Ning Xu1,*,Yaoguo Dang1,and Jie Cui

1.College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing 211100,China; 2.College of Economics and Management,Huai Yin Institute of Technology,Huaian 223001,China

Comprehensive optimized GM(1,1)modeland application forshortterm forecasting of Chinese energy consumption and production

Ning Xu1,*,Yaoguo Dang1,and Jie Cui2

1.College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing 211100,China; 2.College of Economics and Management,Huai Yin Institute of Technology,Huaian 223001,China

In order to improve prediction accuracy of the grey prediction model and forecast China energy consumption and production in a short term,this paper proposes a novel comprehensively optimized GM(1,1)model,also named COGM(1,1), based on the grey modeling mechanism.First,the relationship of the background value formula and its whitenization equation is analyzed and a new method optimizing background values is proposed to eliminate systemic errors in the modeling process. Second,the solving process of the new model is derived.For parameter estimation,a set of auxiliary parameters are used to change grey equation’s form.Then,original parameters are restored by an equations system.After solving the whitenization equation,initialvalue in time response function is established by least errors criteria.Finally,a numerical case and comparison with other grey prediction models are made to testify the new model’s effectiveness,and the computational results show that the COGM(1,1)modelhas a better property and achieves higher precision.The new modelis used to forecast China energy consumption and production,and the ability ofenergy self-sufficiency is further analyzed.Results indicate that gaps between consumption and production in future are predicted to decline.

COGM(1,1),grey prediction,energy consumption, background value.

1.Introduction

Grey systems theory is a new branch of uncertain system science,and the prediction theory is one of its main parts[1].Over the last two decades,grey prediction theory has caught much attention of scholars all over the world and got a rapid development.A group of prediction models have been put forward and many new algorithms based on GM(1,1)have been proposed to solve practical problems[2,3].

Due to the complexity of uncertain system problems, many scholars are looking for new grey modeling methods to work outthe short-term forecastproblems.A brief summary of the latest research relevantto grey prediction theory is presented as follows.Wang etal.studied the construction method of the background formula in grey equation,and suggested a new method to optimize the background formula,which achieves high accuracy when the modeling condition of quasi-exponentialis satisfied[4,5]. The initial value is another factor impacting the modeling process.Dang and Wang et al.studied the condition oftime response function,and derived an algorithm taking accountofnew information priority[6,7].Tien studied the effectiveness of the first entry in the GM(1,1)model,and put forward a generalized optimization algorithm for establishing the initial value[8].Zeng provided some methods to improve the grey prediction model,and the background value was changed into an interval grey number, then a whitenizaiton function was used to sublimate the system’s uncertainty;the provided method is fairly appropriate for more complex systems[9].Li et al.studied the modeling mechanism of the grey model,and constructed a new modelcombining the GM(1,1)modelwith an adaptive controlmethod;a controlparameterin the background value was used to adjust the grey equation adapted to the original sequence’s characteristics[10].Some intelligent algorithms can be used to constructa combined grey model,such as particle swarm algorithm.Zhang testified the effectiveness of the optimization method using the particle swarm algorithm,the combining modelcan gethigher precision than the traditional GM(1,1)[11].Mark combined the statistics method to provide a new grey verhulst model,and the new modelwas used to predictsteelintensity of use in the UK[12].

This paper focuses on the optimization problem of the grey system prediction model,and proposes a new prediction model based on the grey prediction principle which is named as comprehensive optimized GM(1,1)model,or COGM(1,1).The purpose of this new modelis to improve precision of short-term prediction.It achieves higher accuracy and better unbiased property than the traditional GM(1,1)model and other optimized methods.The new model is suitable for sequences satisfying the grey modeling condition of quasi-exponential and the smooth condition.Then the COGM(1,1)model is applied to forecast Chinese energy prediction and consumption.

The structure of this paper is organized as follows. Section 2 provides an analysis of the background value formula,and investigates the optimization method under the grey modeling condition.Section 3 constructs the new model COGM(1,1),and derives its solving process.In this part,a set of auxiliary parameters are proposed to change the form of grey equation,and they are evaluated by the least square method.Then, the original parameters are restored by the equation system.After that,the time response formula is derived from whitenization equation with the initial term established by least error criteria.Section 4 provides the overall modeling procedure and error-checking methods.In Section 5,a numericalcase is made to testthe new model’s property and compare it with other optimization methods of GM(1,1).The computationalresults show that the COGM(1,1)model can keep unbiased property even for sequence with a high growth rate,and the precision is better.Furthermore,the new modelis used to forecastChinese energy consumption and production.The gaps of Chinese energy consumption and production willkeep slowly declining in recentyears.

2.Optimization method ofbackground value formula

Definition 1Let

be the system action sequence,and X(1)={x(1)(1), x(1)(2),...,x(1)(n)}be the first-order accumulated generated operation(AGO)sequence,in which x(1)(k)=

Definition 2For the sequence X(0),X(1)is the AGO sequence.Constructequation

which is called the grey equation ofthe GM(1,1)model,or the basic form.z(1)(k)denotes the background value.The whitenization differentialequation is constructed as

Itis importantto investigate the relationship of the grey equation and the whitenization equation,within which the background value formula is a key role.The traditional GM(1,1)model gives z(1)(k)a formula z(1)(k)=By integrating the whitenization equation in each interval,it is possible to reveal the relationship of grey equation and its whitenization equation.

First,for discretizing the whitenization equation,integrate(2)in interval[k?1,k]and we willobtain

The optimization idea of the background value formula should eliminate the deviation and converge to the realvalues of sequence Z(1)according to its geometric meaning.

First,the optimization way of the background value formula in a single intervalis described as in Fig.1.Fig.1(a) is the geometric meaning of the traditional background value and Fig.1(b)is the optimized method according to discretization of the whitenization equation.

Then,integrated in each interval,the whitenization differentialequation(2)is discretized as

We construct the linear combination of X(1)as the background value formula to link the geometric meaning with actualdata.Suppose the background value formula iswhereα∈[0,1].For(2),the weight coefficientαshould have a certain relationship with the development coefficient.

Fig.1 Geometric meaning of background value z(1)(k)in single interval

Theorem 1Let

be the original sequence,and X(1)={x(1)(1), x(1)(2),...,x(1)(n)}be the first-order accumulated sequence.Z(1)is the sequence of the background value. If X(0)satisfies the quasi-exponential law x(0)(k)= c e?a(k?1),the relationship of the weightcoefficientαand the developmentcoefficient a is

ProofGenerated from X(0)by the AGO operator, X(1)indicates the accumulative exponentialtendency,and is denoted as a non-homogeneous exponential formation x(1)(k)=C e?a(k?1)+D[5].

According to the integral meaning in(3),the background value formula can be derived into

Further,(6)gives the background value’s linear combination formula in whichα∈[0,1]is the weightcoefficient, and the formula can be expressed as

So we can obtain

From(8)and(10),the relation can be derived as

Theorem 1 is proved.?

The optimal background value formula should take overall consideration of z(1)(k)in each interval according to(5).We could adjust combination coefficientαto find the optimalbackground sequence.In Fig.2,sequence Z(1)describes the overallgeometric meaning of the optimalbackground sequence,and the simulation curve of the grey modelshould converge to the bestposition with least systemic errors.

Fig.2 Global geometric meaning of the optimized background sequence

3.Noveloptimized GM(1,1)modeland its solving process

3.1 Optimization method and parameters evaluationThe analysis of the background value gives a way to improve the GM(1,1)model.We construct the optimized GM(1,1)modelwith a new background value formula and an optimalresponse function.

Based on the GM(1,1)model in Definition 2,background value z(1)(k)is constructed as(6).This optimized modelis called COGM(1,1).

Then,estimate the parameters a,b,α.First,substitute the new background formula(6)into(1):Equation(11)can be transformed into

The least square method is used to evaluate the auxiliary parameters r=(a?,b?)T.We can obtain?r= (BTB)?1BTY,where

Theorem 2Assume that auxiliary parameter vector r=(a?,b?)Tis evaluated,the originalparameters can be restored by the following formula:

ProofThe relations of a?,b?and a,b are denoted as Theorem 1 has achieved the relation expression of a andα.Therefore,an equation system can be constructed as

The evaluated values of auxiliary parameters are substituted into(14),and the equations system is solved.We can approach the results

3.2 Establishment approach of the initialterm

After original parameters restored,the continuous form of the time response function can be derived from the whitenization differential equation(2).By discretization, X(1)’s time response function is derived as

where the variable c is given by a certain method.Then reduce(15)to simulate the originalsequence:

Criteria 1Variable c should be given by the condition thatsimulation data have minimalaccumulation of the errorsquare,and could be described as

Theorem 3Assume(16)is the time response function of the COGM(1,1)model,and the variable c which satisfies Criteria 1 should be

ProofAccording to Criteria 1,when F?(c)=0,the minimal value condition of F(c)is described as

4.Checking methods and precision evaluation for COGM(1,1)

The model’s precision could be checked by the following steps.

Step 1Absolute errors scale the difference between the original data and the simulation curve,and AE(k)is defined as

Step 2Relative percentage errors compare the original data and simulation values to evaluate the precision of a specific point.RPE(k)is defined as

Step 3Average relative percentage error(ARPE)can evaluate the model’s total precision,and the computation method can be shown as follows:

5.Case study

5.1 Numericalcase for testing the property of

COGM(1,1)

The traditional GM(1,1)model does not satisfy the unbiased property.While the growth rate is high,the errors accumulate quickly.In this part,we testthe simulation effect when the growth rate?a is high.

An exponential sequence as in Table 1 is used to testify the new model’s effectiveness.After modeling the process,the auxiliary parametersUsing Theorem 2,the values of the originalparameters are solved as

The traditional GM(1,1)modelis used to modelthis sequence as a comparison.Results of the two models are listed in Table 1.

Table 1 Checking results of white exponentialsequence law

It can be observed from Table 1 that the new method satisfies the unbiased property even when the growth rate is high,and has broken the limitation of the high development rate.While the traditional GM(1,1)model has obvious deviation from the originalsequence when the growth rate is high.

5.2 Forecasting of Chinese energy consumption and production

Nowadays,China has become a big energy consumer and also has a high growth rate afterdecades ofhigh speed economic development.Energy demand and supply are important for Chinese macro-economy and have greatinfluence on world market.However,the ability of self-sufficiency may need more attention.Prediction of energy consumption and supply capacity is necessary to evaluate what a degree this country will depend on energy import.The research has studied the prediction of Chinese energy consumption and production in short terms by the new modeling algorithm,namely the COGM(1,1)model.Raw data are listed in Table 2.

Denote column data ofenergy consumption as sequence X(0),and those of energy production as sequence Y(0). Forchecking the characteristics ofthe raw data,the smooth condition is used to determine if the originalsystem could be modeled by the grey prediction model[2].

Table 2 China energy consumption and production(2008–2012) (Unit:Million tons of standard coal)

Letρ(k)denote smooth ratio x(0)(k):

If the ratio satisfiesρ(k)∈[0,ε],k=2,3,...,n,in whichεis a small number withinε＜0.5.In general,if ε=0.5 the originalsystem has a quasi-energetic development tendency,and the grey modelcan be used to predict it.

For checking characteristics of the raw data,the smooth ratios of X(0)areρx(3)=0.369 8,ρx(4)=0.289 1,ρx(5)=0.233 1.The last three smooth ratios of X(0)satisfyρx(k)＜0.5 and have a decreasing tendency.The smooth ratios indicate thatChinese energy consumption has a quasi-energetic growth tendency.We can construct a grey model to simulate the development and forecastthe future values in shortterms.Analyze sequence Y(0),ρy(3)=0.379 5,ρy(4)=0.294 6,ρy(5)=0.237 5 also satisfyρx(k)＜0.5.

The calculation is described step by step as follows.

Step 1Generate AGO sequence X(1)from original sequence X(0),and constructgrey equation as(13).

Step 2Calculate the val?ues of auxiliary parameters by the leastsquare method asding to Theorem 2,restore the originalparameters and get

Step 3Solve the grey model,and the time response formula could be derived and implemented from the whitenization equation.After discretization,it is changed into ?x(0)(k+1)=0.052c e0.0535(k?1).

Step 4Calculate and obtain c=53 199 according to Theorem 3,substitute c into the time response formula,and obtain simulation and prediction values.

We compare the simulation with that of the traditional GM(1,1).The computationalresults are listed in Table 3.

Results shown in Table 3 indicate thatbetter simulation could be approached by COGM(1,1),comparing with the traditionalGM(1,1)model.

Table 3 Simulation results of energy consumption(Unit:Million tons of standard coal)

Simulation is described in Fig.3.The chart is divided into two parts,one of which is the fitting area including data of 2008—2012 and the other is the prediction area including the predicted values of three years in future.

Fig.3 Simulation and prediction of Chinese energy consumption

By the same way,we can construct the COGM(1,1) modelforthe sequence Y(0).Parameters in the process are calculated.Auxiliary parameters areoriginalparameters arein time response formula is established by the optimized condition of initialvalue.Then simulation results could be obtained and shown in Table 4 and Fig.4.

Fig.4 Simulation and prediction of Chinese energy production

Table 4 Simulation results of energy production(Unit:Million tons ofstandard coal)

Results shows that precision of COGM(1,1)is higher than that of GM(1,1).The new model is more suitable to predict Chinese energy production.

Considering Chinese developing environment and policy stability during the Twelve Five-year Plan until2015, it is reasonable to forecastby the COGM(1,1)model.The gapsbetween energy consumption and production in future years are calculated as follows:

Denote Gi(i=2013,2014,2015)as the difference between consumption and production.Estimate the dependence degree ofenergy import,which means whatpercentage the energy consumption gap accounts in native energy production power.The computation formula is given as

where i=2013,2014,2015,xiand yiis the predicted values of energy consumption and production in year i (i＞2012),respectively.

Computing the data of the prediction area,we can obtain R2013=8.146 8%,R2014=7.294 9%,R2014= 6.449 8%.The results indicate the dependence degree of energy import is declining slowly.In 2015,the last year of China’s Twelve Five-year Plan,there will be 6.449 8% of energy consumption gap which cannot be provided by native producers.

6.Conclusions

The background value formula is the key to improve the GM(1,1)model’s accuracy.The discretization of the whitenizaton equation reveals the overall geometric meanings of background values.

The new method COGM(1,1)can improve prediction accuracy.In this model,background value formula is constructed as a linear combination with a constrained condition as in Definition 2.A solving process for COGM(1,1) is derived.Research indicates thatthe new modelhas better precision than that of other models and breaks the confine of high growth sequence for grey models.

The COGM(1,1)model is applied to analyze Chinese energy consumption and production,and achieves better simulation than conventionalGM(1,1)models.The prediction shows thatthe gaps between energy consumption and production are declining in future years.

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Biographies

Ning Xuwas born in 1983.He is a Ph.D.candidate of College of Economics and Management of Nanjing University of Aeronautics and Astronautics and majors in management science and engineering.He received his B.E.and M.E.degrees from Nanjing University of Aeronautics and Astronautics in 2006 and 2011,respectively.His research focuses on grey system theory and prediction modeling algorithm,which is applied in energy system prediction.He is also interested in quantitative economics,especially in macro economy.

E-mail:xuning@nuaa.edu.cn

Yaoguo Dangwas born in 1964.He is currently a professor and Ph.D.supervisor with College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing,China.He is an academic leader in management science and engineering with College of Economics and Management in Nanjing University of Aeronautics and Astronautics.His major research interests are grey system theory,quantitative economics,and post project evaluation.He is now undertaking a project of national natural science foundation of China,and a major projectof key research base of philosophy and social science in Jiangsu colleges.

E-mail:iamdangyg@163.com

JieCuiwas born in 1978.He is currently an associate professor and Ph.D.supervisor with College of Economics and Management,Huai Yin Institute of Technology,Huaian,China.He is an academic leader of Qinglan project in Jiangsu province.His major research interests are grey system theory and prediction and decision methods for management. He is now undertaking a project of national natural science foundation of China. E-mail:nuaacui2008@163.com

10.1109/JSEE.2015.00087

Manuscript received May 13,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(71071077;71301060;71371098).