The Optimized White Differential Equation Based on the Original Grey Differential Equation

Автор: ZHOU Rui, LI Ren-guo, CHEN Yao

Журнал: International Journal of Education and Management Engineering(IJEME) @ijeme

Статья в выпуске: 6 vol.1, 2011 года.

Бесплатный доступ

This paper starting from the original grey differential equations, through finding the relationship between the raw data and the derivative of its , constructed a new white differential equation which equal to the original grey differential equation, at the same time, getting the new GM(1,1)model which closer to the changes of data. Through the modeling and prediction of the standard index series, this model not only adapts to low growth index series, but also adapts to high-growth index series, and the simulation accuracy and prediction accuracy are high.

GM (1, 1), Original Grey Differential Equation, Equivalent, White Differential Equation, Optimization

Короткий адрес: https://sciup.org/15013633

IDR: 15013633

Текст научной статьи The Optimized White Differential Equation Based on the Original Grey Differential Equation

Since Mr Deng Ju-long found the grey system theory in the eighties of last century, after almost three decades of development, the theory has been widely used in various areas of national product [1]. As the important part of the grey system theory, the GM(1,1) model has become the research focus, many scholars have further investigated in improving the model’s precision and widening the model’s applicable scope[3-10]. But when the white differential equation founded, Mr Deng Ju-long made that:" GM(1,1) white model itself and all results derived out from the white model just establish only when they are not contradictory with the defining type, otherwise invalid”[2]. Based on this idea, this paper through finding the relationship of the raw

data

x (0)( k )

and the derivative of its 1 - AGO

constructed a new kind white differential equation which

equaled with the original grey differential equation, at the same time, getting the new GM(1,1)model which closer to the changes of data.

Corresponding author:

2. The Optimization of GM(1,1) Model

Let X     { x  (^ x (2),“ * x  ( n )} is the original series, and the 1 AGO series of X (0) is

k m m m       m                 x (1)( k) = V x (0)( i)

X(1) = {x (1)(1), x (1)(2),---x (1)( n )}         .

i       ,    \ /,       \ n , among them

.        x(1)(t) = CeA + Dz       „     .  „                         .

Theorem 1 When                    (among them A C D are all the constants), the original grey

( eA — 1) dx (1)      (1)    ,

(0)      (i)                                                     ------a+ ax') = b differential equation x + ax =b and the white differential equation Ae      dt equivalent.

x (1)( t ) = CeA + D ,       .    .  „  _                                    .

Proof: Because                      (among them A、C、D are all the constants), according with the relevant definitions we can get ,

x (0) ( t ) = x °) ( t ) x (1) ( t 1) = CeA ( t 1) ( eA 1)

( x <1> ( t ))' = d— = ACe A = Ce A ( t 1) Ae A

Comparing (1) and (2) can get

x <°»( k ) = (eA-!) dx^

Ae    dt

t = k

And wether get which values, this equality always set up.

Put (3) into x ) + ax = b b , getting :

( eA — 1) dx (1)      (I)   ,

-----^--+ ax (V = b

AeA    dt

z .  .    x (1)( t ) = CeAt + D                                                                               .

And when                    , (4) is the equally white differential equation with grey equation differential.

End

So based on the original grey differential equation x + ax = b b , getting the optimized GM (1,1) model as follow:

x ( 0 ) = f x ( 0 ) ( 1 ) , x ( 0 ) ( 2 ) , ■■■ , x ( 0 ) ( n ) 1

Theorem 2 Let

\-AGO • xx(0)  я x(1)(t) = BeA + C series of x ,and

(1)- be nonnegative pre-smooth series, x is

a = ( a, b) T,             x be the parameter of the equation, and

(among them A C D are all the constants). If

' x ( 0 )( 2 ) !

f- A( 2)

1'

Y =

x ( 0 )( 3 )

B =

- x1" (3)

1

V x ( 0 )( n )>

V- x 111 ( n )

1v

, then the

А parameters of the                                                  a =

( a , b ) T  ( B T B ) " * B T Y

=

x (1)( t ) = [ x (0)(1) - b ] e

The new continuous solution of the white equation is                     a

- a ( t - !) A eA

eA - 1

b

+ - a

;

x (1>( k ) = [ x (0)(1) - b ] e

The new discrete solution of the white equation is                     a

- a ( k - 1) A eA

eA - 1

b

+ - a

,

k = 2,3, "•;

А

л

•   x ( 0 )( k + 1 ) = x ( 1 )( k + 1 )

The restore value is

А

- x "’( k ), k = 23.-;

  • 3.    The Process of Modeling Optimized GM(1,1) Model

x (1)( t ) = CeA + D,       .,     .

The precondition of establishing optimized GM (1,1) model is                    (among them A, C ,D x (0)( k ) = PeA (k-1) are undetermined constants )that means the original data must be expressed by                     (among

A      x (0)( k )

x <0)( k -1).

them P is a constant ). Under this precondition,        x (     ) is always a constant. But in the practical

x (0)( i )

x ^ i - Dis not a

A ei application of grey system theory, the original data are the similar index series and constant. So the new model is not practical. To this problem, the model must be optimized again.

Tf th           1     •     •   X (0) = { x (0)(1), x (0)(2),"- x (0)( n )}   th 1-ЛСО • f X(0)-

If the original series is                     ,         ,               , then              series of X is

X (1) = { x (1) (1), x (1) (2), • • • x (1) ( n )}          ,.

,       ,            ,among them

k

X ( 1)( k ) = X x (0 )( i )

e A =   (x ( 0)( i )                      A = ln

Let         x ( i 1) , i=2 , 3_n , then

A e and A which in theorem 2.

Now getting the new optimized GM (1,1) as follow:

i = 1

x (0)( i )

x (0)( i - 1)

.

  i=2 3…n. And

eAi,A i replace the

x ( 0 ) = { x ( 0 ) ( 1 ) , x ( 0 ) ( 2 ) , ••• , x ( 0 ) ( n ) }                                                         ( 1 )

Theorem 3 Let                                  be nonnegative pre-smooth series, x be the

' x ( 0 )( 2 ) x ( 0 )( 3 )

Y =

1 AGO series of x ^. if a = ( a ’^)  the parameter of the equation, and

( " ' ( n )

B =

- x ( 1 ) ( 2 )    1

- x ( 1 ) (3)   1

- x ( 1 ) ( n )   1

A a=

( a , b ) T  ( B T B )' * BTY

=                ,then

, then the parameters of the least square estimate in the grey differential equation.

  • (1)    The       new       continuous       solution       of      the       white       equationis

    - a•(t-1). x (0)( t)

x (1)( t) = [ x (0)(1) - b ]e+ aa

- a.(t-1). x (0)( k)

x (1)( k) = [ x (0)(1) - b ]e     x,0,(k)-x,0,(k-1)

  • (2)    The new discrete solution of the white equation is               aa

k = 2 , 3 ,   ••• n ;

(3) According to “The principle of new

- a ( t - 1) . x (0) ( k ) . ln(

information priority” getting the prediction

x (1)( k +1) = [ x (0)(1) - b ]e equation:                        a

x (0) ( n ) - x (0) ( n - 1)

x (0) ( n )

x ( 0)( n - 1))

b

+ —,  7          , 1

a k = n , n + 1---

.

A

x ( 0

(4) The restore value is

A

A

- x ( 1 )( k ) k =2  3

      ,,    

  • 4.    Comparison of the Precision of Data Simulation

    x

    Take


= e~ „a,b,„ ,b   ,x ™= { x m(')- x ! ”’<2)-''А( n )} 4 = 1,2,3-6

.

as an example, and then we have                                ,     , ,

And use the original GM (1,1) model as M1, model of reference [2] as M2and model of this paper as M3 to predict.

Table 1. The Original Series

- a

i

xi (0)(1)

xi (0)(2)

xi (0) (3)

xi (0)(4)

xi (0) (5)

xi (0)(6)

0.1

1

1.0

1.1052

1.2214

1.3499

1.4918

1.6487

0.3

2

1.0

1.3499

1.8221

2.4596

3.3201

4.4817

0.5

3

1.0

1.6487

2.7183

4.4817

7.389

12.1825

0.8

4

1.0

2.2255

4.953

11.0232

24.5325

54.5982

1.0

5

1.0

2.7183

7.389

20.0855

54.5982

148.4132

1.5

6

1.0

4.4817

20.0855

90.0171

403.4288

1808.0424

1.8

7

1.0

6.0496

36.5982

221.1064

1339.4308

8103.0839

2.0

8

1.0

7.3891

54.5982

403.4287

2980.9579

22026.4657

3.0

9

1.0

20.08

403.42

8103.08

162754.79

3269017.37

Table 2. Comparison of the Simulation Precision

-a

1

2

3

4

5

6

Average error %

M1

0

0.0876625

0.0940814

0.1066281

0.111198

0.120834

0.104008

0.1

M2

0

0.4729089

0.4072969

0.3355724

0.271905

0.203195

0.338175

M3

0

0.0016535

0.0006224

0.0000653

0.0012815

0.000827

0.000741

M1

0

0.8623506

1.0792191

1.300537

1.5200909

1.740273

1.300494

0.3

M2

0

1.3160696

1.2340048

1.1469605

1.0612932

0.974550

1.146575

M3

0

0.0012404

0.0017363

0.0008671

0.0000458

0.000436

0.000721

M1

0

2.5688891

3.5558678

4.5305732

5.494881

6.451143

4.520271

0.5

M2

0

2.0655219

2.0094938

1.9559263

1.9029764

1.848208

1.956425

M3

0

0.0001108

0.0005422

0.001017

0.0007982

0.000928

0.000566

M1

0

7.1362838

10.787576

14.295032

17.663597

20.90019

14.15653

0.8

M2

0

3.0625163

3.0445781

3.0269682

3.0106039

2.993673

3.027668

M3

0

0.0009656

0.002194

0.001413

0.0001291

0.001006

0.000951

M1

0

11.543467

17.996636

23.980549

29.527595

34.66966

23.54358

1.0

M2

0

3.645538

3.6405011

3.6333885

3.6265874

3.620138

3.633230

M3

0

0.0000603

0.0004651

0.001391

0.0008119

0.000229

0.000493

M1

0

26.417176

41.518302

53.520549

63.059523

70.64080

51.03127

1.5

M2

0

4.7467368

4.7467189

4.7460967

4.7455926

4.745129

4.746054

M3

0

0.0006301

0.0003254

0.001015

0.0007916

0.000700

0.000577

M1

0

36.832816

56.255155

69.705425

79.020111

85.47080

65.45686

1.8

M2

0

5.1338165

5.1330021

5.1328225

5.1327235

5.132638

5.133000

M3

0

0.004412

0.005303

0.004418

0.004257

0.004223

0.003769

M1

0

43.862241

65.151782

78.367616

86.571496

91.66413

73.12345

2.0

M2

0

5.2674478

5.2679511

5.268046

5.2679988

5.267970

5.267882

M3

0

0.003801

0.004514

0.003903

0.003615

0.00366

0.003249

M1

0

73.21316

91.84847

97.51940

99.24512

99.770

76.933

3.0

M2

0

5.08825

5.0881

5.0881

5.0881

5.0881

5.08813

M3

0

0.0028098

0.0031082

0.002784

0.002790

0.00278

0.002380

From Table 2 and Table 3, when the original series is a low growth series, the simulation accuracy and prediction accuracy of all models are high, but as the development coefficient become larger, the he simulation

Table 3. Comparison of the Forecasting Precision (Relative error %)

a 0.1 0.3 0.5 0.8 1.0 M1 1step error 0.1289 1.9604 7.3970 24.0093 39.4369 M2 1step error 0.1333 0.8890 1.7940 2.9772 3.6135 M3 1step error 0.00135 0.00071 0.00121 0.00126 0.00025 M1 2step error 0.1367 2.1791 8.3332 26.9963 43.8559 M2 2step error 0.0650 0.8032 1.7400 2.9606 3.6070 M3 2step error 0.00225 0.000746 0.00144 0.00143 0.00026 M1 5 step error 0.1601 2.8322 11.0855 35.2711 55.2708 M2 5 step error 0.1394 0.5462 1.5784 2.9105 3.5874 M3 5 step error 0.00493 0.000846 0.00214 0.00194 0.00027 M1 10 step error 0.1991 3.9110 15.4903 47.0312 69.3755 M2 10 step error 0.2808 0.1193 1.3096 2.8272 3.5548 M3 10 step error 0.00943 0.001013 0.00331 0.00278 0.00028 -a 1.5 1.8 2.0 3.0 M1 1step error 76.6670 89.9372 94.8254 99.9300 M2 1step error 4.7213 5.1325 5.2679 5.0880 M3 1step error 0.0007017 0.003937 0.003664 0.0027891 M1 2step error 81.4556 93.0312 96.7878 99.9787 M2 2step error 4.7171 5.1324 5.2679 5.0880 M3 2step error 0.0007033 03003938 0.003661 0.0027891 M1 5 step error 90.6903 97.6854 99.2316 99.9994 M2 5 step error 4.7045 5.1320 5.2678 5.0880 M3 5 step error 0.0007082 0.003941 0.003653 0.0027891 M1 10 step error 97.0478 99.6313 99.9291 99.9999 M2 10 step error 4.6836 5.1315 5.2677 5.0880 M3 10 step error 0.0007164 0.003946 0.003640 0.0027892 accuracy and prediction accuracy of original model become lower. Especially, when

Ia\ > 2

, the error is 99%!

Although the simulation accuracy and prediction accuracy of M2 is higher than M1, the overall effect is not

very ideal. But for the model of this paper, even if are all higher than 99.99%, the effect is good.

|a| > 2

, the simulation accuracy and prediction accuracy

  • 5.    Conclusions

In this paper, starting from the original grey differential equations, through finding the relationship of the

x (0) ( k ) raw data

and the derivative of its 1 AGO, constructed a new kind white differential equation which equal with the original grey differential equation, at the same time, getting the new GM(1,1)model which closer to the changes of data. Through the modeling and prediction of the standard index series, this model not only adapts to low growth index series, but also adapts to high-growth index series, and the simulation accuracy and prediction accuracy are almost 100%. At the same time, the models remains simple calculation steps of the original GM (1, 1) model, and broaden the scope of application of the model.

Список литературы The Optimized White Differential Equation Based on the Original Grey Differential Equation

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