MODELS ON ROAD TRAFFIC FORECASTING: IDENTIFICATION AND DISCUSSION OF DIFFERENT TIME SERIES MODELS FERNANDO FERNANDES NETO Insiuo de Pesquisas Tecnológicas do Esado de São Paulo IPT / Secrearia do Planejameno e Desenvolvimeno Regional do Esado de São Paulo Palácio dos Bandeiranes - Av. Morumbi, 4500, º Andar, Sala 42, Morumbi, São Paulo/SP E-mails: fernandofernandes@planejameno.sp.gov.br / nando.fernandes.neo@gmail.com CLAUDIO GARCIA Escola Poliécnica da Universidade de São Paulo Deparameno de Engenharia de Telecomunicações e Conrole Avenida Prof. Luciano Gualbero, rav. 3, 58, Buanã, São Paulo/SP, Brasil - 05508900 E-mails: clgarcia@lac.usp.br Absrac In his paper are discussed and calibraed univariae models (scalar approach, SARIMA) and mulivariae models (vecor approach, VAR and VEC) aiming raffic forecass of equivalen axles in he Anchiea-Imigranes sysem. The bes performance models in he backesing procedure were hose of he second ype (vecor), having a mean absolue error of approximaely 3, in a monhly frequency. Keywords VAR, VEC, SARIMA, idenificaion, ime series, oll roads Resumo Nese arigo são discuidos e calibrados modelos univariados (abordagem escalar, SARIMA) e mulivariados (abordagem veorial, VAR e VEC) para a previsão de ráfego em eixos equivalenes no sisema Anchiea-Imigranes. Os modelos que iveram melhor desempenho no backesing foram os do segundo ipo (veorial), endo erro médio absoluo de aproximadamene 3 em uma frequência mensal. Palavras-chave VAR, VEC, SARIMA, idenificação, séries emporais, rodovias Inroducion One of he main problems in he oll road secor is he cash flow planning and is forecasing, due o is idiosyncraic complexiy, e.g. levels of service, seasonal effecs and he inerial evoluion of he raffic; and o he impac of oher variables like he Gross Domesic Produc (). There is a wide range of mehods applied o raffic forecasing, from Time Series models, Kalman Filer based models, Neural Neworks; o Markov Chain models, simulaion models (muli-agen based) and linear regression models, as shown by Bolshinsky and Freidman (202), or a combinaion of hem according o Fillare e al. (2005), varying from highfrequency o low-frequency daa. Also, i is imporan o noice ha despie he rich exising lieraure on raffic forecasing, lile aenion has been paid o he predicion abiliy of mos of hese mehods, as can be seen in (Bain, 2009). In fac, here is a considerable error range in he U.S. raffic forecass, as poined by he same auhor: acual raffic urned ou o lie beween 86 below forecas o 5 above forecas. This considerable error range illusraes he possible magniude of uncerainy when raffic risk is passed o he privae secor. Hence, planning and forecasing play a fundamenal role in his field, in he sense ha mos of he necessary invesmens and, consequenly, heir respecive decision-makings and cash ouflows, mus ake ino accoun a very long imeline concepion, consrucion, mauraion of he projec unil plain capaciy, ec. Thus, he main goal of his paper is he discussion of an alernaive raffic forecasing mehod in oll roads in his case Vecorial Auoregressive models (namely VAR and VEC) and Univariae ime series based on Seasonal ARIMA (SARIMA) models, discussed in he nex session illusraing one of he mos imporan highway sysems in Brazil, he Anchiea-Imigranes Sysem. This paper is divided ino he following secions: inroducion, mehodology, presenaion of he prob- 625
lem, resuls, analysis of he resuls and conclusion. 2. Univariae Models 2 Mehodology The Univariae approach in he presen paper is based on SARIMA models, which are a naural exension o he classical ARIMA models, which is a produc of wo ARIMA polynomials, one wih he regular srucure of he ime series, and he oher one wih he seasonal srucure of he ime series, as can be seen in (Box and Jenkins, 976; Hamilon,994 and Morein and Tolói, 2004). 2.2 Mulivariae Models The Mulivariae Models are mainly based on Vecor Auoregression models. These are nohing more han a mulivariable exension of he classical scalar auo regression models (AR), in he sense ha he process is described in erms of marices and vecors, insead of scalars. Thus, here is a muual causaliy relaionship beween all variables in his dynamic sysem. For example, a VAR(p) process can be wrien as: = φ +φ 2 2 +... +φ p p + a () where he φ i erms are square marices of order n ; n are x n vecors of endogenous variables; a is a x n vecor of uncorrelaed residuals; n is he endogenous variable number and p is he number of lags. In addiion o ha, as he classical scalar auo regression models (AR), if all variables are saionary, his model can be esimaed using he Ordinary Leas Squares (OLS) mehod. On he oher hand, when one or more variables in VAR models are non-saionary, he OLS resuls may be no valid anymore. Consequenly, he Theory of Coinegraion was developed in order o analyze hese possible relaionships beween non-saionary ime series. Furhermore, Granger and Newbold (974) discussed and exposed he problems of spurious regressions over non-saionary ime series. They also verified ha given wo series compleely uncorrelaed and non-saionary, he regression beween hem may produce a significan apparen relaionship. Therefore, if wo variables are non-saionary and have a long-run equilibrium relaionship, hey may be coinegraed ha is, boh are uncorrelaed, nonsaionary, bu wih a relaionship beween hem as exposed by Ashley and Granger (979), Engle and Granger (987) and Johansen (988). Thus Vecor Error Correcion Models (VEC) were developed, which can be seen as exensions o VAR according o Hendry and Juselius (2000, 200) and Lükepohl (99), where i is inroduced an error correcion erm. In order o verify he coinegraion assumpion, in he curren paper he approach ha was made is he verificaion ha all variables are non-saionary, using he Augmened Dickey-Fuller (979) es, using a 95 confidence inerval; hen if and only if he variables are non-saionary following Engle and Granger (987), he coinegraion residuals are obained by running a regression over he variables and hese residuals are esed for saionariy. If hese residuals are saionary (esed using he Augmened Dickey-Fuller es again) he ime series are coinegraed, oherwise hey are no coinegraed. In order o explain how he VEC model srucure is obained, one can sar from a wo variable dynamic sysem, where boh are coinegraed (by hypohesis), following (Hendry and Juselius, 2000, 200; Lükepohl, 99, 2004 and Morein, 20). Be, and 2, wo non-saionary coinegraed variables, and assume ha here is an equilibrium relaion beween hem given by:, β 2, = µ ~ N(0,σ ) (2) If considered ha he variaions in, and 2, depend on he deviaions of his equilibrium in -, i follows ha: Δ, = α (, β 2, )+ a, : a, ~ N(0,σ ) (3.) Δ 2, = α 2 (, β 2, )+ a 2, : a 2, ~ N(0,σ 2 ) (3.2) One can generalize his error correcion model ino a more general form, where hese correcions in he equilibrium may depend on previous changes in he equilibrium due o possible auocorrelaions, like: Δ, = α (, β 2, )+φ, Δ, +φ,2 Δ 2, + a, : a, ~ N(0,σ ) (4.) Δ 2, = α 2 (, β 2, )+φ 2, Δ, +φ 2,2 Δ 2, + a 2, : a 2, ~ N(0,σ 2 ) (4.2) where his model acually is a VAR() model. In order o verify ha, one can simply pu hese pair of equaions ino marix form, resuling in (5) and (6). where: Δ = αβ ' + AΔ + a (5) 626
α = α $ α 2, β ' = β $, A = φ, φ $,2 φ 2, φ 2,2 or rewriing as: = ( αβ ' + A + I) A 2 + a (7) (6) Acually, according o Gujarai e al. (20) such relaionship can be generalized and guaraneed by he Granger Represenaion Theorem, which shows ha any VAR(p) can be wrien as a VEC(q) and viceversa. Depending on he auocorrelaion srucure, one migh find ineresing having a VEC(q) model and is respecive VAR(p). More deails can be found in (Greene, 2005). 3 Presenaion of he Problem In his paper, i is considered a VAR and a VEC model wih he following variables: raffic and Gross Domesic Produc () all of hem endogenous, and wo kinds of univariae SARIMA models, one wih a seasonal difference plus an sochasic seasonal shock, and anoher one wih an auoregressive seasonal erm. The is available a IPEA ( Insiuo de Pesquisas Econômicas Aplicadas Brazilian Insiue of Applied Economic Research) sie, while he oher series are publicly available upon reques o ARTESP Transporaion Regulaory Agency of São Paulo Sae, Brazil ( Agência Reguladora de Transpores do Esado de São Paulo ). The ime series encompasses monhly observaions from March 3 s, 998 unil July 3 s, 203. The las six observaions are lef o es he prevision accuracy of he model. In addiion o ha, i is possible o poin ou as a main concern he fac ha considering he Gross Domesic Produc as an endogenous variable may be couner-inuiive. However, i is known ha raffic can ac as a leading indicaor for he behavior, and acually, such assumpion is esed in his paper, hrough he verificaion of coinegraion beween boh variables. The raffic was normalized under an equivalen vehicle basis, in order o ransform differen ypes of vehicles in cars, e.g. a heavy ruck is equivalen o n cars, while a ligh ruck is equivalen o n-2 cars. The Seasonaliy in he vecor models was considered by including a vecor of dummy variables, since he daa is on a monhly basis. Then, having all he ime series normalized, considered he seasonal effecs, he rank of coinegraion and he number of lags mus be esablished. In his case, he rank of coinegraion is he number of coinegraing vecors which is esed according o (Johansen, 988) and he leas Informaion Crierion number deermines he number of lags, in boh univariae and mulivariae models, as suggesed in (Lükepohl and Kräzig, 2004). For mulivariae models, Bayesian Informaion Crierion was chosen, due o he fac ha i imposes sronger penalies for he inclusion of new parameers, as his kind of model naurally happens o have a larger number of parameers. On he oher hand, for univariae models, Akaike Informaion Crierion was used, due o he fac ha hese models generally have less parameers han he mulivariae ones. The esimaion of he parameers and all ess menioned are compued using GRETL Gnu Regression, Economerics and Time Library (for mulivariae models) and R (univariae models). 4 Resuls In Table, are presened he resuls of he Bayesian Informaion Crieria lag-search for mulivariae models. Table. Bayesian Informaion Crierion of he Lag Search. lags BIC 46.740746* 2 46.874 3 46.868567 4 46.9584 5 46.97029 6 47.06696 So, as can be seen in his able, he mulivariae models mus have only one lag. For he univariae models, i was esed down for he mos common lag composiions over shocks and auoregressive erms, according o he auo.arima funcion, provided in forecas package, wihin he R saisical sofware, o check he opimal ARIMA regular srucure. I resuled in an ARIMA polynomial of he form ARIMA (p=, d=, q=4). In words, a firs-order auo-regressive par; a firs-order difference over he original series; and four lags over he innovaions (shocks). Then, he wo mos usual seasonal polynomials were calibraed, SARIMA (p=, d=0, q=0) and SARIMA (p=0, d=, q=), following he same noaion above. The Rank of coinegraion was deermined according o he Johansen es (988), and for a null rank marix (null hypohesis), here is a p-value of 0.03. So, he saisical evidence poins ou ha here is no coinegraing relaionship beween he variables. De- 627
spie ha, in his paper he VEC model was sill esimaed for comparison purposes. Thus, 4 differen models were obained as follows. Seasonal Model wih Seasonal Difference: Thus, if he monh o be prediced is January, one mus sum up he coefficien S plus he consan, and so on according o he respecive prediced monh. Finally, he VEC model wih seasonal dummies is presened as follows. Δ Δ 2 + 0.0447 a - 0.0978 a - 0.6753 a = 0.4864 Δ - -3-2 - 0.225 a - 0.554 a -2-4 + 3470.72 (8) Δ$ ' = $ + $ 0.7479 9.769 0.009 0.0247 9.769 0.0247 ' $ ' 0.0765 ' $ [ ] + K $ K 2 () ' ' Seasonal Model wih Auoregressive Seasonal componens: Δ = 0,5280 Δ 0.023 a - 0.227 a - -3 + 0.84 Δ - 0.2902 a - 0.564 a 2 VAR Model wih Seasonal Dummies: (0) $ = 0.2523 9.7520 0.009 0.9735 $ -2-4 + 2679.039 $ + K $ K 2 (9) where K and K are he seasonal dummies, as follows in Table 2 2. Table 2. Seasonal Parameers Esimaes of he VAR Model. K K2 S 8443-7270.76 S2-623254 - 4400.54 S3-2487 29.9 S4-460430 4863.24 S5-560794 2545.8 S6-653837 5743.7 S7-29643 263.97 S8-48878 5324.9 S9-4552 - 374.43 S0-95290 3806.3 S - 395400 662.35 Consan 259780-5468.02 where K and K are he seasonal dummies, as follows in Table 2 3: Table 3. Seasonal Parameer Esimaes of he VEC Model. K K2 S 8590-7256.59 S2-622809 - 4357.79 S3-24438 2228.3 S4-46073 4887.3 S5-560589 2565.6 S6-653788 5748.4 S7-296368 268.3 S8-48723 5339.82 S9-4548 - 364.57 S0-95095 3825 S - 395336 668 Consan 977-494.86 5 Analysis of he Resuls Aiming he selecion of he bes model, he ou-ofsample forecasing accuracy is measured in erms of he absolue error mean, as follows. Table 4. Ou-of-sample Errors of he Models. Model ARIMA(,,4) - Seasonal IMA() ARIMA(,,4) - Seasonal AR() VAR() VEC() Mean Absolue Error.28 0.70 3.23 3.4 Thus, he very surprising resul is ha he VEC() model, ha shouldn be even esimaed according o he exising lieraure, is he bes model in erms of ou-of-sample performance, despie he fac ha only six samples ou of he validaion se were used due o sampling issues, which may influence hese resuls. Noneheless, i was already expeced ha a mulivar- 628
Anais do XX Congresso Brasileiro de Auomáica iae model should perform beer han an univariae model, due o he fac ha more informaion is being included. Anoher ineresing fac is ha he loglikelihood of he univariae models are far worse han he mulivariae ones, as can be seen in Table 5 he model which has he larges log-likelihood is he bes one. he oher hand, vecor based models (Figure ) rely on seasonal deerminisic dummy variables. Thus, despie pas values are unknown o he auoregressive par, here are already values being insered in he model, providing esimaes of he seasonal flucuaions. Anoher ineresing poin is he fac ha, despie having a larger number of variables (mulivariae), hey had a poorer performance wihin he sample, so basically, he models which were acually overfied were he univariae ones. Finally, here i is shown he mos imporan feaure of vecor models in erms of policy analysis, which is he impulse response srucure ha can be rerieved of he sysem, following (Sims, 980). This mehod is based on he decomposiion of he covariance marix using a Cholesky algorihm, o obain wha is called a Srucural VAR/VEC. Table 5. Log-Likelihood of he Models. Model ARIMA(,,4) - Seasonal IMA() ARIMA(,,4) - Seasonal AR() VAR() VEC() Log- Likelihood - 2272.78-245.2-268.54-268.54 Hence, based on hese resuls, i seems ha he backesing procedure is a very imporan par of he modeling process, since he log-likelihood esimae does no provide all necessary informaion o analyze which model is he bes. When analyzing he models fied values agains he observed values ( Obs in Figures and 2), i is possible o see ha SARIMA (Figure 2) models converge slower owards o he observed values han he vecor based models. I can be explained due he fac ha hese univariae seasonal models rely on pas observed values o forecas he seasonal facors. On Considering i as a VAR wih conemporaneous relaionships, as in he following expression. φ0 = φ + φ2 2 +... + φn n + K + a (2) Muliplying he whole equaion by he inverse of φ0 one ges a VAR as in Equaion (), ha can be esimaed using he radiional OLS algorihm. 629
Anais do XX Congresso Brasileiro de Auomáica Therefore, afer decomposing he covariance marix, i is possible o impose causal resricions, in order o rerieve he conemporary relaionship marix. So, for example, if hough ha he economy () is expeced o cause he raffic in he road, one may infer how he dynamics beween he ime series may behave wih an impulse-response of he raffic agains he. This is a powerful ool ha enables he researcher o verify dynamic effecs insead of jus applying a firs-order (linear), as in he radiional simple linear regression over he logarihms of he variables (his procedure is acually called elasiciy calculaion ). 6 Conclusion In his paper i was shown ha i is possible o build an auoregressive mulivariable model o describe he raffic daa in one of he mos imporan Toll Road in Brazil, wih significan seasonal effecs and a large amoun of vehicles. Then, four kinds of models were esimaed: a VAR, a VEC and wo kinds of Seasonal ARIMA models. Furhermore, i were discussed mehodologies for esing he coinegraion beween he variables, uniary roo and opimal lag srucure obenion. Thus, i is possible o observe ha boh mulivariae mehodologies produced very similar forecass beween hem, as occurred beween boh univariae models oo. Despie ha, boh kinds of models were significanly differen in he long-run and in he shor-run, being he firs kind (mulivariae) he bes of hem, producing reasonable forecass 3 mean absolue error. Noneheless, i is imporan o noice ha his paper shows he usefulness of impulse-response analysis, which seems o be far more reasonable han he radiional elasiciy measures applied over simple linear regression based models in policy analysis. As perspecive for fuure analysis and work, i is suggesed expanding his analysis o oher large road sysems in Brazil and oher counries, coninuing o updae he exising daabase and verifying possible srucural and parameer changes in hese models, and include in his comparison he performance of NARX models (nonlinear auoregressive models) and sandard neural-nework based models, using only auoregressive componens of he dependen variable, or evaluae he inclusion of oher possible Figure 3. Impulse-Response of Trafego o a Shock in. As can be seen in Figure 3, a sandard shock (a uniary shock in erms of he covariance marix rerieved in he VAR/VEC models) in he evoluion of he causes an increase of 50 housand vehicles, afer 4 monhs and reaches sabiliy afer 5 monhs. 630
candidae independen variables (e.g. ). 7 References ASHLE, R.A., GRANGER, C.W.J. (979). Time series analysis of residuals from S. Louis model. In Journal of Macroeconomics,, 373-394. BAIN, R. (2009). Error and opimism bias in oll road raffic forecass, Working Paper, RePEC. BOLSHINSKI, E., FREIDMAN, R. (202). Traffic flow forecas survey. Tech. rep., Technion Israel Insiue of Technology. BOX, G.E.P., JENKINS, G.M. (976). Times Series Analysis: Forecasing and Conrol. s Ediion, San Francisco Holden Day. DICKE, D.A., FULLER, W.A. (979) Disribuion of he esimaors for auoregressive ime seires wih a uni roo. In European Journal of Finance, vol. 5, p. 69-637. ENGLE, R.F., GRANGER, C.W.J. (987). Coinegraion and error correcion: Represenaion, esimaion and esing. In Economerica, vol. 55, 25-276. FILLATRE, L., MARAKOV, D., VATON, S. December (2005). Forecasing Seasonal Traffic Flows. Workshop EuroNGI, Paris. GRANGER, C.W.J., NEWBOLD, P. (974). Spurious Regressions in Economerics, Journal of Economerics, vol. 2, -20. GREENE, W.H. (2002). Economeric Analysis, 5 h Ediion, Upper Saddle River, New Jersey, Prenice Hall. GUJARATI, D.N., PORTER, D.C. (20) Economeria Básica, Ediora Bookman, São Paulo. HAMILTON, J.D. (994). Time Series Analysis, s Ediion, Princeon, New Jersey, Princeon Universiy Press. HENDR, D.F., JUSELIUS, K. (2000). Explaining Coinegraion Analysis: Par. In The Energy Journal, Inernaional Associaion for Energy Economics, vol. 0 (Number ), -42 HENDR, D.F., JUSELIUS, K. (200). Explaining Coinegraion Analysis: Par 2. Em The Energy Journal, Inernaional Associaion for Energy Economics, vol. 0 (Number ), 75-20. IPEADATA, no síio hp://www.ipeadaa.gov.br, visiado em 0//203. JOHANSEN, S. (988). Saisical Analysis of coinegraion vecors. In Journal of Economic Dynamics and Conrol, vol. 2, 23-254. LÜTKEPOHL, H. (2004). Applied Time Series Economerics, s Ediion, New ork, Cambridge Universiy Press. LÜTKEPOHL, H. (99). Inroducion o Muliple Time Series Analysis, Heidelberg, Springer Verlag. MORETTIN, P.A. (20). Economeria Financeira: Um Curso em Séries Temporais Financeiras, ª Edição, São Paulo, Ediora Edgar Blücher. MORETTIN, P.A., TOLÓI, C. (2004). Análise de Séries Temporais, ª Edição, São Paulo, Ediora Edgar Blücher. SCHWARZ, G. (978). Esimaing he dimension of a model. In The Annals of Saisics, vol. 6, 46-464. SIMS, C. (980). Macroeconomics and Realiy. In Economerica, vol. 48, no., -48. 63