Multi-variable weakening buffer operator and its application

Transcription

1 Multi-variable weakeig buffer operator ad its applicatio Lifeg Wu a,c, Sifeg Liu b, Yigjie Yag b, Lihua Ma c, Hogxia Liu c a College of Ecoomics ad Maagemet, Najig Uiversity of Aeroautics ad Astroautics, Najig 6, Chia b Cetre for Computatioal Itelligece, De Motfort Uiversity, Leicester, LE 9BH, UK c College of Ecoomics ad Maagemet, Hebei Uiversity of Egieerig, Hada, 5638, Chia c Art College, Hebei Uiversity of Egieerig, Hada, 5638, Chia Abstract: To weake the disturbaces of multi-variable ad reveal the real situatio, it is proved that the essece of the weakeig buffer operator (abbreviated as WBO) ca weake the disturbace of oe variable Accordig to this, the multi-variable weakeig buffer operator is put forward The multi-variable weakeig buffer operator ca satisfy the desire to use the freshest data ad its buffer effect is obvious whe the sample size is small Four real cases show that the proposed multi-variable weakeig buffer operator has higher forecastig performaces Keywords: forecastig; grey system theory; weakeig buffer operator; multiple liear regressio; eergy demad forecastig Itroductio Forecastig the future values of time series data plays a very importat role i our research Superior forecastig ability is a importat characteristic of successful maagers i complex ad ucertai eviromets With the rapid developmets of sciece ad techology, maagers ca better uderstad the future situatio ad make right strategies ad plas by gettig more iformatio i advace Thus, some forecastig methods have bee developed, for example, movig Correspodig Author: address: wlf6666@6com

2 average, support vector clusterig, eural etworks ad time series [7, 4] However, these methods require a large amout of data ad must meet some assumptios, ad they are ivalid whe the observed data available is of a small sample size Moreover, i some cases, the time series may exhibit a jumpig pheomeo due to the chagig of policy Therefore, it is difficult to fit a reasoable mathematic model I such a case, it is most importat to weake the impact of policy Cosiderig the above reasos, based o the fact that the recet data could provide more iformatio tha the distat data (More recet data are typically more relevat, especially for short-term forecasts [9]), more emphasis has bee placed o data which is more recet Take the eergy demad i Chia from 985 to 6 as a example Actually, the growth rate of eergy demad i Chia from 985 to is tempered From 3 to 6, Chiese total eergy demad icreased abruptly(durig this period, Chia has already etered ito the heavy chemical idustry stage, becomig oe member of World Trade Orgaizatio The rapid developmet of heavy ad chemical idustry accelerates Chiese ecoomic growth The growth rate of Chiese ecoomy was above % The rapid developmet of ecoomy accelerates Chiese eergy demad growth) To obtai better forecastig results from 4 to 6, we must give more weight to the data of 3 Grey weakeig buffer operator is proposed to cope with the disturbace It was first itroduced i early 99s by Liu [3] Sice the, WBO has bee widely ad successfully applied to various systems [5,5,6,7,6] Its improved form appeared simultaeously [,3,4,6,8,3,4,9,,5] However, traditioal WBOs ca oly deal with a sigle variable, ad little research has bee coducted o multiple variables The multi-variable weakeig buffer operator model is put forward by givig more weight to ewer iformatio i this paper The rest of this paper is orgaized as follows I Sectio, the essece of a variable WBO is itroduced I Sectios 3, the multi-variable weakeig buffer operator model is proposed I Sectio 4, the advatages of the ew model over the traditioal model is clarified by four real cases

3 The coclusios of this work are discussed i Sectio 5 The essece of a variable WBO With the chages of ecoomic developmet, the historical data teds to deviate from the curret situatios Grey buffer operator ca weake these disturbaces ad reveal the real situatio Its defiitio is as follows GM(,) model with the WBO Defiitio [3] Give a raw data sequece X () = {x () (), x () (),, x () ()}, X () D = {x () ()d, x () ()d,, x () ()d}, where D is a first order WBO x () (k)d = x() (k) + x () (k + ) + + x () (), () k + The sequece {x () ()d, x () ()d,, x () ()d} is give by WBO of Eq() A ew series X () D = {x () ()d, x () ()d,, x () ()d} ca be geerated by the first-order accumulated geeratig operator as x () (k)d = k x () (i)d, k =,,, Sice the origial form of GM(,) model i= x () (k)d + az () (k) = b, where z () (k) = x() (k)d+x () (k )d, k =, 3,, [, ] The least squares estimate of a ad b ca be obtaied by where Y = [ â ˆb x () ()d x () (3)d x () ()d ] = (B T B) B T Y,, B = z () () z () (3) z () () The predictio expressio of GM(,) model ca be obtaied as ˆx () (t) = [x () () ˆb â ]e ât + ˆb â by solvig the differetial dx() dt +ax () = b The predictio at k+ ca be obtaied as ˆx () (k+) = ˆx () (k + ) ˆx () (k) The essece of WBO A lemma is give i order to discuss the essece of WBO 3

4 Lemma [] Assume that A C, δa C, b C, δb C, vector orm ad matrix orm are tolerat If a matrix orm followed A δa <, the the solutios of liear system equatios AX = b ad (A + δa)(x + δx) = b + δb satisfy δx x A A A A δa A ( δa A + δb b ) Theorem For the GM(,) model of origial data {x () (), x () (),, x () ()} based o WBO, if the rth data is disturbed, that is ˆx () (r) = x () (r) + ɛ r, r =,,, L r is the δb resulted from this disturbace, T r is the δy resulted from this disturbace, the relative perturbatio boud of the parameter estimatio is B B B B Lr B ( Lr B + Tr Y ) by Lemma The the relative perturbatio boud of the parameter estimatio is larger while the more recet data is perturbed ad Proof = Sice B = z () () z () (3) z () () = + ( ) + + Y = x () ()d x () (3)d x () ()d = x () ()d x () ()d x () ()d x () () x () () x () () + ( ) + ( ) + + ( ) + + ( ) + + ( ) = k= k + x () () x () () x () () x () () x () () x () () 4

5 If it happes that disturbace ˆx () () = x () () + ɛ, matrix B becomes ˆB = + ( ) ( ) + ( ) + + ( ) + + ( ) + + ( ) = B k= ( + ( ) )ɛ ( + )ɛ ( + )ɛ ( + ( ) )ɛ ( the δb = + )ɛ, L = δb = ( + )ɛ vector Y becomes Ŷ = ɛ x () () x () () + ɛ x () () k +, x () () x () () + ɛ x () () ( ( ) + ) + ( )( + ) ɛ ; = Y + the δy =, T = δy = ɛ Similarly, if it happes that disturbace ˆx () (r) = x () (r)+ɛ r, r = 3, 4,,, we obtai L r ad T r It is easy to fid that L r ad T r from B B ( Lr B B Lr B + Tr Y B So the relative perturbatio boud of the parameter estimatio ( ɛ ) are icreasig fuctios of r B B ( Lr B B Lr B + Tr Y B a icreasig fuctio of r, that is to say, the relative perturbatio boud of parameter estimatio is larger, while the more recet data is perturbed ad the perturbatio of each data is equable The essece of WBO ca attach more importace to the more recet data Attachig more importace to the more recet data is likely to have better forecastig performace, which is cosistet with the priority theory of ew iformatio i grey system theory Accordig to the essece of WBO, the multi-variable weakeig buffer operator is put forward )) is 3 Multi-variable weakeig buffer operator 5

6 The sequece {x (k)d, x (k)d,, x m (k)d, y(k)d} is give by the weakeig buffer operator of the observatios {x (k), x (k),, x m (k), y(k)} Let {y()d, y()d,, y(k)d} be the sequece from the regressio model give by y(k)d = β + β x (k)d + β x (k)d + + β m x m (k)d () where β, β, β,, β m are the regressio parameters, Eq() is called as multi-variable weakeig buffer operator The ordiary least squares estimators of the parameters are ˆβ ˆβ ˆβ = (XT X) X T Y, where x ()d x m ()d x ()d x m ()d X =, Y = x ()d x m ()d y()d y()d y()d The the model y(k) = ˆβ + ˆβ x (k) + ˆβ x (k) + + ˆβ m x m (k) is used to predict the variable y(k) The flow chart of the multi-variable weakeig buffer operator model is show i Fig Fig The flow chart of the multi-variable weakeig buffer operator model Because all matrix orms are equivalet, they are substatially cosistet The matrix m orm ad its tolerat vector orm l are used i order to aalyze clearly Theorem For the multi-variable regressio model with WBO y(k)d = β + β x (k)d + β x (k)d++β m x m (k)d, if the rth data is disturbed, that is ˆx i (r) = x i (r)+ɛ i (i =,,, m), ŷ(r) = y(r) + ɛ (r =,,, ) L r is the δx m resulted from the disturbace ˆx i (r) = x i (r) + ɛ i (i =,,, m), T r is the δy resulted from the disturbace ŷ(r) = y(r) + ɛ (r =,,, ), the relative perturbatio boud of the parameter estimatio is X X ( Lr X X Lr X + Tr Y X ) by Lemma The the relative perturbatio boud of the parameter estimatio is larger while the more recet data is perturbed 6

7 Proof Sice x ()d x m ()d x () x m () x ()d x m ()d X = = x () x m (), x ()d x m ()d x () x m () y()d y() y()d Y = = y() y()d y() If it happes that disturbace ˆx i () = x i () + ɛ i (i =,,, m), ŷ() = y() + ɛ, matrix X becomes the x () x m () x () x m () ˆX = x () + ɛ x m () + ɛ m = X +, ɛ ɛ m δx = ɛ ɛ m ɛ ɛ m ɛ ɛ m, δx m = ( + + ) ɛ + ( ) ɛ + + Ŷ = the δy = k= ɛ k y() y() y() + ɛ = Y + δy = ɛ ɛ ɛ k=, k ɛ m ; vector Y becomes ɛ = Y + Similarly, if it happes that disturbace ˆx i ( ) = x i ( ) + ɛ i (i =,,, m), ŷ( ) = y( ) + ɛ, we obtai L = δx m = ( + ) ɛ + ( + + ) ɛ + + T = δy = k= ɛ k 7 k= ɛ ɛ ɛ k ɛ m,

8 If it happes that disturbace ˆx i (r) = x i (r) + ɛ i (i =,,, m), ŷ(r) = y(r) + ɛ, r =,,,, we obtai L r = δx m = ( r ) ɛ + ( r ) ɛ + + k= r+ ɛ m k, T r = δy = k= r+ ɛ k If it happes that disturbace ˆx i (r) = x i (r) + ɛ i (i =,,, m), ŷ(r) = y(r) + ɛ, r =,,,, L r ad T r from relative perturbatio boud, X X ( Lr X X Lr X + Tr Y X X X ( Lr X X Lr X + Tr Y X ) are icreasig fuctios of r So the ), is a icreasig fuctio of r, that is to say, the relative perturbatio boud of the parameter estimatio is larger, while the more recet data is perturbed ad the perturbatio of per period data is equable If the perturbatio of per period data is equable, the relative perturbatio boud of the parameter estimatio is larger while the more recet data is perturbed It meas that the role of ew iformatio o cogitio is better tha the old iformatio, that is to say the impact of data far from the predictio poit o future predictio is gradually weakeed or eve oexistet, while the data close to the predictio poit will have a sigificat impact o predictio It also meas that this method gives ew iformatio more weight As ca be see, L r+ L r ad T r+ T r mea the chage of relative perturbatio boud, whe L r+ L r ad T r+ T r are larger, the relative perturbatio boud chaged greatly, the differece betwee the weight of r + th data ad the weight of rth data is larger The size of L r+ L r ad T r+ T r is determied by the size of sample, if is smaller, the L r+ L r ad T r+ T r are larger; if is larger, the L r+ L r ad T r+ T r are smaller That is to say, whe is small, the differece betwee the weight of old data ad the weight of ew data is more obvious; whe is large, the differece is ot obvious, it ca ot highlight the role of WBO So Eq() is suitable for the small-data-set problems 4 Experimetatio results I this sectio, to testify the proposed model, three criteria are used to evaluate the forecastig precisio They are mea absolute percetage error (MAPE = % k= x() (k) ˆx () (k) ), absolute x () (k) 8

9 mea error (AME = x () (k) ˆx () (k) ) ad mea square error (MSE = k= (x () (k) ˆx () (k)) ) Case : Eergy demad forecastig i Chia [] We cosider a example from paper [] which provides the sample data As the discussio i the itroductio, data from 985 to 3 are used to costruct the multi-variable weakeig buffer operator model The data from 4 to 6 are predicted Actual values ad the forecastig values of four compared models are preseted i Table The fittig ad predictio results of four models are plotted i Fig Fig The fittig ad predictio values by differet models k= Table Forecastig results from differet models (millio to) Year Actual value GM(,) BP etwork support vector regressio multi-variable WBO MAPE AME MSE As ca be see from Table, the multi-variable weakeig buffer operator provides the lowest errors This implies that the multi-variable weakeig buffer operator ca improve the predictio accuracy of the other models Case : Costructio lad demad forecastig example i Kumig[8] We cosider a example from paper [8] The same sample is applied here to compare the precisio Actual values ad fittig values of three compared models are preseted i Table The fittig ad predictio results of three models are plotted i Fig3 As ca be see from Table, from a short-term forecastig viewpoit, multi-variable WBO has 9

10 the lowest errors from 8 to 9 compared with the other models Therefore, we ca coclude that the multi-variable WBO sigificatly ehaces the precisio i out-of-sample Because the costructio lad demad i Kumig exhibited a jumpig pheomeo whe Qiu He took up the mayor post i 7 Qiu supplied lots of lad for the muicipal costructio To predict the tred of costructio lad demad, it is most ecessary to weake the impact of policy The multi-variable WBO made full use of the data from 7, thus the impact of Qiu is elimiated Fig 3 The fittig ad predictio values by differet models Table The fitted values ad errors of differet models Year Actual value GM(,) multivariate regressio [8] multi-variable WBO MAPE AME MSE Case 3: The commercial housig average price forecastig example i Guagdog provice [] We cosider a example from paper [] which provides the sample data The commercial housig average price i Guagdog provice exhibited a jumpig pheomeo due to the recessio i 8 To predict the tred of the commercial housig average price, the multi-variable WBO ca weake the impact by makig full use of the data from 8 Thus, the data from to 8 (i-sample data) are used to costruct the differet models The the value i 9 (out-of-sample) is predicted Actual values ad the forecastig values of three compared models are preseted i Table 3 The fittig ad predictio results of three models are plotted i Fig4 Fig 4 The fittig ad predictio values by differet models

11 Table 3 The fitted values ad errors of differet models Time Actual value GM(,) multivariate regressio [] multi-variable WBO MAPE AME MSE As ca be see from Table 3, from a short-term forecastig viewpoit, the multi-variable WBO has better forecastig performace tha the other models, it idicates that the multi-variable WBO ca obtai accurate forecastig Case 4: The freight volume forecastig example i Hada city [] We cosider a example from paper [] which provides the sample data I 9, the govermet carried out a scheme for the developmet of logistics i Hada, thus the freight volume exhibited a jumpig pheomeo To predict the tred of freight volume, the multi-variable WBO will weake the impact by makig full use of the data from 9 Therefore, the data from 999 to 9 (i-sample data) are used to costruct the differet models The the value from to (out-of-sample) are predicted Actual values ad the forecastig values of three compared models are preseted i Table 4 The fittig ad predictio results of three models are plotted i Fig5 Fig 5 The fittig ad predictio values by differet models Table 4 The fitted values ad errors of differet models

12 Time Actual value GM(,) eural etwork [] multi-variable WBO MAPE AME MSE As ca be see from Table 4 ad Fig5, from a short-term forecastig viewpoit, the multivariable WBO ca effectively reduce the forecastig errors, which meas that the multi-variable WBO reaches the objective of accurate forecastig 5 Coclusio As to the predictio, the most recet data is likely to carry more iformatio tha the older data Therefore, the multi-variable WBO is costructed with preferece for recet data, the predictio results must be very accurate From the real cases show above, it is foud that the multi-variable WBO always has a better forecastig performace tha the covetioal GM(,) model ad multivariate regressio model I this sese, it is suggested that it would be better to use the multi-variable WBO for practical small sample problems Ackowledgemets The authors are grateful to aoymous referees for their helpful ad costructive commets o this paper This work was supported by a Marie Curie Iteratioal Icomig Fellowship withi the 7th Europea Commuity Framework Programme (NoFP7-PIIF-GA-3-695), Natioal Natural Sciece Foudatio of Chia (No745), the youth foudatio of Humaities ad Social Sciece of the Educatio Departmet Hea provice (5-QN-), the Humaistic ad Social Sciece Foudatio of Miistry of Educatio (5YJA637) ad Natural Sciece Fouda-

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