IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 3, November 0 ISSN (Online: 694-084 www.ijcsi.org 443 Improved grey derivative of grey Verhulst model and its application Yi-Zhang College of Mathematics and information institute, China West Normal University, Nanchong 63700,China Abstract Based the principle and characteristic of grey Verhulst mode, the cause of grey Verhulst model s inaccuracy is analyed, the new formula of grey derivative is strutted and the unbiased grey Verhulst model is given in this paper. he new modeling method improves the simulation precision and etends the application scope of grey. Some eamples are also given to show that the precision of the new model is very high. Keywords: grey Verhulst model, grey derivative, unbiased, application. Introduction he grey model applied in many areas since it was introduced by professor Deng. As this model needs fewer samples and the computation is simple, it has superiority than the traditional estimate method. As the grey Verhulst model modeling by the -AGO series, the scope of the traditional Verhulst model has been epanded to the series of approimal single pe. If the original series is an s-type form, and the sie of sample is small, we can build a grey Verhulst model directly by the original series. We can see that the grey Verhulst model is superior to the traditional Verhulst model in the terms of scope, hence, the grey Verhulst model has been widely applied in recent years. Many scholars have conducted a study on grey Verhulst model, mainly on the several improvements to the model, the optimiation of modeling, the optimiation of grey derivative, the optimiation of background value, the optimiation of initial value and the optimiation of calculation method of parameters, so as to further improve the accuracy and scope of models. Based the principle and characteristic of grey Verhulst mode, the cause of grey Verhulst model s inaccuracy is analyed, the new formula of grey derivative is strutted and the unbiased grey Verhulst model is given in this paper. he new modeling method improves the simulation precision and etends the application scope of grey.. he definition of grey Verhulst model Let, (,, ( n be nonnegative series,, (,, ( n - AGO series of and of k i ( i, k,,, n ( (, (3,, ( n MEAN series and. Copyright (c 0 International Journal of Computer Science Issues. All Rights Reserved.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 3, November 0 ISSN (Online: 694-084 www.ijcsi.org 444 0.5( ( k, k,3,, n [Key research projects] A Project Supported by Scientific Research Fund of China West Normal University (A07 Definition : Let - AGO series of, we call a b( ( k MEAN series of, differential equation of grey Verhulst model. Definition : Let - AGO series of d dt a b( equation of grey Verhulst model. heorem : Let - AGO series of, a,b and Let Y grey, we call whitened differential MEAN series of parameter of the equation. ( (3 ( n ( (3, B ( n ( (3 ( n hen, from to the method of least square, we ˆ ( B B B Y have ; he continuous solution of the equation is ( k b a ( a b ˆ ( e 3 he discrete solution of the equation is a ( k k,3,... ; ( b ( a b e ˆ 4 he original value is ˆ ( ( k ˆ ( k ˆ,,... k. 3. he construction of grey derivative From theorem, we know that based on the whitened d differential equation a b(, professor dt Deng derived the grey differential equation k a b( ( by replacing the derivative of discrete series at point k by the difference (that is: d ( k dt tk and the background value by (. However, this kind of approimate method caused much great simulation of grey Verhulst model. hus, now to deal with the derivative of discrete series plays a key role in improving the simulation precision of grey Verhulst model. Based on the continuous solution of the equation get ˆ ( k b ˆ ( k b b ( e a a reasonable that the -AGOseries a ( a b k e, we, so we think it is is a countdown form of a homogeneous eponential series. herefore, let. It suffices to consider the Be C discrete variable k as a continuous one and let the derivative of series grey derivative of the grey Copyright (c 0 International Journal of Computer Science Issues. All Rights Reserved.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 3, November 0 ISSN (Online: 694-084 www.ijcsi.org 445 Verhulst model. he improved grey derivative is: d dk Be C A( C BAe ( Be C A k C k A ( C ( ( ( We only need to find the value of A and C, now, establishing the following equations: A( k Be ( k Be C A( k Be ( k From (3 ( ( C C ln ( k ( k A ( ( k From ( we have: then: C Be ( C ( k ( (3 we have: C, put it into, k ( k e ( e A A ( k ( k ( ( k ( k ( k So, he improved grey derivative is: d dk,and ln ( ( ( ( k k k k ( ( ( ( ( ( ( k ( k k k k k k ( k ( k ( k ( k ( k ( k ( ( ln ( ( k k k k ( k ( k ( k ( k ( k ( k ln ( k heorem : Let - AGO series of the equation. Let ( (3 B ( n and a,b ( (3 Y ( n ( (3 ( n parameter of hen, from to the method of least square, we have ˆ ( B B B Y ; 3 he continuous solution of the equation is ( k b a ( a b ˆ ( e 3 he discrete solution of the equation, is ˆ a ( k k,3,... ; b ( a b e Copyright (c 0 International Journal of Computer Science Issues. All Rights Reserved.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 3, November 0 ISSN (Online: 694-084 www.ijcsi.org 446 5 he original value is ˆ ( ( k ˆ ( k ˆ k,,.... heorem 3: he improved grey derivative of grey is unbiased(that is :simulate s-curve is completely coincidence by this model Proof: let -AGO series be ( k=, De C,, n hen d A ( C dk From heorem, we know ( (3 Y ( n ( (3 B ( n And B ( BB n k B n k ( (3 ( n 3 ˆ ( B n k n k B 4 3 B n n n 4 3 k k k n n 4 3 k k n n 3 k k ( Y,and, n k B Y n k So ˆ ( BB BY n n n 4 3 k k k n n n n 4 4 k k k k n n n n k k k ( put 3 k d A ( C dk into a A the above equation, we get ˆ, from b AC theorem, b b ( e a a ( Ce C De C he improved grey derivative of grey is unbiased 4. Data simulation and application Eample : [] (,.08909, 3.66633, 3.07 he simulation of traditional grey is: ˆ ( k (0.30399 0.69670e.853493k he simulation of improved grey derivative of grey is: Copyright (c 0 International Journal of Computer Science Issues. All Rights Reserved.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 3, November 0 ISSN (Online: 694-084 www.ijcsi.org 447 ˆ ( k (0.086364 0.93636e.347845k From table, we know: the simulation of traditional grey Verhulst model with less precision, the average of fitting accuracy is only 86%, the simulation of improved grey derivative of grey Verhulst model consistent with the original data, this model eliminates the model s of traditional grey Verhulst, is unbiased. Cases, this optimiation is reasonable, effective. Eample : the torpedo development cost in 995-003 for eample he simulation of traditional grey is: ˆ ( k (0.0005 0.007936e.0075k he simulation of improved grey derivative of grey is: ˆ ( k (0.00035 0.00797e able : Comparison of the Simulation Precision.06787k data traditional grey improved grey Simulated value Relative (% Simulated value Relative (% 0 0.08909.43076 5.99.08909 0 3.66633 3.0344-4.89 3.66633 0 3.07 3.680 8.5 3.07 0 year Cumula tive costs able : Comparison of the Simulation Precision traditional grey improved grey Simulat ed value Relati ve (% Simulat ed value Relati ve (% 995 496 496 0 496 0 996 75 4.7 0.37 35.6 3.09 997 46 74..69 333.7 5. 998 3487 337. 7.6 3360.0 3.64 999 3975 3937.8.94 404.3-0.99 000 430 474.3 -.05 4304.3 -.76 00 4387 44.4-0.56 444.5-0.63 00 4497 4463.5 0.74 4454.0 0.96 003 4584 448.8. 4467.8.54 From table, we know: the average of fitting accuracy of traditional grey Verhulst model is only 86%, and the maimum relative has reached.69%. the simulation of improved grey derivative of grey Verhulst model have been enhanced, the average of fitting accuracy of it is 98%, and the maimum relative is only 5.%. Cases, this improved gray Verhulst model enhanced the fitting accuracy, and the optimal method have higher application value. 5. Conclusion Based the principle and characteristic of grey Verhulst mode, the cause of grey Verhulst model s inaccuracy is analyed, the new formula of grey derivative is structed and the unbiased grey Verhulst model is given in this paper. he new modeling method improves the simulation precision and etends the application scope of grey. References [] Deng Julong (00. Estimate and Decision of Grey System. Huahong University of Science& echnology Press, Wuhan. (In Chinese [] Wang Zheng-in, Dang Yao-guo, Liu Si-feng, Unbiased grey Verhulst model and its application.j. Systems Engineering-theory & Practice, 009,9(0:38-44. [3] Dai Wen-han, Xiong wei, Yang Ai-ping, Improved grey Verhulst model and its application. J.CIESC Journal, 00,6 (8:097-00. [4] Chen Lu, Zhang Ling-shuang, Combination grey Verhulst model based on initial value modification.j. Mathematics in Practice and heory, 00,40(:60-64. Copyright (c 0 International Journal of Computer Science Issues. All Rights Reserved.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 3, November 0 ISSN (Online: 694-084 www.ijcsi.org 448 [5] Wang Zheng-in, Dang Yao-guo,Shen Chun-guang, Improvement on derivative item in grey Verhulst model, J.Statistics & Information Forum, 00,5(6:9-. [6] an Guanjun (000. he Structure Method and Application of Background Value in Grey System GM(, Model (I.J. Systems Engineering-theory & Practice, 4, 98-03(In Chinese [7] Zhang Yi, Wei Yong, Xiong Chang-wei, one new optimied method of GM (, model. J.Systems Engineering-theory & Practice, 007, 4: P4-46 [8] Ping Zhou, Yong Wei, he Optimiation of Background Value in Grey Model GM(,,J.Journal of Grey System (W,006,. 39-4 [9] Yi Zhang,Yong Wei, Ping Zhou,he Improved Approach of Grey Derivative in GM(, Model,J.he Journal of Grey System(UK006, 4: P375-380. [0] Yi Zhang,Yong Wei, Chang-wei Xiong, An Approach of GM(, Based on Optimum Grey Derivative,J.he Journal of Grey System(UK007.Vol.9 No.4. [] Yi Zhang,Chang-huan Feng,Research on the Background Value of Non-equidistance GM(, Model,J.International Confeience on Computer Application and System Modeling,00,0,P: 60- -6 Zhang Yi received the degree of master of science in 007. Now he is working in the College of Mathematics and Information of China West Normal University. And he has a project supported by scientific research fund of China West Normal University (A07. He has published papers more than a doen. Now, his research is on grey system, time series analysis, and the uncertainty information system analysis. Copyright (c 0 International Journal of Computer Science Issues. All Rights Reserved.