The optimized white Differential equation of GM(1,1) based on the. original grey differential equation. Rui Zhou Jun-jie Li Yao Chen

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1 pplied Mechanics and Materials Online: ISSN: , Vols , pp doi:0.4028/ 202 Trans Tech Pulications, Switzerland The optimized white Differential equation of GM(,) ased on the original grey differential equation Rui Zhou Jun-jie Li Yao Chen Department of Mathematics and Information China West Normal University, Nanchong, Sichuan, P.R.China Key words: GM (,); original grey differential equation; equivalent; white differential equation; optimization stract: This paper starting from the original grey differential equations, through finding the relationship etween the raw data ( k ) and the derivative of its GO, constructed a new white differential equation which equal to the original grey differential equation, at the same time, getting the new GM(,)model which closer to the changes of data. Through the modeling and prediction of the standard inde series, this model not only adapts to low growth inde series, ut also adapts to high-growth inde series, and the simulation accuracy and prediction accuracy are high. INTRODUCTION 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 een widely used in various areas of national product []. s the important part of the grey system theory, the GM(,) model has ecome the research focus, many scholars have further investigated in improving the model s precision and widening the model s applicale scope [3-0]. But when the white differential equation founded, Mr Deng Ju-long made that:" GM(,) white model itself and all results derived out from the white model just estalish 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 ( k ) and the derivative of its GO, constructed a new kind white differential equation which equaled with the original grey differential equation, at the same time, getting the new GM(,)model which closer to the changes of data. THE OPTIMIZTION OF GM(,) MODEL LetX = { (), (2), } is the original series, and the GO series of X is = { (), (2), }, among them () () () () X n = k () ( k) ( i) i= ll rights reserved. No part of contents of this paper may e reproduced or transmitted in any form or y any means without the written permission of Trans Tech Pulications, (ID: , Pennsylvania State University, University Park, US-09/04/6,04:0:54)

2 2972 Progress in Structures Theorem When () t ( t) = Ce + D (among them, C, D are all the constants), the original grey differential equation () + a = and the white differential equation ( e ) e dt () d + () a = equivalent. () So ased on the original grey differential equation + a =, getting the optimized GM (,) model as follow: ( 0) ( 0 Theorem 2 Let ) ( 0 ) ( 0 {, ( 2 ),, ) } = e nonnegative pre-smooth series, ( ) is GO series of ( 0),and () t ( t) = Be + C (among them, C, D are all the constants). If T aˆ = a, e the parameter of the equation, and ( 2) ( 3) 2 ( ) (3) Y =, B=, then the ( 0 ) ( ) parameters of the least square estimate in the grey differential equation T T a = T B B B Y a =(, ),then ()The new continuous solution of the white equation is a ( t ) e () e ( t) = [ () ] e + ; (2)The new discrete solution of the white equation is a ( k ) e () e ( k) = [ () ] e +,k = 2,3,..; (3)The restore value is 0 k k k + = +,k = 2,3, ; THE PROCESS OF MODELING OPTIMIZED GM(,) MODEL The precondition of estalishing optimized GM (,) model is () t ( t) = Ce + D (among them, C,D are undetermined constants )that means the original data must e epressed y ( k ) ( k) = Pe (among them P is a constant ). Under this precondition, e = k k is always a ( ) constant. But in the practical application of grey system theory, the original datre the similar inde series and e i = i i prolem, the model must e optimized again. is not a constant. So the new model is not practical. To this ( )

3 pplied Mechanics and Materials Vols If the original series is X n = { (), (2), },then GO series of X is () () () () X n k =. i= () = { (), (2), },among them ( k) ( i) Let e i = i i,i=2,3 n,then ( ) i ( i) = ln, i=2,3 n. nd e i, i replace the ( i ) e and which in theorem 2. Now getting the new optimized GM (,) as follow: { } Theorem 3 Let ( 0) ( 0 ) ( 0 =, ( 2 ),, ) e nonnegative pre-smooth series, ( ) e the GO series of ( 2) ( 3) ( ) ( 2) ( ) Y =, (3) B= ( 0 ) ( ) differential equation a =(, ) ( 0). If a = (, ) T a the parameter of the equation, and, then the parameters of the least square estimate in the grey T T a = T B B B Y,then ()The new continuous solution of the white equation is ( t) a ( t ) ( t) ln ( t ) () ( t) ( t ) ( t) = [ () ] e + ; (2)The new discrete solution of the white equation is ( k ) a ( t ) ( k ) ln ( k ) ( 0) () ( k ) ( k ) ( k) = [ () ] e +,k = 2,3, n; (3) ccording to The principle of new information priority getting the prediction a ( t ) ( k ) ln ( n ) equation: () ( n ) ( k+ ) = [ () ] e +, k = nn, +. (4)The restore value is 0 k k k + = +,k = 2,3, ; COMPRISON OF THE PRECISION OF DT SIMULTION Take ( ) { } k+ = e ak as an eample, and then we have ( 0 = (), (2), ), 2,3 6 i i i i k=, nd use the original GM (,) model as M, model of reference [2] as M2and model of this paper as M3 to predict.

4 2974 Progress in Structures Tale : The Original Series a i i () i (2) i (3) i (4) i (5) i (6) Tale 2: Comparison of the Simulation Precision -a verage error % M M M M M M M M M M M M M M M From Tale 2 and Tale 3, when the original series is a low growth series, the simulation accuracy and prediction accuracy of all models are high, ut as the development coefficient ecome larger, the he simulation accuracy and prediction accuracy of original model ecome lower. Especially, when a > 2, the error is 99%! lthough the simulation accuracy and prediction accuracy of M2 is higher than M, the overall effect is not very ideal. But for the model of this paper, even if a > 2, the simulation accuracy and prediction accuracy are all higher than 99.99%, the effect is good. a M step error M2 step error M3 step error M 2step error M2 2step error M3 2step error

5 pplied Mechanics and Materials Vols M 5 step error M2 5 step error M3 5 step error M 0 step error M2 0 step error M3 0 step error Conclusions In this paper, starting from the original grey differential equations, through finding the relationship of the raw data ( k ) and the derivative of its GO, constructed a new kind white differential equation which equal with the original grey differential equation, at the same time, getting the new GM(,)model which closer to the changes of data. Through the modeling and prediction of the standard inde series, this model not only adapts to low growth inde series, ut also adapts to high-growth inde series, and the simulation accuracy and prediction accuracy are almost 00%. t the same time, the models remains simple calculation steps of the original GM (, ) model, and roaden the scope of application of the model. Key research projects project supported y Scientific Reserch Fund of SiChuan Provincial Education Department(2ZB47) and fund of China West Normal University (005) REFERENCES []Liu Si-feng, Dang Yao-guo, Fang Zhi-geng(2004). The Grey System Theory and Its pplication [M]. Science Press. Beijing. [2] Deng Ju-long (2002). Estimate and Decision of Grey System. Huazhong University of Science& Technology Press, Wuhan. [3] Luo Dang, Liu Sifeng, Dang Yaoguo(2003). The Optimization of Grey Model GM(,).Engineering Science, 8, 50~53 [4]Wang Hong. Optimized GM(,)model and Its Prediction ccuracy nalysis[j]. Statistics and Decision 2008,23: [5] Dong Fen-yi, TIN Jun.Optimization integrated ackground value with original condition for GM(,)[J]. Systems Engineering and Electronics Vol.29. No [6] Wang Yinao, Liu Kaidi, Li Yingchuan (200). GM(,) modeling method of optimum the whiting values of grey deriative. Systems Engineering-theory & Practice, 5, 24~28. [7]Hu Da-hong, Wei Yong. pplicaility of Grey model to the monotonic decreasing sequences and the approimate estimate of parameters[j].systems Engineering and Electronics :6-20. [8] Liu Sifeng, Deng Julong (2000). The range suitale for GM (,). Systems Engineering-theory & Practice, 5, 2~24. [9]Dang Yao-guo, Liu Si-feng. Liu Bin. The GM Models that () (n) e Taken as Initial Value[J]. Chinese Journal of Management Science. 2005,2():

6 Progress in Structures / The Optimized White Differential Equation of GM(,) Based on the Original Grey Differential Equation /