Comparison Between Regression and Arima Models in Forecasting Traffic Volume

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Ausralian Journal of Basic and Applied Sciences, (): 6-36, 007 ISSN 99-878 007, INSIne Publicaion Comparison Beween Regression and Arima Models in Forecasing Traffic olume M. Sabry, H.

1 Ausralian Journal of Basic and Applied Sciences, (): 6-36, 007 ISSN , INSIne Publicaion Comparison Beween Regression and Arima Models in Forecasing Traffic olume M. Sabry, H. Abd-El-Laif and N. Badra Deparmen of Public Works, Faculy of Engineering, Ain Shams Universiy, Cairo, Egyp. Deparmen of Physics and Engineering mahemaics, Faculy of Engineering, Ain Shams Universiy, Cairo, Egyp. Absrac: This paper invesigaes he applicaion of wo ime series forecasing echniques, namely logisic regression and univariae auo regressive inegraed moving average (ARIMA), o predic daily raffic volume for he Egypian inerciy roads. The daa of hree permanen raffic volume saions was seleced in his paper. A oal of 4-year raffic daa for he seleced saions (from 990 o 003) were used in he developmen of he wo models o predic raffic volume. All of he daa ses were bes represened by boh logisic regression model and an auoregressive inegraed moving average (ARIMA) model. The daa se conaining average annual, average monhly and average weekly daily raffic volume for years 990 hrough 00 was used o fi a ime series model. The resuling models were used o forecas raffic volumes for year 00 and 003. The forecased raffic volumes for he wo models were hen compared wih he acual raffic volumes. According o he analysis, he ARIMA model seems o be he bes forecasing mehod especially for he average monhly and average weekly daily raffic volume. Key words: Traffic volume, ARIMA model, Traffic volume, Regression model, Logisic funcion, Forecasing mehod. INTRODUCTION Sudies on ime-series and forecasing procedures have been developed since he recen 50 years. Forecasing plays a major role in raffic engineering. Considering he amoun of ime, effor and cos involved in model building, i would be reasonable o expec a high degree of accuracy in he forecass made. Time series is a se of saisical observaions, arranged in chronological order. An observed series heoreically consiss of wo pars: he series generaed by real process and he noise, which is he resul of ouside disurbances. Eliaion of his noise is he main aim of ime series analysis. One of he problems creaed in forecasing is he selecion of a suiable model for a specific purpose. Selecion of model depends on several facors including he conex of forecass, he availabiliy of hisoric daa, he degree of desirable accuracy, he cos of evaluaion and he ime period o be forecased. The main purpose of his research is o analyze he daily raffic volume daa colleced auomaically and o evaluae predicive models and hen produce a se of forecass for he sie a which he daa was colleced. I has become apparen ha he degree of accuracy has rarely been achieved wih convenional models and invesigaions ino heir performance in ranspor sudies have resuled in considerable criicism. Akins (977), Blanchard (983) and Horowiz and Enslie (978) saed ha he amoun of daa available o he ranspor planner is limied due o ime and financial resricions. This lack of relevan daa could be one of he main conribuions o he poor performance of accurae forecasing echniques. Time series analysis has been used in ransporaion sudies o reduce he dependence on convenional models. These simplified models are relaively easy o build and are used when several years of observed informaion is available. They are paricularly suiable for shor - erm forecasing and accuracy of he resuls is impressive. Comparaive sudies of forecasing echniques have been developed by Layon, Defris and Zehnwirh (986). I is clear from hese sudies ha he Box - Jenkins model (994) is he mehod providing bes forecass for he majoriy of he series esed. Box - Jenkins produced he bes forecass for 74% of he series used in a sudy by Reid (97). The cos associaed wih he Box - Jenkins approach in a given siuaion is generally greaer han many oher quaniaive mehods. Williams and Leser (003) have sudied he heoreical basis for modeling univariae raffic condiion daa sreams as seasonal auoregressive inegraed moving average processes. Corresponding Auhor: N. Badra, Deparmen of Physics and Engineering mahemaics, Faculy of Engineering, Ain Shams Universiy, Cairo, Egyp. 6

2 Aus. J. Basic & Appl. Sci., (): 6-36, 007 This paper invesigaes he applicaion of wo ime-series forecasing echniques: namely logisic regression and univariae ARIMA (Auo regressive inegraed moving average) o predic daily raffic volume for hree saions, Zazeeq Ismeilia agriculural road (Saion 5), Tana Mansoura road (Saion 9) and Mee Ghamr Aga road (Saion ). Regression analysis and ARIMA are wo commonly used saisical ime series forecasing echniques. A oal of 4 years raffic volume daa for he hree saions from 990 o 003 were used in he developmen of he wo models o predic raffic volume. All of he daa ses were calibraed by boh logisic regression model and ARIMA model. The daa se conaining average annual, average monhly and average weekly daily raffic volume on hree saions for years 990 hrough 00 is used o fi a ime series model. The resuling model is used o forecas raffic volumes for year 00 hrough 003. The forecas raffic volumes for he wo models are hen compared wih he acual raffic volumes. MATERIALS AND METHODS This research uses boh regression and ARIMA approaches in forecasing he road raffic volume. The following describes he heoreical background of hese wo approaches. Regression: Regression analysis is used o uilize a relaion beween a variable of ineres, called he dependen or response variable and one or more independen variables. Regression analysis is used o predic he response variable from knowledge of he independen variables. A oher imes, regression analysis is uilized primarily for exaing he naure of he relaionship beween he independen variables and he response variable (Neer, Wasserman and Whimore (988). Regression models conaining one independen variable are called simple regression models. While, regression models conaining wo or more independen variables are called muliple regression models. The general form of he muliple regression model can be se in he following form: y = f (x, x,, x n) () y x, x,, x n is he response variable and are he independen variables. This form can be linear or nonlinear funcions of he independen variables. The simple nonlinear regression model can be se in he following form: 3 y = A o + A x + A x + A 3 x +.. () A, A, A, A are he regression model parameers. o 3 These model parameers are esimaed using he leas square error approach. Goodness of fi measures available for he leas squares regression analysis such as coefficien of deeraion (R ) and - es should be also esimaed. ARIMA: ARIMA models are flexible and widely used in ime series analysis. ARIMA combines hree processes: auoregressive (AR), differencing o srip off he inegraion (I) of he series and moving averages (MA). Each of he hree process ypes has is own characerisic way of responding o a random disurbance. The resuling general linear model is in he form (Pankraz 983): Z = φ Z + φ Z + + φ p Z p + a - θ a - θ a - - θ q a q (3-a) φ i, i=,,,p are he auoregressive parameers, θ j, j=,,,q are he moving average parameers, Z is obained by differencing he original ime series d imes, (equal - if d=), 7

3 a is he whie noise componen a, p is he order of auoregressive parameers, q is he order of moving average parameers and d is he order of differencing. Aus. J. Basic & Appl. Sci., (): 6-36, 007 Idenificaion is a criical sep in building an ARIMA ( p, d, q ) model. To esimae he parameers φ i and θ j for fixed p and q, perform he linear muliple regression indicaed by he model: \ Z = Z Φ - T θ + µ (3-b) \ Z is he original ime series realizaion ( Z, Z +, ), Z is he vecor of lagged regressors, ( Z, Z,, Z p ), Φ is he vecor of unknown lagged parameers, ( φ, φ,, φ p ) and θ is he vecor of unknown lagged parameers, ( θ, θ,, θ ). q The variables φf and θ are called lagged variables. i j Equaion (3-a) can be wrien as: p q ( - φ B - φ B - - φ p B ) Z = ( - θ B - θ B - - θ q B ) a (4) d φ (B) ( - B) = θ ( B ) a (5) - d = φ ( B ) ( - B ) θ ( B ) a (6) B is he backward shif operaor, i. e., Z = B Z. Idenificaion is a criical sep in building an ARIMA ( p, d, q ) ( sp, sd, sq ) L model. p is he AR order which indicaes he number of parameers of φ, d is he number of imes he daa series mus be differenced o induce a saionary series, q is he MA order which indicaes he number of parameers of θ, sp is he seasonal AR order which indicaes he number of parameers of φ, sq is he seasonal MA order which indicaes he number of parameers of θ and sd is he number of imes he daa series needs o be seasonally differenced o induce a seasonally saionary series. Idenificaion mehods are rough procedures applied o a se of daa o indicae he kind of represenaional model ha is worhy of furher invesigaion. The aim of his mehod is o obain some idea of he values of p, d and q needed in he general linear ARIMA model and o obain bes esimaes for he parameers. Parameer esimaion algorihm ries all combinaions of parameers, which are limied o an ineger lying beween zero and wo. The combinaion wih he leas fiing error will be searched. The range limiaions of he parameers are se o resric he search o a reasonable scope. Parameers greaer han wo are rarely used in pracice. This principle reflecs no only a view of naure bu also a view of he relaionship beween a ime series model and naure. In almos all cases, a social science ime series process can be modeled as a probabilisic funcion of a few pas inpus and ime series observaions. Daa Collecion: Daily raffic volume saisics of hree saions (Saion 5, Saion 9 and Saion ) from a period of January 990 o December 003 were considered for models' calibraion and validaion in his paper. The average annual, monhly and weekly daily raffic volumes were calculaed for he daa. The colleced daa were divided ino wo ses, calibraion daa and esing daa, in order o esify he performance of he wo suggesed forecasing mehods. To achieve a more reliable and accurae resul, a long period (from January 990 o 8

4 Aus. J. Basic & Appl. Sci., (): 6-36, 007 December 00) was seleced as he calibraing period, while, he period from January 00 o December 003 was considered as he esing period. RESULTS AND DISCUSSIONS Regression Approach: Several forms of regression models have been ried for calibraion in order o ge he bes one. For insance, logisic, simple linear and quadraic models were ried. The logisic form was found he bes resuls in erms of R and es values. In addiion, Benja recommended using his model form, due o permiing accurae models when here are emporary flucuaions in raffic volume (Benja 986). The logisic model can be se in he following form (Benja 986): 3 4 ln = A 0 + A + A + A 3 + A (7) max is he curren average daily raffic volume, is he imum raffic volume, is he maximum raffic volume, is he ime (year or monh or week number) and A o, A, A and A 3 are he coefficiens o be deered by he logisic regression model. The parameers A o, A, A, A 3 and A 4 are calculaed by using he leas squares regression analysis 3 4 wih ln ( / + max - ) as he dependen variable and,, and as he independen variables. The saisical package SPSS was used for he analysis work of his paper. The model parameers were esimaed using he leas square error approach. Differen logisic models were also calibraed aking ino consideraion he power of he ime value (e. g. firs order, second order, ec) and he bes model in erms of he coefficien of deeraion R was also chosen. The calibraing period was used o esimae he parameers using he logisic regression echnique. The values of he parameers are shown in Tables, and 3 for he hree cases of average annual, monhly and weekly daily raffic volume, respecively. (Appendix A) Table : Resuls of logisic regression analysis for average annual daily raffic. Parameers Saion 5 Saion 9 Saion Asympoes: max Coefficiens: A A A R Table : Resuls of logisic regression analysis for average monhly daily raffic. Parameers Saion 5 Saion 9 Saion Asympoes: max Coefficiens: A A R Table 3: Resuls of logisic regression analysis for average weekly daily raffic. Parameers Saion 5 Saion 9 Saion Asympoes: max Coefficiens: A A R Resuls of observed and esimaed values of average annual daily raffic volume in he period from 990 o 003 a saions 5, 9 and are shown in Figures, and 3, respecively. 9

5 Aus. J. Basic & Appl. Sci., (): 6-36, 007 Fig. : Observed and forecased raffic volume for 00 and 003 a Saion5. Fig. : Observed and forecased raffic volume for 00 and 003 a Saion9. Fig. 3: Observed and forecasing raffic volume for 00 and 003 a Saion. ARIMA: The calibraing period was used o esimae he parameers using he described searching algorihm. The values of he calibraed parameers are shown in Tables 4, 5 and 6 for he average annual, monhly and weekly daily raffic, respecively. (Appendix B) Table 4: Resuls of parameer esimaes for average annual daily raffic volume using ARIMA ARIMA (p,d,q) Parameers µ Lag Lag Lag 3 R Saion 5 (, 0, 0 ) AR Saion 9 ( 3, 0, 0 ) AR Saion (,, 0 ) AR Table 5: Resuls of parameer esimaes for average monhly daily raffic volume using ARIMA. ARIMA (p,d,q) (sp,sd,sq) S Parameers µ Lag Lag Lag R Saion 5 (, 0, 0) (, 0, 0) Non-Seasonal AR Seasonal AR Saion 9 (3, 0, 0) (, 0, 0) Non-Seasonal AR Seasonal AR Saion (,, 0) (,, 0) Non-Seasonal AR Seasonal AR

6 Aus. J. Basic & Appl. Sci., (): 6-36, 007 Table 6: Resuls of parameer esimaes for average weekly daily raffic volume using ARIMA. ARIMA (p,d,q) (sp,sd,sq) S Parameers µ Lag Lag Lag 4 R Saion 5 (, 0, 0) (, 0, 0) 50 Non-Seasonal AR Seasonal AR Saion 9 (, 0, 0) (, 0, 0) 5 Non-Seasonal AR Seasonal AR Saion (,, 0) (,, 0) 5 Non-Seasonal AR Seasonal AR Comparison Beween he Two Forecasing Mehods: The predicions of raffic volume in he esing period are done using he wo forecasing mehods. Figures 4 o 9 illusrae he comparison beween he acual and forecased values for boh monhly and weekly daily raffic volume in he hree saions, respecively. To compare he predicion performance of he wo approaches for he period from January 00 o December 003, Sandard Error Deviaion (STDE) was calculaed. The sandard deviaion error (STDE) is defined as follows (Gaynor and R. Kirkparrick 994): (8) Resuls wih imal errors, seems o give he bes performance of he wo mehods. n is he number of esing poins and e ( - ) is he error in forecasing raffic volume. - acual - forecas Fig. 4: Observed and forecased monhly raffic volume for 00 and 003 a Saion5. Fig. 5: Observed and forecased monhly raffic volume for 00 and 003 a Saion9. Fig. 6: Observed and forecased monhly raffic volume for 00 and 003 a Saion. 3

Aus. J. Basic & Appl. Sci., (): 6-36, 007 Fig. 7: Observed and forecased weekly raffic volume for 00 and 003 a Saion 5. Fig. 8: Observed and forecased weekly raffic volume for 00 and 003 a Saion 9.

7 Aus. J. Basic & Appl. Sci., (): 6-36, 007 Fig. 7: Observed and forecased weekly raffic volume for 00 and 003 a Saion 5. Fig. 8: Observed and forecased weekly raffic volume for 00 and 003 a Saion 9. Fig. 9: Observed and forecased weekly raffic volume for 00 and 003 a Saion. Fig. 0: STDE of forecasing average annual daily raffic volume. Fig. : STDE of forecasing average monhly daily raffic volume. 3

8 Aus. J. Basic & Appl. Sci., (): 6-36, 007 Fig. : STDE of forecasing average weekly daily raffic volume. Resuls of he sandard deviaion error (STDE) in he esing period using he wo forecasing mehods for average annual, monhly and weekly daily raffic volumes are shown in Figures 0, and, respecively. I can be concluded from he figures ha he ARIMA sandard deviaion is generally less han ha of he logisic regression for esimaing eiher annual, monhly or weekly daily raffic volume. Conclusion: This paper invesigaes he applicaion of wo ime-series forecasing echniques: namely regression and univariae ARIMA o predic raffic volume a hree Egypian inerciy roads, Zazeeq Ismeilia agriculural road (Saion 5), Tana Mansoura road (Saion 9) and Mee Ghamr Aga road (Saion ). The period of colleced daa is from January 990 o December 003. The average annual, monhly and weekly daily raffic volumes are calculaed for he daa. The colleced daa were divided ino wo ses, calibraing and esing daa, in order o esify he performance of he wo suggesed forecasing mehods. The period from January 990 o December 00 was he calibraing period, while; he period from January 00 o December 003 was he esing period. Boh logisic regression model and an auoregressive inegraed moving average (ARIMA) model were calibraed using he daa. The saisical package SPSS was used for he analysis work of his paper. The forecased raffic volumes for he wo models were hen compared wih he acual raffic volumes. To compare beween he predicions performances of he wo approaches, he sandard error deviaion (STDE) was calculaed. According o he analysis, he ARIMA model seems o be he bes forecasing mehod for raffic volume. APPENDIX A (LOGISTIC MODELS) The logisic models ha were used in forecasing average annual, monhly and weekly daily raffic volume are as follows: forecasing = ( + ( + max) ea 0 + A + A ) / ( + ea 0 + A + A ) Taking ino consideraion he calibraed parameers shown in Tables, and 3: he above formula can be rewrien as follows: Saion 5: Average annual daily raffic volume: () = ( = year - 989) Average monhly daily raffic volume: () = ( = (year 990)* + monh order) 33

9 Average weekly daily raffic volume: Aus. J. Basic & Appl. Sci., (): 6-36, 007 () = ( = 66 + (year 00)*5 + week order) Saion 9: Average annual daily raffic volume: () = ( = year - 989) Average monhly daily raffic volume: () = ( = (year 990)* + monh order) Average weekly daily raffic volume: () = ( = 66 + (year 00)*5 + week order) Saion : Average annual daily raffic volume: () = ( = year - 989) Average monhly daily raffic volume: () = ( = (year 990)* + monh order) Average weekly daily raffic volume: () = ( = 66 + (year 00)*5 + week order) I should be noed ha he forecased raffic volumes ( ) ha are esimaed by hese logisic models should never exceed +. max APPENDIX B (ARIMA MODELS) Taking ino consideraion he calibraed parameers shown in Tables 4, 5 and 6: he ARIMA models can be rewrien as follows: Saion 5: Average annual daily raffic volume: ARIMA (, 0, 0) =

10 Aus. J. Basic & Appl. Sci., (): 6-36, 007 Average monhly daily raffic volume: ARIMA (, 0, 0) (, 0, 0) = Average weekly daily raffic volume: ARIMA (, 0, 0) (, 0, 0) 50 = Saion 9: Average annual daily raffic volume: ARIMA (3, 0, 0) = Average monhly daily raffic volume: ARIMA (3, 0, 0) (, 0, 0) = Average weekly daily raffic volume: ARIMA (, 0, 0) (, 0, 0) 5 = Saion : Average annual daily raffic volume: ARIMA (,, 0) = +, () = () - ( -) = Average monhly daily raffic volume: ARIMA (,, 0) (,, 0) = +, () = () - ( -) = Average weekly daily raffic volume: ARIMA (,, 0) (,, 0) 5 = +, () = () - ( -) = REFERENCES Akins, S., 977. Transporaion Planning: Is here a Road Ahead?, Traffic Engineering and Conrol, ol. 8, pp: Blanchard, R., 983. Transporaion and he New Planning, Traffic Engineering and Conrol, ol. 4, pp: Benja, J., 986. A Time Series Forecas of Average Daily Traffic olume, Transporaion Research, ol. 0, 986, pp: Box, G., G. Jenkins and G. Reinsel, 994. Time Series Analysis: Forecasing and Conrol, Prenice Hall, Chen, C. and S. Gran, 00. Use of Sequenial Learning for Shor Term Traffic Flow Forecasing, Transporaion Research, ol. 9, pp: Davis, C., 00. Saisical Mehods for he Analysis of Repeaed Measuremens, New York. Gaynor, P. and R. Kirkparick, 994. Inroducion o Time Series Modeling and Forecasing in Business and Economics, McGraw-Hill, Inc., New York. Horowiz, J. and R. Enslie, 978. Comparison of Measured and Forecas Traffic olume on Urban Inersae Highways, Transporaion Research, ol., pp:

11 Aus. J. Basic & Appl. Sci., (): 6-36, 007 Kim, J., 003. Forecasing Auoregressive Time Series wih Bais- correced Parameer Esimaors, Inernaional Journal of Forecasing, ol. 9, pp: Lain, J., J. Carrol and P. Green, 003. Analyzing Mulivariae Daa, U. K. Layon, A., L. Defris and B. Zehnwirh, 986. An Inernaional Comparison of Economic Leading Indicaors of Telecommunicaion Traffic, Inernaional Journal of Forecasing, ol., pp: Neer, J., W. Wasserman and G. Whimore, 988. Applied Saisics, New York. Pankraz, A., 983. Forecasing Wih Uniariae Box-Jenkins Models, Jon Wiley & Sons, New York. Reid, D., 97. Forecasing in Acion A Comparison of Forecasing Techniques in Economic Time Series, Join Conference of OR Sociey s group on Long Term Research Planning and Forecasing, Smih, B., B. Williams and R. Oswald, 00. Comparison of Parameric and Nonparameric Models for Traffic Flow Forecasing, Transporaion Research, ol. 0, pp: Williams, B., 00. Mulivariae ehicular Traffic Flow Predicion: Evaluaion of ARIMAX Modeling, Transporaion Research Board, Washingon, pp: Williams, M. and A. Leser, 003. Modeling and Forecasing ehicular Traffic Flow as a Seasonal ARIMA Process: Theoreical Basis and Empirical Resuls, Journal of Transporaion Engineering, ol. 9, pp: Yaffee, R. and M. McGee, 000. Inroducion o Time Series Analysis and Forecasing, San Diego: Academic Press, Inc., Yam, R., R. Whifield and R. Chung, 000. Forecasing Traffic Generaion in Public Housing Esaes, Journal of Transporaion Engineering, ol. 6, pp:

Time series Decomposition method

Time series Decomposition method Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,