Comparison of Parametric and Nonparametric Techniques for Non-peak Traffic Forecasting

Transcription

1 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Mathematca and Computatona Scences Comparson of Parametrc and Nonparametrc echnques for Non-peak raffc Forecastng Yang Zhang, Yunca Lu Internatona Scence Index, Mathematca and Computatona Scences waset.org/pubcaton/000 Abstract Accuratey predctng non-peak traffc s cruca to day traffc for a forecastng modes. In the paper, east squares support vector machnes () are nvestgated to sove such a practca probem. It s the frst tme to appy the approach and anayze the forecast performance n the doman. For comparson purpose, two parametrc and two non-parametrc technques are seected because of ther effectveness proved n past research. Havng good generazaton abty and guaranteeng goba mnma, perform better than the others. Provdng suffcent mprovement n stabty and robustness reveas that the approach s practcay promsng. Keywords Parametrc and Nonparametrc echnques, Non-peak raffc Forecastng I. INRODUCION HE success of ntegent transportaton system (IS) much depends on the provson of accurate rea-tme nformaton and predctons of traffc status. Due to the mportance of traffc forecasts, more research attenton has been focused on ths subject n recent decades. Some papers such as extensve revew appear []-[3], whch attract sgnfcant scentfc nterest n more fexbe approaches. After summarzng former works, traffc forecastng s cassfed qute dfferenty on the bass of dverse cassfcaton standards. he type of forecastng depends on the foowng groups of factors: snge road nk or transportaton network, freeways or urban streets, physca modes or mathematca methodooges, unvarate or mutvarate method, etc. As ustrated n the statement, traffc predcton can be separated nto two paradgms: the emprca based, ncorporatng fary standard statstca methodoogy on one hand, and that based on traffc process theory, ether of demand or of suppy, on the other []. By appyng statstca methodoogy or heurstc methods n traffc forecastng, the emprca approaches can be cassfed nto two categores: parametrc and nonparametrc technques. o obtan accurate forecasts, snce the eary 980s, extensve varety of parametrc approaches has been empoyed rangng from hstorca average agorthms, smoothng technques, near and nonnear regresson, fterng technques, to autoregressve near processes ncudng autoregressve hs work was supported by the Natona Scence & echnoogy Supportng Program durng the th Fve-year Pan Perod (No. 006BAJ8B0), Chna. Y. Zhang and Y. Lu are wth the Insttute of Image Processng and Pattern Recognton, Shangha Jao ong Unversty, Shangha, 0040, Chna. (phone: ; fax: ; e-ma: zhang-yang@sjtu.ed u.cn, whomu@sjtu.edu.cn). movng average (ARMA) famy that s regarded as a mestone n forecastng fed. Latey extraordnary deveopment of dstnct nonparametrc technques, ncudng nonparametrc regresson, neura networks, etc., has shown ther great potenta aternatve to ther parametrc counterparts. In essence, nonparametrc statstca regresson can be regarded as a dynamc custerng mode that rees on the reatonshp between dependent and ndependent traffc varabes. It attempts to dentfy past nformaton that are smar to the state at predcton tme, whch eads to easy mpemented nature. Some researchers demonstrated that nonparametrc technques generay perform we due to ther strong abty to capture the nondetermnstc and compex nonnearty of traffc tme seres [4]-[]. Many computatona ntegence (CI) technques ncudng fuzzy systems, machne earnng, and evoutonary computaton have been successfuy adopted n the fed. Partcuary, artfca neura networks (ANNs) such as the rada bass functon neura network (RBF-NN) [], [3], have been successfuy apped. he theory of support vector regresson (SVR) has aso been ntroduced by severa researchers to mode traffc characterstcs and predct traffc states [7], [4], [5]. he recent appcatons of varous CI technques and hybrd ntegent systems have shown ther good potenta on traffc forecastng. Least squares support vector machnes () proposed by J. A. K Suykens [6], [7] are cosey reated to reguarzaton networks and Gaussan processes, but emphasze and expot prma-dua nterpretatons addtonay. he eary appcaton of the method to fnanca tme seres forecastng has obtaned breakthroughs and pausbe performance [8]. In studes of k-nearest neghbor (k-nn) nonparametrc approaches to traffc forecastng, state vectors are found to be essenta to ensure more accurate predcton [9]. he traffc n non-peak perod s the essenta part of day traffc. Unpredctabe congeston occurred n the reatvey stabe traffc nevtaby brngs some dffcuty for a predctors. hus, the quaty of forecasts n non-peak hours s fundamenta to traffc predcton. And the method s proposed to predct and anayze traffc status n non-peak perod. Meanwhe, a hybrd state space method s apped to determne the approprate nput and output dmensons. Due to the smpcty of the parametrc technques and the effectveness of the nonparametrc ones, both types of technques are chosen for comparson: hstorca-mean (HM), autoregressve movng average (ARMA), rada bass functons (RBF) networks, and support vector regresson (SVR) modes. Internatona Schoary and Scentfc Research & Innovaton 3(3) schoar.waset.org/ /000

2 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Mathematca and Computatona Scences Internatona Scence Index, Mathematca and Computatona Scences waset.org/pubcaton/000 o vadate our method, the forecast performance s measured by dfferent ndces of forecast accuracy. he resuts show that the proposed approach s generay better than the other modes n both effectveness and robustness through the comparatve case anayss. II. PROPOSED SCHEME A. Least Squares Support Vector Machnes () Compared wth standard support vector machnes (SVMs) [0], appy near east squares crtera to the oss functon nstead of tradtona Quadratc Programmng (QP) method, whch eads to the advantages of fast convergence, hgh accuracy and ow computatona efforts [6], [7]. he agorthm s ntroduced smpy and mathematcay. Suppose a tranng data set s {( ):,, L, } D = x,y = () where x n s the nput varabe set; y s the output varabe set; corresponds to the sze of the tranng data. In the weght space (prma space), the formuaton can be descrbed as: mn J ( we, ) = w + ζ e, w,b,e = s.. t y = w Φ ( x) + b + e, =,, L, where Ф: L = n H s a nonnear mappng functon that maps the nput vector x nto a hgher (possby nfnte) dmensona feature space H; w H s the weght vector; e s the error varabe; e=[e,,e ] s the error vector and b a bas term. In addton, J s the oss functon, and ζ s the adjustabe error term correspondng to a weghted east squares cost functon. o sove the above mnmzaton probem, the Lagrangan functon s defned by: = () L( w, b, e, a) = J( w, e) α [ w Φ( x ) + b+ e y ] (3) where α are named Lagrange mutpers or support vaues that consttute the support vector α (α=[α,,α ] ). he condtons for optmaty are gven by: L = 0 w = α ( ), Φ w x = w L = 0 α 0, = = (4) b L = 0 α = e ζ, =,, L,, e L = 0 Φ ( ) + b + e y = 0, =,,,. w x L α In standard SVMs, w and Ф(x ) are never cacuated. After varabes w and e are emnated, the foowng near system can be easy obtaned: 0 0 b (5) = Ω + I α y ζ where y=[y,,y ], =[,,], and Ω={Ω j }. Here Mercer s condton s apped wthn the matrx Ω Ω = Φ ( x ) Φ ( x ) = K( x, x ) (6) j j j In the optmum the weght vector can be denoted by w= αφ( ) = x, and the regressor s obtaned by appyng the Mercer s condton: = f ( x) = w Φ( x) + b= α K( x, x ) + b (7) where α and b can be obtaned by sovng the above matrx equaton. For postve defnte kerne functon K(x, x j ), three typca kernes are commony used. hese are near kerne wth formua K(x, x j )=x x j, poynoma kerne of degree d wth formua K(x, x j )=(x x j +) d and rada bass functon (RBF) kerne wth formua ( ) K ( x, x ) = exp x x /σ (8) j j where σ s a tunng parameter. he paper focuses on the use of the RBF kerne for ts good performance and advantages n tme seres forecastng probem proved n past research. he precson and convergence of are both affected by ζ and σ. B. State Space Method he state space methodoogy has a ong hstorca background [9]. here the state contans vectors that ncude a the nformaton about a certan system that carres over nto the future. Specfcay, the measurements durng each tme nterva t, t-,, t-d compose a state vector where d s an approprate number of ags (d ). Suppose there are N weeks chosen for tranng n the traffc data obtaned. For a partcuar day n a week (Monday, for exampe), the state vector X k (t, d) of the traffc parameters measured every hour can be descrbed by: where [ ] X (, td ) =V(), t x (, td) (9) k k k [ L ] x ( td, ) =V( t ),, V( t d), k [, N], t [, 4] (0) k k k can be chosen as the nput varabes n tranng the RBF-NN, SVR and modes; V k (t) denotes the traffc parameter Internatona Schoary and Scentfc Research & Innovaton 3(3) schoar.waset.org/ /000

3 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Mathematca and Computatona Scences Internatona Scence Index, Mathematca and Computatona Scences waset.org/pubcaton/000 durng the current tme nterva t n the week k; V k (t-) represents the one durng the prevous -hour nterva, etc. A ndces n x k (t, d) are cosey reated to V k (t). Partcuary, when t d, x k (t, d) contans the ast d-t+ parameters measured n the day before the chosen partcuar day. After d s appropratey determned, the vectors {x k (t, d),v k (t)}, k [, N] can be used as nput-output pars n the tranng process for each t, t [, 4]. And the number of each group of tranng sampes s N- (4 groups). hen for each tme nterva t n the week N+, the vector x N+ (t, d) s used to obtan the fna forecastng resut V N+ (t). It s usuay compared wth the orgna measurement V N+ (t) that means the parameter measured durng the correspondng tme nterva t n the week N+. In practce, x k (t, d) may not ony contan the d agged vaues but aso be suppemented wth the hstorca nformaton V hst,k- (t) that represents the hstorca ndex at the tme of day and day of the week assocated wth the tme nterva t n the week k- aong the cycca curve. he combnaton forms a hybrd state space. he comparson of dfferent nonparametrc forecastng resuts s nvestgated after specfyng the varabe d. C. Measures of Forecast Accuracy wo statstcs are used to assess the quaty of forecastng. he mean absoute percentage error (MAPE) and the varance of absoute percentage error (VAPE) are rreevant to the unt of the measures and nsenstve to the changes n the magntude of forecasts: MAPE = VO() t V P ()/ t VO() t 00% () t= O P VO() t V P() t V () t V () t t= VO() t t= VO() t VAPE = 00% ( ) () where V O (t) s the observed vaue of the measure durng the tme nterva t (V O (t) 0); V P (t) s the predcted vaue of the measure; and s the number of forecastng perods (=0 n the experment). Specfcay, the MAPE cacuates the average reatve error between the forecast output and actua observed data, whch refects the accuracy of the forecastng. he VAPE cacuates the sum of the devatons from the average performance durng the forecastng n a perods, whch represents the stabty of a forecastng mode. Meanwhe, the percentage error (PE) between our predcton and the orgna data s aso apped: ( O P ) PE() t = V () t V () t / VO() t 00% (3) III. OHER FORECASING MEHODS wo parametrc and two nonparametrc methods are used as representatve technques to suppy overa comparsons. A. Hstorca-mean (HM) Mode HM mode, a smpe conventona parametrc technque, s descrbed as beow: K V () t = V () t (4) Aver, K hst, w+ K = where V Aver, K (t) s the average whch represents the forecastng resut V w + (t) and K s the number of the hstorca weeks before the week w+. Namey, the forecastng resut s obtaned from the average of the hstorca traffc data at the same tme of day and day of the week. hs method s used as a representatve of parametrc technques to suppy a comparson. B. Autoregressve Movng Average (ARMA) Modes Autoregressve movng average (ARMA) modes, ncudng purey autoregressve (AR) and purey movng-average (MA) modes as speca cases, are one of the most popuar casses of near tme seres modes []. he modes are frequenty apped to mode near dynamc structures, to depct near reatonshps among agged varabes, and to provde effectve near forecastng. he AR and MA casses can be further extended to modeng more compcated dynamcs of tme seres. Combnng AR and MA forms together yeds the ARMA mode defned as Vk+ () t =bv k+ ( t ) + L+ bv p k+ ( t p) + ε + aε + L+ a ε t t- q t-q (5) where {ε t } s a smpe type of stochastc process, denoted as {ε t } WN(0, σ ); p, q 0 are ntegers, and (p, q) represents the order of the mode; V k+ (t) denotes the traffc parameter durng the current tme nterva t n the week k+; V k+ (t-) represents the one durng the prevous -hour nterva, etc. he whte nose {ε t } serves as a budng bock n defnng more compex near tme seres processes and refects nformaton that s not drecty observabe. ARMA mode s one of the most frequenty used fames of parametrc modes n the tme seres anayss. hs s due to ther fexbty n approxmatng many statonary processes. C. Rada Bass Functon Neura Network (RBF-NN) As a compettve nonparametrc regresson approach, the RBF-NN mode for predctng traffc data seres s apped n ths paper. he method s dfferent from the conventona mutayer perceptrons (MLPs) approach n whch the nonnearty of the mode s ony embedded n the hdden ayer of the network []. here are many addtona advantages n RBF-NN mode, whch have been proved recenty. Compared wth MLPs, the network can be we deveoped to become more adaptve to unversa approxmatons wth more accuracy and ess tme. Moreover, the RBF network outputs become near Internatona Schoary and Scentfc Research & Innovaton 3(3) schoar.waset.org/ /000

4 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Mathematca and Computatona Scences Internatona Scence Index, Mathematca and Computatona Scences waset.org/pubcaton/000 functons of the output ayer weghts when the bass functons are appropratey fxed. he above two advantages make us seect the hghy deveoped too for comparson. D. Support Vector Regresson (SVR) SVR has been ntroduced wthn the context of statstca earnng theory and structura rsk mnmzaton (SRM) prncpe. Researchers regard t as a powerfu methodoogy for near and nonnear regresson. Beneftng from the SRM prncpe, the SVR can gan a much better abty on generazaton whch s especay mportant for machnng earnng agorthms. In bref, the SVR maps the nputs nto a hgher dmensona feature space wth an approprate kerne nner product, then n the mapped space mnmzng the oss vaue wth Quadratc Programmng (QP) technques can determne some parameters whch excusvey denotes a regresson functon [0], [3]. ogether wth the determned parameters and the fxed regresson formuaton, the regresson functon can be ascertaned. N.B. before the SVR processng, an approprate kerne functon and oss functon must be chosen to get a better souton for our probem. For the proved effcency of SVR n the eary research, and both and SVR predctors beongng to the same SVMs famy, SVR predctor s aso compared wth ours. IV. EXPERIMENAL RESULS Data for ths study come from the Performance Measurement System (PeMS), whch can be accessed through the Internet [4]. he trave tme ndex (I) s commony used n the anayss of traffc status. It expresses the average amount of extra tme t takes to trave n the peak reatve to free-fow trave. And t can present congeston eves n a format that s easy to understand and communcate to the genera pubc. Consderng t from a practca perspectve, the PeMS suppes the I drecty on ts webste to the pubc for reference and evauates the traffc stuaton n the whoe freeway network. he traffc data of 4 weeks from May to Oct. 5, 006 are used n the paper. he -hour ane-aggregated average I data are downoaded because of our access to mted traffc data. he data for a partcuar day start every hour between 00:00 am and 3:00 pm. Our attenton s focused on predctng the I of the ast week based on the former. rave me Index Week Fg. Pot of a traffc data obtaned from the transportaton network (4 weeks) Fg. shows the tota 4 weeks traffc data houry. In ths secton, the out-of-sampe forecastng abty of the modes s evauated. Fg. ustrates the above houry average I data of,680 tme ponts. he data are contnuousy recorded over a perod of the seected 70 days (the frst 0 weeks). Wth the pan axes of 4 hours and 70 days, the three-dmensona graph shows the perodca pattern of 4 hours wth two peak perods. rave me Index Day Fg. hree dmensona graph of the frst 0 weeks Based on the anayss of the I of each day, the non-peak perods from the data are seected. Specfcay, the tme ntervas (6:00 am 0:00 am, 5:00 pm 9:00 pm on weekdays; 3:00 0:00 pm on weekends) contan the mornng and evenng peak perods. herefore, there are 4 and 6 tme ponts n the non-peak perod for each weekday and each weekend day respectvey. Namey, there are totay 0 tme ponts yng n the non-peak hours of one week. And the am of the paper s to predct these ponts n the 4th week. Dfferent parameters (K, d, etc.) are apped for the predctors n the experments. Due to ther genera performance n our other studes, the most representatve modes wth better performance measured n MAPE are chosen for smpcty: HM (K=3), ARMA (,), RBF-NN (d=), SVR (d=3), and LS-SVR (d=4). For the three nonparametrc technques, the tranng process s based on the nformaton n the non-peak perods of the former 3 weeks. Specfcay, the former weeks are seected for tranng, whe the atter are chosen for vadaton n the proposed mode (N=3). Parameters adjusted appropratey hep to obtan better performance. Due to dstnct dfferences among the 0 tme ponts of the week, 0 (ζ, σ) pars wth ζ rangng from 0.05 to 0000 and σ from 0.5 to 00 hep producng the fna forecast. An software kt s apped n the experments [5]. Fg. 3 compares the forecast performance of the HM, ARMA, RBF-NN, and SVR modes wth that of the mode n two groups. Fg. 3(a) presents the orgna data and the forecasts from the parametrc technques at 0 tme ponts separatey, whe Fg. 3(b) dspays the resuts from the nonparametrc technques for comparson. Correspondngy, Fg. 4 compares the PEs of these predctors pont by pont. 0 4 Internatona Schoary and Scentfc Research & Innovaton 3(3) schoar.waset.org/ /000

5 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Mathematca and Computatona Scences Internatona Scence Index, Mathematca and Computatona Scences waset.org/pubcaton/000 rave me Index rave me Index Orgna HM ARMA (a) HM, ARMA and Orgna RBF-NN SVR (b) RBF-NN, SVR and Fg. 3 Comparson of forecasts from the parametrc and nonparametrc modes Percentage Errors (%) Percentage Errors (%) HM ARMA (a) HM, ARMA and -6 RBF-NN SVR (b) RBF-NN, SVR and Fg. 4 Comparson of PEs from the parametrc and nonparametrc modes From the fgures, t s obvous that most PEs of the nonparametrc predctors are ess than 4%. hs ndcates that the nonparametrc technques outperform the parametrc ones. It aso can be found that the smpe HM mode can provde reatvey stabe and accurate forecasts. Meanwhe, the two SVMs famy members generay perform better, and neary a the absoute vaues of PEs of our mode are ess than %. A predcted vaue from the RBF-NN mode s vsby naccurate (:00 pm, Frday), whch makes ts PE up to about -8%. And the SVR mode aso produces such an naccurate vaue wth the PE up to 4% (0:00 pm, Frday). Moreover, t seems that more dffcutes exst for a modes to provde accurate forecasts on weekends. Showng the predcton resuts from dfferent modes, abe I ays strong emphass on the study usng the measures of forecast accuracy: MAPE and VAPE. It can be easy notced that the performance of the two SVMs famy members s better than that of the other modes. Moreover, compared wth the HM, ARMA, RBF-NN, and SVR predctors, our mode reduces 8.3%, 34.8%, 3.85%, and 6.6% n MAPE respectvey. Meanwhe, t aso reduces 5.8%, 49.73%, 38.3%, and 7.93% n VAPE respectvey. hs can fuy demonstrate that our mode s more accurate and robust than the other four. ABLE I COMPARISON OF PREDICION PERFORMANCE IN MAPE & VAPE USING DIFFEREN PREDICORS FOR NON-PEAK HOURS (%) Predctor MAPE VAPE HM (K=3) ARMA (, ) RBF-NN (d=) SVR (d=3) (d=4) In order to compare the modes from a hostc perspectve, the anayss usng another measure of forecast accuracy, the PE, s further produced. After the PEs are cacuated for each mode, the numbers of the predcted tme ponts yng n dfferent ranges of PEs (the absoute vaues of PEs) are statstcay anayzed. he range boundares are set as %, %, and 4%. For the fve groups of non-peak perod forecasts, abe II ceary shows the comparsons of the numbers computed from each mode. It can be seen that ony the two SVMs famy members have no PEs above 4%. Smpe comparsons show that our mode has more predcted ponts yng n the range ess than % and fewer ponts n the range more than %. ABLE II HE NUMBERS OF PREDICED POINS LYING IN DIFFEREN RANGES OF PES Predctor [0, %] (%, %] (%,4%] Above 4% HM (K=3) 76 3 ARMA (, ) RBF-NN (d=) SVR (d=3) (d=4) 78 0 Internatona Schoary and Scentfc Research & Innovaton 3(3) schoar.waset.org/ /000

6 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Mathematca and Computatona Scences Internatona Scence Index, Mathematca and Computatona Scences waset.org/pubcaton/000 rave me Index Δ Δ Orgna I Search wndow mts Fg. 5 Orgna I and search wndow As shown n Fgure 5, the observed I V I (t) (t N) and a varant Δ (Δ>0, Δ R) can produce a search wndow that s two Δ wde. At each tme pont t, the upper boundary of the wndow s V I (t)+δ and the ower one s V I (t)-δ. he use of the wndow can determne the dstrbuton of the predcted I and evauate the accuracy of each mode n another way. Obvousy, when the search wndow expands wth Δ ncreasng, more predcted ponts e nsde t. abe III sts the specfc numbers of the predcted I yng nsde dfferent ranges (Δ, Δ ] usng the modes. In the tabe, Δ and Δ determne the search wndow W wth wdth=δ and W wth wdth=δ (Δ <Δ ). he predcted ponts yng n the range (Δ, Δ ] means that these ponts e nsde wndow W and outsde W. Examnng the presented numbers, t can be seen that 5.96% forecasts from the mode e nsde the wndow wth wdth=0.00 (Δ =0.005). Partcuary, there are no predcted ponts yng outsde the search wndow wth Δ= hs proves the robustness and accuracy of the proposed mode from another pont of vew. ABLE III HE NUMBERS OF PREDICED POINS LYING IN DIFFEREN RANGES (Δ, Δ ] (Δ, Δ ] HM ARMA RBF-NN SVR LS-SVR (0, 0.005] (0.005, 0.00] (0.00, 0.05] (0.05, 0.040] (0.040, 0.060] (0.060, 0.075] Measurng the forecast accuracy n four ways, the anayses show that the proposed mode generay outperforms the others n non-peak traffc forecastng. It demonstrates the effectveness and robustness of the mode apped n the speca stuaton. V. CONCLUSION Frst appyng the n non-peak perod forecastng, the case studes comprehensvey compare the performance of two parametrc and two nonparametrc technques. 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