A New Optimization Approach for Grey Model GM(1,1) - Advanced Research on Computer Education, Simulation and Modeling - page 120

n

∑

(0)

xk ()

⎛

(0)

⎞

n n

c x () k cn ( − )

1

∑ ∑

Let C =

z

(1)

() k

= − = −

k = 2

, then

⎜ ⎟

e

a

− 1 a e

a

− a

⎝ ⎠

k = 2 k = 2

2

2 2

⎛

n

⎞

c 2( c n − 1)

n

1 ⎛

n

⎞

⎛ ⎞

∑

(0)

xk ()

∑ ∑

C =

⎜ ⎟

2 (0) (0)

= ( ) n − − x () k + x () k

cn ( − )

1

⎜ ⎟

⎜ ⎟ ⎜ ⎟

−

k = 2

a a 2

e − 1 a e ( − 1) a

⎝ ⎠ ⎝ ⎠

a

k = 2 k = 2

⎜ e − a ⎟

⎜ ⎟

⎝ ⎠

⎛

(0)

⎞

n n

cxk xk
a

()

c

n

1

n

∑ ∑ ∑ ∑

(0)

E =

zkx k

(1)

()

( 0 )

()

= − () =

xk

(0)

() − xk

(0)

()

2

⎜ ⎟ a

e − 1

e

a

− 1 a

⎝ ⎠

k = 2 k = 2 k = 2 k = 2

2

2

⎛

(0)

⎞

n n

cxk
a

()

c 2 c

n

1

n

⎛ ⎞

∑ = ∑ ∑∑

(1) 2

F = zk () − =

( ) n − − x

(0)

() k + x

(0)

() k

2

⎜ ⎟ ⎜ ⎟

−

a

e − 1

e

a

1( a e

a

− ) a

2

⎝ ⎠

⎝ ⎠

k = = 2

2 k k = 2 k = 2

⎛

2

⎞

1

n

⎛

n

⎞

∑ ∑

(0) 2 (0)

CD −− ( ) n E =

⎜ ( n − ) x ( k ) − x ( k ) ⎟

, and

⎜ ⎟

⎜ ⎟

a

⎝ ⎠

⎝ ⎠

k = 2 k = 2

⎛

2

⎞

1

n

⎛

n

⎞

∑ ∑

2
( nFC

(0) 2 (0)

−− = ⎜ ( n − ) x ( ) k − x ( ) k ⎟

. From (1.1), we have

⎜ ⎟

⎜ ⎟

2

a

⎝ ⎠

⎝ ⎠

k = 2 k = 2

⎛

2

⎞

1

n

⎛

n

⎞

∑ ∑

(0) 2 (0)

⎜ ( n − ) x ( k ) − x ( k ) ⎟

⎜ ⎟

⎜ ⎟

a

⎝ ⎠

CD n E
nFC

( )
( )

−−
−−

⎝ ⎠

k = 2 k = 2

a = =

= a .

2

⎛

2

⎞

1

n n

⎛ ⎞

∑ ∑

(0) 2 (0)

⎜ ( n − ) x ( k ) − x ( k ) ⎟

⎜ ⎟

⎜ ⎟

a

2

⎝ ⎠

⎝ ⎠

k = 2 k = 2

The above proof shows that it reaches precision of 100 percent to simulate the normal
exponential series whether the development Coefficient a is large or small. However,
it's not enough that the exponential series are of such characteristics. If the simulation
precision of coefficient c is not high, it will also lead to a larger error. We can also
fix the coefficient of the recuperating values by an appropriate way to realize that
there is not error about the simulation of coefficient, thus there is not model error
wholly for the normal exponential series. For the coefficient c , according to the
smallest principle that the difference quadratic sum between actual values and
recuperating values, we could utilize the least square method to obtain c .
At present, there are only two kinds of methods to verify the superiority of grey
prediction model: one is to test the error size by normal exponential series simulation;
another is to test the error size by the simulation about the data from actual production. In
view of the fact that this paper already strictly proved that this new model has the white
exponential law of coincidence, which guarantees that the error is not from the model
but from the calculation process because of rounding, so we do not need to use the
normal exponential series to illustrate, only need to give several examples to explain it.
Example [4] The data of financial input in science and technology of 1997~2003 years of
Jiangsu Province are given as follow: