交通流量預測是交通工程中一個活躍的研究課題。大致上，交通流量預測模型可以分為三種，第一種是有母數方法，第二種是無母數方法，最後一種是基於PDE的模擬。另外還有有母數方法和無母數方法的混合方法。在這項工作中，我們建議將數據驅動的仿真技術與機器學習工具相結合，以減少預測誤差，並使用卡爾曼濾波器（KF）實現數據同化。

KF包括兩個步驟：預測和校正。在預測步驟中，我們使用EX方法來離散LWR模型，其中MacNicholas模型作為速度和密度之間的基本關係。由於預測未來需要邊界點，因此我們使用具有注意力機制的LSTM獲得的預測值來設置邊界條件，在更新步驟中，我們使用具有注意力機制的LSTM獲得的預測值來當作觀測值，它用來跟我們的預測值進行權重並得到更新後的預測值。

在本研究中，我們使用SARIMA和具有注意力機制的LSTM作為基線方法。自回歸整合移動平均線（ARIMA）是用於單變量時間序列數據預測的最廣泛使用的預測方法之一。SARIMA是ARIMA的延伸，有季節性的成分。長短期記憶網絡(LSTM NN)是一種可以學習長期依賴關係的特殊RNN。此外，加入注意機制可以幫助我們更好的預測未來。我們將它們與我們提出的方法比較，實驗結果表明，該方法優於SARIMA和具有注意機制的LSTM。

Traffic flow prediction an active research topic in transportation engineering. In general, the traffic flow prediction model can be divided into three categories, one is PDE-based simulation, another one is parametric approaches, and the other is non-parametric approaches. There are further hybrid approaches to parametric approaches and non-parametric approaches. In this work, we propose combining the data-driven simulation technique with machine learning tools to decrease prediction error, and use the Kalman Filter (KF) on this basis to achieve the effect of data assimilation.

The KF consists of two steps: prediction and correction. In the prediction step, we use the EX method to discretize the LWR model where the MacNicholas model is used as the fundamental relation between the velocity and density. Since the data at the boundary points in the future period are not available. The predicted values obtained by using LSTM with the attention mechanism are used for setting the boundary condition. In the correction step, we use the predicted value obtained by the LSTM with attention mechanism as the observation value, which is used to weight our predicted value and get the correction predicted value.

In this study, we use SARIMA and the LSTM Attention as the baseline methods. Autoregressive Integrated Moving Average (ARIMA) is one of the most widely used methods of prediction for university time series data prediction. SARIMA is an extension of ARIMA with seasonal components. Long Short Term Memory networks (LSTMs) is a special kind of RNN that can learn long-term dependencies better than RNN. In addition, adding attention mechanisms can help us better predict the future. We compare them with our proposed method. The experimental results demonstrate that our method outperforms SARIMA and LSTM Attention.

[1] M. J. Lighthill and G. B. Whitham. On kinematic waves ii. a theory of traffic
flow on long crowded roads. Proceedings of the Royal Society of London. Series A.
Mathematical and Physical Sciences, 229(1178):317–345, 1955.
[2] P. I. Richards. Shock waves on the highway. Operations Research, 4(1):42–51, 1956.
[3] J.-P. Lebacque. The Godunov scheme and what it means for first order traffic flow
models. In Proceedings of the 13TH International Symposium on Transportation and
Traffic Theory, pages 24–26, 1996.
[4] F. Van Wageningen-Kessels, H. Van Lint, S. P. Hoogendoorn, and K. Vuik. Implicit
and explicit numerical methods for macroscopic traffic flow models: Efficiency and
accuracy. 2009.
[5] J. Thai, B. Prodhomme, and A. M. Bayen. State estimation for the discretized LWR
PDE using explicit polyhedral representations of the Godunov scheme. In American
Control Conference, pages 2428–2435. IEEE, 2013.
[6] M. S. Ahmed and A. R. Cook. Analysis of freeway traffic time-series data by using
Box-Jenkins techniques. 1979.
[7] V. D. V. Mascha, M. Dougherty, and S. Watson. Combining kohonen maps with
ARIMA time series models to forecast traffic flow. Transportation Research Part C:
Emerging technologies, 4(5):307–318, 1996.
[8] S. Lee and D. Fambro. Application of subset Autoregressive Integrated Moving
Average model for short-term freeway traffic volume forecasting. Transportation
Research Record, 1678(1):179–188, 1999.
[9] B. Williams and L. Hoel. Modeling and forecasting vehicular traffic flow as a seasonal
ARIMA process: Theoretical basis and empirical results. Journal of Transportation
Engineering, 129(6):664–672, 2003.
[10] S. Kumar and L. Vanajakshi. Short-term traffic flow prediction using seasonal
ARIMA model with limited input data. European Transport Research Review, 7(3):
21, 2015.
[11] B. L. Smith and M. J. Demetsky. Short-term traffic flow prediction models-a comparison
of neural network and nonparametric regression approaches. In Proceedings of
IEEE International Conference on Systems, Man and Cybernetics, pages 1706–1709.
IEEE, 1994.
[12] C.-H. Wu, J.-M. Ho, and D.-T. Lee. Travel-time prediction with Support Vector
Regression. IEEE Transactions on Intelligent Transportation Systems, 5(4):276–281,
2004.
[13] Y. Lv, Y. Duan, W. Kang, Z. Li, and F.-Y. Wang. Traffic flow prediction with big
data: A deep learning approach. IEEE Transactions on Intelligent Transportation
Systems, 16(2):865–873, 2014.
[14] X. Ma, Z. Tao, Y. Wang, H. Yu, and Y. Wang. Long short-term memory neural network
for traffic speed prediction using remote microwave sensor data. Transportation
Research Part C: Emerging Technologies, 54:187–197, 2015.
[15] Y. Tian and L. Pan. Predicting short-term traffic flow by Long Short-Term Memory
Recurrent Neural Network. In IEEE international conference on smart city/
SocialCom/SustainCom (SmartCity), pages 153–158. IEEE, 2015.
[16] R. Fu, Z. Zhang, and L. Li. Using LSTM and GRU neural network methods for
traffic flow prediction. In Youth Academic Annual Conference of Chinese Association
of Automation (YAC), pages 324–328. IEEE, 2016.
[17] Y. Jia, J. Wu, and M. Xu. Traffic flow prediction with rainfall impact using a deep
learning method. Journal of Advanced Transportation, 2017.
[18] Y. Liu, H. Zheng, X. Feng, and Z. Chen. Short-term traffic flow prediction with
Conv-LSTM. In International Conference on Wireless Communications and Signal
Processing (WCSP), pages 1–6. IEEE, 2017.
[19] X. Ran, Z. Shan, Y. Fang, and C. Lin. An LSTM-Based method with attention
mechanism for travel time prediction. Sensors, 19(4):861, 2019.
[20] I. Okutani and Y. J. Stephanedes. Dynamic prediction of traffic volume through
Kalman filtering theory. Transportation Research Part B: Methodological, 18(1):
1–11, 1984.
[21] S. I.-J. Chien and C. M. Kuchipudi. Dynamic travel time prediction with real-time
and historic data. Journal of Transportation Engineering, 129(6):608–616, 2003.
[22] J. W. C. Van Lint. Online learning solutions for freeway travel time prediction. IEEE
Transactions on Intelligent Transportation Systems, 9(1):38–47, 2008.
[23] T. Schreiter, C. P. I. J. Van Hinsbergen, F. S. Zuurbier, J. W. C. Van Lint, and S. P.
Hoogendoorn. Data-model synchronization in extended Kalman filters for accurate
online traffic state estimation. In TFTC Summer Meeting, 2010.
[24] S. V. Kumar. Traffic flow prediction using Kalman filtering technique. Procedia
Engineering, 187:582–587, 2017.
[25] S. Kim and H. Kim. A new metric of absolute percentage error for intermittent
demand forecasts. International Journal of Forecasting, 32(3):669–679, 2016.