Highway Passenger Traffic Volume Prediction of Cubic Exponential Smoothing Model Based on Grey System Theory

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1 Inernaonal Conference on on Sof Compung n Informaon Communcaon echnology (SCIC 04) Hghway Passenger raffc Volume Predcon of Cubc Exponenal Smoohng Model Based on Grey Sysem heory Wenwen Lu, Yong Qn, Honghu Dong, Yanfang Yang, Zhao an School of raffc and ransporaon, Bejng Jao ong Unversy Bejng esearch Cener of Urban raffc Informaon Inellgen Sensng and Servce echnologes Bejng, Chna Emal: 30869@bju.edu.cn Absrac In consderaon of hghway passenger ransporaon sysem s a Grey sysem of ncomplee nformaon; hs paper consrucs a Grey model and a cubcexponenal-smoohng model a frs. In order o avod he shorcomngs of a sngle model, consrucs a cubcexponenal-smoohng combnaon model based on Grey sysem heory by usng he opmal weghed prncple. Compared wh he sngle model, he predcon accuracy of he cubc-exponenal-smoohng combnaon model s hgher, and he predced resuls are much more relable. So s beer suable for hghway passenger raffc volume predcon. A las, hs paper gves he predcon of he Naonal hghway passenger raffc volume n he frs fve monhs of 04. Keywords- hghway passenger raffc volume predcon; Grey model; cubc exponenal smoohng model; combnaon model I. INODUCION o predc he raffc volume and s developmen rends, characerscs and regulary n perspecve and scenfcally, s an mporan heory bass of esablshng he hghway passenger ranspor developmen plan as well as for road bus feld. Many scholars have carred on varous researches of raffc volume predcon and proposed a varey of predcon mehods. Such as me seres mehod, regresson analyss, Gray sysem mehod, neural newor, adapve flerng mehod, he mehod of lnear rend model, parcle swarm opmzaon algorhm and suppor vecor machne (SVM), ec. Alhough hese mehods are able o forecas he hghway passenger raffc volume, he effec s no deal. he combnaon model has been appled n recen years, for example, combnaon predcon based on Grey Marova, combnaon predcon mehod based on mulvarae sascal and he mproved BP newor, combnaon predcon mehod based on exponenal smoohng and Marova model, ec. hs paper s on he bass of fully sudes on he worng prncple of each sngle model and applcably esablshed n he Grey model and cubc exponenal smoohng model, and hen esablshed he combnaon model.in order o conras wh he cubc exponenal smoohng model, he arcle adoped he relave error nspecon n Gray model error nspecon. Moreover, hs paper calculaed and compared he predcon resuls n cubc exponenal smoohng model by usng dfferen smoohng coeffcen, raher han he subjecve choce, whch maes calculaon much more accurae. he predcon resuls show ha he combnaon predcon can mprove he predcon accuracy effecvely. II. HEE EXPONENIAL SMOOHING MODEL PEDIC POCESS Maerals used n he exponenal smoohng model [] s no so much and s operaon s smple, and he exponenal smoohng s he correcon process, by whch can auomacally correc predcon error. herefore, can be used for hghway passenger raffc volume forecas. Is compuaon formula s: $ y a b c + = + +. () Suppose me seres s y, y, K y, he I mes of phase exponenal smoohng value s S, hen he exponenal smoohng calculaon formula s: () () S = αy + ( α) S. () () () () S = αs + ( α) S. (3) (3) () (3) S = αs + ( α) S. (4) A he above formula, α s he smooh coeffcen. Equaon () s he frs exponenal smoohng value, Equaon (3) s he secondary exponenal smoohng value and Equaon (4) s he hrd exponenal smoohng value. Model coeffcen's calculang formula s: a = 3S 3S + S. (5) () () (3) 04. he auhors - Publshed by Alans Press 86

2 α b = [(6 5 α) S (5 4 α) S ( α) () () (3) +(4 3 α) S ] c α S S S ( α) III. ( () () (3) ) = + GAY MODEL FOECASING POCESS. (7). (6) Grey forecasng mehod [-3] does no consder he nfluence facors of he research objec, bu raher o fnd useful nformaon n s own me seres. Grey forecasng model s se up under he condon of ncomplee nformaon or message ha no now for sure, herefore can be used for raffc volume forecasng, whch s feasble and has ceran praccal sgnfcance. A. he Process of he Consrucon of he Gray Model ) Suppose X s a negave orgnal sequence, has he form of X = ( x (), x (), K x ( n)),we mae an accumulaon on X, and hen ge he generae seres s: () () () () X = ( x (), x (), K x ( n)), can be ge from he formula as follows: () X ( ) = x ( ) =,, K n. (8) = ) GM model s dfferenal equaon n he form of an albno s: dx d Parameer [ ] () () + ax = u. (9) aˆ = a u = ( B B) B Y N. 3) Consruc he daa marx B and daa vecory N, mae () () () Z ( ) = 0.5 X ( ) X ( ), =,3,... n, ge daa marx B and YN are as follows: () z () () z (3) B =, () z ( n) = (), (3), (4),... ( ). YN x x x x n 4) Deermne he dscree soluon: $ () u a u x ( + ) = ( x () ) e +. () a a 5) esore o he orgnal daa: () () x$ ( + ) = x$ ( + ) x$ ( ),And he resul s: $ x ( + ) = a( x () u/ a) e a. () B. Grey Model Error Inspecon he precson of he Grey model es [4] generally has hree mehods: relave error sze es, correlaon es and poseror error mehod. In order o compare he predcon resuls wh he cubc exponenal smoohng model, hs paper uses relave error sze es. Calculae resdual error s: E = ((),(), e e e()) n = X X K. (3) ( ) ( ) $ ( ) he formula of relave error s: e = x x. (4) e ( ) rel( ) = 00%,,, n x ( ) = K. (5) IV. HE CONSUCION OF COMBINAION MODEL Combnaon model [5-6] means o forecas he same objec a frs by usng dfferen forecas mehods, and hen n order o mprove he predcon precson, consrucs a new model hrough weghed combnaon. he opmal weghng mehod [7] s based on ceran sandards o consruc he objecve funcon Q, calculae he weghed coeffcens of combnaon model under he consrans (such as he wegh sum o ) as well as mnmze Q. Suppose raffc volume forecas [8-9] problem have sngle forecasng models y ( =,,... ), and o forecas he n perods. ecord y s he frs I nd of forecasng model s exac number n me, y s he frs I nd of forecasng model s predcaon, e s he frs I nd of forecasng model s forecasng error, p s he frs I nd of forecasng model s weghng coeffcen and deermne ^ 87

3 = p =, hrough whch we can ge he combned forecasng model form: Y p $ y p $ y p $ y p $ y = + + =. (6) = Assume ha a sngle forecasng model fng error s: e $ = y y ( =,,... ; =,,... n ) hen he predcon model can form fng error marx: n n n e ee L ee = = = n n n ee e ee E L =. (7) = = = M M M n n n ee ee e = = = he process of solvng he combnaon model s o ge he exreme value n he objecve funcon under he consran condon by usng leas square mehod. he consran condons s: n n mn Q= e, p = = =. (8) Defne s column vecor wh componen of he whole of, namely = [, L,], record P= [ p, p, L pm ], so by he consran condons can oban: n mn Q= e = P EP, P= = =. (9) Calculae he above formulas by usng he Lagrange mulpler mehod, he opmal wegh vecor s obaned as follows: E E P =. (0) he mnmum value of he objecve funcon s: mn Q =. () E V. EXPEIMENAL ANALYSIS he daa s from he naonal sascs bureau's monhly sascs, hs paper seleced he ranspor of passengers nsde. Mae use of hghway passenger raffc of January - December, 03 a frs, ulzed wo nds of predcon model [0] for modelng calculaon respecvely. ABLE I 03 NAIONAL HIGHWAY PASSENGE AFFIC (EN HOUSAND). monh Jan. Feb. Mar. Apr. May. Jun. raffc monh Jul. Aug. Sep. Oc. Nov. Dec. raffc A. Exponenal Smoohng Model Predcon ABLE II PEDICED ESULS COMPAED WIH HEE KINDS OF SMOOHING COEFFICIEN. Monh raffc Predcon Absolue error elave error (%) Mean

4 he ey n calculaon of exponenal smoohng s he selecon ofα. In general, f he daa s volale, value should be larger, by whch can ncrease he nfluence of he predcon resuls of recen daa. If he daa flucuae smoohly, value of α should be smaller. heorss hn ha when me seres have flucuaons, bu he long-erm rend change s no bg, we can pc and choose he larger value, n whch usually ranges from 0. o 0.5. hs paper seleced hree values (0.3, 0.4, and 0.5) and calculaed he predcon, hen analyzed he error. By analyze he resuls n able, we can ge ha when α equal o 0.4, he average absolue error and relave error are he mnmum, whch has a beer predcon resuls. So, he smoohng coeffcen s 0.4. When α ae 0.4, we can calculaed a = , b = , c = , and hen obaned he predcon model s: $ y+ = B. Grey Model Predcon By usng Grey heory o deal wh he orgnal daa, he GM (, ) model s esablshed. Among hem a = 0.006, u = hen he predcon model s: $ x ( + ) = e, predced value can be obaned by usng Grey model n able 3. ABLE III HEE KINDS OF PEDICION MODEL PEDICION VALUE AND EO. Monh raffc GM GM Exponenal Exponenal Porfolo Porfolo Porfolo value E predcon elave error predcon Absolue error elave error % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % C. Combnaon Model Predcon Suppose y s he predcon of GM (, ) model, y s he predcon of cubc exponenal smoohng model. Usng he wo models of he nown daa, you can ge E =, by he above equaon; he opmal wegh vecor s calculaed P = [ ] hen ge he opmal combnaon model: y = 0.7y+ 0.89y. By analyze of able 3 we can concluded ha he combnaon model has an average relave error absolue value 0.95%, whch s far less han he gray model MAPE.0% and he cubc exponenal smoohng model MAPE 3.3%. herefore, he combnaon forecas model of predcon effec s beer han a sngle model, he predcon error s smaller. 3.5 x raffc GM predcon Exponenal predcon Porfolo predcon Fg. Comparng he resuls of several nds of predcon. By analyzng he combnaon model and fgure,we can drawn ha has he hghes precson,as well as has obvous advanages n raffc volume forecas, so he use of combnaon model o predc can ge much more accurae resuls. Usng he combnaon model o forecas naonal 89

5 hghway passenger raffc volume of January-May, 04, he resuls are as follows: ABLE IV HE PASSENGE AFFIC VOLUME OF COMBINAION FOECAS MODEL IN 04. Monh Jan. Feb. Mar. Apr. May. Predcon (en housand) VI. CONCLUSION ) he opmal weghed combnaon model comprehensvely ulzed he effecve nformaon of he GM (, ) model, he cubc exponenal smoohng model, by whch avoded he lmaon of sngle heory, mproved he forecasng precson, reduced he predcon error, and made he predcon effec beer. )In hs paper he choce of smoohng coeffcen n cubc exponenal smoohng model s based on general experence, only seleced hree coeffcens o analyze, so here sll have shorcomngs need o mprove. 3)he raonaly and feasbly of he model s verfed, and he combnaon model can mprove he predcon accuracy accordng o he naonal hghway passenger raffc daa colleced from he naonal bureau of sascs. 4 ) Predced he naonal hghway passenger raffc volume of 04 by usng he combnaon model, whch has an mporan reference value o ransporaon managemen deparmen n arrangng he passenger ranspor managemen reasonable and srenghenng he managemen of bus passenger ermnals. ACKNOWLEDGEMEN Manuscrp receved Aprl, 04. hs wor s suppored by he Naonal 863 program (0AA40) and Scence and echnology Supporng Program (04BAG0B0), and s also suppored by he Naonal Naure Scence Foundaon under Gran EFEENCES [] XIE Anoxa-u, JIANG Hu-yuan, SHEN Yao-we. Passenger raffc Volume Forecas based on Gray-lnear egresson Combnaon Model, [J]. AILWAY ANSPO AND ECONOMY, 008, 30(8). [] LIU S-eng, DENG J-long.he ange Suable for GM (, ) [J].SYSEMS ENGINEEING HEOY&PACICE, 000, 0(5) [3] BAO Beng, ANG Chen-mn. Modfed Grey Model of Passenger Volume of C y Publc raffc [J]. JOUNAL OF BEIJING JIAOONG UNIVESIY, 005, 9(3). [4] MA Anoxa-ae, WANG C-Guangzhou. he Applcaon of he Cubc Exponenal Smoohng Mehod on Volume Forecasng of Da- An alway,[j].jounal OF EAS CHINA JIAOONG UNIVESIY, 005,(3). [5] ZHANG Lng-gang, NIU De-nng, MENG Zhao-mn, FAN Ja-dong. Applcaon of Combnaon Forecasng Model n raffc Volume Forecasng,[J]. ECHNOLOGY & ECONOMY IN AEAS OF COMMUNICAIONS, 00, (5). [6] HOU L-mn, MA Guo-feng. Forecas of alway Passenger raffc Based on a Grey Lnear egresson Combned Model,[J]. COMPUE SIMULAION, 0, 8(7). [7] ZHAO Lng, XU Honge, CHENG Honglang.oad raffc accdens predcon based on opmal weghed combned model, [J]. Compuer Engneerng and Applcaons, 03, (4). [8] Xna Wua, Henry X. A shocwave profle model for raffc flow on congesed urban arerals, [J]. ransporaon esearch Par B: Mehodologcal. Volume 45, Issue 0, December 0, Pages [9] Kran Kumara, M. Pardab, V.K. Kayar. Shor erm raffc Flow Predcon for a Non Urban Hghway Usng Arfcal Neural Newor Proceda, [J]. Socal and Behavoral Scences.Volume 04, December 03, Pages [0] Xaojan HU, We WANG, Hu SHENG. Urban raffc Flow Predcon wh Varable Cell ransmsson Model,[J]. Journal of ransporaon Sysems Engneerng and Informaon echnology.volume 0, Issue 4, Augus 00, Pages