Predicting Model of Traffic Volume Based on Grey-Markov

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1 Vo. No. Modern Apped Scence Predctng Mode of Traffc Voume Based on Grey-Marov Ynpeng Zhang Zhengzhou Muncpa Engneerng Desgn & Research Insttute Zhengzhou 5005 Chna Abstract Grey-marov forecastng mode of traffc voume was founded by appyng the mode of GM () and Marov random process theory. The mode utzes the advantages of Grey-marov GM () forecastng mode and Marov random process n order to dscover the deveopng and varyng tendency of the forecastng data sequences of traffc voume. The anayss of an exampe ndcates that the grey-marov mode has good forecastng accuracy and exceent appcabty n predctng traffc voume. Keywords: Grey theory Grey-marov mode Predcton of traffc voume. Introducton Generay the pannng of a hghway s desgned on the bass of the traffc voume predcton. The so-caed defnton of traffc voume predcton s to study and cacuate the nsde ncrease and change of traffc and to obtan a voume n terms of desgn years accordng to the varety of transportaton capacty and the deveopment of economy and socety n the past present and future etc. Athough many earners have processed arge quantty of researches for predctng the traffc voume the resut s st bad.the transportaton engneerng s a compcated system whch ncudes many factors many structura ayers and many targets. The traffc nformaton contans the obvousy ayer compexty of structure the fuzzy reaton of constructon the varety of deveopment and the ndetermnaton of coeffcents and data. Because of the nfuence of some artfca factors unaffectedy envronmenta change and the restrcton of the technque methods at present t eads to the resut that the statstc or forecast data embrace some errors mstaes scarcty or faes. So the compcated system of traffc voume predcton s a representatvey grey probem (Zhang and Luo 00). The grey mode has been apped n the traffc voume predcton and prmary maes use of mode GM () to perform the forecast (Wen et a.006; Xue and Zeng 006). Because the souton of mode GM () s an exponenta curve that s smooth t doesn t match wth those data that are vbraton sequences and ts forecast accuracy s ower. The study object of marov transton mode s a dynamc system whch forecasts the future by anayzng the nsde reguaton of deveopment n tme to come and t refects the nfuence degree and aws whch es n the transton process of factors from one state to the other. The marov transton mode s sutabe for the souton to predct these stochastc data sequences that are steady but n the reastc word these raw sequences are vbratng and changng n a certan varety trend. From the anayss above we now that the mode GM () and the marov mode coud be ntegrated wth each other to forecast by ther advantages. That s: mode GM () can be used to forecast the change trend of data sequences whe the marov mode can be used to decde the vbraton reguaton of ther deveopment and both can be joned together to become a grey-marov forecast mode. Snce t maes fu use of the od nformaton gven from these raw data and ncreases the forecast accuracy the appcaton of the grey-marov forecast mode whch provdes a new method to predct these greaty stochastc data sequences has been mproved further.. Estabshment of the Mathematcs Mode. Mode of GM() The grey GM () mode can mae use of the dscrete data seres to estabsh a equaton of grey contnuous dfferenta equaton by addng these data from the frst n Accumuatng Generaton Operator (AGO) and the equaton can be soved to perform the forecast (Deng 990). Let x be a raw seres whch s as foow: = { () L } = L n X x x x n Let x accumuated addng once and the accumuated generatng seres s obtaned: x = x x () L x ( n) () { } 6

2 Modern Apped Scence March 00 where By dfferentatng () ( 0) x = x ( ) = L n = x a whtened dfferenta equaton s obtaned dx dt The whtened tme-response of Eq. () s as foow (Deng 990): + ax = u () ( 0 ) () u a a xˆ + = x e + Let the souton u a ˆx accumuate subtratng once and the accumuated subtraton seres s obtaned: xˆ + = xˆ + xˆ (5) The curve of ˆx refects the vbraton trend of the raw seres. Fnay we can adopt the method of Deng (Deng 990) to chec the mode accuracy.. Grey marov chan Let { X n n T} be a marov chan where mn T( n ) and j L (L s caed status seres then the expressng (Sha 99) ( + ) p = P X = j X = (6) m n m s caed the n th step transton probabty and the matrx composed by transton matrx of Marov chan whch s expressed as: P ( n) ( n) ( n ) () p s caed the nth step probabty = p (7) If the eements transton probabtes of marov chan are grey t w be caed a grey marov chan and can be made up of a grey transton matrx (He and Bao 99). In the actua appcaton we now that t s dffcut to mae certan the vaues of transton probabty for t acs some nformaton but t s easy to have the nformaton of grey zone p by studyng the transton probabty. When the transton matrx s a grey matrx t s requred that the eements of whtenzaton matrx P % ( ) = p % s provded that p ( ) 0 % = L. () p j= % j L ; When the premnary dstrbuton of a marov mted chan s P = ( p p L p ) and the whtenzaton transton probabty matrx s P % ( ) = p % then we can get the next step dstrbuton of the chan: P = P P% (8) The second step can be expressed as: P = P P% = P 0 P% (9) The rest may be deduced by anaogy and the n th step dstrbuton s shown as: Pn = P P% n From Eq. t can be seen that we can easy forecast any future dstrbuton of the system f we have aready nown the raw dstrbuton and the grey transton probabty matrx.. Grey-marov mode Let x { x x () x ( n) } = L be a raw data seres. After we have checed the mode accuracy we get the 7

3 Vo. No. smuaton sequence as: xˆ { xˆ xˆ () xˆ ( n) } 8 = L by mode GM(). Let ˆ Modern Apped Scence ( 0 ) y xˆ = for a vbraton sequence Yˆ whch s a marov chan we can dvde t nto states accordng to the concrete crcumstance and ts any state can be expressed as: = % % ˆ = y + A % y ( ) % = + B = L () ˆ where A and B are constant whch can be decded by the dfference between the forecast vaue and the raw data. Yˆ s a functon whch s changed n tme and so are the grey whtened eements of % %. If N ( m ) s the data number of the raw seres whch transfer m step from to j and N s the number of data that are n the grey zone then we ca: N p ( m) = j = L () N the m th step transton probabty. The transton matrx R( m ) s as foow: p( m) p( m) L p ( m) p( m) p( m) p ( m) Rm L = () M M M M p( m) p( m) L p( m) R( m ) refects the transton reguaton between dfferent states and s the foundaton of the forecast mode of grey marov. We can predct the future trend of the system by studyng the stochastc transton matrx R( m ). In practca appcaton f the forecast vaues s to be paced n the zone then nvestgate the th ne of the matrx R nsde and f max{ pj } = pr we can concude the next state of the system may transfer ts state from j to r. If R has more than two nes whose probabty vaues are same ae or cose to each other and t s dffcut to decde the next drecton of the system wth certan t s needed to study and chec the matrx R() or R( m )(m ). At the same tme t can decde the transton of the system by checng R or R( m )(m ) and ~ ~ aso be made sure the forecast zone [ ]. Fnay the eventua forecast s n the mdde pont of the grey zone then got: Yˆ = ( % +% ) () whch aso can be expressed as: Yˆ = yˆ + ( A + B) (5). Exampe Anayss The data of a hghway s traffc voume through years are sted n Tab... Estabshment of GM () mode From tabe we get x ={ }. After do them n AGO we obtan x ={ }. Then we can have two constants: a = u = 65.. dx By combnng wth Eq. () we can estabsh the mode GM(): x = 65.. dt After sovng the equaton tme-response functon can be obtaned as:

4 Modern Apped Scence March xˆ ( + ) = 97.e From Eq. (5) t can be got: yˆ xˆ ( ) xˆ ( ) xˆ = + = +. The examnaton resut of the predcton accuracy s as foow: x = 96.7 S = 06.0 q =.5 S = 7.. The post-examnaton margn rato s as foow: C = S / S = < 0.5. The probabty of tte error s as foow: P{ q q < 0.675S} = P{ q q < 80.0} = > 0.95 The accuracy grade of the forecast s exceent (Deng 990).. Compartmentazaton of the predcton Accordng to the raw traffc voume and for smpfcaton the predcton vaues can be dvded nto four states by Eq. () as foows: =[ % % ]: % ˆ = y 0.x % ˆ = y 0.05x =[ % =[ % =[ % % ]: % = y x % = ˆ 0.05 ŷ % ]: % = % y ( ) ŷ ˆ = x % ]: % = y + x % y ( ) ˆ 0.05 ˆ = + 0.x =[ % ˆ = y x % ˆ = y + 0.x where y ˆ s the forecast traffc as mode GM () and x s the annua average traffc voume. If we show the vaues of the fact the predcton yˆ and four states through these years we w obtan a dagram sted as Fg. n whch there are four parae and symmetry band dstrcts form the top to the bottom.. Cacuaton of the transton probabty From Fg. we now that the number of raw sequence whch s n the zone of s N = N = N = N = and s the number of raw data from to respectvey by a step. If the rest may be deduced n the same way we can cacuate the number of raw transton data. Fnay we have % ]: % / / / / p whch maes up of the matrx R = 0 / 0 / 0 / / 0 forecast the transton state of the traffc voume n the future.. Decson of the predcton and vbraton zone by Eq. (). Accordng to R we can By studyng R we now that the average predcton of 00w mosty be n the vbraton zone whch s [ ].Then usng formua () or (5) we have ˆ Y (00) = ( )=65. In the same way we can get Y ˆ (00) =58 Y ˆ (00) =6660 Y ˆ (00) =76 Y ˆ (005) =99. Concuson The grey mode GM () refects the macroscopca reguaton the marov mode shows the vbraton deveopment of the mcrocosmc system and both not ony have the mutua advantage but aso can mae fu use of the nformaton whch s ncuded n these raw data. Therefore the forecastng grey-marov mode has much hgher accuracy reabty and appcaton n the traffc voume predcton. On the other hand because the predcton accuracy s n ne wth the raw data seres and the dvded states but there s not a gven standard that can reay unfy and sette these probems and the appcaton of the mode st needs a further research and mprovement. 9

5 Vo. No. Modern Apped Scence References Deng J. L. (990). Grey system theory tutora Huazhong Unversty of Scence and Technoogy Press Wuhan Chna. He Y. and BaoY. D. (99). Grey marov chan predcton mode and the mpcaton. Systems Engneerng Theory and Practce 99(): -7. Sha J. Z. (99). Marten decson programmng and ts appcaton n management Natona Defence Industry Press Beng Chna. Wen K. G. Qu S. R. and Wang J. (006). An urban traffc fows predcton mode based on system grey theory. Transactons of Shenyang Lgong Unversty 006 5(): -. Xue C. M. and Zeng Y. K. (006). On grey predcton mode for road traffc freght voume. Journa of Kunmng Unversty of Scence and Technoogy ( Scence and Technoogy) 006 (): Zhang X. T. and Luo X. H. (00). The appcaton of grey theory and mode n traffc voume predcton. Hghway 00(8): -7. Tabe. Hstorca Traffc Voume Year AADT (n/d) AADT=Annua Average Day Traffc Fgure. Annua Average Traffc Voume 50

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