Naiming Xieand Sifeng Liu

1.College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China;

2.Institute of Grey System Studies,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China

Interval grey number sequence prediction by using non-homogenous exponential discrete grey forecasting model

Naiming Xie1,2,*and Sifeng Liu1,2

1.College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China;

2.Institute of Grey System Studies,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China

This paper aims to study a new grey prediction approach and its solution for forecasting the main system variable whose accurate value could not be collected while the potential value set could be defned.Based on the traditional nonhomogenous discrete grey forecasting model(NDGM),the interval grey number and its algebra operations are redefned and combined with the NDGM model to construct a new interval grey number sequence prediction approach.The solving principle of the model is analyzed,the new accuracy evaluation indices,i.e.mean absolute percentage error of mean value sequence(MAPEM)and mean percent of interval sequence simulating value set covered (MPSVSC),are defned and,the procedure of the interval grey number sequence based the NDGM(IG-NDGM)is given out.Finally,a numerical case is used to test the modelling accuracy of the proposed model.Results show that the proposed approach could solve the interval grey number sequence prediction problem and it is much better than the traditional DGM(1,1)model and GM(1,1)model.

grey number,grey system theory,interval,discrete grey forecasting model,non-homogeneous exponential sequence.

1.Introduction

In the system forecasting process,information that can be obtained is usually limited and incomplete.Different types of uncertainty sets,such as fuzzy set,interval,rough set, and grey set,are often used to describe such uncertainty information[1].Fuzzy sets are used to describe the vague information[2],rough sets are used to characterize the inconsistent and incomplete information[3],interval numbers are used to defne the boundary information[4]while grey numbers are used to describe the partially known information which means the certain number could not be known while its potential value set can be defned[5,6]. These kinds of uncertainty information bring troubles to the system analysis process.Especially in solving system forecasting problems,it is very diffcult to catch the developing trend with the uncertainty information.The grey forecasting model,proposed by Julong Deng,is defned to solve grey uncertainty problems which lack information [5].The original idea of the grey forecasting model is to improvethe forecasting capability under the condition that the original data sequence is limited while traditional statistical models could not be constructed effectively.Traditional forecasting models,like linear regression model and time series model,usually require at least 30 observations.However,many forecasting problems,e.g.socioeconomic forecasting,are impossible to collect so many observations.

The grey forecasting model(GM(1,1))is frstly proposed to predict and control the long-term output of grain in China[7].The mechanism of the proposed model is to dig as much as possible modelling information by accumulating generation of the original sequence.In the process of accumulating,the random disturbance of the sequence is dramatically weakened and the trend of accumulated sequence is embodied clearly.Wang et al.considered that the grey system theory was the main theoretical breakthrough of management science in China[8]. As a novel forecasting method,grey forecasting models are theoretically extended and applied widely in various real applications.In theoretical aspects,some scholars focused on improving the GM(1,1)model by combining the model with other methods or constructing expansion models[9–12].Others focused on developing new grey forecasting models for special developing rules,like S-curve and non-equidistance[13,14].For solving real applications,the grey forecasting model has been widely applied in economic developing forecasting,energy forecasting,etc[15–23].The results of these real applications showed that grey forecasting can catch the developing trend well in different cases.However,some scholarstook the view that the GM(1,1)model can only apply in the short-term and low developing ratio sequence forecasting while not adapt to high developing ratio sequences. We found that the GM(1,1)model’s parameters estimation adopted discrete equation while simulative and predictive values calculation adopted continuous equation.The difference between the two equations resulted in low simulative precision.Then we proposed discrete grey models to improve the forecasting and simulative accuracy of sequences.Hereafter,we developed the optimized form of the single variable discrete grey model and the discrete grey model approximate to the non-homogeneous exponential trend[24,25].Yao and Wang applied optimization theories to establish a generalized discrete GM(1,1)model in which the time response of simulation sequence was established directly[26–28].

The above references mainly focused on the grey forecasting modelling and their applications based on the limited data.In fact,similar with the fuzzy set,rough set or interval number,the grey number is the true expression of grey uncertainty and it is the basic element of a grey system.Unfortunately,the grey number has not been effectively included in the grey forecasting modelling process. The grey number is defned as a number whose certain value is unknown while the potential value set is known, denoted as“⊗”[29].We proposed a new method on comparing grey numbers with different kinds of grey number types[30].Yang and Liu discussed the operation of grey numbers based on the kernel and greyness of different grey numbers[31].Some scholars applied grey numbers in constructing the grey decision-making model and grey relational model[32–34].However there is no paper focusing on combining grey numbers with forecasting models.This paper mainly focuses on creating such a forecasting model. We redefne the interval grey number,combine the interval grey number with non-homogenous exponential sequence to construct the novel grey forecasting model and discuss the solutions of the proposed model.

The rest of this paper is structured as follows.In Section 2,with a brief review of the traditional nonhomogenous discrete grey forecasting model(NDGM) and redefnition of interval grey number,a novel grey forecasting model is constructed by combining the NDGM with the interval grey number sequence which approximates to the non-homogenous exponential rule.Section 3 discusses the algorithm of the proposed novel grey forecasting model.Section 4 demonstrates a numerical example of the proposedmodel and compares it with DGM(1,1) model and GM(1,1)model.Finally,Section 5 concludes the paper.

2.Construction of interval grey number sequence based NDGM

The NDGM model means that the discrete grey forecasting model approximates to non-homogeneous exponential developing rules.We have constructed its model formula and discussed the solution and properties of the model[35].

2.1NDGM

Assume that the sequence

is an original data sequence or system main variable data sequence,and the sequence

is the accumulated generation sequence of X(0),where

The equation

is called the NDGM.ˆx(1)(k)is the simulative or forecasting value of x(1)(k)and β1,β2,β3are parameters.If k=1,2,...,n−1 and(4)can be rewritten into the matrix form as

Apply the least square method with input data X(1),parameters β1,β2and β3satisfy the matrix equation

where

and BTis the transpose matrix of B.The equation

is called the recursive function.ˆx(1)(k)in(9)is the simulative value of x(1)(k).Consider the equation

Similarly,ˆx(0)(k)in(10)is thesimulativevalueofx(0)(k). Align all x(0)(k)at k=1,2,...,n,we can get the simulative sequence

With the simulative sequenceˆX(0)and the original sequence X(0),the mean absolute percentage error(MAPE) is defned as

2.2Defnition of interval grey number

Defnition 1 Grey numberThe grey number,marked as“⊗”,is defned as a number whose certain value is unknown but the possible range of the true value can be got. The set D is defned as the information background of a grey number⊗.d∗is the true value of the grey number⊗. Then

(i)⊗is a grey number under the information background D.

(ii)D is a potential value-covered set of⊗.

(iii)d∗is the true value of the grey number⊗.

The grey number is marked as

Generally,grey number could be abbreviated writing in⊗.

Defnition 2Interval grey numberAssume upper and lower limitation values can be got in a continuous value-covered set D,i.e.the potential value-covered set is an interval number,and then we call⊗as an interval grey number and D as the interval covered set,marked as∀⊗⇒d∗∈D,D=[a,¯a]or abbreviated as⊗=[a,¯a], both the lower boundary a and the upper boundary¯a are conventional real numbers.

Defnition 3 Grey number operationLet Diandbe the value-covered set and true value of grey number⊗i, Djand d∗jbe the value-covered set and true value of grey number⊗j,°be an operation,⊗ijbe the result of⊗iand⊗jon the°operation.Let Dijbe the value-covered of the grey number⊗ij.Then we have the general°operation formula: abbreviated as⊗i°⊗j=⊗ij.

Since the grey numbers⊗i,⊗jand their true values d∗i, d∗jare unknown,thus the most important operation is the operation of Diand Djin the general°operation equation.Specially,if interval grey number⊗i=[ai,¯ai]and⊗j=[aj,¯aj]are independent of each other,we can get

Obviously,⊗ijis composed by⊗iand⊗j,so⊗ijis not independent of⊗i,then we can defne

2.3Interval grey number sequence based NDGM

As the grey number can closely express the real system, the grey number should be combined with grey forecasting models to construct novel grey forecasting models based on grey number sequences.

Defnition 4Similar with the NDGM,assume the original grey number sequence of the main system variable is

where x(0)(⊗i)(i=1,2,...,n)is the interval grey numbers.And upper and lower limitations are collected,i.e. x(0)(⊗1)∈[a1,¯a1],x(0)(⊗2)∈[a2,¯a2],...,x(0)(⊗n)∈[an,¯an].Then,accumulated generation sequence X(1)(⊗) is

where

Then the equation

is called interval grey number sequence based the NDGM (IG-NDGM).β1,β2and β3are model parameters.

Defnition 5Set Skas the set of value yk,yk,meanas the mean value of yk,ˆSkas the set of valueˆyk,andˆyk,meanas the mean value ofˆykaccordingly,i.e.

then defne

as the absolute percentage error of mean value(APEM)of ykand

as the mean absolute percentage error of sequences of simulating mean value with original mean value (MAPEM).Defne

as the percent of interval sequence simulating value set covered(PSVSC)of ykand

as the mean percent of PSVSC(MPSVSC).

3.Solution of IG-NDGM

3.1Principle of solving IG-NDGM

To the interval grey number sequence forecasting problem, assume the original sequence is X(0)(⊗)and its accumulated sequence is X(1)(⊗).That is

where

Then,the most important work is to solve the simulative interval grey number sequences:

and

As is shown in Fig.1,the accumulated sequence X(1)(⊗) is divided into two sequences,i.e.upper boundary sequence

and lower boundary sequence

Then two separated NDGMs are constructed with ¯X(1)(⊗)and X(1)(⊗).According to the modelling process of the NDGM,we can get the simulative sequences

and

which compose the simulative interval grey number sequenceˆX(1)(⊗).According to(15),ˆx(1)(⊗k)is not independent ofˆx(0)(⊗1),ˆx(0)(⊗2)andˆx(0)(⊗k),so we can get

Fig.1 Simulative curve of IG-NDGM

3.2Steps of IG-NDGM

According to the process of NDGM and the principle of IG-NDGM,the IG-NDGM procedure is expressed as follows.

Step 1Collect data and form the original interval grey number sequence X(0)(⊗).

Step 2Generate accumulated sequence X(1)(⊗)by one accumulated generating operation(AGO)as shown in (18).

Step 3Extract the modelling sequences,i.e.the upper boundary sequence¯X(1)(⊗)and the lower boundary sequence X(1)(⊗).

Step 4Separately construct the NDGM equations(as shown in(4))of¯X(1)(⊗)and X(1)(⊗),i.e.

and

Step 5Apply the least square method to solve the model parameters¯β1,¯β2,¯β3and β1,β2,β3with(6).

Step 6Apply the recursive function(9)to solve ˆ¯X(1)(⊗)andˆX(1)(⊗),where

Step 7Solve the simulative sequence

whereˆx(0)(⊗k)is solved with(28).

Step 8Solve the simulative mean value sequence

where

Step 9Calculate MAPEM(%)and MPSVSC(%)with (21)and(23),analyze the simulative accuracy and apply the proposed model for forecasting.

4.Number example illustration

Assume that the collected original interval grey number sequence is

And we can get the lower boundary sequence,the upper boundary sequence,and the mean value sequence:

According to Section 3.2,we can calculate the parameters’values:

Then we could calculate the simulative sequence as shown in Table 1.dL represents the lower boundary value and dU represents upper boundary value.The MAPEM and MPSVSC values are calculated accordingly.

Table 1 Simulative value of IG-NDGM

Similarly,we separately construct DGM(1,1)model and GM(1,1)model with the lower boundary sequence X(0)(⊗)and the upper boundary sequence ¯X(0)(⊗). These two models are used to simulate and forecast the lower boundary and upper boundary trends accordingly. Considering the difference of model forms,we sepa-rately calculate the model parameters with the least square method.The values of parameters and simulative equations are shown in Table 2.The results of simulative sequence and forecasting sequence are shown in Table 3.

Table 2 Parameter values of DGM(1,1)model and GM(1,1)model

Table 3 Simulative values of DGM(1,1)model and GM(1,1)model

As is shown in Table 1 and Table 3,we know that all of the IG-NDGM,DGM(1,1)model and GM(1,1)model have got high simulative accuracy of MAPEM.Especially the higher accuracy is reached by the proposed IG-NDGM. The MAPEM of IG-NDGM is only 0.14%while results of DGM(1,1)model and GM(1,1)model are nearly 3%. However,the mean value is the average status of each sequence.To the interval grey number sequence,if the original grey number sequences are covered by the simulative sequences well,it illustrates that simulative results are good.We defne MPSVSC to evaluate the accuracy of sets covered.As is shown in Table 1 and Table 3,the results show that IG-NDGM can get a better simulative result while DGM(1,1)model and GM(1,1)model are inferior to the proposed IG-NDGM.The MPSVSC of IG-NDGM is only 1.73%while the results of other two models are more than 30%.

5.Conclusions

Considering one could only collect the interval information rather than the accurate values of the variable in the system forecasting process,we adopt interval grey number to express such uncertain information and construct a novel grey forecasting model.Firstly,we redefne grey number and interval grey number,and then we construct the IG-NDGM.Subsequently,the solving principle of IGNDGM is analyzed and the procedure is given.Next,an illustration example is used to simulate the accuracy of IG-NDGM and compared with the traditional DGM(1,1) model and GM(1,1)model.The results indicate that IGNDGM is much better than DGM(1,1)model and GM(1,1) model.It not only better simulates the mean value of sequence but also better covers the value sets of interval.Finally,the proposed approach will be a useful tool when the forecaster could not collect the certain values and would like to avoid the forecasting results deviating the development trend.

[1]S.F.Liu,Y.Lin.Grey information theory and practical applications.London:Springer-Verlag,2006.

[2]L.A.Zadeh.Fuzzy sets.Information and Control,1965,8(3): 338–353.

[3]Z.Pawlak.Rough sets.International Journal of Information& Computer Sciences,1982,11(5):341–356.

[4]R.E.Moore.Method and application of interval analysis. Philadelphia:Society for Industrial and Applied Mathematics, 1979.

[5]J.L.Deng.The control problem of grey systems.System& Control Letters,1982,1(5):288–294.

[6]J.L.Deng.Grey hazy sets.The Journal of Grey System,1992, 4(1):13–30.

[7]J.L.Deng.Grey fuzzy forecast and control for grain.Journal of Huazhong University of Science&Technology,1983,11(2): 1–8.(in Chinese)

[8]J.Wang,R.Yan,K.Hollister,et al.A historic review of management science research inChina.OMEGA-The International Journal of Management Science,2008,36(6):919–932.

[9]J.Wen,Y.Huang,J.Chen,et al.Comments on the minimum number of data required for GM(1,1)modeling.The Journal of Grey System,1999,11(3):229–243.

[10]F.M.Tseng,H.C.Yu,G.H.Tzeng.Applied hybrid grey model to forecast—seasonal time series.Technological Forecasting and Social Change,2001,67(2/3):291–302.

[11]W.Y.Peng,C.W.Chu.A comparison of univariate methods for forecasting container throughput volumes.Mathematicaland Computer Modelling,2009,50(7/8):1045–1057.

[12]L.C.Hsu.Forecasting the output of integrated circuit industry using genetic algorithm based multivariable grey optimization models.Expert Systems with Applications,2009,36(4):7898–7903.

[13]Y.M.Cheng,C.H.Yu,H.T.Wang.Short-interval dynamic forecasting for actual S-curve in the construction phase. Journal of Construction Engineering and Management-ASCE, 2009,137(11):933–941.

[14]X.P.Xiao,K.Peng.Research on generalized non-equidistance GM(1,1)model based on matrix analysis.Grey Systems:Theory and Application,2011,1(1):87–96.

[15]S.C.Chang,H.C.Lai,H.C.Yu.A variablePvalue rolling grey forecasting model for Taiwan semiconductor industry production.Technological Forecasting and Social Change, 2005,72(5):623–640.

[16]L.L.Ku,T.C.Huang.Sequential monitoring of manufacturing processes:an application of grey forecasting models.International Journal of Advanced Manufacturing Technology,2006, 27(5/6):543–546.

[17]R.C.Tsaur.Forecasting LCD TV demand using the fuzzy grey model GM(1,1).International Journal of Uncertainty Fuzziness and Knowledge-based Systems,2007,15(6):753–767.

[18]C.Chen,H.L.Chen,S.P.Chen.Forecasting of foreign exchange rates of Taiwan’s major trading partners by novel nonlinear grey Bernoulli model NGBM(1,1).Communications in Nonlinear Science and Numerical Simulation,2008,13(6): 1194–1204.

[19]Z.Zhao,J.Z.Wang,J.Zhao,et al.Using a grey model optimized by differential evolution algorithm to forecast the per capita annual net income of rural households in China. OMEGA-International Journal of Management Science,2012, 40(5):525–532.

[20]D.Akay,M.Atak.Grey prediction with rolling mechanism for electricity demand forecasting of Turkey.Energy,2007,32(9): 1670–1675.

[21]I.Lu,C.Lewis,S.J.Lin.The forecast of motor vehicle,energy demand and CO(2)emission from Taiwan’s road transportation sector.Energy Policy,2009,37(8):2952–2961.

[22]U.Kumar,V.K.Jain.Time series models(grey-Markov,grey model with rolling mechanism and singular spectrum analysis) to forecast energy consumption in India.Energy,2010,35(4): 1709–1716.

[23]H.W.Tang,M.S.Yin.Forecasting performance of grey prediction for education expenditure and school enrollment.Economics of Education Review,2012,31(4):452–462.

[24]N.M.Xie,S.F.Liu.Discrete grey forecasting model and its optimization.Applied Mathematical Modelling,2009,33(2): 1173–1186.

[25]C.Y.Zhu,N.M.Xie.Difference-ratio-based NDGM interpolation forecasting algorithm and its application.Grey Systems: Theory and Application,2012,2(1):70–80.

[26]T.X.Yao,J.Forrest,Z.W.Gong.Generalized discrete GM(1,1)model.Grey Systems:Theory and Application,2012, 2(1):4–12.

[27]Y.H.Wang,Q.Liu,J.R.Tang,et al.Optimization approach of background value and initial item for improving prediction precision of GM(1,1)model.Journal of Systems Engineering and Electronics,2014,25(1):77–82.

[28]Y.H.Wang,Y.G.Dang,X.J.Pu.Improved unequal interval grey model and its application.Journal of Systems Engineering and Electronics,2011,22(3):445–451.

[29]J.L.Deng.The elements on grey theory.Wuhan:Press of Huazhong University of Science&Technology,2002.(in Chinese)

[30]N.M.Xie,S.F.Liu.Novel methods on comparing grey numbers.Applied Mathematical Modelling,2010,34(1):415–423.

[31]Y.J.Yang,S.F.Liu.Reliability of operations of grey numbers using kernels.Grey Systems:Theory and Application,2011, 1(1):57–71.

[32]E.K.Zavadskas,A.Kaklauskas,Z.Turskis,et al.Multiattribute decision-making model by applying grey numbers. Informatica,2009,20(2):305–320.

[33]S.F.Liu,N.M.Xie,J.Forrest.Novel models of grey relational analysis based on visual angle of similarity and nearness.Grey Systems:Theory and Application,2011,1(1):8–18.

[34]N.M.Xie,S.F.Liu.A novel grey relational model based on grey number sequences.Grey Systems:Theory and Application,2011,1(2):117–128.

[35]N.M.Xie,S.F.Liu,Y.J.Yang,et al.On novel grey forecasting model based on non-homogeneous index sequence.Applied Mathematical Modelling,2013,37(7):5059–5068.

Biographies

Naiming Xie was born in 1981.He received his B.S.,M.S.and Ph.D.degrees in grey systems theory from Nanjing University of Aeronautics and Astronautics(NUAA).Now he is an associate professor and dean assistant in the College of Economics and Management of NUAA.His research interests include grey systems theory and management science.

E-mail:xienaiming@nuaa.edu.cn

Sifeng Liu was born in 1955.He received his B.S. degree in Henan University,M.S.degree in Shandong University and Ph.D.degree in grey systems theory from Huazhong University of Science& Technology.Now he is the dean of Institute of Grey System Studies of NUAA and a professor in the College of Economics and Management of NUAA.He is also the chairman of IEEE Grey System Society and vice chairman of IEEE SMC China(Beijing)Branch and editor-inchief of The Journal of Grey System.His research interests include grey systems theory,management science and project management.

E-mail:sfiu@nuaa.edu.cn

10.1109/JSEE.2015.00013

Manuscript received March 17,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(70901041;71171113)and the Aeronautical Science Foundation of China(2014ZG52077).