Analysis on the Strengthening Buffer Operator Based on the Strictly Monotone Function

Transcription

1 International Journal o Applied Physics and Mathematics Vol. 3 No. March 03 Analysis on the Strengthening Buer Operator Based on the Strictly Monotone Function Hu Xiao-Li Wu Zheng-Peng and Han Ran Abstract We construct our kinds o new strengthening buer operators by using inverse unction theorem based on the aiom system o buer operator. And demonstrate the GuanShi strengthening buer operator which we compare with is a special case o our new operators. Ater studying the inner link and characteristics between the GuanShi and our new buer operators we greatly develop the application scope o strengthening buer operator. This paper researches on buer operators construction with unctions and gives a new direction or construction o buer operators. Inde Terms Strengthening buer operator grey system monotone increasing unction. also slows down the development trend o data or make data sequence o smaller amplitude. In order to eclude the intererence o these actors reerences [4]-[8] put orward a practical weakening buer operator and reerences [9]-[4] construct strengthening buer operators. In this paper we propose a new strengthening buer operator and study its characteristics and internal relationship on the basis o the above work according to the buer operator three aioms and thoughts o the average development rate o generalized time series. It is irst time to research on buer operators construction with unctions and gives a new direction or construction o buer operators. I. INTRODUCTION Both or eperimental or statistical data we must analyze the data beore modeling. Otherwise there will be not consistent between quantitative prediction and qualitative analysis conclusion. The cru o the problem doesn t depend on the advantages or disadvantages o the selected models. System behavior data is distorted by some eternal impact because o itsel. Thereore seeking combining site between quantitative orecast and qualitative analysis trying to eliminate shock intererence o system data restoring the true data so as to improve the accuracy o prediction is an important task or each prediction workers. Grey system theory is a kind o theory systems which ind the law rom the data itsel. It seeks the change rule through original data mining and arrangement based on society economy and ecology system and so on. Although the representation o objective system and data are comple the data has a whole unction. Thereore it contains some inherent law. The key lies in how to choose the appropriate approaches to mining and use it in the grey system theory. Liu SF proposed buer operator theory which eliminate the impact o eternal shock intererence successully through the sequent ways o generation weakening its randomness and showing its regularity o the data sequence []-[3]. And it could get the data sequence relecting the changing regular patten o system. Intererence rom shock disturbed actors to system behavior data sequence will speed up the development trend o data or make the amplitude wider. On the other hand it Manuscript received October 5 0; revised January This work was supported in part by Communication University o China under Grant XNL3. The authors are with School o Science Communication University o China Beijing 0004 China ( huiaoli@cuc.edu.cn wuzhengpeng@6.com hanran@cuc.edu.cn). II. CONSTRUCTION OF STRENGTHENING BUFFER OPERATOR Liu SF Dang YG & Guan YQ etc. construct strengthing buer operator as ollows. Set X ( () () ( n)) to be a system data sequence. Set among XDi ( () di ( n) di ) i 34 ( k ) ( k) d ( k) ( n) nk ( n k ) ( k) ( k) d ( k) ( n) ( k ) ( k) d 3 [ ( k) ( n)] nk ( k ) ( k) d 4 nk ( k) ( n) ( k n) Then when X is a monotone increasing sequence monotone attenuation or oscillation sequence D D D 3 D 4 are strengthening buer operators. Theorem : Set X ( () () ( n)) non-negative system behavior data sequence i ( ) 0 then ( k) d ( k) d ( k) d3 ( k) d4. I and () () (3) (4) DOI: /IJAPM.03.V3.9 3

2 International Journal o Applied Physics and Mathematics Vol. 3 No. March 03 only () () ( n) all the equals sign was established. Proo: see reerence [3]. Here we use the monotone unction theory to construct a new strengthening buer operator based on the strengthening buer operators D D D 3 D 4. Theorem : Set X ( () () ( n)) non-negative system behavior data sequence and ( i) 0 0. is a strictly monotone increasing unction whose inverse unction is g. Among them ) I X is a monotone attenuation sequence because o ( k) ( n) 0 we get ( ( k)) 0 ( ( )) k ( ( k )) ( nk) ( ( k)) ( ( k)) ( nk) ( ( k )) ( k) e g{ } ( ( k)) nk (5) ( ( k )) ( ( k)) ( ( k)) nk Set XE ( () e ( n) e ) Then when X is a monotone increasing sequence monotone attenuation sequence or oscillation sequence E is the strengthening buer operator. Proo: Easy to veriy that ( ( n )) ( n) e g{ } ( n) ( ( n )) nn So E meets the buer operator ied point aiom. And it also meets the buer operator aiom o ully using inormation and the analytic change standardization aiom. So E is a buer operator. Because is a strictly monotone increasing unction ) I X is a monotone increasing sequence because o 0 ( k) ( n) so 0 ( ( k)) 0 ( ( k)) 0 ( ( k)) ( ( k)) ( nk) ( ( k)) ( nk) ( ( k )) 0 ( ( k)) ( ( k)) nk ( ( k )) ( k) e g g( ( ( k))) ( k) ( ( k)) nk thereore E is a strengthening buer operator. (c) i X is an oscillation sequence set ( k ) ma in ( i ) ( h ) min i n ( i ) ( k) ( k) ( n) ; ( h) ( h) ( n) ( ( k)) ( ( k)) 0 0 ( ( h)) ( ( h)) ( ( )) k ( ( k )) ( nk) ( ( k)) ( ( k)) ( nk) ( ( k )) ( ( k)) ( ( k)) nk ( ( k )) ( k) e g g( ( ( k))) ( k) ( ( k)) nk ( ( k )) ( k) e g g( ( ( k))) ( k) ( ( k)) nk thereore E is a strengthening buer operator. 0 ( (h)) 0 ( (h)) ( (h)) ( nh) ( (h)) ( nh) 33

3 International Journal o Applied Physics and Mathematics Vol. 3 No. March 03 ( ( h )) 0 ( (h)) ( ( h)) nh ( ( h )) ( h) e g g( ( (h))) (h) ( ( h)) nh ma in ( i) ma i n ( i) e min in ( i) min i n ( i) e thereore E is a strengthening buer operator. Theorem 3: Set X ( () () ( n)) non-negative system behavior data sequence and ( i) 0 0. is a strictly monotone increasing unction whose inverse unction is g.among them ( k) g{ ( ( k )) nk } ( ( k)) Set XE4 ( () ( n) ) I X is a monotone increasing sequence attenuation or oscillation sequence E4 is the strengthening buer operator. Proo: easy to veriy ( ( n )) ( n) g{ } g( ) ( n) nn So E 4 meets buer operator ied point aiom. And it also meets the buer operator aiom o ully using inormation and the analytic change standardization aiom. So E 4 is a buer operator. Because is a strictly monotone increasing unction ) I X is a monotone increasing sequence because o 0< ( k) ( n) so 0 ( ( k)) 0 ( ( k)) 0 ( n k) ( ( k)) ( ( k)) ( nk) ( ( k)) 0 ( ( k)) 0 ( ( k )) nk ( ( k)) ( ( k)) ( k) g{ ( ( k )) nk } g( ( ( k))) ( k) ( ( k)) thereore E4 is a strengthening buer operator. ) I X is a monotone attenuation sequence because o 0< ( k) ( n) so ( ( k)) 0 0 ( ( k)) ( n k) 0 ( ( k)) ( ( k)) ( nk) 0 ( ( k)) ( ( k)) ( ( k )) ( ( k)) 0 nk ( ( k)) ( k) g{ ( ( k )) nk } g( ( ( k))) ( k) ( ( k)) thereore E4 is a strengthening buer operator. 3) i X is an oscillation sequence set ( k ) ma in ( i ) ( h ) min i n ( i ) ( k) ( k) ( n) ; ( h) ( h) ( n) ( ( k)) ( ( k)) 0 0 ( ( h)) ( ( h)) 0 ( ( k)) ( n k) 0 ( ( k)) ( ( k)) ( nk) 0 ( ( k)) ( ( k)) ( ( k )) ( ( k)) 0 nk ( ( k)) ( k) g{ ( ( k )) nk } g( ( ( k))) ( k) ( ( k)) 34

4 International Journal o Applied Physics and Mathematics Vol. 3 No. March 03 0 ( ( h)) 0 ( n h) ( ( h)) ( ( h)) ( nh) ( ( h)) 0 ( ( h)) 0 ( ( h )) nh ( ( h)) ( ( h)) ( h) g{ ( ( h )) nh } g( ( ( h))) ( h) ( ( h)) ma in ( i) ma i n ( i) min in ( i) min i n ( i) thereore E4 is a strengthening buer operator. Theorem 4: Set X ( () () ( n)) non-negative system behavior data sequence and ( i) 0 0. is a strictly monotone increasing unction whose inverse unction is g.among them ( n k ) ( ( k)) ( k) e g{ } ( ( k )) ( ( n )) Set XE ( () e ( n) e ). I X is a monotone increasing sequence attenuation or oscillation sequence E is the strengthening buer operator. Proo: see reerence [4]. Theorem 5: Set X ( () () ( n)) non-negative system behavior data sequence and ( i) 0 0 is a strictly monotone increasing unction whose inverse unction is g. Among them ( ( k )) ( k) e3 g{ } [ ( ( k)) ] nk Set XE3 ( () e3 ( n) e3 ). Then when X is a monotone increasing sequence monotone attenuation or oscillation sequence E 3 is the strengthening buer operator. Proo: see reerence [4]. Theorem 6: Set X ( () () ( n)) non-negative system behavior data sequence and ( i) 0 0. is a strictly monotone increasing unction whose inverse unction is g. Then ( k) e ( k) e ( k) e3 ( k). I and only i () () ( n) all the equals sign (6) (7) was established. Proo: see the theorem and g is strictly monotone increasing unction so we get the result. The results above are also established or weight vector w ( w w n ) w i 0. The deducing process is similar. I ( ) g( ) buer operators E E E 3 E 4 are the strengthening buer operators separately in reerence [4]. D D D 3 D4 are the strengthening buer operators. And D D D 3 D4 are the special cases. III. APPLICATION We set the system data sequence based on the monthly sales rom January to July o the new product in reerence [3] (see Table I) to eplain the application o the strengthening buer operators in process o data modeling prediction. The unit o sales in the table is ten million yuan. TABLE I: MONTHLY SALES OF NEW PRODUCT Month Sales According to the Table I we calculated the growth rate o monthly sales separately were.96% 4.95% 7.% 9.94%.0% and.65%. It is easy to see that the irst hal o them grew ecessively slower than the latter. We eliminated the intererence rom the impacting disturbance actors to the original data sequence. Take ( ) g( ) to construct the buer operator and get the calculation results are as ollows: When no buer operator to eect ( k ) e 0.084k 693.9; When buer operator D to eect ( k ) e 0.5k ; When buer operator D to eect ( k ) e 0.7k ; TABLE II: GM () MODEL IN THREE CONDITIONS Buer Operator Predicted Value Prediction Relative Error(%) Non D Take D ( ) g( ) 0.5 to construct the buer operator and get the calculation results are as ollows: When buer operator E to eect ( k ) e 0.65k ; When buer operator E to eect ( k ) 359.6e 0.3k ; 35

5 International Journal o Applied Physics and Mathematics Vol. 3 No. March 03 TABLE III: GM() MODEL IN THREE CONDITIONS Buer Operator Predicted Value Prediction Relative Error(%) Non E E IV. CONCLUSION Based on the researches o others sequence o the ront part grow or decay too slowly while the latter too quickly. It makes inconsistent results o quantitative prediction and qualitative analysis. The new strengthening buer operators could solve the problems eectively in the process o constructing models to predict. In the process o constructing buer operators we used to construct one by one. This is the irst time to connect unction with buer operators to construct one broad class o buer operators. It supplies several choices to resolve the problems in shock data sequence modeling and gives the new direction to construct buer operators by unction. But research o how to optimize is underway. [5] N.-M. Xie and S.-F.Liu A New Weakening Buer Operator. Chinese Journal O Management Science 003 vol. pp [6] Y.-G. Dang S.-F. Liu and B. Liu The Research On Weakening Buer Operators Chinese Journal O Management Science 004 vol. no. pp [7] Y.-Q. Guan and S.-F. Liu Study On Weakening Buer Operator Sequence Chinese Journal O Management Science 007 vol. 5 no. 4 pp [8] J. Cui and Y.-G. Dang A Kind O New Weakening Buer Operators Structure And Its Application Control And Decision 008 vol. 3 no. 7 pp [9] Y.-G. Dang B. Liu and Y.-Q. Guan The Research On Strengthening Buer Operators Control And Decision 005 vol. 0 no. pp [0] N.-M. Xie and S.-F. Liu Strengthening Buer Operator's Characteristics And Some Practical Construction O Strengthening Operator Statistics and Decision 006 no. 4 pp [] Y.-Q. Guan and S.-F. Liu Study On Strengthening Buer Operator Sequence And m Order Operator Role Journal O Yunnan Normal University 007 vol. 7 no. pp [] Y.-G. Dang S.-F. Liu and C.-M. Mi Research On Strengthening Buer Operators Characteristics Control And Decision 007 vol. no. 7 pp [3] Y.-Q. Guan and S.-F. Liu Based on the ied point o strengthening buer sequence operator and its application Control And Decision 007 vol. no. 0 pp [4] Z.-P. Wu S.-F. Liu and C.-M. Mi Study On The Strengthening Buer Operators Based On The Strictly Monotone Function Journal o Grey System 008 vol. no. pp REFERENCES [] S.-F. Liu Y.-G. Dang and Z.-G. Fang Grey System Theory And Its Application Beijing: Science Press 004. [] S.-F. Liu The Three Aioms O Buer Operator And Their Applications The Journal o Grey System 99 vol. 3 no. pp [3] S.-F. Liu Buer Operator and Its Application Grey System Theory And Practice 99 vol. no. pp [4] S.-F. Liu Shock Disturbed System Prediction Trap And Buer Operator Journal O Huazhong University O Sci. & Tech. 997 vol. 5 no. pp and Econometrics. Hu Xiaoli was born on 4th October 978 in Yidu city Hubei province. Now she is a PhD candidate majoring in Economics at Beijing Normal University. And also she is a lecturer at Communication University o China. She is a lecturer o Department o Statistics in School o Science at Communication University o China in Beijing rom Sep. 005 to present. She is interested in Financial Engineering 36