One new gray differential equation of gm(1,1) model by menez

Scientific Journal of Information Engineering December 2013, Volume 3, Issue 6, PP.104-110

One New Gray Differential Equation of GM(1,1) Model* Mei Fan, Yong Wei† College of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637002, China Email: 3306866@163.com

Abstract Through analyzing GM(1,1) original model and white differential equation, it deduces one gray differential equation to match white differential equation and prove the model with white exponential law coincidence property and coefficient coincidence property. At last two examples were given to verify the model’s efficiency and high precision and prove that the model could conduct interim and long-term forecasts for the growth index sequence. Keywords: GM(1,1) Model; Gray Differential Equation; Precision; White Exponential Law

1. Introduction Gray system theory [1-3] researches how to analyze, built model, decide and control. It was pioneered by Professor Deng. Among gray GM(1,1) forecast model is the one of the core context. In the use of reality, it has many successful examples to use GM(1,1) model to forecast but there exist this situation that simulation precision is not good. So many learners put up with the suitable range[4] and defect theory[5] though analyzing GM(1,1) modeling mechanism[6], and by optimization background values[7-10] and gray derivative[11-12] and improving parameters[13] and gray differential equation[14] to improve model’s precision. The paper analyzes the relation of GM(1,1) original model and white equation and use the Lagrange mean value theorem to deduced and get a new gray differential equation to match the white equation.

2. White equation derivative’s analysis





Non-negative original sequence X  0  x(0) (1), x(0) (2), , x(0) (n) , 1-AGO sequence X (1)  x(1) (1), x(1) (2), , x(1) (n) , 1 k dx   t  1 x(1) (k )   x(0) (i), k  1, 2, , n . GM(1,1) model’s white equation  ax   t   b , its continuous-time dt i 1 response function: b 1 x   t   ceat  a From (1) it could get original sequence prediction formula:





0 x   t   c 1  ea eat

And because of

1 1 x   k   x   k  1 0  x k   k   k  1

⑴

⑵ ⑶

1 and x   t  is a continuous function. In (k-1, k) it can be derivative. So by using Lagrange mean value theorem we *

This project is supported by Sichuan Science and Technology Hall(2008JY0112); Sichuan higher quality of personnel training and teaching reform projects (P09264); Science and Technology Fund of Sichuan Personnel Department(2010)32 - 104 http://www.sjie.org/

can get that:    k  1, k  1 1 x   k   x   k  1 0 x   k    [ x(1) (t )]' |t  , k   k  1

There is no harm in hypothesis that   1    k  1   k ,    0,1 However, (4) could be written that 1 1 x   k   x   k  1  1  ' 0 x   k    x t    k   k  1

t 1  k 1 k

   k  1, k 

' 1   x   t    

⑷

t k 1

With k continuing to get: 1 dx   t  1    0 x   t   dt

⑸

Assuming t1  t  1  , then t  t1  1   dx 0 So Eq.(5) could be written that x   t1  1    

 

1 t 1 dt

dx   t  0 x   t  1     dt 1

So 0 Then x   t  and

⑹

1 x   t  ‘s expression is (2) and (1), with them into Eq. (6), it can get: 1 a   ln a 1  e a

While combining equation

1 dx   t 

⑺

1  ax   t   b with Eq. (6) then it can get:

0 1 x   t  1     ax   t   b

⑻

0 1 (the different from the traditional method is not x   k   ax   k   b ) 0 From the other hand, combining (2) with (7) we can get x   t  1    

a 1 e

a

0 x   t  1 and get it into Eq.(8)

then can get the gray differential equation:





0 1 x   t  1  1  ea x   t  

Making it discrete and we get:





0 1 x   k  1  1  e a x   k  



b 1  e a





b 1  e a



3. Model’s construction 1)









Assuming original sequence X  0  x(0) (1), x(0) (2), , x(0) (n) , 1-ago sequence X (1)  x(1) (1), x(1) (2), , x(1) (n) , k

x(1) (k )   x(0) (i), k  1, 2,

, n .then the new GM(1,1) model:

i 1

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



0 1 x   k  1  1  e a x   k  

1 dx   t 



b 1  e a



1  ax   t   b

b The discrete-time responsive function: x1  k   ceak  , k  1, 2, , n. a 0 The regressive restore sequence: x   k   c 1  ea eak , k  1, 2, , n. Among c is the constant to be determined.





2) Parameter Identification





0 1 x   k  1  1  ea x   k  

Demand 1  1  e

a

, 2 



b 1  e a a



b 1  e a a

, n 1

⑼

 , then Eq.(9) can be written:

0 1 x   k  1  1x   k   2 , k  1, 2,

 x 0   2     x1 1   0   Y   x  3  , Z    x1  2       0   1 x n       x  n  1

Demand

 , k  1, 2,



It can be obtained 1, 2 by using least squares method a   Z ' Z 

, n 1

1   1 , a   1  .   2   1 1

Z 'Y .

n n  1 n  0 1 0 1  x  k   x   k  1   x   k  x   k  1  n  1 k 2 k 2 k 2   2  1 2 n  1  n 1 1  So   x  k  1    x  k  1  n  1  k 2  k 2  n   1  n  0 1  2    x  k   a  x  k  1  n  1  k 2 k 2  







a   ln 1  1  , b 

a2

1 . For parameter c, it can be get by using least original sequence’s mean square error. 2

  0   n 0 0 That is F  c     x  k   x   k     ceak 1  ea  x   k    k 1  k 1  n

Demand F '  c   0 .then

c

e

So x

 k   1  ea 

2ai

n 0  x   i  eai

i 1 n

e

i 1

2ai



0 in1 x   i  eai i 1

 0 



1  e  a



, so F(c) get minimum.

1  e  a

n 0  x   i  eai

e ak  i 1

e

2ai

e ak , k  1, 2,

i 1

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⑽

3) New model has white exponential law coincidence property and coefficient coincidence property. Theorem: If original sequence strictly obey the exponential law, that is x 0  k   c1ea1  k 1 , k  1, 2, 



, n , .then it

has a  a1 , c1  c1 with calculating from the model. 0 0 a k 1 Proof: the original sequence x  has white exponential law x   k   c1e 1   , k  1, 2, 1 to generate sequence x   k  

1 

, n .Its time accumulate

1  ea k  , then from (10) can get: 1

1  ea1

a  k  1 1  c1e 1  n 1

 a  k  1  a  k  1 c a  k  1  c1  1 1  e 1  1  e 1    ce 1 a  a      1 e 1  1 e 1  2 2 2      a  k  1  a  k  1   c c 1 1 1  1  e 1    1  e 1   a   a    n 1     1 e 1    1 e 1 

a  k  1  a  k  1  a  k  1  a  k  1  c12 c2  1 1  e 1  e 1  1  e 1 e 1 a     a       1 1 e 1  e 1   n  1      2 2    a  k  1 2a  k  1  a  k  1   c12 c 1   1  e 1   e 1  1  2e 1 2 2      a  a         1 1 1  e  1  e   n  1         e 



2 a  k  1 2a  k  1 1  a1  k  1  e   e 1  e 1  n 1  

a  k  1 2a  k  1 a  k  1 n 1 2 1 n 1 2 1      e 1 e 1 e 1 a a a a a a  1 e 1 1 e 1 1 e 1 1 e 1 1 e 1  n  1 1  e 1 

e 

a  k  1

2a1  k  1

1  a1  k  1  e    n 1  

 a  k  1  e 1        

2   e2a1  k  1  1  ea1  k  1     a    n 1    1 1  e  1

1 e



Then



a   ln ea1  a1 , c 1 

c1 e 1 a1

 0 

(ea1  1)  c1 , x

k  

a k 1 1  ea  ea k  c1e   ,  1

c1 e

so the model has white exponential law coincidence property and coefficient coincidence property.

4. Case analysis and conclusion To facilitate the description, the model from literature [1] writes to be original GM(1,1) model, the model from - 107 http://www.sjie.org/

①

literature [2] writes to be GM(1,1) model and this article’s model writes to be new GM(1,1) model. Eg1:1) original sequence: x 0  k  1  eak , k  0,1, 2,3, 4,5 .Separately get –a=0.1, 0.5, 1.0, 2.0, 2.1, original sequence statistics is in Table 1. TABLE 1 THE ORIGINAL SERIES -a

xi(0)(1)

xi(0)(2)

xi(0)(3)

xi(0)(4)

xi(0)(5)

xi(0)(6)

0.1

1.1052

1.2214

1.3499

1.4918

1.6487

0.5

1.6487

2.7183

4.4817

7.389

12.1825

1.0

2.7183

7.389

20.0855

54.5982

148.4132

2.0

7.3891

54.5982

403.4287

2980.958

22026.4657

2.1

8.1662

66.6863

544.5719

4447.067

36315.5027

2) We can get these time response functions by using this paper’s method to original series.

1 x1  k   9.5049e0.1k  9.9974 1 x2  k   1.5415e0.5k  1.5415 1 x3  k   0.582ek  0.581

1 x4  k   0.156496e2.00002249k  0.175761 1 x5  k   0.139545e2.1k  0.139545 3) comparison of three kinds of GM(1,1) models’ simulation precision TABLE 2 COMPARISON OF THE SIMULATION PRECISION -a

①

Original GM(1,1) model

GM(1,1) model

New GM(1,1) model

1 6  i 5 i 2

 1 6 1 6   i   0  100    i  5 i 2 5 i 2  x  i   

 1 6   i   0  100   5 i 2  x  i   

1 6  i 5 i 2

 1 6   i   0  100   5 i 2  x  i   

0.1

0.0035

0.1054

0.0016

0.1072

6.0e-005

4.3634e-003

0.5

0.3068

4.5206

0.0101

0.1833

4.0e-005

4.3484e-004

1.0

14.8072

23.5440

0.0104

0.0492

0.0011

3.0661e-003

2.0

-----

0.02786

0.0041

0.018554

4.619411e-003

2.1

-----

0.04332

0.00438

0.0262709

2.73047e-004

From table 2, regardless of high growth or low growth index sequence or mean absolute error or mean relative error, the new GM(1,1) model has high simulation precision. Although a=2.0,the new GM(1,1) model’s mean absolute ① error is a little lower than GM(1,1) model, but it’s higher than the original series. Because the model has white exponential law coincidence property and coefficient coincidence property, the simulated error is just calculated error. 4) comparison of three kinds of GM(1,1) model’s forecasting precision TABLE 3 COMPARISON OF THE FORECASTING PRECISION -a 1

Original GM(1,1) model

0.1

0.5

1.0

0.1289

7.3970

39.4369

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①

Step

GM(1,1) model

0.2964

0.03535

0.01584

New GM(1,1) model

0.032929

0.00049787

4.0024e-003

Original GM(1,1) model

0.1367

8.3332

43.8559

GM(1,1) model

0.3675

0.01782

0.009529

New GM(1,1) model

0.039726

0.00030197

4.0024e-003

Original GM(1,1) model

0.1601

11.0855

55.2708

GM(1,1) model

0.5702

0.1766

0.009363

New GM(1,1) model

0.036788

0.0003369

4.0024e-003

Original GM(1,1) model

0.1991

15.4903

69.3755

GM(1,1) model

0.8949

0.4407

0.04086

New GM(1,1) model

3.5701e-003

3.8388e-004

4.002e-003

error

2 Step

①

error

5 Step

①

error

10 Step error

①

From the upper statistics, when the developed coefficient is very small, new model’s forecasting precision is very high and 10 step’s forecasting error is also very small, so the new model apply to middle-term and long-term forecasts. Eg2[7]:The paper takes Analysis Ltd’s 2004-2008 European cell phone game market scale’s statistics example to proof the new model’s practicality.

[15]

as an

From these statistics, built follow model: 

a  0.4447,



b  4.0574,

1 x   k   9.0472e0.4447k  9.1234

TABLE 4 COMPARISON OF PRACTICAL PROBLEM’S SIMULATION PRECISION Year Original sequence

2004

2005

2006

2007

2008

7.9

12.3

19.3

Original

Simulated value

7.7521

12.0072

18.5978

28.8060

GM(1,1)model

error

0.1479

0.2928

0.7022

1.1940

Relative error(%)

1.8721

2.3808

3.6384

3.9801

Simulated value

7.9498

12.3879

19.3038

30.0806

error

0.0498

0.0879

0.0038

0.0806

Relative error(%)

0.6822

0.7146

0.0197

0.2687

New

Simulated value

7.9084

12.332

19.2387

30.0135

GM(1,1)model

error

0.0048

0.032

0.0613

0.0135

Relative error(%)

0.0006

0.0026

0.0032

0.0005

①

GM(1,1) model

From table 4,new model’s simulation precision is very higher than original GM(1,1) model and GM(1,1) practical problem and show fully good simulation.

①

model to

5. Conclusion Though analyzing the gray GM (1,1) modeling mechanism and combined white equation and Lagrange mean value theorem, it gets a new gray differential equation and proves that the new model has the white exponential rate and - 109 http://www.sjie.org/

white coefficient coincidence. Then using the raw data that is homogeneous index series to simulate, when a>2.0, the new model can still predict. So it extends the classic gray GM (1,1) model of scope, and has some value to study.

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AUTHORS Mei Fan (1989- ) is master graduate

Yong Wei (1957- ) received his DEA

student of Department of Mathematics and

degree in France Orleans University in

Information,

Normal

1992, and his Eng. D. Degree in

University. Her research areas are focused

China

West

Southwest Petroleum University, China, in

on the gray systems theory and its

1997, Standing Director of Mathematics

application.

Association of Sichuan Province, Standing

Email: dielianhuasushi5@163.com

Director of Gray System Major of Chinese Society of Optimization, Overall Planning and Economic Mathematics,

master tutor, Professor of Department of

Mathematics and Information, China West Normal University, major study is Gray System Analysis. Email: 3306866@163.com

- 110 http://www.sjie.org/