Expert Systems with Applications 37 (2010) 5640–5644

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Expert Systems with Applications

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An approach to increase prediction precision of GM(1,1) model basedon optimization of the initial condition

Yuhong Wang a,b,*, Yaoguo Dang a, Yueqing Li b, Sifeng Liu a

a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Chinab Department of Industrial Engineering, University of Arkansas, Fayetteville, AR 72701, USA

a r t i c l e i n f o

Keywords:GM(1,1) modelOptimizationInitial conditionFirst-order accumulative generationoperator

0957-4174/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.eswa.2010.02.048

* Corresponding author. Address: College of EcNanjing University of Aeronautics and Astronautics,+86 25 84893774; fax: +86 25 84896149.

E-mail address: wyh2003@gmail.com (Y. Wang).

a b s t r a c t

We propose a novel approach to improve prediction accuracy of GM(1,1) model through optimization ofthe initial condition in this paper. The new initial condition is comprised of the first item and the last itemof a sequence generated from applying the first-order accumulative generation operator on the sequenceof raw data. Weighted coefficients of the first item and the last item in the combination as the initial con-dition are derived from a method of minimizing error summation of square. We can actually find that thenewly modified GM(1,1) model is an extension of the original GM(1,1) model and another modifiedmodel which takes the last item in the generated sequence as the initial condition when weighted coef-ficients takes distinctly specific values. The new optimized initial condition can express the principle ofnew information priority emphasized on in grey systems theory fully. The result of a numerical exampleindicates that the modified GM(1,1) model presented in this paper can obtain a better prediction perfor-mance than that from the original GM(1,1) model.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Grey systems theory was proposed by professor Deng firstly in1982 (Deng, 1982, 1989). From then on the new discipline has beenconcentrated on by a large number of researchers and utilized in avariety of fields such as natural science, social science and engi-neering science etc. Grey systems theory focuses mainly on suchsystems as partial information known and partial information un-known. With the rapid development of science and technologymore and more systems have the same characteristic as grey sys-tems. Solutions to these uncertain systems have become a greatchallenge for further development in associated fields. However,one of possible means can be provided by some approaches in greysystems theory.

GM(1,1) model is one of important models in grey models group.From the procedure of construction of GM(1,1) model we can findthat the model is neither a differential equation nor a differenceequation. In fact, it is an approximate model which has the charac-teristic both differential equation and difference equation. It is inev-itable for the approximate model to result in some errors in practicalapplications. To increase prediction accuracy using GM(1,1) model alarge number of researchers concentrate upon improvements of

ll rights reserved.

onomics and Management,Nanjing 210016, China. Tel.:

GM(1,1) model mainly in three aspects. On the one hand, a numberof researchers focus mainly on improvements of grey derivative.Wang proposes a GM(1,1) direct modeling method with a step bystep optimizing grey derivative’s whitened values to unequal timeinterval sequence modeling. He has also proves that the new meth-od still has the same characteristic of linear transformation consis-tency as the old method (Wang, Chen, Gao, & Chen, 2004). Mupresents a method to optimize the whitened values of grey deriva-tive and constructs an unbiased GM(1,1) model. He also proves thatthe new model has the characteristic of law of whitened exponent(Mu, 2003a, 2003b).

On the other hand, a number of researchers focus mainly onimprovements of background value. Tan presents a new methodto construct the formulation of background value. The new modelwith the application of a new formulation of background value cansimulate an exponential sequence well (Tan, 2000). Mu points outthe reasons that the GM(1,1) model can not predict an exponentialfunction perfectly through the analysis of mechanism of construct-ing the GM(1,1) model. And he proposes a new method to con-struct GM(1,1) model based on the improvement of backgroundvalue (Mu, 2002). Mao utilizes a modified GM(1,1) model basedon background value to predict vehicle fatality risk and obtain abetter prediction performance (Mao & Chirwa, 2006).

Finally, improvements of the initial condition in the timeresponse function are focused on by some researchers gradually.Dang proposes a method to improve grey models using the nthitem of Xð1Þ as the starting condition of the grey differential model

Y. Wang et al. / Expert Systems with Applications 37 (2010) 5640–5644 5641

to increase prediction precision (Dang, Liu, & Chen, 2004). Xie pro-poses discretely grey prediction models and corresponding param-eter optimization methods. And he also illustrates three classes ofgrey prediction models such as the starting-point fixed discretegrey model, the middle-point fixed discrete grey model and theending-point fixed discrete grey model (Xie & Liu, 2008). Liu alsopresents a method to improve the prediction precision by optimi-zation of the coefficient of exponential function (Liu & Lin, 2006). Inaddition to these three classes of improvements on GM(1,1) model,Hsu presents a grey model improved by the Bayesian analysis topredict output of integrated circuit industry (Hsu & Wang, 2007).Li also presents a method of combining GM(1,1) model and markovchain to predict the number of Chinese international airlines (Li,Yamaguchi, & Nagai, 2007).

These types of improvements for grey models may increase pre-diction precision in some practical applications. However, therestill exists some space to improve prediction precision ofGM(1,1) model. We present a novel approach to optimize the ini-tial condition in time response function. The new optimization ap-proach of the initial condition is comprised of the initial item in thetime response function from the original GM(1,1) model (Deng,2002; Liu & Lin, 2006) and the nth item of a sequence generatedfrom applying the first-order accumulative generation operatoron a sequence of raw data (Dang et al., 2004). And the correspond-ing weighted coefficients of the two parts forming the initial con-dition in new optimization method are derived from minimizingerror summation of square in terms of the time response function.The combination of newly initial condition can express the princi-ple of new information priority emphasized on in grey systemstheory fully. Hence, we can make full use of new pieces of informa-tion in raw data to improve prediction precision with applicationof the new initial condition. Meanwhile, the new optimized modelis also an extension of the original GM(1,1) model. And the coeffi-cients of weight in the optimized initial condition can change inthe interval of zero and one. Especially, the newly optimized modelcould also be degenerated into the improved model presented byprofessor Dang (Dang et al., 2004) and the original GM(1,1) modelwhen the coefficients take zero or one respectively. Then we be-lieve that the newly optimized model can be helpful to improvethe prediction precision of GM(1,1) model.

The remaining parts of this paper are organized as following. Abrief introduction to the original GM(1,1) model and the optimiza-tion method of the initial condition are given in Section 2. Section 3will illustrate the application of new optimized method through anumerical example. Conclusions and the work focused mainly onin the future will be given in Section 4.

2. Optimization method of the initial condition

2.1. A brief introduction to the original GM(1,1) model

Assume that

Xð0Þ ¼ ðxð0Þð1Þ; xð0Þð2Þ; . . . ; xð0ÞðnÞÞ

is a non-negative sequence of raw data, and its sequence of the first-order accumulative generation operator on Xð0Þ (Deng, 2002; Liu &Lin, 2006) is denoted as following,

Xð1Þ ¼ ðxð1Þð1Þ; xð1Þð2Þ; . . . ; xð1ÞðnÞÞ; where

xð1ÞðkÞ ¼Xk

i¼1

xð0ÞðiÞ; k ¼ 1;2; . . . ; n:

From the sequence Xð1Þ obtained from applying the first-order accu-mulative generation operator on sequence Xð0Þ, we can derive the

sequence of generated mean value of consecutive neighbors in thefollowing,

Zð1Þ ¼ ðzð1Þð2Þ; zð1Þð3Þ; � � � ; zð1ÞðnÞÞ; where

zð1ÞðkÞ ¼ 12ðxð1ÞðkÞ þ xð1Þðk� 1ÞÞ; k ¼ 2;3; . . . ;n

Definition 1(Deng, 2002; Liu & Lin, 2006): For a non-negative se-quence of raw data Xð0Þ;Xð1Þ is a newly generated sequence withthe application of the first-order accumulative generation operatoron Xð0Þ; Zð1Þ is a new sequence with the application of the generatedmean value of consecutive neighbors operator on Xð1Þ, then the fol-lowing equation

xð0ÞðkÞ þ azð1ÞðkÞ ¼ b

is a grey differential equation, also called GM(1,1) model. And theequation

dxð1Þ

dtþ axð1Þ ¼ b

is the whitened equation of GM(1,1) model. If we let

B ¼

�zð1Þð2Þ 1�zð1Þð3Þ 1

..

. ...

�zð1ÞðnÞ 1

266664

377775; Y ¼

xð0Þð2Þxð0Þð3Þ

..

.

xð0ÞðnÞ

266664

377775; a ¼ ½a; b�T

In terms of the least square estimate method we can obtain theparameter estimators from the GM(1,1) model in the following,

a ¼ ½a; b�T ¼ ðBT BÞ�1BT Y

Proof is omitted (Liu & Lin, 2006).The time response function of the whitened equation yields,

xð1ÞðtÞ ¼ ðxð1Þð1Þ � b=aÞe�aðt�1Þ þ b=a:

And the time response equation of GM(1,1) is the followingequation,

xð1Þðkþ 1Þ ¼ ðxð0Þð1Þ � b=aÞe�ak þ b=a; k ¼ 1;2; . . . ;n:

The restored values of raw data is given below,

xð0Þðkþ 1Þ ¼ xð1Þðkþ 1Þ � xð1ÞðkÞ ¼ ð1� eaÞðxð0Þð1Þ � b=aÞe�ak;

k ¼ 1;2; . . . ;n:

Professor Deng (2002) points out that the whitened equationand its corresponding time response function can not be derivedfrom the GM(1,1) model directly, which are some approximatereplacement. In fact, the time response function is a solution ofthe whitened function when the initial condition is xð0Þð1Þ. Theseapproximate processing procedures also bring hints for us to im-prove prediction precision of GM(1,1) model. As the principle ofnew information priority is emphasized on in grey systems theoryto improve prediction performance of data of small sample sizes.So how to improve performance in utilizing new information andkeep consistent with the original GM(1,1) model are main contentsin Section 2.2

2.2. Optimization of the initial condition to improve predictionprecision

From the description of the original GM(1,1) model in Section2.1 we can find that the general solution of the whitened equationcan be expressed as following,

xð1ÞðtÞ ¼ ce�at þ b=a;

Table 1The sequence of simulation data from the exponential equation.

t f(t)

1 2.98362 4.45113 6.64024 9.90615 14.77816 22.04647 32.8893

5642 Y. Wang et al. / Expert Systems with Applications 37 (2010) 5640–5644

where c is a constant, a and b are parameters can be derived fromthe least square estimation method.

For the general solution if we let t ¼ 1 and t ¼ n, respectively,then we can obtain the following Eqs. (1) and (2),

xð1Þð1Þ ¼ ce�a þ b=a; ð1Þxð1ÞðnÞ ¼ ce�an þ b=a ð2Þ

We also assume another parameter b, where b 2 ½0;1�. Then wemultiply b and ð1� bÞ at the both sides of Eqs. (1) and (2), respec-tively, then Eqs. (3) and (4) are derived as following,

bxð1Þð1Þ ¼ bce�a þ bb=a; ð3Þð1� bÞxð1ÞðnÞ ¼ ð1� bÞce�an þ ð1� bÞb=a ð4Þ

From (3) and (4), we can get the following expression

bxð1Þð1Þ þ ð1� bÞxð1ÞðnÞ ¼ bce�a þ bb=a

þ ð1� bÞce�an þ ð1� bÞb=a

And the constant c can be expressed in the following,

c ¼ bxð1Þð1Þ þ ð1� bÞxð1ÞðnÞ � b=abe�a þ ð1� bÞe�an

ð5Þ

To improve prediction precision of GM(1,1) model, we cite themethod proposed by professor Liu (Liu & Lin, 2006) to derive theexpression of c through minimizing the error summation of squarein terms of the time response function. Construct a function f ðcÞ,yielding

f ðcÞ ¼Xn

k¼1

ðxð1ÞðkÞ � xð1ÞðkÞÞ2;

where xð1ÞðkÞ ¼ ce�ak þ b=a, representing the prediction values ofxð1ÞðkÞ; xð1ÞðkÞ is a generated datum with the application of first-or-der accumulative generation operator on Xð0Þ; k ¼ 1;2; . . . ;n. Hence,

f ðcÞ ¼Xn

k¼1

ce�ak þ ba� xð1ÞðkÞ

� �2

;

let df ðcÞdc ¼ 0, then

2Xn

k¼1

ce�ak þ ba� xð1ÞðkÞ

� �e�ak ¼ 0

c ¼Pn

k¼1½ðxð1ÞðkÞ � b=aÞe�ak�Pnk¼1e�2ak

ð6Þ

From expressions (5) and (6) we can get that

bxð1Þð1Þþ ð1�bÞxð1ÞðnÞ�b=abe�aþð1�bÞe�an

¼Pn

k¼1½ðxð1ÞðkÞ�b=aÞe�ak�Pnk¼1e�2ak

b¼ ðxð1ÞðnÞ�b=aÞPn

k¼1e�2ak� e�anPnk¼1½ðxð1ÞðkÞ�b=aÞe�ak�

ðe�a� e�anÞPn

k¼1½ðxð1ÞðkÞ�b=aÞe�ak�þ ðxð1ÞðnÞ� xð1Þð0ÞÞPn

k¼1e�2ak

Then we can obtain the optimized time response function (7) asfollowing,

xð1ÞðtÞ ¼ ½bxð1Þð1Þ þ ð1� bÞxð1ÞðnÞ � b=a�½be�a

þ ð1� bÞe�an��1e�at þ b=a: ð7Þ

The restored values can be derived in terms of application of first-order inverse accumulative generation operator on the optimizedtime response function,

xð0ÞðtÞ ¼ xð1ÞðtÞ � xð1Þðt � 1Þ¼ ð1� eaÞ½bxð1Þð1Þ þ ð1� bÞxð1ÞðnÞ � b=a�½be�a

þ ð1� bÞe�an��1e�at:

From the expression above we can see that in fact the time re-sponse function is the solution of the whitened equation ofGM(1,1) model when the initial condition is bxð1Þð0Þ þ ð1�bÞxð1ÞðnÞ. Through optimization of the initial condition predictionprecision of GM(1,1) model can be improved. Moreover, the princi-ple of new information priority can be expressed fully and the orig-inally initial condition is also embodied through the optimizedinitial condition. Particularly, when b ¼ 0, the optimized initialcondition is deteriorated to be xð1ÞðnÞ, and the time response func-tion is to be

xð1ÞðtÞ ¼ ½xð1ÞðnÞ � b=a�e�aðt�nÞ þ b=a;

which is just the optimized result of the initial condition proposedby professor Dang (Dang et al., 2004); when b ¼ 1, the optimizedinitial condition is deteriorated to be xð1Þð1Þ, and the time responsefunction is to be

xð1ÞðtÞ ¼ ½xð1Þð1Þ � b=a�e�aðt�1Þ þ b=a;

which is the originally initial condition proposed by professor Deng(Deng, 2002; Liu & Lin, 2006). Hence, the modified GM(1,1) modelbased on the optimized initial condition is an extension of the ori-ginal GM(1,1) and the improved GM(1,1) proposed by professorDang et al. (2004). And the new optimized model has a better pre-diction precision than that of the original model. We will illustratethe optimized method through a numerical example in Section 3and compare prediction precision from the optimized model andthe original model.

3. Numerical example

In this section we illustrate the application of the optimized ini-tial condition in GM(1,1) model. We take an exponential functionf ðtÞ ¼ gekt . If we let parameters g and k take a fixed value, respec-tively, then we can obtain a sequence of simulation data whenparameter t takes different values. Here we let g ¼ 2 and k ¼ 0:4,then the exponential equation is f ðtÞ ¼ 2e0:4t . If we let t ¼ 1;2;. . . ;7 respectively, then the sequence of simulation data obtainedfrom the function is shown as Table 1.

To compare with prediction performances between the originalGM(1,1) model and the modified GM(1,1) model proposed in thispaper, we utilize the first five data in the sequence of simulationdata to construct the original GM(1,1) model and the modifiedGM(1,1) model, respectively.

Xð0Þ ¼ ðxð0Þð1Þ; xð0Þð2Þ; xð0Þð3Þ; xð0Þð4Þ; xð0Þð5ÞÞ¼ ð2:9836;4:4511;6:6402;9:9061;14:7781Þ

First, the parameters are estimated and the original GM(1,1) modelis constructed as following,

a ¼ �0:3948; b ¼ 2:3948; xð1Þðkþ 1Þ ¼ 9:0495e0:3948k � 6:0659

When k ¼ 1;2; . . . ;6, the prediction values are

Y. Wang et al. / Expert Systems with Applications 37 (2010) 5640–5644 5643

bX ð0Þ ¼ ðxð0Þð2Þ; xð0Þð3Þ; xð0Þð4Þ; xð0Þð5Þ; xð0Þð6Þ; xð0Þð7ÞÞ¼ ð4:3807;6:5014;9:6486;14:3194;21:2512;31:5387Þ

Second, the parameter is derived and the modified GM(1,1) model isconstructed yielding, a ¼ �0:3948; b ¼ 2:3948

b¼ ðxð1ÞðnÞ�b=aÞPn

k¼1e�2ak�e�anPnk¼1½ðxð1ÞðkÞ�b=aÞe�ak�

ðe�a�e�anÞPn

k¼1½ðxð1ÞðkÞ�b=aÞe�ak�þðxð1ÞðnÞ�xð1Þð0ÞÞPn

k¼1e�2ak¼0:5224

xð1ÞðtÞ¼ ½bxð1Þð1Þþð1�bÞxð1ÞðnÞ�b=a�½be�aþð1�bÞe�an��1e�atþb=a:

¼½0:5224�2:9836þð1�0:5224Þ�P5

i¼1xð0ÞðiÞ�b=a�½0:5224e�aþð1�0:5224Þe�5a� e�atþb=a

¼6:2025e0:3948t�6:0659

When k ¼ 2;3; . . . ;7, the prediction values by the modified GM(1,1)model are in the following,

Fig. 2. Comparison of relative errors of prediction data from the original GM(1,1)model and the modified GM(1,1) model.

Fig. 1. Comparison of prediction performance from the original GM(1,1) model andthe modified GM(1,1) model.

bX ð0Þ ¼ ðxð0Þð2Þ; xð0Þð3Þ; xð0Þð4Þ; xð0Þð5Þ; xð0Þð6Þ; xð0Þð7ÞÞ¼ ð4:4561;6:6132;9:8146;14:5657;21:6168;32:0812Þ

In fact, the data xð0Þð2Þ; . . . ; xð0Þð5Þ are the simulation data de-rived from the original GM(1,1) model or modified GM(1,1) modelwhich are constructed in terms of the sequence of dataXð0Þ ¼ ðxð0Þð1Þ; xð0Þð2Þ; xð0Þð3Þ; xð0Þð4Þ; xð0Þð5ÞÞ; xð0Þð6Þ and xð0Þð7Þ arethe prediction data according to the two types of newly con-structed models. The comparison of simulation and prediction databy the two newly constructed models mentioned above is shownas Table 2.

Fig. 1 shows that both the original GM(1,1) model and the mod-ified GM(1,1) model have perfect prediction performance whenparameter t is much smaller in the function f ðtÞ ¼ 2e0:4t . However,with the increment of parameter t, prediction errors from the twotypes of GM(1,1) model are gradually increasing. However, fromFig. 1 we can find that prediction performance from the modifiedGM(1,1) model is much better than that from the originalGM(1,1) model.

Fig. 2 provides a means of evaluating how well the predictionvalues tracking the function f ðtÞ ¼ 2e0:4t . For relative errors, it isdesirable to have relative error as close to zero as possible. AndFig. 2 also shows that relative errors from the modified GM(1,1)model are much closer to zero than those from the originalGM(1,1) model. Likewise, we can find that the prediction valuesfrom the original GM(1,1) model are fully underestimated throughparameter t and the relative errors range in the interval of�4.1065% and �1.5816%. Prediction values from the modifiedGM(1,1) model are slightly overestimated in the first datum pointand underestimated in the subsequent data points and the relativeerrors range in the interval of �2.4570% and 0.1123%. From thecomparison analysis of Figs. 1 and 2 it is evident that the modifiedGM(1,1) model obtain a better prediction performance.

Table 2The comparison of prediction performance from the original GM(1,1) model and the mod

t f(t) Original GM(1,1) model prediction Relative er

1 2.98362 4.4511 4.3807 �1.58163 6.6402 6.5014 �2.09034 9.9061 9.6486 �2.59945 14.7781 14.3194 �3.10396 22.0464 21.2512 �3.60697 32.8893 31.5387 �4.1065

Note: Relative error ¼ ðf ðtÞ � f ðtÞÞ � 100%=f ðtÞ; t ¼ 2;3; . . . ;7.

4. Conclusion and future work

In this paper we propose an optimized method of the initialcondition in GM(1,1) model to improve its prediction accuracy.We have the initial condition of GM(1,1) model consisted of thefirst item xð1Þð1Þ and the last item xð1ÞðnÞ of a new sequence Xð1Þ

ified one.

ror (%) Modified GM(1,1) model prediction Relative error (%)

4.4561 0.11236.6132 �0.40669.8146 �0.9237

14.5657 �1.437321.6168 �1.948632.0812 �2.4570

5644 Y. Wang et al. / Expert Systems with Applications 37 (2010) 5640–5644

generated from applying the first-order accumulative generationoperator on the sequence of raw data Xð0Þ. From minimizing errorsummation of square of the sequence Xð1Þ we can derive theweighted coefficients b and 1� b for xð1Þð1Þ and xð1ÞðnÞ in the initialcondition, respectively. The result of numerical example indicatesthat the modified model can improve prediction accuracy ofGM(1,1) model significantly. We can also find that the modifiedoptimized GM(1,1) model is an extension of the original GM(1,1)model proposed by professor Deng (Deng, 1982, 1989, 2002), pro-fessor Liu (Liu, Dang, & Fang, 2004; Liu & Lin, 2006) and anothermodified GM(1,1) model presented by professor Dang et al.(2004) actually when b takes zero and one respectively. The mod-ified GM(1,1) model can also express the principle of new informa-tion priority emphasized on in grey systems theory fully. Thenadequate information can be extracted from the sequence of smallsample sizes when we apply the modified GM(1,1) model.

Some modifications such as optimization of the grey derivative,optimization of the background value and combinations of distinctmethods will be focused mainly on to improve prediction accuracyof GM(1,1) model in future work.

Acknowledgement

This work is partially supported by National Science Foundationof China (70473037); Program of High-level Government-spon-sored overseas studies for postgraduates by China ScholarshipCouncil; Project of Soft Science Plan of Jiangsu Province(BR2006025); Scientific Innovation Foundation of Nanjing Univer-sity of Aeronautics and Astronautics (Y0811-091); Social ScienceFoundation of Nanjing University of Aeronautics and Astronautics(V0852-091); Social Science Foundation of Jiangsu Province(08EYB005). We would also like to acknowledge the supervision

and insightful remarks of Dr. Ed Pohl in Department of IndustrialEngineering, University of Arkansas.

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