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 %)
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.
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