Sesors & Trasducers, Vol. 59, Issue, November 0, pp. 7-76 Sesors & Trasducers 0 by IFSA http://www.sesorsportal.com New Expoetial Stregtheig Buffer Operators ad Numerical Simulatio Cuifeg Li, Huajie Ye, Zhegguo Weg Zhejiag Busiess Techology Istitute, No. 988, Jichag Road, Nigbo, Zhejiag 50, Chia Tel: 5768958 E-mail: cuicui07@hotmail.com Received: September 0 /Accepted: 5 October 0 /Published: 0 November 0 Abstract: Based o the theory of buffer operators i the grey system, some ew expoetial stregtheig buffer operators are established, ad the relevat theorems are proved accordig to the axiomatic of buffer operators. The problem that there are some cotradictios betwee quatitative aalysis ad qualitative aalysis i pretreatmet for vibratio data sequeces is resolved effectively. A example simulatio shows compared with the existig stregtheig buffer operators, the id of ew stregtheig buffer operators icreases the forecast precisio of GM (, ) remarably. Copyright 0 IFSA. Keywords: Grey system, Buffer operator, Expoet, Simulatio, Expoetial.. Itroductio The grey system theory has bee caught great attetio by researchers sice 98 ad has already bee widely used i may fields, such as idustry, agriculture, zoology, maret ecoomy ad so o. GM (, ) has bee high improved by may scholars from home ad abroad. The grey system theory ca effectively deal with icomplete ad ucertai iformatio system. Although the data of the objective system is scattered, they always have their overall fuctio, ad ievitably cotais a certai rule. The ey is how to choose the appropriate methods to miig ad utilizatio. Professor Liu Sifeg put forward the cocept of impact disturbace buffer operator, ad costructs a id of buffer operator extesively. The weaeig buffer operator ad stregtheig buffer operator are researched deeply by some scholars. The cocept of buffer operator is put forward, ad some widely used buffer operator is structured by paper [- 5]. The properties of stregtheig buffer operator are studied by paper [6-7]. A ew stregtheig buffer operator x( ) + x( ) d = x( ) is put forward by x( ) paper [8], the average stregtheig buffer operator, weighted average stregtheig buffer operator ad the geometric mea stregtheig buffer operator are structured. At last, electricity cosumptio per capita is costructed through two order ehacemet treatmet by this method. It has improved the precisio of fittig ad predictio accuracy. A ew x( ) buffer operator x( ) d = x( ) is x ( ) x( ) obtaied by paper [9], ad it is used to simulate to examples. The result shows good predictio accuracy of GM (, ) model. Several ids of variable weight stregtheig buffer operator are preseted by paper [0], the problem of traditioal buffer operator which iteract too strogly or too wealy is solved. Article umber P_57 7
Sesors & Trasducers, Vol. 59, Issue, November 0, pp. 7-76. Several Cocepts Defiitio Suppose that X = { x(), x(),..., x( ) } is a o-egative data sequece, we ca obtai that: ) If for =,,...,, there exists x ( ) < x ( + ), the data sequece X is defied as a mootoe icreasig sequece; ) if for =,,...,, there exists x ( ) > x ( + ), the data sequece X is defied as a mootoe decreasig sequece; ) if for ', {,,..., }, there exists ' ' x ( ) < x ( + ), x ( ) < x ( + ) the data sequece X is defied as a oscillatory sequece. M = max{ x ( ),,..., } m = mi{ x( ),,..., } The value of M-m is called the amplitude of sequece X. Defiitio Suppose that X is a o-egative data sequece, D is the operators of X, the data sequece which acts by D is gotte as follows. = ( x() d, x() d,..., x( ) d) The, D is called sequece operators. The data sequece is acted cotiuously, two step operator, three step operator, eve r step operator are gotte as follows. r,,..., Axiom (Fixed-poit axiom) Suppose that X is a o-egative data sequece, D is the sequece operators, the D meets that: x( d ) = x ( ) Axiom (Full use of iformatio axiom) Each data x( )( =,,..., ) i system behavior data sequece X should be the whole process fully participate i the operator fuctio. Axiom (Aalytical, ormative axiom) Ay x( ) d( =,,..., ) ca be expressed by a uified elemetary aalytic type x(), x(),..., x( ). The three axioms above are called three axioms of Sequece buffer which satisfies three axioms of buffer operator is defied as Defiitio Suppose X = { x(), x(),..., x( ) } is a o-egative data sequece, the X D = ( x() d, x() d, is..., x ( ) d ) called ) Whe X is a mootoe icreasig sequece, D is stregtheig buffer operator xd ( ) x ( ), =,,,...,. ) Whe X is a mootoe decreasig sequece, D is stregtheig buffer operator xd ( ) x ( ), =,,,..., ) Whe X is a oscillatory sequece, D is stregtheig buffer operator { x } { x mi { x( ) } mi { x( ) max ( ) max ( ) From above, the data of mootoe icreasig sequece uder stregtheig operator is shru. The data of mootoic decay sequece uder stregtheig buffer operator is expaded.. The Costructio of a New Idex Stregtheig Buffer Operator Theorem Suppose that X = { x(), x(),..., x( ) } is a o-egative data sequece, = ( x() d, x() d,..., x( d ) ) is defied as buffer sequece. Where x( ) x( ) x( ) x( ) xd ( ) = xe ( ), =,,..., D is defied as a stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. Prove: It is easily gotte that D satisfies three axioms of buffer operator, so that D is Suppose X is a mootoe icreasig sequece, thus x ( ) x ( ) xx ( ) ( ) x ( ) x ( ) 0, e x( ) x( ) x( ) x( ) xd ( ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a mootoe decreasig sequece, thus The, x ( ) x ( ) xx ( ) ( ) x ( ) x ( ) 0, e x( ) x( ) x( ) x( ) x( d ) = xe ( ) x ( ) We ca obtai that x( ) d x( ), so D is stregtheig 7
Sesors & Trasducers, Vol. 59, Issue, November 0, pp. 7-76 Suppose X is a oscillatory sequece, thus x( a) = max x( ) =,,..., x( b) = mi x( ) =,,..., x( a) x( ) x( a) x( ) x( a) d = x( a) e max { x( ) } max { x( ) d }. I the same way that mi { x( ) } mi { x( ) ca be obtaied, thus D is stregtheig Corollary : Two step stregtheig buffer operator ca be obtaied based o stregtheig buffer operator D defied i Theorem. = D = ( x() d, x() d,..., x( ) d ) xd ( ) = xde ( ) x( ) d x( ) d d x( ) x( ) It is easily obtaied that d is two step stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. X = x(), x(),..., x( ) is Theorem Suppose that a o-egative data sequece, = ( x() d, x() d,..., x( d ) ) is defied as buffer sequece. Where xd ( ) xe ( ) ( + )( x ( ) x xix () ( ) =, =,,..., D is defied as a arithmetic mea stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. Prove: It is easily gotte that D satisfies three axioms of buffer operator, so that D is Suppose X is a mootoe icreasig sequece, thus ( + )( x( ) x xix () ( ) x ( ) x ( ) 0, e ( + )( x( ) x xix () ( ) xd ( ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a oscillatory sequece, thus x( a) = max x( ) =,,..., x( b) = mi x( ) =,,..., ( + )( x( ) x xix () ( ) xad ( ) = xae ( ) max { x( ) } max { x( ) d } I the same way that mi { x( ) } mi { x( ) ca be obtaied, thus D is stregtheig Corollary : Two step stregtheig buffer operators ca be obtaied based o stregtheig buffer operator D defied i Theorem. = D = ( x() d, x() d,..., x( ) d ) = xd ( ) = xe ( ) ( + )( x( ) d x( ) d) d x() i x( ) i It is easily obtaied that d is two step stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. X = x(), x(),..., x( ) Theorem Suppose that is a o-egative data sequece, = ( x() d, x() d,..., x( d ) ) is defied as buffer sequece. Where ( + )( x ( ) x xix () ( ) x ( ) x ( ) 0, e ( + )( x ( ) x xix () ( ) xd ( ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a mootoe decreasig sequece, thus xd ( ) xe ( ) ( + )( + )( x ( ) x i x( i) x( ) =, =,,..., D is defied as a weighted arithmetic mea stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. Prove: It is easily gotte that D satisfies three axioms of buffer operator, so that D is Suppose X is a mootoe icreasig sequece, thus 7
Sesors & Trasducers, Vol. 59, Issue, November 0, pp. 7-76 ( + )( + )( x( ) x i x( i) x( ) x ( ) x ( ) 0, e ( + )( + )( x( ) x i x( i) x( ) x( d ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a mootoe decreasig sequece, thus ( + )( + )( x ( ) x i x( i) x( ) x ( ) x ( ) 0, e ( + )( + )( x( ) x i x( i) x( ) x( d ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a oscillatory sequece, thus xa ( ) = max x ( ) =,,..., xb ( ) = mi x ( ) =,,..., ( + )( + )( x ( ) x i x( i) x( ) xad ( ) = xae ( ) max { x( ) } max { x( ) d } I the same way that mi { x( ) } mi { x( ) ca be obtaied, thus D is stregtheig Corollary : Two step stregtheig buffer operator ca be obtaied based o stregtheig buffer operator D defied i Theorem. = D = ( x() d, x() d,..., x( ) d ) = xd ( ) = xe ( ) ( + )( + )( x( ) d x( ) d) id x( i) x( ) i It is easily obtaied that d is two step stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. Theorem Suppose that X = { x(), x(),..., x( ) } is a o-egative data sequece, = ( x() d, x() d,..., x( d ) ) is defied as buffer sequece. Where ( x( ) x xix () + xd ( ) = xe ( ), =,,..., D is defied as a geometric average stregtheig operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece. Prove: It is easily gotte that D satisfies three axioms of buffer operator, so that D is Suppose X is a mootoe icreasig sequece, thus ( x ( ) x ( xix ( ) + x ( ) x ( ) 0, e ( x( ) x ( xix ( ) + x( d ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a mootoe decreasig sequece, thus ( x ( ) x ( xix ( ) + x ( ) x ( ) 0, e ( x ( ) x ( xix ( ) + x( d ) = xe ( ) x ( ) We ca obtai that x( d ) x ( ), so D is stregtheig Suppose X is a oscillatory sequece, thus x( a) = max x( ) =,,..., x( b) = mi x( ) =,,..., ( x ( ) x ( xix ( ) + xad ( ) = xae ( ) max { x( ) } max { x( ) d } I the same way that mi { x( ) } mi { x( ) ca be obtaied, thus D is stregtheig Corollary : Two step stregtheig buffer operators ca be obtaied based o stregtheig buffer operator D defied i Theorem. = D = ( x() d, x() d,..., x( ) d ) 7
Sesors & Trasducers, Vol. 59, Issue, November 0, pp. 7-76 xd ( ) = xde ( ) ( x( ) d x( ) d) ( d x( i) x) + It is easily obtaied that d is two step stregtheig buffer operator whe X is a mootoe icreasig sequece, or a mootoe decreasig sequece, or eve a oscillatory sequece.. Example We tae the umber of oe city s mobile phoe users (uit: millio) as a example to validate the stregtheig buffer operator i GM (, ) proposed by this paper. We select the umber of 996-00 mobile phoe X = (65.0 67.6 70.6 75.66 8. 9. 0.9 6.9) as the origial data. Where the umber of 996-00 is used as modelig data, the umber of 00-00 is used as simulatio test data. Average aual growth rates of the user umber of mobile telephoe were. %, 5.5 %, 7.6 %, 0.0 %,.0 % from the origial data i996-00. Thus it ca be see, the growth of origial data sequece before half part slows tha the latter part. Usig the data to predict is difficult to believe. Aalysis of this situatio, from 000, i order to expad the amout of mobile phoe users, Chia telecommuicatio eterprises lauched a series of services that stimulated the demad for mobile phoe. I order to mae a reasoable forecast of the umber of mobile phoe users, the origial data should be stregthe firstly. The origial data which was treated with the oe step stregtheig buffer operator proposed by this paper firstly establishes predictio model as show i Table. Table. Model GM (, ) produced by differet Stregtheig butter Operators. Seq X GM (, ) model x(996 + ) = 7.0e -675.90 x(996 + )=6.56e 7.7 x(996 + )=0.65e 9.988 x(996 + )= 00.50e 65.96 x(996 + )= 775.509e 70. 0.085 0.7 0.679 0.69 0.085 Predict Relative error 00 00 00 00 99.70 08.9.07 7. 0.65 6.5 0.7 0. 0.8 0.5.0.0 0.9.6 0.7 6.6 0.8.7 0..95 The relative error which the origial data sequece modelig uder stregtheig buffer operator D, D, D, D is much smaller as show i Table. The predictio relative error is the best uder D. The predict values of mobile phoe were 0.65, 6.57 for 00 ad 00, which approach the actual values 0.9, 6.9. Relative errors were oly 0.7 % ad 0. %, predictio accuracy is very good. Compared with paper [9], the practical applicatio results show the effectiveess of the proposed approach. 5. Coclusios Based o the theory of buffer operators i the grey system, some ew expoetial stregtheig buffer operators are established, ad the relevat theorems are proved accordig to the axiomatic of buffer operators. The problem that there are some cotradictios betwee quatitative aalysis ad qualitative aalysis i pretreatmet for vibratio data sequeces is resolved effectively. A example simulatio shows compared with the existig stregtheig buffer operators, the id of ew stregtheig buffer operators icreases the forecast precisio of GM (, ) remarably. Acowledgemets This paper is supported by the Natural Sciece Foudatio of Zhejiag Provice, P. R. Chia (No. 6006). Refereces []. Liu Si-Feg, Dag Yao-Guo, Fag Zhi-Geg, Xie Nai-Mig, Grey System Theory ad Its Applicatio (Fifth editio), Sciece Press, Beijig, 00. []. Liu Si-Feg, The trap i the predictio of a shoc disturbed system ad the buffer operator, Joural of Huazhog Uiversity of Sciece ad Techology, 5, 997, pp. 5-7. []. Wu Zhegpeg, Liu Si-Feg, Mi Chuami, Dag Yaoguo, Cui Lizhi, Study o stregtheig buffer 75
Sesors & Trasducers, Vol. 59, Issue, November 0, pp. 7-76 operator based o fixed poit, Cotrol ad Decisio, 5, 00, pp. 8-. []. Dag Yaoguo, Liu Sifeg, Liu Bi, Tag Xuewe, Study o the buffer weaeig operator, Chiese Joural of Maagemet Sciece,, 00, pp. 08-. [5]. Liu S. F., Buffer operator ad its applicatio, Theories ad Practices of Grey System,, 99, pp. 5-50. [6]. Dag Yaoguo, Liu Bi, Gua Ye-Qig, O the stregtheig buffer operators, Cotrol ad Decisio, 0, 005, pp. -6. [7]. Dag Yao-Guo, Liu Si-Feg, Mi Chua-Mi, Study o characteristics of the stregtheig buffer operators, Cotrol ad Decisio,, 007, pp. 70-7. [8]. Cui Lizhi, Liu Sifeg, Wu Zhegpeg, New stregtheig buffer operators ad their applicatios, Systems Eegieerig-Theory&Practice,, 00, pp. 8-89. [9]. Cui Jie, Dag Yaoguo, Liu Sifeg, Xie Naimig, Study o a id of ew stregtheig buffer operators ad umerical simulatios, Chia Academic Joural Electroic Publishig House,, 0, pp. 08-. [0]. Yua Liju, Yao Tiaxiag, Costructig methods of stregtheig buffer operators, Cotrol ad Decisio, 6, 0, pp. 5-7. 0 Copyright, Iteratioal Frequecy Sesor Associatio (IFSA). All rights reserved. (http://www.sesorsportal.com) 76