j.cie.2018.02.042 Ye J. Dang Y. Li B. Grey-Markov prediction model based on background value optimization and central-point triangular whitenization weight function Communications in Nonlinear Science and Numerical Simulation 2018 54 320 330 10.1016/j.cnsns.2017.06.004 MR3671419 2-s2.0-85020381185 Zeng B. Liu S. F. A self-adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application Mathematical Methods in the Applied Sciences 2017 23 1 7843 7857 MR3742164 Ma X. Liu Z. B. The GMC (1, n) model with optimized parameters and its application The Journal of Grey System 2017 29 4 122 138 2-s2.0-85040313685 Hsin P.-H. Chen C.-I. Application of game theory on parameter optimization of the novel two-stage Nash nonlinear grey Bernoulli model Communications in Nonlinear Science and Numerical Simulation 2015 27 1-3 168 174 10.1016/j.cnsns.2015.03.006 MR3341552 2-s2.0-84928657748 Wei B. Xie N. M. Hu A. Q. Optimal solution for novel grey polynomial prediction model Applied Mathematical Modelling 2018 62 717 727 10.1016/j.apm.2018.06.035 MR3834167 Ma X. Xie M. Wu W. Q. Zeng B. Wang Y. Wu X.-X. The novel fractional discrete multivariate grey system model and its application Applied Mathematical Modelling 2019 70 402 424 10.1016/j.apm.2019.01.039 MR3908745 Wu L. F. Zhang Z. Y. Grey multivariable convolution model with new information priority accumu lation Applied Mathematical Modelling 2018 62 595 604 10.1016/j.apm.2018.06.025 MR3834159 Zeng B. Duan H. M. Yun B. Wei M. Forecasting the output of shale gas in China using an unbiased grey model and weakening buffer operator Energy 2018 151 238 249 10.1016/j.energy.2018.03.045 2-s2.0-85046016469 Ma X. Liu Z. B. Application of a novel time-delayed polynomial grey model to predict the natural gas consumption in China Journal of Computational and Applied Mathematics 2017 324 17 24 10.1016/j.cam.2017.04.020 MR3654731 Wang Z. X. Li Q. Pei L. L. A seasonal GM(1,1) model for forecasting the electricity consumption of the primary economic sectors Energy 2018 154 522 534 10.1016/j.energy.2018.04.155 Zeng B. Tan Y. T. Xu H. Quan J. Wang L. Y. Zhou X. Y. Forecasting the electricity consumption of commercial sector in Hong Kong using a novel grey dynamic prediction model Journal of Grey System 2018 30 1 157 172 2-s2.0-85042387448 Hamzacebi C. Es H. A. Forecasting the annual electricity consumption of Turkey using an optimized grey model Energy 2014 70 3 165 171 10.1016/j.energy.2014.03.105 2-s2.0-84901684155 Li D. C. Chang C. J. Chen C. C. Chen W. C. Forecasting short-term electricity consumption using the adaptive grey-based approach-an Asian case Omega 2012 40 6 767 773 10.1016/j.omega.2011.07.007 2-s2.0-84858745881 Wu L. F. Zhao H. Y. Using FGM(1,1) model to predict the number of the lightly polluted day in Jing-Jin-Ji region of China Atmospheric Pollution Research 2019 10 2 552 555 10.1016/j.apr.2018.10.004 Wang Z. X. Li Q. Modelling the nonlinear relationship between CO2 emissions and economic growth using a PSO algorithm-based grey Verhulst model Journal of Cleaner Production 2019 207 214 224 10.1016/j.jclepro.2018.10.010 Meng W. Yang D. L. Huang H. Prediction of China's sulfur dioxide emissions by discrete grey model with fractional order generation operators Complexity 2018 2018 1 13 8610679 2-s2.0-85042187541 Wu L. F. Li N. Yang Y. J. Prediction of air quality indicators for the Beijing-Tianjin-Hebei region Journal of Cleaner Production 2018 196 682 687 10.1016/j.jclepro.2018.06.068 Wang C.-H. Hsu L.-C. Using genetic algorithms grey theory to forecast high technology industrial output Applied Mathematics and Computation 2008 195 1 256 263 10.1016/j.amc.2007.04.080 MR2379212 Zbl1163.91536 2-s2.0-36749064932 Zhao Z. Wang J. Zhao J. Su Z. Using a Grey model optimized by Differential Evolution algorithm to forecast the per capita annual net income of rural households in China Omega 2012 40 5 525 532 10.1016/j.omega.2011.10.003 2-s2.0-80755125913 Bezuglov A. Comert G. Short-term freeway traffic parameter prediction: Application of grey system theory models Expert Systems with Applications 2016 62 284 292 2-s2.0-84976533386 10.1016/j.eswa.2016.06.032 Xiao X. P. Yang J. W. Mao S. H. Wen J. H. An improved seasonal rolling grey forecasting model using a cycle truncation accumulated generating operation for traffic flow Applied Mathematical Modelling 2017 51 386 404 10.1016/j.apm.2017.07.010 2-s2.0-85028954335 Dang Y. G. Liu S. F. Chen K. J. The GM models that be taken as initial value Kybernetes 2004 33 2 247 254 Cui J. Liu S. F. Zeng B. Xie N. M. A novel grey forecasting model and its optimization Applied Mathematical Modelling 2013 37 9 4399 4406 10.1016/j.apm.2012.09.052 MR3020581 Zhou W. He J. M. Generalized GM (1,1) model and its application in forecasting of fuel production Applied Mathematical Modelling 2013 37 9 6234 6243 10.1016/j.apm.2013.01.002 MR3039004 Chen C. I. Huang S. J. The necessary and sufficient condition for GM(1,1) grey prediction model Applied and Computational Mathematics 2013 219 11 6152 6162 10.1016/j.amc.2012.12.015 MR3018459 Li G.-D. Masuda S. Yamaguchi D. Nagai M. A new reliability prediction model in manufacturing systems IEEE Transactions on Reliability 2010 59 1 170 177 2-s2.0-77949270763 10.1109/TR.2009.2035795 Xie N. M. Liu S. F. Discrete grey forecasting model and its optimization Applied Mathematical Modelling 2009 33 2 1173 1186 10.1016/j.apm.2008.01.011 MR2468514 Wu L. F. Liu S. F. Yao L. G. Discrete grey model based on fractional order accumulate System Engineering-Theory & Practice 2014 34 7 1822 1827 Wu L. F. Liu S. F. Yao L. G. Yan S. L. Liu D. L. Grey system model with the fractional order accumulation Communications in Nonlinear Science and Numerical Simulation 2013 18 7 1775 1785 10.1016/j.cnsns.2012.11.017 MR3021642 Wu L. F. Liu S. F. Cui W. Liu D. L. Yao T. X. Non-homogenous discrete grey model with fractional-order accumulation Neural Computing and Applications 2014 25 5 1215 1221 10.1007/s00521-014-1605-1 Liu J. F. Liu S. F. Wu L. F. Fang Z. G. Fractional order reverse accumulative discrete grey model and its application Systems Engineering and Electronics 2016 38 3 720 724 Zhang K. Liu S. F. Linear time-varying parameters discrete grey forecasting model Systems Engineering-Theory & Practice 2010 30 9 1650 1657 Stewart G. W. On the perturbation of pseudo-inverses, projections and linear least squares problems SIAM Review 1977 19 4 634 662 10.1137/1019104 MR0461871 Sun J. G. Matrix Disturbance Analysis 1987 Beijing, China Science Press Zeng L. A new time-varying parameter grey model and its application Journal of Systems Science and Mathematical Sciences 2017 37 1 143 154 MR3671848