当前位置：搜档网 › A novel fractional grey system model and its application

A novel fractional grey system model and its application

Applied Mathematical Modelling40(2016)5063–5076

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage:https://www.sodocs.net/doc/fa16067972.html,/locate/apm

Shuhua Mao a,b,Mingyun Gao a,?,Xinping Xiao a,Min Zhu a

a College of Science,Wuhan University of Technology,Wuhan430070,China

b National Engineering Research Center for Water Transport Safety,Wuhan University of Technology,Wuhan430063,China

a r t i c l e i n f o

Article history:

Received29September2014

Revised17November2015

Accepted7December2015

Available online19December2015

Keywords:

Fractional grey model

Fractional differential equation

Fractional accumulation

Matrix decomposition

Particle swarm optimization

a b s t r a c t

Of the grey models proposed for making predictions based on small sample data,the

GM(1,1)model is the most important because of its low demands of data distribution,

simple operation,and calculation requirements.However,the classical GM(1,1)model has

two disadvantages:it cannot re?ect the new information priority principle,and,if it is

necessary to obtain the ideal effect of modeling,the original data must meet the class ra-

tio test.This paper presents a new fractional grey model,FGM(q,1),which is an extension

of the GM(1,1)model in that?rst-order differential equations are transformed into frac-

tional differential equations.Decomposition of the data matrix parameters during the pro-

cess of solution shows that the new model follows the new information priority principle.

For modeling,the mean absolute percentage error(MAPE)is established as the objective

function of the optimization model,and a particle swarm algorithm is used to calculate

the accumulation number and the order of the differential equation that can minimize the

MAPE.Finally,the results from three groups of data modeling show that,compared with

other classical grey models,FGM(q,1)has higher modeling precision,can overcome the

GM(1,1)model class ratio test restrictions and has a wider adaptability.

1.Introduction

Grey system theory was proposed by Deng[1,2]to solve problems for which the information is poor,incomplete,or uncertain.The GM(1,1)model,the foundation of this theory,was established based on the nature of differential properties for a discrete sequence of data of one variable.As shown in Fig.1,the?rst“1”refers to a?rst-order differential equation, and the second“1”indicates that there is only one variable in the model.With the advantage of a simple modeling process and low demands regarding the distribution pattern of the data sequence,the GM(1,1)model has been widely applied to predictions of complex systems,such as vehicle fatality risk[3],Lorenz chaotic system[4],labor formation[5]the moving path of the typhoon[6]and energy consumption[7].Although these successes have been achieved,to establish the satisfac-tory models still need some precondition.Ying[8]proposed the theory of the admissible region of class ratio,which states that for a satisfactory GM(1,1)model to be established,the class ratio of the original sequence should be in(e?2n+1,e2n+1), where n is the length of the original sequence.Deng[9]summarized three necessary condition about grey modeling and thought the theory of the admissible region of class ratio could help to judge the feasibility of GM(1,1)modeling.

The priority of new information is a signi?cant principle in forecasting theory,which means that latest information has most reference value for modeling.However,the classical GM(1,1)model did not materialize this principle.Scholars have

?Corresponding author.Tel.:+86189********;fax:+8602786535250.

E-mail addresses:maosh_415@https://www.sodocs.net/doc/fa16067972.html,(S.Mao),wh14_gao@https://www.sodocs.net/doc/fa16067972.html,(M.Gao),xiaoxp@https://www.sodocs.net/doc/fa16067972.html,(X.Xiao),zm1023636681@https://www.sodocs.net/doc/fa16067972.html,(M.Zhu).

https://www.sodocs.net/doc/fa16067972.html,/10.1016/j.apm.2015.12.014

5064 S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

Fig. 1. GM(1,1) model.

found that improvement of the accumulated generating operation could emphasize it, such as the buffer operator proposed

by Liu [10,11] , improved buffer operator proposed by Dang [12,13] and Li [7] . Although emphasizing the principle of new information priority, the subjectivity limits application of these models when determining the weighting e?cient in buffer operator. Expanding ?rst accumulated generating operation into fractional accumulated generating operation, Wu proposed the fractional accumulation GM(1,1) model(the FAGM(1,1) model) and thought this model also embodied the new infor- mation priority [14] . Successfully applied into the maintenance cost of weapon system [15] and the gas emission [16] , this model showed the remarkable predication. Xiao [17,18] regarded the fractional accumulated generating matrix as a speci?cal buffer operator, a kind of generalized accumulated generating. Mao [19] thought the fractional accumulated generating as a special data transformation, and he proved that the FAGM(1,1) could overcome the restrictions regarding class ratio of the GM(1,1) model.

The current fractional grey model just used fractional order accumulated generating, while the whitening differential equation of the model is still the classic integer-order differential equation not the fractional differential equation. Although this incomplete combination has obtained some achievement, there is still room for improvement in forecasting.

The idea that an integer-order derivative can be extended to a non-integer order can be traced to a communication between Leibniz and LHopital in the 1600s. Since the ?rst monograph about fractional calculus was published in 1974 [20] , fractional model has rapidly developed. The fractional differential equation(FDE) is the core of these models. Considering the admirable memory principle [21] in FDE, these models have been successfully applied into chemical processing systems [22] , hard drive design [23] , and denoising for texture images [24] . However, there is still not any rational and effective way to obtain analytic solution of general FDE. In fact, the FDE is usually deemed to be the limit form of fractional difference equation [25] . The numerical simulation of FDE could be obtain when FDE is transformed into fractional difference equation [20] (i.e., ?nite element method or ?nite difference method). Although the development in numerical solution boom the application of FDE, Although the development in numerical solution boom the application of FDE, the speed and accuracy usually cannot be both guaranteed in the same numerical techniques for the solution [26,27] .

Considering the memory principle in FDE, based on the original GM(1,1) model combined with the fractional accumu- lated generating matrix, and further extending the whitening differential equation of the model to the fractional differential equation, this paper proposes the fractional grey model, FGM(q,1), which has a more complete combination and a wider range of applications. Though there exit some errors when adopt the numerical simulation, the variable and data in our model are pitifully small. Its effective to solve the equation by transforming FDE into difference equation system.

The remainder of this paper is organized as follows. In Section 2 , the fractional accumulation and inverse matrices, which are based on matrix theory, are introduced, and the fractional grey model FGM( q , 1), in which the differential equation is expanded from ?rst order to fractional order, is proposed. In Section 3 , we discuss the transformation of a fractional differential equation to a fractional difference equation to calculate predicted and reverted values via fractional accumulation matrices. In Section 4 , two examples and a real-life case are used to compare the new FGM( q , 1) model with the three previous models: GM(1,1), DGM(1,1), and FAGM(1,1).

2. Fractional accumulated generating matrices and their inverse matrices

Accumulated generating operation (AGO) is an important original component of grey system theory, the main purposes of which are to reduce the volatility of raw data and improve the grey exponential rate, the theoretical basis for modeling grey differential equations. If a single AGO is not enough, multiple operations can be performed, but using accumulated generating too many times may damage the grey exponential rate of a series [11,17] . Thus, determining the order of an accumulation matrix is very important for an ideal model.

For an original sequence x (0) = ( x (0) (1) , x (0) (2) , . . . , x (0) ( n )) T

, a r t h or d er accumulat e d generating operation( r -AGO) can

be de?ned as follows.

De?nition 1. Let sequence x ( r ) be the r -AGO sequence of x (0) , where x (r ) (k ) = k i =1

x (r ?1) (i ) ,k = 1 , 2 , . . . , n . In particular, when r = 1 , x (1) (k ) = k i =1

x (0) (i ) , and the expression x (1) = A x (0) can be obtained from matrix operation theory, where A is a 1-AGO matrix, and

A = ?

? ? ? 1 0 ···0 1 1 ···0 . .

. . . . . . . 0 1

1 (1)

? ? ? .

S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076 5065

Similarly, x (r ) = A x (r ?1) = AA x (r ?2) = A 2 x (r ?2) = ···= A r x (0) . A r is a r -AGO matrix, and A r = (a r i j

) n ×n

, where

(a r

i j ) n ×n

C r ?1

i ?j + r ?1 = (i ?j + r ?1)! (r ?1)!(i ?j )! i ≥j

i < j

(1)

It can be determined that

A r = ?

? ? ? ?

1 0

···

1 0 ···0 r (r +1)

r 1 ···0 . . .

. . .

. . . . . . r (r +1) ···(r + n ?2)

(n ?1)!

r (r +1) ···(r + n ?3) (n ?2)!

r (r +1) ···(r + n ?4) (n ?3)!

···

? ? ? ? ? ?

. (2)

De?nition 2. For n ∈ N , the general rising factorial function of a rational number is

1 n = 0

(x +1) ···(x + n ?1) n !

n ∈ N

+ .

(3)

Using De?nition 2 , Eq. (2) can be converted to

A r = ? ? ? ? ? ? ? ? r 0

0 0 ···0 r 1 r 0 0 ···0 r

r 1

r 0

···0

. . . .

. .

. . .

. . . . . .

r n ?1

n ?2

n ?3

···

r 0

? ?

? ? ? ? ? ?

. (4)

Based on Eq. (2) , and combined with the promotion of number, by letting r extend from positive integers to rational

numbers (fractions) we can obtain the fractional accumulated generating matrix. Correspondingly, the inverse operation of accumulated generating is called inverse accumulated generating and can be expressed as

x (r ?1)

(k ) = k m =1

x (r ?1) (m ) ?k ?1

m =1

x (r ?1) (m ) = x (r ) (k ) ?x (r )

(k ?1)

(5)

When r = 1 , x (0) (k ) = x (1) (k ) ?x (1) (k ?1) , in accordance with the matrix operation x (0) = A ?1 x (1) , where

A ?1 = ?

? ?

? 1

0 0 ···0 ?1 1 0 ···0 0 ?1 1 ···0 . . . . . . . . . . . . . . . 0

0 0 (1)

? ? ? ? ? . (6)

Similarly, x (0) = A ?1 x (1) = A ?2 x (2) = ···= A ?r x (r ) . We refer to A

?r as the r-AGO matrix. Theorem 1. According to Eq. (2)

, the r-AGO matrix, A r , satis?es ( A r ) ?1

= A ?r . Proof. For the lower triangular matrix A r , it is easy to obtain det ( A r ) = 1 , so A r is reversible. According to De?nition 1 , it is

straightforward to obtain

A ?r = ?

? ? ? ?

1 0 0 ···

?r 1 0 ···0 ?r (?r +1)

r 1 ···0 . . .

. . .

. . . . . . . . . ?r (?r +1) ···(?r + n ?2) (n ?1)! ?r (?r +1) ···(?r + n ?3)

(n ?2)!

?r (?r +1) ···(?r + n ?4)

(n ?3)!

···1

? ? ? ? ? ?

(7)

Therefore, A r A

?r = E and the original propositions are proved.

5066 S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

Using De?nition 2 , Eq. (7) can be expressed as

A ?r =

? ?

? ? ?

? ? ? ? ?

? ? ? ?

?r 0

0 0 0

?r 1 ?r

0 ···0 ?r 2

?r 1

?r 0

···0 . . . . . . . . . . . . . . . ?r

n ?1

?r n ?2

?r n ?3

···

?r 0 ?

? ? ? ? ?

? ? ? ? ? ? ? ? ? ?

. (8)

According to Theorem 1 :

x (0) = A ?r A r x (0) = A ?r x (r )

(9)

For example, when n = 4 , a 2

-AGO matrix and its inverse matrix are obtained as follows: A

2 7 = ?

? ? ? ? 1 0 0 0

0 0

7 1

0 48 343

9 49

2 7

? ? ? ? ? , and A ?2

7 = ?

? ? ? ? 1 0 0 0

7 1

0 0

?5 49

0 ?20 343

?5 49

?2 7

? ? ? ?

3. GM(1,1) model and fractional grey model 3.1. GM(1,1) model

Let x (0) = ( x (0) (1) , x (0) (2) , . . . , x (0) ( n )) T

be an original data sequence, x (1) be a 1-AGO sequence, where x (1) = A x (0) , and

z (1) = ( z (1) (2) , z (1) (3) , . . . , z (1) ( n )) T

be the mean generated sequence, where z (1) (k ) = 0 . 5 x (1) (k ?1) + 0 . 5 x (1) (k ) . Compared with x (0) , x (1) is monotonically increasing, similar to the accumulation of energy in a system and can be used to establish

the GM(1,1) model with differential properties. The speci?c de?nition of the model is as follows.

De?nition 3. Let Eq. (10) be the de?nition formula of the GM(1,1) model,

x (0) (k ) + a z (1) (k ) = b

(10)

where k = 2 , 3 , . . . , n , a is the development coe?cient of the model, and b is the grey input of the model.

De?nition 4. Let the differential Eq. (11) be de?ned as the whitenization differential equation of the GM(1,1) model.

d x (1) dt

+ a x (1)

= b (11)

Combined with the initial condition x (1) (1) = x (0) (1) , the solution of Eq. (11) is

? x (1)

(k ) =

x (0) (1) ?b a

e ?a (k ?1) + b

(12)

The reduced value of Eq. (12) can be obtained using Eq. (9) :

?x (0) = A ?1 ?x

(1)

. (13)

For a given original data sequence, upon using 1-AGO, the modeling sequence satis?es the grey exponential law so that it

can be ?tted by the grey differential equation, which is an important concept of the original grey system method. If 1-AGO is insu?cient, we can perform multiple accumulations or even extend to fractional accumulation to better ?t the sequence of the grey exponent law. Extending 1-AGO to fractional accumulated generating yields the fractional accumulation GM(1,1) model, i.e, FAGM(1,1) model. 3.2. Fractional grey model

As a generalization of integer-order differential equations, fractional differential equations simplify complex system mod- eling and allow more accurate description, among other advantages, and therefore they are widely used in practice. Incor- porating fractional differential equations into the grey model, this paper proposes the fractional grey model as follows.

S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

5067

For an original data sequence x (0) = (x (0) (1) , x (0) (2) , . . . , x (0) ( n )) T

, x ( r ) is the r-AGO sequence, where x (r ) = A r x (0) .

z (r ) = ( z (r ) (2) , z (r ) (3) , . . . , z (r ) ( n )) T

is the mean generating sequence of the r-AGO sequence, and z (r ) (k ) = 0 . 5 x (r ) (k ?1) +

0 . 5 x (r ) (k ) . The fractional grey model is then de?ned as follows.

De?nition 5. Let the differential Eq. (14) be the whitenization formula of the FGM(q,1) model.

d q x (r ) d t

q + a x (r )

= b . (14)

For a differentiable function x ( t ), its derivative can be approximated as a difference when the step size h is small: x k

= x k ?x k ?1 h

, where x k = x (kh ) . Theorem 2. When h → 0, the n-order derivative function x (?q )

of a differentiable function x ( t ) at t = kh is given by

x (?q )

k = 1 h q k

i =1

?q k ?i

x i . (15)

Proof.

(1) When q = 1 , it is obvious that when i = 1 , 2 , . . . , k ?2 , [ ?q

k ?i

] = 0 , so

1 h q k

i =1

?q k ?i

x i = x k ?x k ?1 h = x (?1)

k , (16)

and clearly Eq. (15) holds.

(2) Assuming that the previous formula holds when q = m , ( m ≥1), then when q = m + 1 ,

x (?m ?1)

= x (?m )

(?m )

k ?1 h

m +1

k i =1

(?1)

k ?i

m ?k + i + 1

k ?i

x i ?k ?1 i =1

(?1)

k ?1 ?i

m ?k + i + 2 k ?i ?1

x i

m +1

k ?1 i =1

(?1)

k ?i

m ?k + i + 1

k ?i

m ?k + i + 2 k ?i ?1

x i + x k

, (17)

x (?m ?1)

= 1

m +1

k ?1 i =1

(?1)

k ?i m + 1 ?k + i + 1

k ?i x i +

m + 1 ?k + k + 1 k ?k x k

(18)

m +1

k i =1

(?1)

k ?i

m + 1 ?k + i + 1

k ?i

x i

(18)

That is, when q = m + 1 , the last equation holds, so for any q ∈ N + , Eq. (15) holds.

By extending q from positive integers to rational numbers, we have

d q x (r ) d t

t = k h

≈1 h

q k

i =1

?q k ?i

x (r ) i .

(19)

Theorem 2 shows that the q -order derivatives of a sequence x

( r ) can be approximated by as d q x (r ) d t q = 1 h q x (r ?q )

= 1 h

q A ?q x (r ) . (20)

In combination with the de?nite integral principle, it is possible to compute the approximate q -order integral of d q x

(r ) d t

over the interval [ (k ?1) h , kh ] ,

··· kh

(k ?1) h

d q x d t q d t q = k i =1

?q k ?i

x (r )

i .

(21)

Given points x (r ) ((k ?1) h ) and x ( r ) ( k h ), it is possible to compute the approximate value of x ( r ) ( t ) at t

···

(k ?1) h

x (r ) (t ) d t q ≈

q 2

( x ((k ?1) h ) + x (kh ) ) . (22)

5068 S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

When q = 1 , Eq. (22) is actually a trapezoidal integral formula so it can be viewed as a generalization of the trapezoidal

rule, here called a q -order trapezoidal integral formula. De?nition 6. If

x (r ?q ) (k ) + a z (r )

(k ) = b ,

(23)

where x (r ?q ) = A ?q x (r ) , then term Eq. (23) the de?nition formula of the FGM( q , 1) model.

Theorem 3. The difference form of the fractional differential Eq. (14) is given by Eq. (23) .

Proof. Compute the q -order integral for both sides of Eq. (9) at t over the interval [ k ?1 , k ] ,

··· k

k ?1 d q x (r ) d t q d t q + a ··· k k ?1 x (r ) (t ) d t q = b ··· k k ?1

d t q

(24)

Combine Eqs. (20) and (22) with the generalization of the de?nite integral to obtain

x (r ?q ) (k ) + a z (r )

(k ) = b .

(25)

Theorem 3 shows that Eq. (23) is the difference form of Eq. (14)

, so that (14) is approximately equivalent to (23) . Then, one can complete parameter estimation and solve of Eq. (14) using Eq. (23) . To estimate the model parameters, we have the following theorem 4 .

Theorem 4. Assuming x (r ) = x (r ) (1) , x (r ) (2) , ... , x (r ) (n )

T , and z

( r ) is its mean generating sequence, the q-order inverse accu- mulated generating sequence is x (r ?q ) , where x (r ?q ) = A ?q x (r ) , if ? P = ( a b

) is a parameter sequence, and

Y = ? ? ? ? ? ? x (r ?q ) (2) x (r ?q ) (3) . . . x (r ?q ) (n ) ? ? ? ? ? ? , B = ? ? ? ? ? ? ?z (r )

(2) 1

?z (r ) (3) 1 . . .

. . . ?z (r ) (n )

? ? ? ? . Then, the least square estimation of the model x (r ?q ) (k ) = ?a z (r ) (k ) + b is given by

? P = ( B T B ) ?1 B T Y .

(26)

Proof. Applying the equation x (r ?q ) (k ) = ?a z (r ) (k ) + b to data yields,

x (r ?q ) (2) = ?a z (r ) (2) + b (27a) x (r ?q ) (3) = ?a z (r )

(3) + b

(27b)

. .

x (r ?q ) (n ) = ?a z (r ) (n ) + b

(27c)

Namely Y = B ? P for one estimate of a, b . By replacing x (r ?q ) (k )(k = 2 , 3 . . . , n ) on the left side of the equation by ?a z (r ) (k ) + b , the error sequence ε = Y ?B ? P is obtained. Assuming that

s = ε T

ε = (Y ?B ? P ) T (Y ?B ? P ) = n k =2

( x (r ?q ) (k ) + a z (r ) (k ) ?b ) 2 ,

the a, b that minimize s satisfy

? ? ? ? ?s ?a

= 2 n k =2

( x (r ?q ) (k ) + a z (r ) (k ) ?b ) z (1) (k ) = 0 ?s ?b = ?2 n k =2

( x (r ?q ) (k ) + a z (r ) (k ) ?b ) = 0 (28)

That is

? n

k =2

( x (r ?q ) (k ) + a z (r ) (k ) ?b ) z (r ) (k ) = 0 n

k =2

( x (r ?q ) (k ) + a z (r ) (k ) ?b ) ·1 = 0 , (29)

which, written in matrix form, is B T (Y ?B ? P ) = 0 , so B T B ? P = B T Y and ( B T B ) ?1 B T B ? P = ( B T B ) ?1 B T Y . Thus ? P =

B T B

?1 B T

Y .

S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076 5069

Theorem 5. For the parameter matrix of Theorem 4 , the decomposition form is obtained as follows:

B = ?

? ? ? ? ? ?z (r ) (2)

?z (r ) (3)

1 . . .

. . . ?z (r ) (n )

? ? ? = B 1 A r M , (30)

where,

B 1 = ?

? ? ? ? ? ?1 2 ?1 2 0

···0 0 0

?1 2

···0 0 .

. .

. . . .

. . . . . .

. .

. 0

···

?1 ?

? ?

? ? ? (n ?1) ×n

, M = ?

? ? ? ? ?

? ?

? ? ?

? ? ? ?

x (0) (1)

1 ?r 0

x (0) (2) ?

1 ?r 1 x (0) (3) ?

1 ?r

2 . . . . . .

x (0) (n )

1 ?r n ?1

? ? ?

? ?

? ? ?

? ? ? ? n ×2

And

Y = ? ? ? ? ? x (r ?q ) (2)

x (r ?q ) (3) . . . x (r ?q ) (n )

? ?

? ? ?

= Q A r x (0) ,

(31)

where,

Q = ? ? ? ? ? ?

? ? ? ? ? ?

1 ?q 0

···0

?q 2

?q 1

?q 0

···

0 .

. . . . . . . . . . . . .

. ?q

n ?1

n ?2 ?q

n ?4

···

?q 0

? ? ? ?

? ? ? ? ? ?

(n ?1) ×n

Proof. Let e 1 = ?

? ? ? ? ? ? ? ? ? ? ? 1 ?r

0 1 ?r 1 . . . 1 ?r n ?1

? ?

? ? ? ? ? ?

? ?

n ×1

, I = ? ? ? ? ? 1 1 . . . 1 ? ? ? ? ? (n ?1) ×1 , I ?= ? ? ? ? ? 1 1 . . . 1 ? ? ? ? ? n ×1 , x (0) = ? ? ? ? ? x (0) (1) x (0) (2) . . . x (0) (n ) ? ? ? ? ? , z (r ) = ? ? ? ? ? z (r ) (2) z (r ) (3) . . . z (r ) (n ) ? ? ? ? ? , then, the original data matrix M can be written as M = ( x (0) ?e 1 ) . According to the above de?nition, we can obtain

?z (r ) = B 1

A r x (0) . Only the following is needed to prove I =

B 1 A r (?e 1 ) , I * = A r e 1

. According to Theorem 1, the inverse matrix of A r is A ?r , which is all that is needed to prove e 1 = A ?r I * . Eq. (7) can be used to prove

1 ?r

k ?1

k i =1

?r k ?i

, k = 1 , 2 , . . . , n . (32)

(1) When k = 1 , [ 1 ?r 0 ] = 1 = [ ?r 0

] , Eq. (32) obviously holds;

5070 S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

(2) Assuming that Eq. (32 ) holds when k = m , then when k = m + 1 ,

m +1 i =1

m + 1 ?i

m i =1

m ?i

?r m

?r + 1 m ?1

?r m

(33)

and then [ ?r +1 m ?1 ] + [ ?r m ] = [ ?r +1 m ]

, thus Eq. (33) can be written as m +1 i =1

?r m

1 ?r

(m + 1) ?1

(34)

Therefore, when k = m + 1 Eq. (32) is satis?ed.

In summary, for any k ∈ N

+ Eq. (32) holds, and the decomposition of parameter matrix B is proved. Similarly , we can obtain the decomposition of the parameter matrix Y .

The decomposition of a parameter matrix, as described in Theorem 5, provides a method to calculate the parameters in

the model matrix using Eq. (26) . Matrix decomposition also shows us that the modeling mechanism, data matrix B , can be

decomposed into three matrices: B 1 , A r , and M , where B 1 is the mean generating matrix, A r is the r-AGO matrix, and M is

the original data matrix. B 1

, used in the method of background value selection, is a constant matrix. The background value is essentially the q -order integral of x ( r ) ( t ) in the interval [ k ?1 , k ] using a quasi-trapezoid formula. A r , the r -times accumulated

generating matrix for the accumulated generating operation, is designed to weaken the randomness of the sequence and to strengthen the modeling rules of data sequence, and thus can use grey fractional differential equations for good ?tting, which is an extension and expansion of the important ideas of the grey system method. M is a data matrix related to the original sequence and the order of accumulated generating operations. At the same time, Y can also be decomposed into

the product form of three matrices: Q, A r , and x (0) , where Q is the q -order difference matrix, and x

(0) is the original data matrix. Q shows the difference form of the differential equations converted into difference equations; it is not only a data matrix determined by the order of the albinism differential equation but also the foundation of the fractional order grey model.

Obviously, when q = 1 the decomposition of a parameter matrix is consistent with the fractional accumulation grey model in references [14,18] : that is, the fractional grey model degenerates into the fractional accumulation grey model, the FAGM(1,1) model for short. If q = 1 and r = 1 , the decomposition of a parameter matrix is completely consistent with the decomposition of GM(1,1) in references [17,28] .

Theorem 6. For the difference equations shown in (23) , if the given original value x ( r ) (1) satis?es x (r ) (1) = x (0) (1) , the solution

of sequence x (1) satis?es

?x (r )

(k ) =

2 b ?2

k ?1 i =1

k ?i

x (r ) (i ) ?a x (r ) (k ?1)

a + 2

, k = 2 , 3 , . . . , n . (35)

Proof. For x (r ?q ) = A ?q x (r ) , using Eq. (4) yields

x (r ?q ) (k ) =

k i =1

?q k ?i

x (r ) (i ) = x (r )

(k ) +

k ?1 i =1

?q k ?i

x (r )

(i ) .

(36)

Additionally, z (r ) (k ) = 1

2 x (r ) (k ) + x (r ) (k ?1)

, so Eq. (23) can b e written as

x (r )

(k ) +

k ?1 i =1

?q k ?i

x (r ) (i ) + a 2 ( x (r ) (k ) + x (r )

(k ?1)) = b .

(37)

Then,

+ 1

x (r )

(k ) = b ?

k ?1 i =1

?q k ?i

x (r )

(i ) ?

a 2

x (r )

(k ?1) , (38)

x (r ) (k ) =

b ?

k ?1 i =1

k ?i

x (r ) ( i ) ?a x (r ) ( k ?1) 1 + a

, (39)

to obtain Eq. (35) . Thus Theorem 6 is proved.

If q = 1 and r = 1 , Eq. (35) can be converted to

x (1)

(k ) =

2 b + 2 x (1) (k ?1) ?a x (1) (k ?1)

a + 2

(40)

S.Mao et al./Applied Mathematical Modelling40(2016)5063–50765071

Ifβ1=1?0.5a

1+0.5a

=2?a

2+a

,β2=b

1+0.5a

=2b

2+a

,Eq.(40)can be converted to

x(1)(k)=β1x(1)(k?1)+β2,(41) which is the equation of the DGM(1,1)model[29].

We can obtain the predicted values of the FGM(q,1)model using Eq.(35),and we can obtain the original values of the model similarly according to Eq.(9)as follows

?x(0)=A?r?x(1).(42)

3.3.Determination of the order of the accumulation matrix and differential equation

We have discussed the FGM(q,1)model and the main steps of https://www.sodocs.net/doc/fa16067972.html,pared to the classical GM(1,1)model, this paper makes two improvements.First,the accumulated generating method of sequences uses fractional accumulated generating instead of the classic one-time accumulated generating of the original model.Second,the order of the differential equation of the model is given by a fractional differential equation instead of the classic?rst-order differential equation.For this extended model,one must determine the order r of the accumulated generating operation and the order q of the differential equation.

The value of the mean absolute percentage error(MAPE)of a model is often used to judge the merits of modeling;we choose the r and q that minimize the MAPE as the optimal model parameters.Thus,we regard the minimum MAPE as the optimization target.The MAPE optimization model based on r and q is as follows:

min MAPE(r,q)=

n?1

k=2

|?x(0)(k)?x(0)(k)|

x()()

×100%.(43)

And

APE(k)=|?x(k)?x(k)|

x k

×100%.(44)

The MAPE calculation process involves the symbolic computation of absolute value.MAPE is not differentiable on r and q so it is di?cult to use the formula to?nd the optimal solution of the model.This paper uses a particle swarm algorithm to determine the optimal solution of r and q.The speci?c procedures are as follows:

MAPE is a two-dimensional function dependent only on r and q.In the two-dimensional search space,distribute N par-ticles at random for T cycles.

In the k t h(k=1,2,...,T)cycle of the i th particle,let V k

i,j be the velocity in dimension j(j=1,2),X k

be r,the order

of the accumulated generating matrix,and X k

i2be q,the order of the differential equations.The values pbest k i1and pbest k i,2

correspond to the values of r and q,respectively,that minimize MAPE for the i particle of the search,i.e.,the individual

extremum,and gbest k

1and gbest k

correspond to the values of r and q,respectively,that minimize the value of MAPE for a

particle swarm by the k th cycle of the search,i.e.,the entire extreme.The position and velocity of a particle are updated as follows:

V k+1 i,j =V k

i,j

+c1λ1(p b e st k i,j?X k i,j)+c2λ2(g b e st k j?X k i,j)(45)

X k+1 i,j =X k

i,j

+V k+1

i,j

i=1,2,...,N j=1,2k=1,2,...,T?1,(46)

where c1and c2are constants called the acceleration coe?cients,c1=c2=1.4962,andλ1andλ2are random numbers in [0,1].The output for the minimum value of the MAPE and the corresponding values of r and q are obtained after T cycles of searching.

In summary,the detailed modeling steps of FGM(q,1)can be described as follows:

Step:1Determine the original data sequence according to the actual application background x(0);

Step:2With r and q as parameters,obtain the fractional accumulated generating matrices A r,A?q and the fractional accu-mulated generating sequence x(r),the mean generating sequence z(r),and the fractional order differential sequence x(r?q);

Step:3Based on the least square method,given by Eq.(26),obtain the model parameter estimates,?P=(a,b);

Step:4Obtain the predicted value sequence?x(r)of the model using Eq.(35),and obtain the reverted value sequence?x(0) of the model using Eq.(42);

Step:5Determine the parameters r and q to minimize the MAPE using Eq.(43)with particle swarm optimization algo-rithm;

Step:6Put r into the accumulated generating matrix A r,and calculate the sequences x(r),z(r),and x(r?q).Repeat the Step:3 and Step:4steps and calculate the predicted values and the reduce values until the model is complete.

5072 S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

Table 1

Forecasts obtained using different grey models. Actual GM(1,1) DGM(1,1) FAGM(1,1) FGM( q , 1) values Forecasts APE( k ) (%) Forecasts APE( k ) (%) Forecasts APE( k ) (%) Forecasts APE( k ) (%) 0.26 ––

––

––0.73 0.24 67.12 0.29 60.45 0.94 28.37 0.76 3.78 2.07 0.67 67.63 0.90 56.49 1.80 13.25 2.14 3.23 7.08 1.88 73.45 2.81 60.32 5.26 25.69 6.64 6.23 21.22 5.27 75.16 8.76 58.70 14.72 30.65 20.61 2.88 63.98 14.75 76.95 27.34 57.27 41.06 35.83 63.98 0 202.51 41.27 79.62

85.29 57.88

114.5 43.46

198.61 1.92

MAPE 73.32% 58.52% 29.54% 3.01% r 1 1 2.5164 1.9233 q

0.6901

number

v a l u e

Fig. 2. Forecasts obtained using different grey models.

4. Case study 4.1. Examples

4.1.1. Example A

The GM(1,1), DGM(1,1, FAGM(1,1), and FGM( q , 1) models are applied to the original sequence

x (0)

= ( 0 . 26 , 0 . 73 , 2 . 06 , 7 . 08 , 21 . 22 , 63 . 98 , 202 . 51 ) T .

The following class ratio sequence is obtained:

σ(0) = { 0 . 3562 , 0 . 3544 , 0 . 2910 , 0 . 3336 , 0 . 3317 , 0 . 3159 } .

(47)

According to the reference [9] , the length of sequence is n = 7 , so the admissible region of the class is (e ?2

n +1 , e 2

n +1 ) , i.e.

(0.7788, 1.2840). It is found that no part of this sequence is within this region, which means that a satisfactory GM(1,1) model cannot be established. The sequence of Table 1 is monotonically increasing. The forecasts and absolute percentage errors (APE) yielded by the above four models are presented. Furthermore, the absolute values of the relative error (MAPE), the order r of the accumulated generating matrix, and the order q of the differential equation are all calculated. The out- comes of the GM(1,1) and DGM(1,1) models are unsatisfactory because they fail the above testing class ratio. The MAPE of GM(1,1) reaches values as high as 73.32%. The accuracy achieved by FAGM(1,1) is 29.54%, better than for GM(1,1) but still not satisfactory. The FGM( q , 1) model achieves the best accuracy with an error of only 2.64%. In Fig. 2 , the uppermost curve, which represents the actual values, is overlaid by the blue curve, which represents the values predicted by the FGM( q , 1) model. This agreement indicates that the FGM( q , 1) model was able to predict the actual values successfully. However, for the other models there exists a relatively large distance between the predicted and actual values. We can rank the four models from the worst to the best in terms of their ability to ?t the data, as follows: GM(1,1), DGM(1,1), FAGM(1,1), FGM( q , 1). It is also found that the distance between the predictions of other models and the actual values grows with increasing number, which means they can not be applied to long-term predictions.

S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

5073

Table 2

––

–12.74 12.01 5.72 12.32 3.32 12.25 3.88 12.79 0.36 5.77 6.36 10.15 6.37 10.39 5.83 0.97 5.60 2.99 2.76 3.36 21.84 3.29 19.35 3.05 10.66 2.89 4.70 1.75 1.78 1.68 1.70 2.66 1.81 3.50 1.75 0 1.23 0.94 23.45 0.88 28.38 1.22 0.79 1.21 1.76 0.90 0.50 44.65 0.46 49.38 0.91 1.63 0.92 1.85 0.74 0.26 64.38

0.24 68.16

0.74 0.05

0.74 0

MAPE 24.55% 25.95% 3.07% 1.66% r 1 1 0.9248 0.0228 q

0.0420

number

v a l u e

(a)Actual values

number v a l u e (b)Forecasts by GM(1,1)model.

number

v a l u e (c)Forecasts by DGM(1,1)model.

number v a l u e (d)Forecasts by FAGM(1,1)model.

number

v a l u e (e)Forecasts by FGM(q ,1)model.

Fig. 3. Forecasts obtained using different grey models..

4.1.2. Example B

The GM(1,1), DGM(1,1), FAGM(1,1), and FGM(q,1) models are applied to another original sequence x

(0) = ( 34 . 15 , 12 . 74 , 5 . 77 , 2 . 76 , 1 . 75 , 1 . 23 , 0 . 90 , 0 . 74 ) T

in the same way. The class ratio sequence is

σ(0) = { 2 . 6805 , 2 . 2080 , 2 . 0906 , 1 . 5771 , 1 . 4228 , 1 . 3667 , 1 . 2162 } .

(48)

It is di?cult to establish a satisfactory GM(1,1) model because most data in the class ratio sequence are beyond the admissi- ble region of the class (0.8007, 1.2488). In contrast to Table 1 , the sequence of Table 2 is monotonically decreasing. According to Fig. 3 , the growing trends of the four grey model forecasts are all similar to the trend of the actual values. The tails of Fig. 3 b and c are obviously below the tail of Fig. 3 a , which means the forecasting accuracy of the GM(1,1) model in Fig. 3 b and the DGM(1,1) model in Fig. 3 c are both poor. In fact, the MAPE of the GM(1,1) and DGM(1,1) models are both over 20%, as shown in Table 2 . The MAPE of FAGM(1,1) is 3.07%, and the MAPE of FGM( q , 1) is 1.66%, which means that FGM( q , 1) is the best model.

5074

S. Mao et al. / Applied Mathematical Modelling 40 (2016) 5063–5076

number

v a l u e

(a)Actual values

number v a l u e (b)Forecasts by GM(1,1)model.

number

v a l u e (c)Forecasts by DGM(1,1)model.

number v a l u e (d)Forecasts by FAGM(1,1)model.

number

v a l u e (e)Forecasts by FGM(q ,1)model.

Fig. 4. Forecasts obtained using different grey models.

Table 3

Forecasts obtained using different grey models. Year

Actual GM(1,1) DGM(1,1) FAGM(1,1) FGM( q , 1) values

Forecasts APE( k ) (%) Forecasts APE( k ) (%) Forecasts APE( k ) (%) Forecasts APE( k ) (%) 2007 357.4 –

––––

–––2008 164.6 155.7 5.40 158 3.99 164.71 0.06 165 0.24 2009 85.1 97.1 14.11 97.7 14.77 85.51 0.48 85.1 0.00 2010 59.4 60.6 1.95 60.4 1.62 57.63 2.98 57.7 2.86 2011 40.3 37.8 6.28 37.3 7.43 41.65 3.36 41.8 3.73 2012 31.0

23.6 24.02

23.1 25.63

31 0.01

31 0.00

MAPE 10.35% 10.69% 1.38% 1.37% r 1 1 1.6368 1.8398 q

1.1820

4.2. Case study

As shown in Fig. 3 , the price of ?xed broadband as a percentage of Gross National Income (GNI) per capita (p. c.) in developing countries was collected from 2007 to 2012 [30] .

The class ratio sequence is

σ0 = { 2 . 1713 , 1 . 9342 , 1 . 4327 , 1 . 4739 , 1 . 30 0 0 } .

(49)

It is di?cult to construct a satisfactory GM(1,1) model because most data in the class ratio sequence are beyond the

admissible region of the class(0.7515 1.3307). Since the beginning of the 21st century, broadband technology in developing countries has advanced rapidly, and the installation costs for ?xed broadband have been greatly reduced. Forecasting the installation cost of broadband is of great signi?cance for national economies and the development of network technology. Because of the rapid development broadband technology, an emerging technology in developing countries, the reference data are limited. The grey model may be the most e?cient tool for forecasting. As seen in Fig. 4 , the growing trends shown in Fig. 4 b –e are all similar to the trend in Fig. 4 a , the actual values. The MAPEs of the GM(1,1) and DGM(1,1) models are both greater than 10%, while the MAPEs of the FAGM(1,1) and FGM(q,1) models are below 1.5%.

S.Mao et al./Applied Mathematical Modelling40(2016)5063–50765075

Fig.5.Relationship between grey models.

5.Conclusions

1.This paper proposes a novel grey model,FGM(q,1).A fractional accumulation sequence is calculated from a fractional

accumulation matrix,and a fractional differential equation is constructed from the fractional accumulation sequence.By transforming the fractional differential equation to a fractional difference equation,the prediction of a fractional accu-mulation sequence is obtained.Finally,the reduced value is calculated using the inverse fractional accumulation matrix.

2.The FGM(q,1)model is a generalization of the FAGM(1,1)model and the classical GM(1,1)model.The relationship be-

tween GM(1,1),FAGM(1,1),and FGM(q,1)is shown in Fig.5.When fractional accumulation takes the place of1-AGO,the GM(1,1)and DGM(1,1)models can be expanded to the FAGM(1,1)or FAGM(1,1,D)models.Furthermore,considering the fractional differential equation,the FAGM(1,1)model can develop into the FGM(q,1)model.The FAGM(1,1)model is a speci?c FGM(q,1)model when q=1,and the GM(1,1)model is a speci?c FAGM(1,1)model when r=1.

3.The fractional grey differential equation is the continuation of the classical grey differential equation.This paper has

summarized the equations of integer-order derivatives and fractional derivatives and has constructed the fractional grey differential equation.Converting a continuous fractional differential equation into a discrete fractional differential equa-tion enables us to obtain the solution of this model.

4.FGM(q,1)overcomes the limit of class ratio and has strong adaptability.The accuracy of FGM(q,1)is a signi?cant im-

provement,especially when the original sequence does not satisfy the condition of class https://www.sodocs.net/doc/fa16067972.html,ing two examples and

a case study in Section4,we have shown that the MAPE of the FGM(q,1)model is always the lowest of the four grey

models.

Acknowledgments

This work was supported by Natural Science Foundation of China(No.51108465);the General Education Program(GEP) Requirements in the Humanities and Social Sciences project(Grant No.11YJC630155),the Fundamental Research Funds for the Central Universities project(Grant No.2014-Ia-015),the China Postdoctoral Science Foundation project(Grant No. 2012M521487),and the China Postdoctoral Science Foundation Special Foundation project(Grant No.2013T60755).The au-thors thank Xinping Yan of National Engineering Research Center for Water Transport Safety of Wuhan University of Tech-nology for providing the guiding suggestions.

References

[1]J.Deng,Control problems of grey systems,Syst.Control Lett.1(5)(1982)288–294.

[2]J.L.Deng,Introduction grey system theory,J.Grey Syst.1(1)(1989)1–24.

[3]M.Z.Mao,E.C.Chirwa,Application of grey model gm(1,1)to vehicle fatality risk estimation,Technol.Forecast.Soc.Change73(5)(2006)588–605.

[4]Y.G.Zhang,Y.Xu,Z.P.Wang,Gm(1,1)grey prediction of lorenz chaotic system,Chaos Solitons Fract.42(2)(2009)1003–1009.

[5]M.-S.Yin,H.-W.V.Tang,On the?t and forecasting performance of grey prediction models for china’s labor formation,https://www.sodocs.net/doc/fa16067972.html,put.Model.57(3-4)

(2013)357–365.

[6] C.I.Chen,S.J.Huang,The necessary and su?cient condition for gm(1,1)grey prediction model,https://www.sodocs.net/doc/fa16067972.html,put.219(11)(2013)6152–6162.

[7]W.Li,H.Xie,Geometrical variable weights buffer gm(1,1)model and its application in forecasting of china’s energy consumption,J.Appl.Math.(2014)

1–6.

[8]K.C.Ying,C.J.Liao,Y.T.Hsu,Generalized admissible region of class ratio for gm(1,1),J.Grey Syst.12(2)(2000)153–156.

[9]J.L.Deng,The Basis of Grey Theory,Huazhong University of Science&Technology,Wuhan,2002.

[10]S.F.Liu,The trap in the prediction of a shock disturbed system and the buffer operator,J.Huazhong Univ.Sci.Technol.25(1)(1997)25–27.Ab9rr

Times Cited:0Cited References Count:0.

[11] F.Liu S,Y.Dang,Z.Fang,N.Xie,Grey System Theory and its Application,Science Press,Beijing,2010.

[12]Y.Dang,S.Liu,B.Liu,X.Tang,Study on the buffer weakening operator,Chin.J.Manag.Sci.12(2)(2004)108–111.

[13]Y.Dang,B.Liu,Y.Guan,On the strengthening buffer operators,Control Decis.20(12)(2005)1332–1336.

[14]L.F.Wu,S.F.Liu,L.G.Yao,S.L.Yan,D.L.Liu,Grey system model with the fractional order accumulation,Commun.Nonlinear Sci.Numer.Simul.18(7)

(2013)1775–1785.

[15]S.L.Fang,L.F.Wu,Z.G.Fang,X.J.Guo,Using fractional gm(1,1)model to predict the maintenance cost of weapon system,J.Grey Syst.25(3)(2013)

9–15.

[16]L.F.Wu,S.F.Liu,D.Chen,L.Yao,W.Cui,Using gray model with fractional order accumulation to predict gas emission,Nat.Hazards71(3)(2014)

2231–2236.

[17]X.P.Xiao,S.H.Mao,The Method of Grey Prediction and Decision,Science Press,Beijing,2013.

[18]X.P.Xiao,H.Guo,S.H.Mao,The modeling mechanism,extension and optimization of grey gm(1,1)model,Appl.Math.Model.38(5-6)(2014)1896–

1910,doi:10.1016/j.apm.2013.10.004.

5076S.Mao et al./Applied Mathematical Modelling40(2016)5063–5076

[19]S.Mao,M.Y.Gao,J.H.Wen,X.Nie,Generalized admissible region of class ratio for fractional order gm(1,1)model,J.Grey Syst.26(3)(2014)56–68.

[20]K.Oldham,J.Spanier,The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order,Academic Press,New York

and London,1974.

[21]W.Deng,Short memory principle and a predictor-corrector approach for fractional differential equations,https://www.sodocs.net/doc/fa16067972.html,put.Appl.Math.206(1)(2007)174–

188.

[22] A.Flores-Tlacuahuac,L.T.Biegler,Optimization of fractional order dynamic chemical processing systems,Ind.Eng.Chem.Res.53(13)(2014)5110–5127.

[23]Y.Luo,T.Zhang,B.Lee,C.Kang,Y.Q.Chen,Fractional-order proportional derivative controller synthesis and implementation for hard-disk-drive servo

system,Ieee Trans.Control Syst.Technol.22(1)(2014)281–289.

[24]Y.F.Pu,P.Siarry,J.L.Zhou,Y.G.Liu,N.Zhang,G.Huang,Y.Z.Liu,Fractional partial differential equation denoising models for texture image,Sci.

China-Inf.Sci.57(7)(2014)19.

[25]J.Cheng,The Theory of Fractional Difference Equation,Xiamen University Press,Xiamen,2011.

[26]N.J.Ford,A.C.Simpson,The numerical solution of fractional differential equations:Speed versus accuracy,Numer.Algorithms26(4)(2001)333–346.

[27]K.Diethelm,J.M.Ford,N.J.Ford,M.Weilbeer,Pitfalls in fast numerical solvers for fractional differential equations,https://www.sodocs.net/doc/fa16067972.html,put.Appl.Math.186(2)

(2006)482–503.

[28]X.P.Xiao,Y.C.Hu,H.Guo,Modeling mechanism and extension of gm(1,1),J.Syst.Eng.Electron.24(3)(2013)445–453.

[29]N.M.Xie,S.F.Liu,Discrete grey forecasting model and its optimization,Appl.Math.Model.33(2)(2009)1173–1186.

[30] A.Pitt,B.Granger,Measuring the Information Society,Report,International Telecommunication Union,2013.

A novel fractional grey system model and its application

相关文档

最新文档