GM(1,1) Model Based on Improved Euler Formula And Its Application

Yonggang Chen Wenzhan Dai Department of Automatic control Department of Automatic control

Zhejiang Sci-Tech University Zhejiang Sci-Tech University HangZhou, Zhejiang Province, China HangZhou, Zhejiang Province, China

yonggang44@sina.com Correspondence author :dwzhan@zist.edu.cn

Abstract - In this paper, it is analyzed that the background value in grey mode GM(1,1) is an important factor to model precision, and the reconstruction method of background value in grey GM(1,1) model based on Euler formula is pointed out. Simulation examples show the effectiveness of the proposed approach.

Index Terms - GM(1,1) Model. Forecast. Euler formula

I. INTRODUCTION

GM(1,1) proposed by Professor Deng has been used in industry , agriculture, military affaires, science and technology and so on widely[1-5]. There are many successful examples based on GM(1,1) forecasting model, however it also has shortcoming in low precision. It was proved that the precision of grey model GM(1,1) greatly depends on the background value of model.

In this paper , it is pointed out that GM (1,1) model is a differential equation, we can use improved Euler formula differential equations to estimate GM (1,1) model’s parameter to improve the model precision .The reconstruction method of background value in grey GM(1,1) model based on Euler formula is presented. Simulation examples show the effectiveness of the proposed approach.

II. THE TRADITIONAL MODEL GM(1,1) MODELING THEORY

The GM(1,1) is one of the most frequently used grey forecasting model. This model is a time series forecasting model. Its modeling mechanism is as follow:

Supposed original non-minus sequence is: { })(,),2(),1( )0()0()0()0( nxxxX �= �1�

where ( ) 0)(0 >ix ,i=1,2, �,n� The AGO (accumulated generation operation) of data sequence (1) is defined as:

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

where nkixkxk

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

1

)0()1( ==�=

The GM(1,1) model can be constructed by establishing first order differential equation for )()1( tx as :

utaxdt

tdx=+ )(

)( )1()1(

(3)

Solve parameters of vector using the least square method, we can get:

[ ] nTTT YBBBua 1)(ˆˆˆ −==Φ (4)

where

[ ][ ]

[ ] �������

�

�

�������

�

�

+−−

+−

+−

=

1

11

)()1(21

)3()2(21

)2()1(21

)1()1(

)1()1(

)1()1(

��kxkx

xx

xx

B

[ ]TnxxxY )(),...,3(),2( )0()0()0(= So:

( ) ( ) ( ) ( ) ( ) ( )[ ]kxkxkz 111 1211 ++=+

is background value. The discrete solution of Equation (6) is:

aue

auxkx ak

ˆˆ

]ˆˆ

)1([)1(ˆ )1()1( +−=+ − (5)

Where nk ,...,2,1= Applying the inverse AGO, we then get

kaa eauxekx ˆ)1(ˆ)0( ]ˆˆ

)1()[1()1(ˆ −−−=+ (6)

From formula (3) ,we can see that the accuracy of forecasts greatly depends on constants a and u, So we can use improved Euler formula to estimate parameter’s a and u in order to improve GM (1,1) l accuracy.

III. IMPROVED EULER FORMULA[9]

3.1 EULER FORMULA

GM (1,1) model is a differential equation, we can use improved Euler formula to estimate GM (1,1) model’s parameter a and u, the general form of differential equations as follows:

( )yxfy ,' = (7) For equation (7), Euler formula for calculation format:

( )kkkk yxhfyy ,1 +=+ ( )1,,2,1,0 −= nk � (8)

Where ( )hxyy kk +=+1 �

( )kk xyy =

kk xxh −= +1 .

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Proceedings of the 2007 IEEEInternational Conference on Integration Technology

March 20 - 24, 2007, Shenzhen, China

The error of Euler formula is:

( ) ( ) ( )kkk xyhyhyxy ''2

''2

22

11 ≈=− ++ ε (9)

For equation (7), retreat Euler formula is:

( )111 , +++ += kkkk yxhfyy

( )1,,2,1,0 −= nk � The error of retreat Euler formula is:

( ) ( )kkk xyhyxy ''2

2

11 −≈− ++ (10)

3.2 TRAPEZOID FORMULA

From formula(9) and(10),we can see if the arithmetic average of these two methods can eliminate major

errorskyh "

2

2

± ,the accuracy of model can be improved. This

method is commonly called the average trapezoidal method. Its formula for calculating is

( ) ( )[ ]111 ,,2 +++ ++= kkkkkk yxfyxfhyy (11)

Trapezoidal formula is implicit .We used Euler formula for initial iteration, the trapezoidal method of iterative formula is as follow :

( ) ( )( ) ( )[ ]

��

++=

+=

+++

+

mkkkkk

mk

kkkk

yxfyxfhyy

yxhfyy

111

01

,,2

;,

( )�,2,1,0=k (12) 3.3 IMPROVED EULER FORMULA

Although the trapezoidal method can improve the accuracy, its algorithm is complicated such as using iterative formula for the actual calculations .The amount of added, therefore, we need to think of new ways, specifically, we first use Euler

equation to account a value 1

_

+ky , then formula (12) is used

to emendate 1

_

+ky , the establishment of this forecast- correction system is commonly known as the Improved Euler formula

( )kkkk yxhfyy ,1

_

+=+

1+ky = ky +2h

[ ( )kk yxf , + ��

���

+

−

+ 11, kk yxf ]

This formula can be expressed as:

( ) ( )( )[ ]kkkkkkkk yxhfyhxfyxfhyy ,,,21 ++++=+

(13)

3.4 THE GM (1,1) BASED ON IMPROVED EULER FORMULA

Supposed original non-minus sequence is: { })(,),2(),1( )0()0()0()0( nxxxX �=

where ( ) 0)(0 >ix ,i=1,2, �,n. The AGO (accumulated generation operation) of data

sequence (1) is defined as:

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

where nkixkxk

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

1

)0()1( ==�=

The GM(1,1) model can be constructed by establishing a

first order differential equation for )()1( tx as follow :

utaxdt

tdx=+ )(

)( )1()1(

Suppose: ( ) ( )( ) ( ) ( ) utaxtxtf +−= 11, (14)

the formula (14) are incorporated into the formula (13)�and suppose h=1, we can get :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[( ) ]uauaa

txutaxtxtx

+−−∗

+++−+=+

2

1111 1211

= ( ) ( ) ( ) ( )2

122

1 12

1 auutxaaatx −++−+ ��

��� −

Then ( ) ( )( )aatx +−+ 21 21 = ( ) ( )( ) auuatx −+− 221 ( ) ( )( )aatx +−+ 20 21 = ( ) ( )( ) ( )auaatx −+− 2221 The above equation simplifies:

( ) ( ) ( ) ( )a

ua

atxtx+

++

−=+11

1 10

Namely:

( ) ( ) ( ) ( )a

ukxaakx

++

+−=+

111 10 (15)

Suppose

A=aa

+−

1, U=

au+1

(16)

So the formula (15) can turn into formula (17) ( ) ( ) ( ) ( ) UkAxkx +=+ 10 1 ( )1,,2,1 −= nk � (17)

The formula (16) can be written as follow in Matrix: ( ) ( )( ) ( )

( ) ( )�����

�

�

�����

�

�

nx

xx

0

0

0

32

�=

( ) ( )( ) ( )

( ) ( ) �����

�

�

�����

�

�

− 11

1211

1

1

1

nx

xx

��• �

�

���

�UA

Where =1B

( ) ( )( ) ( )

( ) ( ) �����

�

�

�����

�

�

− 11

1211

1

1

1

nx

xx

��,

( ) ( ) ( ) ( ) ( ) ( )( )Tn nxxxY 000 ,,3,2 �=

Using the least square method, we can get A and U:

224

[ ]TUA = ( ) nTT YBBB 1

111

−

So we can further use formula (16) to get a and u , finally we use a and u to build GM (1,1) Model.

IV. APPLIED EXAMPLES

With the global economic recovery, the supply of oil have an important influence on a country's industry. The method advanced by this paper can be applied to build output model of China crude oil (1991-2004) as follows

( ) ( ) ( )aue

auxkx ka

ˆˆ

ˆˆ

)1(1ˆ ˆ11 +���

��� −=+ −

( ) )1(ˆ 0 +kx = kaa ea

uxe^^

^

^

)0( )1(1 −

��

�

�

��

�

�−

��

��� −

( ) )1(ˆ 0x =14.099 Where k=1,2,� ,n.

The prediction of model is shown as table1.

TABLE I MODING AND FORECASTING RESULTS (units:1×108T)

Approach advanced by this paper

Approach advanced by paper[6]

Year Serial

number

0riginal data

)()0( kx

Model Values

Relative error (%)

Model Values

Relative error (%)

1991 1 ������� 14.0990 0 14.0990 0 1992 2 14.2100 14.2888 -0.55 14.4208 1.48 1993 3 14.5240 14.5144 0.07 14.6507 0.87 1994 4 14.6080 14.7435 -0.93 14.8843 1.89 1995 5 15.0050 14.9762 0.19 15.1216 0.78 1996 6 15.7340 15.2126 3.31 14.1219 -3.89 1997 7 16.0740 15.4527 3.87 15.3973 -4.21 1998 8 16.1000 15.6967 2.51 15.6444 -2.83 1999 9 16.0000 15.9444 0.35 16.1092 0.68 2000 10 16.3000 16.1961 0.64 16.3660 0.40 2001 11 16.3959 16.4518 -0.34 16.6269 1.41 2002 12 16.7000 16.7115 -0.07 16.8920 1.15 2003 13 16.9600 16.9753 -0.09 17.1613 1.19 2004 14 17.5000 17.2433 1.47 17.2270 -1.56

From table 1, the precision of model is good�the difference between predictive result and the true value is less than

1.0279% and square error 01 / SSc = =0.1583 (among it, S1 is a relative convergent error of residual error, S0 is a relative convergent error of original sequence ),The error probability is }6745.0|)({| 0

)0()0( SeiePp <−= .After checking statistical charts, we could get that final checkout error of this model is smaller than 0.3640. so we can use them to predict the crude oil, in recent years.

�. CONCLUTION

In this paper, it is analyzed that the background value in grey mode GM(1,1) is an important factor to model precision, and the reconstruction method of background value in grey GM(1,1) model based on Gauss-Legendre formula of numerical value integral is pointed out. Simulation examples show the effectiveness of the proposed approach.

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