A Nonlinear Autoregressive Model with Exogenous Variables for Traffic Flow Forecasting in Smaller Urban Regions

  • Junzhuo Li Guilin University of Electronic Technology
  • Wenyong Li Guilin University of Electronic Technology
  • Guan Lian Guilin University of Electronic Technology
Keywords: intelligent transportation system, traffic flow forecasting, time series, NARX model, traffic data


Data-driven forecasting methods have the problems of complex calculations, poor portability and need a large amount of training data, which limits the application of data-driven methods in small cities. This paper propos-es a traffic flow forecasting method using a Nonlinear AutoRegressive model with eXogenous variables (NARX model), which uses a dynamic neural network Focused Time-Delay Neural Network (FTDNN) with a Tapped Delay Line (TDL) structure as a nonlinear function. The TDL structure enables the FTDNN to have short-term memory capabilities. At the same time, before the data is input into the FTDNN, the use of trend decomposition or differential calculation on the traffic data sequence can make the NARX model maintain long-term predictive ca-pabilities. Compared with common nonlinear models, the FTDNN has structural advantages. It uses a simple TDL structure without the memory mechanism and the gated structure, which can reduce the parameters of the model and reduce the scale of data. Through the four-day data of Guilin City, the traffic volume forecast for five minutes is verified, and the performance of the NARX model is better than that of the SARIMA model and the Holt-Win-ters model.


Crivellari A, Beinat E. Forecasting spatially-distributed urban traffic volumes via multi-target LSTM-based neural network regressor. Mathematics. 2020;8(12): 2233. doi: 10.3390/math8122233.

Qu W, et al. Short-term intersection traffic flow forecasting. Sustainability. 2020;12(19): 8158. doi: 10.3390/su12198158.

Vlahogianni EI, Karlaftis MG, Golias JC. Short-term traffic forecasting: Where we are and where we're going. Transportation Research Part C: Emerging Technologies. 2014;43: 3-19. doi: 10.1016/j.trc.2014.01.005.

Hamed MM, Almasaeid HR, Said ZMB. Short-term prediction of traffic volume in urban arterials. Journal of Transportation Engineering. 1995;121: 249-254. doi: 10.1061/(Asce)0733-947x(1995)121:3(249).

Van der Voort M, Dougherty M, Watson S. Combining kohonen maps with ARIMA time series models to forecast traffic flow. Transportation Research Part C: Emerging Technologies. 1996;4(5): 307-318. doi: 10.1016/S0968-090x(97)82903-8.

Jomnonkwao S, Uttra S, Ratanavaraha V. Forecasting road traffic deaths in Thailand: Applications of time-series, curve estimation, multiple linear regression, and path analysis models. Sustainability. 2020;12(1): 395. doi: 10.3390/su12010395.

Andrysiak T, Saganowski L, Maszewski M. Time series forecasting using Holt-Winters model applied to anomaly detection in network traffic. Advances in Intelligent Systems and Computing. 2018;649: 567-576. doi:10.1007/978-3-319-67180-2_55.

Zhang H, et al. A hybrid short-term traffic flow forecasting model based on time series multifractal characteristics. Applied Intelligence. 2018;48: 2429-2440. doi: 10.1007/s10489-017-1095-9.

Yao RH, Zhang WS, Zhang LH. Hybrid methods for short-term traffic flow prediction based on ARIMA-GARCH model and wavelet neural network. Journal of Transportation Engineering, Part A: Systems. 2020;146(8): 04020086. doi: 10.1061/JTEPBS.0000388.

Huang W, et al. Real-time prediction of seasonal heteroscedasticity in vehicular traffic flow series. IEEE Transactions on Intelligent Transportation Systems. 2018;19(10): 3170-3180. doi: 10.1109/Tits.2017.2774289.

Hafner CM, Kyriakopoulou D. Exponential-type GARCH models with linear-in-variance risk premium. Journal of Business and Economic Statistics. 2019;39(2): 589-603. doi: 10.1080/07350015.2019.1691564.

Bentes SR. Forecasting volatility in gold returns under the GARCH, IGARCH and FIGARCH frameworks: New evidence. Physica A: Statistical Mechanics and its Applications. 2015;438: 355-364. doi: 10.1016/j.physa.2015.07.011.

Krithikaivasan B, Zeng Y, Deka K, Medhi D. ARCH-based traffic forecasting and dynamic bandwidth provisioning for periodically measured nonstationary traffic. IEEE/ACM Transactions on Networking. 2007;15: 683-696. doi: 10.1109/Tnet.2007.893217.

Anand NC, Scoglio C, Natarajan B. GARCH - Non-linear time series model for traffic modeling and prediction. NOMS 2008 IEEE Network Operations and Management Symposium, 7-11 Apr. 2008, Salvador, Brazil. New Jersey: IEEE; 2008. p. 694-697. doi: 10.1109/Noms.2008.4575191.

Ding C, et al. Using an ARIMA-GARCH modeling approach to improve subway short-term ridership forecasting accounting for dynamic volatility. IEEE Transactions on Intelligent Transportation Systems. 2018;19: 1054-1064. doi: 10.1109/Tits.2017.2711046.

Smith BL, Williams BM, Oswald RK. Comparison of parametric and nonparametric models for traffic flow forecasting. Transportation Research Part C: Emerging Technologies. 2002;10(4): 303-321. doi: 10.1016/S0968-090X(02)00009-8.

Bogaerts T, et al. A graph CNN-LSTM neural network for short and long-term traffic forecasting based on trajectory data. Transportation Research Part C: Emerging Technologies. 2020;112: 62-77. doi: 10.1016/j.trc.2020.01.010.

Mahdy B, et al. A clustering-driven approach to predict the traffic load of mobile networks for the analysis of base stations deployment. Journal of Sensor and Actuator Networks. 2020;9(4): 53. doi: 10.3390/jsan9040053.

Mamera M, van Tol JJ, Aghoghovwia MP, Kotze E. Sensitivity and calibration of the FT-IR spectroscopy on concentration of heavy metal ions in river and borehole water sources. Applied Sciences. 2020;10(21): 7785. doi: 10.3390/app10217785.

Zhang SQ, Lin KP. Short-term traffic flow forecasting based on data-driven model. Mathematics. 2020;8(2): 152. doi: ARTN 15210.3390/math8020152.

Cai PL, et al. A spatiotemporal correlative k-nearest neighbor model for short-term traffic multistep forecasting. Transportation Research Part C: Emerging Technologies. 2016;62: 21-34. doi: 10.1016/j.trc.2015.11.002.

Wang ZY, Ji SW, Yu BW. Short-term traffic volume forecasting with asymmetric loss based on enhanced KNN method. Mathematical Problems in Engineering. 2019;2019: 4589437. doi: Artn 458943710.1155/2019/4589437.

Evans J, Waterson B, Hamilton A. Forecasting road traffic conditions using a context-based random forest algorithm. Transportation Planning and Technology. 2019;42: 554-572. doi: 10.1080/03081060.2019.1622250.

Ou JS, Xia JX, Wu YJ, Rao WM. Short-term traffic flow forecasting for urban roads using data-driven feature selection strategy and bias-corrected random forests. Transport Res Rec. 2017;2645: 157-167. doi: 10.3141/2645-17.

Szegedy C, et al. Going deeper with convolutions. 2015 IEEE Conference on Computer Vision and Pattern Recognition, 7-12 June 2015, Boston, Massachusetts. New Jersey: IEEE; 2015. p. 1-9. doi: 10.1109/cvpr.2015.7298594.

Sun T, et al. Bidirectional spatial-temporal network for traffic prediction with multisource data. Transport Res Rec. 2020;2674: 78-89. doi: 10.1177/03611981209273930.

Wang JW, Chen RX, He ZC. Traffic speed prediction for urban transportation network: A path based deep learning approach. Transportation Research Part C: Emerging Technologies. 2019;100: 372-385. doi: 10.1016/j.trc.2019.02.002.

Cho K, et al. Learning phrase representations using RNN encoder-decoder for statistical machine translation. 2014 Conference on Empirical Methods in Natural Language Processing, 25-29 Oct. 2014, Doha, Qatar. Strousburg: Association for Computational Linguistics; 2014. p. 1724-1734. doi: 10.3115/v1/D14-1179.

Guo YP, Zhang DL, Ling YX, Chen HH. A joint neural network for session-aware recommendation. IEEE Access. 2020;8: 74205-74215. doi: 10.1109/ACCESS.2020.2984287.

Cai L, et al. Traffic transformer: Capturing the continuity and periodicity of time series for traffic forecasting. Transactions in GIS. 2020;24(3): 736-755. doi: 10.1111/tgis.12644.

Drakulic D, Andreoli J-M. Structured time series prediction without structural prior. arXiv. 2022;02: 03539. doi: 10.48550/arXiv.2202.03539.

Yu B, Yin HT, Zhu ZX. ST-UNet: A spatio-temporal U-network for graph-structured time series modeling. arXiv. 2019;03: 05631. doi: 10.48550/arXiv.1903.05631.

Liu MH, et al. Time series is a special sequence: Forecasting with sample convolution and interaction. arXiv. 2021;06: 09305. doi: 10.48550/arXiv.2106.09305.

Choi J, Choi H, Hwang J, Park N. Graph neural controlled differential equations for traffic forecasting. arXiv. 2021;12: 03558. doi: 10.48550/arXiv.2112.03558.

Liao BB, et al. Deep sequence learning with auxiliary information for traffic prediction. 24th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 19-23 Aug. 2018, London, England. New York: Association for Computing Machinery; 2019. p. 537-546. doi: 10.1145/3219819.3219895.

Hong W-C. Application of seasonal SVR with chaotic immune algorithm in traffic flow forecasting. Neural Computing and Applications. 2012;21: 583-593. doi: 10.1007/s00521-010-0456-7.

Cai L, et al. A noise-immune LSTM network for short-term traffic flow forecasting. Chaos. 2020;30: 023135. doi: 10.1063/1.5120502.

Duran-Hernandez C, Ledesma-Alonso R, Etcheverry G. Using autoregressive with exogenous input models to study pulsatile flows. Applied Sciences. 2020;10(22): 8228. doi: 10.3390/app10228228.

Tian Y, et al. Integration of a parsimonious hydrological model with recurrent neural networks for improved streamflow forecasting. Water. 2018;10(11): 1655. doi: 10.3390/w10111655.

Jang B, et al. PIMD signal modeling based on FTDNN. 2019 2nd IEEE International Conference on Information Communication and Signal Processing. 28-30 Sep. 2019, Weihai, China. New Jersey: IEEE; 2019. p. 1-4. doi: 10.1109/ICICSP48821.2019.8958484.

Pwasong A, Sathasivam S. A new hybrid quadratic regression and cascade forward backpropagation neural network. Neurocomputing. 2016;182: 197-209. doi: 10.1016/j.neucom.2015.12.034.

Chen HF, Zhao WX. New method of order estimation for Arma/Armax processes. SIAM Journal on Control and Optimization. 2010;48(6): 4157-4176. doi: 10.1137/090768680.

How to Cite
Li J, Li W, Lian G. A Nonlinear Autoregressive Model with Exogenous Variables for Traffic Flow Forecasting in Smaller Urban Regions. Promet [Internet]. 2022Dec.2 [cited 2024Jun.21];34(6):943-57. Available from: http://traffic.fpz.hr/index.php/PROMTT/article/view/4145