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And the decay speed of buffer sequence XD becomes
steady after X affected by weakening buffer operator.
We can get x ( k ) − x ( k − 1) ≤ x ( k ) d − x ( k − 1)
d , then
x(
k)
x(
k)
d , namely x( k − 1) x(
k −1)
d .
≤
x(
k −1)
x(
k −1)
d
x(
k)
≥
x(
k)
d
So we can obtain σ ( k)
≥ σ ( k)
d ≥ 1 because the
buffer sequence and the original sequence maintain the
same monotony.
End.
For the original sequence which possesses
characteristics: the front part grows (weakens)
excessively quickly and the latter part grows (weakens)
excessively slowly, it is not suitable for establishing
model by the traditional class ratio modeling
method [ 2]
directly. In reference [2], any average value of
class ratio in original sequence is always between the
minimum ratio and the maximum ratio, but the latter part
data in original sequence become more steady and its
growth (decay ) speed become more and more slow, then
class ratio in the latter part data is more close to
1.Namely, it can effectively improve the prediction
precision of grey model only if we can make the class
ratio ~ σ in predicting model be more close to 1 than any
class ratio in original sequence. However, the traditional
[2]
class ratio modeling method can not meet this
requirement.
Let X = { x(
k ) x(
k ) > 0, orx ( k ) < 0, k = 1,2,3.... n}
be the
primitive behavior data sequence which possesses
characteristics: the front part grows (weakens)
excessively quickly and the latter part grows (weakens)
excessively slowly. The class ratio of each point
is x(
k − 1)
{ σ ( k ) σ ( k ) = , k = 2,3,...., n
x(
k )
} , let
~
t
t
σ ( n)
− σ (1)
σ = σ ( n)
+ (1 − σ ( n))
, ( t = 1,2,...)
t
ω
ω = max{ σ ( k)},
k = 2,3,.... n
~
~
(3)
a = lnσ
~
=
~
a n
(4)
c
x(
n)
e
And c ~ is the parameter which makes x n)
= x(
n)
.It
further reflects the importance of new information.
Then we can obtain the optimized grey model
~
~
~
−a k
x(
k)
= ce which meet above requirement. And this
model can effectively improve the prediction precision of
grey model.
The method based on (3) and (4) is called weakening
class ratio modeling method.
Theorem4 The weakening class ratio modeling
method realize the results of using weakening buffer
operator preprocessing data in original sequence, namely
the class ratio ~ σ in predicting model is more close to 1
than any class ratio in original sequence.
Proof: Let X = { x(
k)
x(
k)
> 0, orx(
k)
< 0, k = 1,2,3.... n}
be the
primitive behavior data sequence which possesses
characteristics: the front part grows (weakens)
~
(
excessively quickly and the latter part grows (weakens)
excessively slowly.
1) When{ x ( k )} is monotone increasing aboutk, then
the class ratio of each point x(
k −1)
σ ( k)
= ≤ 1, the latter
x(
k)
part data in original sequence become more steady and
its growth speed becomes more close to 1, then we can
let ω = max{ σ ( k )} = σ ( n)
≤ 1 ,and obtain
~
t
t
based on weakening
σ ( n)
− σ (1)
σ = σ ( n)
+ (1 − σ ( n))
,( t = 1,2,...)
t
σ ( n)
~
≤
class ratio modeling method. Thus σ ( n ) ≤ σ 1 ,
namely the class ratio ~ σ in predicting model is more
close to 1 than any class ratio in original sequence.
2) When { x ( k )} is monotone decreasing about k ,
then the class ratio of each point x(
k −1)
σ ( k)
= ≥ 1, the
x(
k)
latter part data in original sequence become more steady
and its decay speed becomes more close to 1, then we
can let ω = max{ σ(
k )} = σ(1)
≥1
,and obtain
~
t
t
based on
σ ( n)
− σ (1)
σ = σ ( n)
+ (1 − σ ( n))
, ( t = 1,2,...)
t
σ (1)
weakening class ratio modeling method.
~
≥
~
Thus σ ( n ) ≥ σ 1 , namely the class ratio σ in
predicting model is more close to 1 than any class ratio
in original sequence.
So the weakening class ratio modeling method
realize the results of using weakening buffer operator to
preprocess data in original sequence based on theorem3.
B. Property of weakening class ratio modeling method
Theorem5 The weakening class ratio modeling
method is conform to the homogeneous white index law.
Theorem6 The simulant sequence produced by
weakening class ratio modeling method and the original
sequence maintain the same monotony.
Theorem7 In weakening class ratio modeling method,
the value of the parameter t in
~
t t
determines closing
σ ( n)
−σ
(1)
σ = σ(
n)
+ (1 −σ(
n))
,( t = 1,2,...)
t
ω
degree between ~ σ and 1, the value of the parameter t is
equivalent to the weakening strength of weakening
buffer operator.
Proof: Let X = { x(
k ) x(
k ) > 0, orx ( k ) < 0, k = 1,2,3.... n}
be
the primitive behavior data sequence which possesses
characteristics: the front part grows (weakens)
excessively quickly and the latter part grows (weakens)
excessively slowly.
1) When{ x()}
k is monotone increasing about k , then
the class ratio of each point x(
k −1)
, and
letσ
( n)
= max{ σ ( k)}
≤ 1, then,
σ ( k)
= ≤ 1
x(
k)
~
t
t
σ ( n)
− σ (1)
⎛ σ (1)
n n
n n
t
n
n
⎟ ⎞
σ = σ ( ) + (1 − σ ( ))
= σ ( ) + (1 − σ ( ))1−
⎜
σ ( )
⎝ σ ( ) ⎠
t
.
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