Amelioration of Grey GM (1, 1) Forecasting Model

Huanyong Zhang Wenzhan Dai Department of Automatio Department of Automatio

Zhejiang Sci-Tech Universit Zhejiang Sci-Tech Universit Hangzhou ,Zhejiang Province,China Hangzhou ,Zhejiang Province,China

zhanghy09@163.com Correspondence e author: dwzhan@zstu.edu.cn

Abstract - First ,the factors which influence forecasting precision of GM(1,1) model is analyzed and it is considered that there is some error between accumulated sequence and discrete solution, which has effect on the precision of general model GM(1,1) ,then amelioration approach of GM (1, 1) forecasting model is proposed. Finally, the method of amelioration GM (1, 1) model is applied in modeling of chemistry fiber output and the results show the effectiveness .

Index Terms - Grey model. Degree of fitting. Forecasting precision

I. INTRODUCTION

Grey system theory has already been widely used in many fields since 1982[1-3]. Although lots of achievements have been made in the grey system theory, there still exist some failures in building grey models. So it is important to make an intensive study of GM(1,1) and find out the crucial factor which influences the precision of GM(1,1) model. Many scholars provided efficient methods in this respect such as error character of GM(1,1) in [5], the background value in [6-8]

and )1()1(x in [9] In this paper, first ,the factors which influence forecasting

precision of GM(1,1) model is analyzed and it is considered that there is some error between accumulated sequence and discrete solution, which has effect on the precision of general model GM(1,1) ,then amelioration approach of GM (1, 1) forecasting model is proposed. Finally The method of amelioration GM (1, 1) model is applied in modeling of chemistry fiber output and the results show the effectiveness .

II. DEFICIENCY OF THE TRADITIONAL GREY FORECASTING MODEL

Once accumulated generating operation (AGO) GM(1,1) model is in common use of the grey forecasting model, this model consists of first-order differential equation with one variable. Followings are the modeling process:

Suppose non-minus data sequence is increasing sequence[3]�

)}(),...,2(),1({ 0)0()0()0( nxxxX ��= (1)

and niix ,...,2,1,0)()0( => Once accumulated generating operation (AGO)

)}(),...,2(),1({ )1()1()1()1( nxxxX = (2)

and nkixkxk

i,...,2,1,)()(

1

)0()1( ==�=

Definition type of GM(1,1) model, namely, grey differential equation of GM(1,1) model is:

nkukazkx ,...,3,2,)()( )1()0( ==+ (3) Corresponding whiting equation is:

utaxdt

tdx =+ )()( )1()1(

(4)

The solution of (4) is

aue

auxtx at +−= −])1([)(ˆ )1()1( (5)

Its discrete solution is

aue

auxkx ak +−=+ −])1([)1(ˆ )1()1( (6)

where ua, are the identification parameters which ca be

estimated as follows: YBBBua TT 1)( −=��

���

�

where [ ]TnxxxY )(),...,3(),2( )0()0()0(= ,

�����

�

�

�����

�

�

−

−−

=

1

11

)(

)3()2(

)1(

)1(

)1(

��nz

zz

B

)]()1([21)1( )1()1()1( kxkxkz ++=+

Its forecasting model is:

aka eauxe

kxkxkx

−−−=

−+=+

])1()[1(

)(ˆ)1(ˆ)1(ˆ

)1(

)1()1()0(

(7)

From (3) to (4), it is a whiting acquiescence, namely, the grey equation is replaced with real differential equation. Due to the non-minus discrete sequence, its once accumulated generating operation (AGO) sequence is monotonically increased and concave-up. So exponent curve is used to fit the figure curve of accumulated sequence.. But not all practical accumulated sequences meet exponent curve, and its discrete solution also can not reflect the change of accumulated sequence accurately, namely, formula (7) need to be mended to improve the precision of grey model.

III. MODIFICATION OF GM(1,1) FORECASTING MODEL

The (7) is changed into follows:

1-4244-1092-4/07/$25.00 © 2007IEEE. 93

Proceedings of the 2007 IEEEInternational Conference on Integration Technology

March 20 - 24, 2007, Shenzhen, China

aue

auxkx ka +−=+ +− )()0()1( ])1([)1(ˆ β (8)

Define the target function as follows:

�

�−

=

+−

−

=

+−+−=

+−+=

1

1

2)1()()0(

1

1

2)1()1(

)}1(])1({[

)]1()1(ˆ[

n

k

ka

n

k

kxaue

aux

kxkxJ

β

�9�

Find out β in order that J is least, we can use the gradient method:

0

)]1([])1([))1((2

))1(()}1(])1({[2

)()1(1

1

)(2)0()0(

)()0(1

1

)1()()0(

=

+−+−−−=

−+−+−−=

+−−

=

+−

+−−

=

+−

�

�ββ

ββ

α

kan

k

ka

kan

k

ka

ekxaue

aux

auxa

eauxkx

aue

auxa

ddJ

(10)

�

�−

=

−−

−

=

−−

=+−+

−

1

1

)1(

1

1

2)0(2

0)]1([

])1([

n

k

aka

n

k

aka

ekxaue

eauxe

β

β

(11)

�

�−

=

−

−

=

−

−

−

−+= 1

1

2)0(

1

1

)1(

])1([

])1([

n

k

ak

n

k

ak

a

eaux

eaukx

e β (12)

let

�

�−

=

−

−

=

−

−

−+= 1

1

2)0(

1

1

)1(

])1([

])1([

n

k

ak

n

k

ak

eaux

eaukx

q and a

qln−=β .

After calculating β , the formula (8) can be used to forecast the trend of sequence.

IV. PRACTICAL APPLICATION

Chemical fiber is important industrial product that reflects the level of country’s chemistry industry and oil industry development. So building the model of output of chemical fiber and forecasting its future developments has the important practical significance. Now build the GM(1,1) model of Chinese output of chemical fiber from 1990 to 2003 by the method proposed in this paper.

First, GM(1,1) model by the traditional method is built as:

42.165)1(ˆ11641.146)1(ˆ

19595.8354.1001)(*ˆ

)0(*

1578.0)0(*

)1(1578.0)1(

=

≥=

≥−= −

xkekx

kekxk

k

��

�

Its amelioration GM(1,1) model by the method proposed

in this paper is as follow:

42.165)1(ˆ11641.146)1(ˆ

19595.8354.1001)(ˆ

)0(

)1142.0(1578.0)0(

)8858.0(1578.0)1(

=≥=

≥−=+

−

xkekx

kekxk

k

��

�

1142.0=β Results are shown in TABLE I.

TABLE I MODEL VALUE�UNIT: TEN THOUSAND TONS�

traditional method

method proposed by this paper

Year sequence number

real value model

value relative errors(%)

model value

relative errors(%)

1990 1 165.42 165.42 0 165.42 0 1991 2 191.03 171.14 10.41 174.26 8.78 1992 3 213.04 200.40 5.94 204.04 4.23 1993 4 237.37 234.64 1.15 238.91 -0.65 1994 5 280.33 274.75 1.99 279.74 0.21 1995 6 341.17 321.70 5.71 327.55 3.99 1996 7 375.45 376.69 -0.33 383.53 -2.15 1997 8 471.62 441.07 6.48 449.08 4.78 1998 9 510.00 516.45 -1.26 525.84 -3.11 1999 10 600.00 604.71 -0.79 615.71 -2.62 2000 11 694.00 708.06 -2.03 720.94 -3.88 2001 12 841.38 829.08 1.46 884.15 -0.33 2002 13 991.20 970.78 2.06 988.43 0.28 2003 14 1181.1 1136.7 3.76 1157.4 2.01

Fig. 1 Fitting curve

From the TABLE I, it is easily seen that the precision of modeling based on the method proposed by this paper is much better than the traditional model .

Mean square error is calculated as follow

21

1

2)0()0( })]()(ˆ[{1 �=

−=n

kkxkx

nMSE ( )(ˆ )0( kx is model

value, )0(x is real value), the mean square error of the traditional method is 9045.41 =MSE , the mean square error of the method proposed in this paper is

8653.32 =MSE . The fitting curve is shown in Figure1.

V. CONCLUSIONS

In this paper, the factors which influence forecasting precision of GM(1,1) model is analyzed and it is considered that there is some error between accumulated sequence and discrete solution, which

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has effect on the precision of general model GM(1,1) ,then amelioration approach of GM (1, 1) forecasting model is proposed. Finally The method of amelioration GM (1, 1) model is applied in modeling of chemistry fiber output and the results show the effectiveness .

REFERENCES [1] Sun Ji-hu, “Quantity Based on Grey System Theory”, Journal of China

University of Mining & Technology, Vol.11 No.2, Dec.2001, [2] R. K. Guo, Charles Ernie Love, “Grey Repairable System Analysis”,

International Journal of Automation and Computing, no. 2, pp. 131-144, February 2006.

[3] S. F. Liu, T. B. Guo, Y. G. Dang, Grey System Theory and Its Application, Beijing: Science Press, 1999.10

[4] X. Luo, Z. K. Feng, Y. Li “Application of gray predicting theory on buildings deformation information system”, Journal of Harbin Institute of Technology, no. 9, pp. 1579-1581, September 2006.

[5] W. S. Huang, P. R. Ji, X. Y. Hu, “Experimental Study on Error Feature of Gray Model GM(1,1)”, JOURNAL OF UNIVERSITY OF HYDRAULIC AND ELECTRIC ENGINEERING/ YICHANG, vol. 22, no. 1, pp. 69-72, January 2000.

[6] G. J. Tan, “The Structure Method and Application of Background Value in Grey System GM(1,1) Model(I) ,” System Engineering Theory and Practice, no.4,pp.98-103, April 2000.

[7] G. J. Tan, “The Structure Method and Application of Background Value in Grey System GM(1,1) Model(II),” System Engineering Theory and Practice,no.5,pp.125-127, May 2000.

[8] G. J. Tan, “The Structure Method and Application of Background Value in Grey System GM(1,1) Model(III),” System Engineering Theory and Practice, no.6,pp. 70-74, June 2000

[9] D. H. Zhang, S. F. Jiang, K. Q. Shi, “Theoretical Defect of Grey Prediction Formula and Its Improvement”, System Engineering Theory and Practice, no.8,pp. 1-3, August 2002.

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