Studies in System Science (SSS) Volume 2, 2014 www.as‐se.org/sss
The Optimization of GOM (1, 1) Model by Derivative Reduction Method Wan Qin Department of Mathematics and Information , China West Normal University, Nanchong, Sichuan,China wanqin1014@126.com Received 8 August 2013; Accepted 24 December 2013; Published 21 April 2014 © 2014 American Society of Science and Engineering Abstract For the reason that there are many difficult parameter estimation problems in traditional GOM (1, 1) model,the author puts forward a new method of parameter estimation, and it is simple in program and easy to operate. Furthermore, this paper uses derivative reduction method for traditional GOM (1, 1) model and which had been optimized, thus gets new predictor formula. Finally two examples are given to show that the new GOM (1, 1) model in this paper improves the simulation prediction precision, so the practicality and effectiveness of the new method in this paper are verified by the examples. Keywords Reverse Accumulate; GOM (1, 1); Derivative Reduction; Least Squares
Introduction Derivative Reduction Method Traditional GM (1, 1) models use the response x(1) (k ) of whiten differential equation to generate the reduction
value x (0) (k 1) x (1) (k 1) x (1) (k ) …(1) through IAGO method [1] . Obviously, there is a premise to the rationality of reduction formula , that is the primary function in the whiten differential equation must be the background value x (1) (k ) of the grey differential equation corresponding to it. When the background value x (1) (k ) of the basic grey differential equation x ( 0) (k ) + a x (1) (k ) = b has been optimized as mean value z(1) (k ) , then if use the reduction formula (1) to differential equation x (0) (k ) + a z(1) (k ) = b ,we may fell farfetched slightly. Especially when there is big difference between the optimized background value (recorded as y(k ) )and x (1) (k ) , reduction formula (1) may even causes many questions to people. So references [2] holds that under the condition that grey derivative is still x ( 0) (k ) , the solution of whitening differential equation is only the function expression of the background value y(k ) , then we cannot continue to use the reduction formula (1) again, and we should use the method of derivative reduction [ y(k )]k = x (0) (k ) . References [3] aims at grey differential equation with grey derivative f ( x ( 0) (k )) and background value y(k ) (it may be the background value z(1) (k ) of the original grey model, and may be the background value x (1) (k ) of the basic grey model, but may also be the background value of the grey model with all kinds of optimized background value.), as long as we can solve x (0) (k ) from f ( x ( 0) (k )) = [ y(k )]k ,then we want to obtain the predicting formula through the method of derivative reduction. Traditional GOM (1, 1) Model [ 4]
n
Let x ( 0) ( K ) be original sequence, x (1) (k ) x ( 0) (k ) is the first time reverse accumulated sequence. First, establish i k
GOM (1, 1) grey model base on x (1) (k ) in this way:
( x (1) (k ) x (1) (k 1)) (1 )( x (1) (k 1) x (1) (k )) ax (1) (k ) b
….(2), which
ae a e a 1 , k 2, 3,..., n . a and b are parameters that awaiting identification. (e a 1)2
Second, by using the simple linear regression method, it can obtain the estimate value of parameter a based on point
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n
range (t, bt ) which bt ln i 1
x (0) (t) ae a e a 1 .By substitution of the estimated value of parameter a into , so can x (0) (i) (e a 1)2
get the value of . Then, take substitution of the value of into traditional GOM (1, 1) Model(2), then obtain the estimated values of develop coefficient a and control coefficient b by using the least squares method. Finally, generate predictor formula through the reduction formula x (0) (k ) x (1) (k ) x (1) (k 1) …(3). In the traditional grey GOM (1, 1) model, the value of parameter a had been twice estimated with simple linear regression method and least square method, and the process of calculation is complicated. So some scholars had put forward the optimized method of estimating the value of parameter, such as reference [5]. And some scholars optimized the background value of traditional grey GOM (1, 1) model in order to improve the simulation and prediction accuracy, such as reference [6]. dx (1) (t) ( x (1) (k ) x (1) (k 1)) dt in a traditional grey GOM In this paper, author considers that the grey derivative is t k (1 )( x (1) (k 1) x (1) (k ))
(1, 1) model, and we can find that
n dx (1) (t) x ( 0) (k 1) (1 )x (0) (k ) f ( x (0) (k )) by x (1) (k ) x (0) (k ) . Then in dt t k i k
the whiten differential equation, the grey derivative is not x ( 0) (k ) but a function f ( x (0) (k )) which takes x ( 0) (k ) for its independent variable. So compared with reduction formula (3) x (0) (k 1) = x (1) (k 1) ‐ x (1) (k ) , the derivative reduction method seems more reasonable than a traditional GOM (1, 1) model. At the same time, the derivative reduction method will be also used for the GOM (1, 1) model with optimized background value. In addition, this article also puts forward a new method of estimating the value of parameter a and b based on least square method, and the process of calculation is relatively simple. The Summary of Derivative Reduction Value Base on GOM (1,1) Model
It attempts to make some improvements based on the traditional GOM (1, 1) model in reference[4] and the GOM (1, 1) model with optimized background value in reference[6], and the effectiveness of derivative reduction method would be shown in results. Optimize the Traditional GOM (1, 1) Model by Derivative Reduction Method.(the Model Generated by the New Method Denoted as GOM’(1,1) Model)
It can obtain the predictor formula of the GOM (1, 1) model by following steps with the method of derivative reduction:
let X (0) x ( 0) (1), x ( 0) (2),..., x ( 0) (n) , x (0) (k ) 0 be original sequence, X (1) x (1) (1), x ( 2) (2),..., x (3) (n) is the first time n
reverse accumulated sequence, and x (1) (k ) x (0) (i) ,k=1,2,…,n. Establishes the GOM (1, 1) grey model based on i k
x (1) (k ) ( denoted as GOM(1,1) model [ 4] ):
( x (1) (k ) x (1) (k 1)) (1 )( x (1) (k 1) x (1) (k )) ax (1) (k ) b
which
, (4)
aea e a 1 ， k 2, 3,..., n . a and b are parameters that awaiting identification. Its whiten differential (e a 1)2
dx (1) (t ) ax (1) (t ) b equation be dt , Then it can obtain predictor formula of the GOM (1, 1) model by following steps. x (1) (0) x (1) (1)
Step1, obtain a and b from the grey differential equation by the least squares method, thus get the value of .If the original sequence has grey exponential law, by (2) it can get
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x (1) (k )
Let 1
e a (0) b x (k ) , a a e 1
ea b , 2 , then a e 1 a
x (1) (k ) 1 x ( 0) (k ) 2 , let k 2, 3,..., n ,thus
x (1) (2) 1 x (0) (2) 2 x (1) (3) 1 x ( 0) (3) 2 ........................
x (1) (n) 1 x ( 0) (n) 2
Based on the least squares method,it can get 1
n
n
n
k 2
k 2
k 2
(n1) x(1) (k )x( 0) (k ) x(1) (k ) x( 0) (k ) n
(n1) x( 0) (k ) k 2
2 n
x( 0 ) (k ) k 2
2
, 2
n
n
k 2
k 2
x(1) (k ) 1 x(0) (k ) n1
^ ^ ^ 1 a ea ea 1 Then a ln , b a 2 , so can calculate ^ 2 a ( ) 1 e 1 1
dx (1) (t ) ax (1) (t ) b based on the value of dt b b , and its discrete response of the whiten differential equation is x (1) (k ) ce ak , k 1, 2,... a a
Step2, obtain the continuous response of whiten differential equation
a and b : x (1) (t) ce ak C is a constant;
Step3, by substituting initial value k 1 ,then x (1) (1) ce a
b b , obtains c ( x (1) (1) )e a . a a
b (Note: The initial value can be x (1) ( j ), j 1, 2,.....n , then c ( x (1) ( j ) )e ja .) a
Step4, obtain the derivative value by the method of derivative reduction, dx (1) (t) ace ak x (0) (k 1) (1 )x (0) (k ) (5), dt t k ^ 1 ae a e a 1 is a constant; so we can get the predictor formula x (0) (k ) ace ak x (0) (k 1) k 2, 3,..., n 2 a 1 (e 1) from (5)
and
Step5, when substituting initial value x (0) (1) , it begins to forecast the results by predictor ^
formula x (0) (k )
1 ace ak x (0) (k 1) , k 2, 3,..., n . 1
Optimize the GOM (1, 1) Model with Optimized Background Value in Reference[6] by Derivative Reduction Method.(the Model Generated by the New Method Denoted as GOM”(1,1) Model)
let X (0) x (0) (1), x (0) (2),..., x (0) (n) ( x (0) (k ) 0 ) be original sequence, X (1) x (1) (1), x ( 2) (2),..., x (3) (n) is the first time n
reverse accumulated sequence and x (1) (k ) x (0) (i) ,k=1,2,…,n. Establishes the optimized GOM (1, 1) model base on i k
(1)
x (k ) ( denoted as GOM(1,1) model
[ 6]
):
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x (1) (k ) x (1) (k 1) b , k 2, 3,...n .Then it can obtain the predictor formula of the GOM (1, 1) ln x (1) (k ) ln x (1) (k 1) model by following steps. x ( 0) (k 1) a
Step1, obtain a and b by the least squares method. x (1) (2) x (1) (1) (1) (1) ln x (2) ln x (1) x (1) (3) x (1) (2) Let B ln x (1) (3) ln x (1) (2) ( 1 ) x (n) x (1) (n 1) (1) (1) ln x (n) ln x (n 1) squares method;
1 x ( 0) (1) ( 0) 1 a a x (2) , P , it can get P ( BT B)1 BT Y by the least , Y b b x ( 0) (n 1) 1
dx (1) (t) ax (1) (t ) b continuous response of whiten differential equation dt Step2, 0btain the based on the value of x (1) (0) x (1) (1)
a and b : x (1) (t) ce ak
b b , and its discrete response of the whiten differential equation is x (1) (k ) ce ak , k 1, 2,... a a
C is a constant;
Step3, obtain the derivative value of the response by the method of derivative reduction, dx (1) (t) ace ak x ( 0) (k 1) (6) dt t k ^
Step4, generate the predicting formula includes the undetermined coefficient C by (6): x ( 0) (k ) ace a(k 1) , k 1, 2,..., n Step 5, by substituting initial value xˆ ( 0) (1) = x ( 0) (1) , obtains c
x ( 0) (1) , so the predicting formula is: a
^
x ( 0) (k ) x ( 0) (1)e a(k 1) . k 1, 2,..., n ^
(Note: the initial value can be x ( 0) ( j ), j 1, 2,.....n , then x ( 0) (k ) x ( 0) ( j )e a(k 1) .) The Examples for the Superiority of New Method Example1:
We use the standard index series x(t ) 0.5e 0.2t ( (t 0,1, 2,..., 7) ) in references [5]as an example. The comparison of precision among the traditional GOM (1, 1) model and the its improved models(See table1). The expression of three models:(1) Model in references[4] is MOD;(2) The improved model of references[5] is GOM1;(3) The new model generated from MOD by the new method in this paper denoted as GOM’(1,1) . Example2:
We take the standard index series x(t ) 10000e 2t (t 1, 2,..., 5) in references [6]for an example. The comparison of precision between the GOM (1, 1) model with optimized background value in reference[6] and its improved models (See table2). The expression of three models:(1) Model in references [6] is MOD2;(2) The new model generated from MOD by the new method in this paper denoted as GOM’(1,1) .(3) The new model generated from MOD2 by the new method in this paper denoted as GOM”(1,1). From the data in Table 1 and Table2, we can find: simulation and prediction accuracy of the improved model is much higher than that of GOM(1, 1) in references [4], [5]and[6].
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TABLE 1 THE PRECISION COMPARISON OF GOM,GOM1 AND GOM’(1,1) MODEL
0.5000 0.4094 0.3352 0.2744 0.2247 0.1839 0.1506 0.1233
GOM Simulated value Relative error % 0.5060 1.21 0.4132 0.94 0.3374 0.67 0.2755 0.40 0.2250 0.13 0.1837 0.14 0.1500 0.40 0.1225 0.67 0.57 GOM1 Simulated value Relative error % 0.4962 0.76 0.4065 0.70 0.3330 0.63 0.2729 0.56 0.2235 0.50 0.1831 0.43 0.1500 0.37 0.1233 0 0.56 GOM’(1,1) Simulated value Relative error % 0.5000 0 0.4095 ‐0.0002 0.3356 ‐0.0011 0.2748 ‐0.0015 0.2252 0.0097 0.1844 ‐0.0028 0.1511 ‐0.0036 0.1238 ‐0.0038
average relative error%
0.0032
original sequence 0.5000 0.4094 0.3352 0.2744 0.2247 0.1839 0.1506 0.1233 average relative error% original sequence 0.5000 0.4094 0.3352 0.2744 0.2247 0.1839 0.1506 0.1233 average relative error% original sequence
TABLE2 THE PRECISION COMPARISON OF GOM2, GOM’(1,1) AND GOM”(1,1) MODEL
original sequence 10000 1353.35 183.16 24.79 3.35 average relative error% original sequence 10000
GOM2 Simulated value Relative error % 10000.17 0.002 1353.38 0.002 183.15 0.006 24.78 0.040 3.15 5.970 1.204 GOM’(1,1) Simulated value Relative error % 0 1.0000* 10 4
1353.35
0.1353* 10 4
‐0.00137
183.16
0.0183* 10 4
‐0.00319
24.79
0.0025* 10 4
‐0.00625
3.35 average relative error%
0.0003* 10 4
original sequence 10000
0.09104 0.0255 GOM”(1,1) Simulated value Relative error % 0 1.0000* 10 4
1353.35
0.1353* 10 4
0.01979
183.16
0.0183* 10 4
0.04196
24.79
0.0025* 10 4
0.06997
3.35
0.0003* 10 4
‐0.05802
average relative error%
0.047
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Conclusions
For the reason that there are many difficult parameter estimation problems in traditional GOM (1, 1) model,the author puts forward a new method of parameter estimation, and it is simple in program and easy to operate. Furthermore, this paper uses derivative reduction method for traditional GOM (1, 1) model and which had been optimized, thus gets new predictor formula. Finally two examples are given to show that the new GOM (1, 1) model in this paper improves the simulation prediction precision, so the practicality and effectiveness of the new method in this paper are verified by the examples. ACKNOWLEDGMENT
[Key research projects] A project supported by youth research fund of China West Normal University (13D018) REFERENCES
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Deng Julong. Grey Predicting and Decision, Huazhong University of Science and Technology Press,Wuhan,China,2002:71‐ 81(In Chinese)
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Wan Qin,Wei Yong. Derivative Reduction Value Base on the GM(1,1) Model with Optimum Grey Derivative [J]. Journal of Grey System 2008,13, (4):181‐186.
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Song Zhongming, Deng Julong. The Accumulated Generating Operation in Opposite Direction and Its Use in Grey Model GOM (1,1) [J]. Systems Engineering. 2001,1, (19) : 66‐69
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Liu Jinying, Yang Tianxing, Wang Shuling. Directed method for computing parameters of GOM(1,1) [J]. Journal of JiLin University(Engineering and Technology Edition) 2003,4(33): 75‐79
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Yang Zhi,Ren Peng, Dang Yaoguo. Grey opposite‐direction accumulated generating and optimization of GOM(1,1) model[J]Systems Engineering‐Theory & Practice 2009,8,(29):160‐164
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