Short-Term Traffic Flow Prediction Using Variational LSTM Networks
Mehrdad Farahani, Marzieh Farahani, Mohammad Manthouri, Okyay Kaynak
SS HORT -T ERM T RAFFIC F LOW P REDICTION U SING V ARIATIONAL
LSTM N
ETWORKS
A P
REPRINT
Mehrdad Farahani
Department of Computer EngineeringIslamic Azad University North Tehran BranchTehran, Iran [email protected]
Marzieh Farahani
Department of Computing ScienceUmeå UniversityUmeå, Sweden [email protected]
Mohammad Manthouri
Department of Electrical and Electronic EngineeringShahed UniverisityTehran, Iran [email protected]
Okyay Kaynak
Department of Electrical and Electronic EngineeringBogazici UniversityIstanbul, Turkey [email protected] A BSTRACT
Traffic flow characteristics are one of the most critical decision-making and traffic policing factorsin a region. Awareness of the predicted status of the traffic flow has prime importance in trafficmanagement and traffic information divisions. The purpose of this research is to suggest a forecastingmodel for traffic flow by using deep learning techniques based on historical data in the IntelligentTransportation Systems area. The historical data collected from the Caltrans Performance Mea-surement Systems (PeMS) for six months in 2019. The proposed prediction model is a VariationalLong Short-Term Memory Encoder in brief VLSTM-E try to estimate the flow accurately in contrastto other conventional methods. VLSTM-E can provide more reliable short-term traffic flow byconsidering the distribution and missing values. K eywords Traffic Flow Prediction · Short-term Prediction · Variational Encoder · Long Short-Term Memory
Urban life has undergone many changes in the development of local communities. This transport transformation andtraffic congestion lead to road-clogging, slower speeds, longer trip times, and increased vehicular queuing in most of theurban and suburban passages in the world. This issue will be the trigger of abundant problems such as air pollution andnoise pollution and in total, has a massive role in quality reductions. Therefore, governors recognize intelligent trafficflow control systems as a priority plan for their countries. The traffic flow forecasting is a crucial step for obtaining timeoptimizers in the public traffic adaptive control system.Traffic flow prediction is a significant issue for both transport management from one side and drivers and ordinarypeople on the other side. These methods help managers to recognize heavy traffics in the countrysides. Using somepredefined paradigms and protocols can avoid the incidence of long traffic jams. On the other hand, drivers and ordinarypeople can also make a better decision based on that prediction and contributing to decreasing traffic levels. Therefore,predicting traffic flow characteristics in a geographical area is one of the most critical decision-making and policymakersthat have a significant effect on urban traffic management. Mainly traffic flow prediction divided into three categories[1]. • Short-term forecasting (the interval is 5 minutes to 30 minutes) • Medium-term forecasting (a time interval of 30 minutes to several hours) a r X i v : . [ c s . L G ] F e b hort-Term Traffic Flow Prediction Using Variational LSTM Networks A P
REPRINT • Long-term forecasting (ranges of one day to several days)The ultimate goal in this domain is to evaluate the traffic flow prediction with the historical traffic data in a particularregion before it happens. However, unpredictable disturbances, including internal-events in transportation ways (such asan accident, falling part of the route) and unexpected external-events (such as a flood, storm) make long-term forecastinginaccurate enough. While medium-term or short-term forecasting can be reliable if they correctly setup.In this research, the short-term case takes into consideration. The hybrid deep learning method predicts the flow basedon a complex generative model from the data, which can recognize the spatial and temporal correlation within thesequence of traffic flows in a particular range. Furthermore, in the following, the recommended model compares toother state-of-the-art models.The contribution of this paper can be summarized as follows: • Presenting a novel hybrid deep learning model based on a Variational Long Short-Term Memory Encoder(VLSTM-E) • The proposed model is considering the distribution of data to forecast short-term traffic flow • Take into consideration the missing data, which occurred by sensors failure by the distributed dataThe paper is segmented as follows; the next section gives a brief description of terminologies, challenges, and othermethods of short-term traffic forecasting research concerning several neural network techniques. In section 3, thebackground of the model is introduced. Then, In section 4, the suggested model is presented. The dataset is denotedin section 5, and the results, and performance evaluation are presented in section 6. Finally, conclusions and futureresearch are stated in section 7.
Traffic flow forecasting is one of the most useful tools in intelligent transportation systems (ITS). It allows the system tobe in a control automatic operation state and anticipates the events before they occur. It can be able to predict and assessthe states and prepare itself for logical decision-making at the machine level, and based on human-made protocols canmanage the condition [2]. Meanwhile, the short-term prediction of the traffic flow is more critical than the other twobefore categories in the field of intelligent transportation systems, in which many research and development are done inboth academically and operationally [2]. A great deal of research on the short-term forecasting model can be classifiedinto two main categories: • Parametric, Including methods such as state-space methods [3], Kalman filter methods [4], spectra analysismethods [5], statistical techniques [6], ARIMA, ARIMAX, and SARIMA models [7, 8, 9], and Markov model[10, 11]. • Nonparametric, In these models, with non-linear backgrounds, we are trying to find the model that hasthe most receptive learning features. Many research has gotten lots of remarkable results with this insight,such as non-parametric regression techniques [12, 13, 14], k-nearest neighbor models [15], fuzzy techniques[16, 17, 18], neural networks [19, 20, 21, 22, 23], and support vector machine [24, 25, 26].The spatial-temporal real-time information by traffic sensors around the country is one of the signs of technologicaladvancement that brings up valuable facilities for the transportation systems of the country. The information provides amassive amount of patterns and paradigms of terrestrial transport in a geographic location. Moreover, the direct andindirect effects of that information present the foundation for the application of deep learning networks. Deep learningis a section of machine learning that grants short-term forecasts of traffic flows to find latent dependence relationshipsin a set of patterns with high dimensions of explanatory variables. This model tries to detect extreme disturbances in thetraffic flow within a pool of latent relations providing by real-time sensors [27, 28]. Nevertheless, there is no clue thatwhich types of deep learning models are the most appropriate model for forecasting traffic flows. All of these modelsare trying to find a part of these latent relations by presenting a different structure.For example, the Stacked Autoencoders model was introduced by considering time and space correlation, was able tolearn the general characteristics of the traffic flow [29]. Another model that was able to achieve better performanceis the Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) networks [30]. These models provided asolution for gaining better results with an increase in the length of the sequences of information. It is necessary to takeinto account the effects of time before, and after more on each day. The performance of these models is significantlydowned due to the accumulation of errors. The LSTM+ model in [31] made it possible to achieve better performanceconsidering these effects. 2hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT
In addition to predicting traffic flow behavior, which is one the importance of the traffic flow prediction, traffic sensorsare usually controlling manually, so these collections of data from sensors accompany with various lengths, irregularsampling, and missing data. These dissonances make this prediction complicated. To solve this challenge, the researcherproposed a model base on Long Short-Term Memory in [32]. Also, Convolutional Neural Network models, whichshowed their abilities to resolve image issues, are used in this domain so that they could provide excellent results inprediction the traffic flow [33].
Since the central core of the proposed mode divided into two parts, variational and Long Short-Term Memory (LSTM).In the following, each section introduced in detail.
Long short-term memory (LSTM), as shown in Fig (1), proposed by [34], is a recursive neural network architecturethat is capable of learning long-term dependencies. This model has been developed to deal with vanishing gradientproblems and considered a deep neural network architecture over time. The main component of the Long short-termmemory layer is the memory cell. Figure 1: Long short-term memory cell.A memory cell consists of four main elements: an input gate, a neuron with reconnection, a forget gate, and anoutput gate. The following equations show step by step operation of a layer of memory cells for input time series as X = ( x , x , x , ..., x n ) , hidden states memory cells H = ( h , h , h , ..., h n ) . i t = σ (cid:0) x t U i + h t − W i (cid:1) (1) f t = σ (cid:0) x t U f + h t − W f (cid:1) (2) o t = σ (cid:0) x t U o + h t − W o (cid:1) (3) ˜ C t = tanh (cid:0) x t U g + h t − W g (cid:1) (4) C t = σ (cid:0) f t ∗ C t − + i t ∗ ˜ C t (cid:1) (5) h t = tanh( C t ) ∗ o t (6)The ∗ sign in this calculation considered as element-wise multiplication, and by refusing the bias terms, it can be shownhow the hidden layer calculated at a time h t . In the calculations above: • i, f, o are called the input, forget and output gates, respectively. • W i , W f , W o the weights connect the recurrence layer at t − to the hidden layer at time t . • U i , U f , U o weights that connect the hidden layer at time t − to the recursive layer at time t .At the end of the weighted non-linear calculation in the gates section, the output enters int a sigmoid activation functionso that it can simulate the gating concept since the sigmoid activation function as shown in Eq (7) with a range from 0to 1 can provide a gateway as an open or closed concept 3hort-Term Traffic Flow Prediction Using Variational LSTM Networks A P
REPRINT σ x = 11 + e x (7)In Long Short-Term Memory networks, the objective function can be different depending on the structure of theproblem, which cross-entropy, softmax, and l quadratic can be called accessible functions. Before paying attention to the variational part, it is necessary to get acquainted with the concept of an Autoencoder[35]. The Autoencoder network is a bipartite neural network that teaches the network to compress the information byforcing an encoder network to the output in that case to a low dimensional representation z , which is then consumed bya decoder network to output the original data as shown in (2).Figure 2: Autoencoder model architecture.However, concerning the variational part [36], we must say that the goal is to achieve a model in which reproduction isnot dependent only on data. Variational Autoencoder tries to decode data from some known probability distribution,in this case, Gaussian distribution that comes from encoding part to produce reasonable outputs even if they are notencoding actual data as shown in Fig (3).Suppose x = x (1) , x (2) , x (3) , ..., x ( N ) be a set of observed variables and z = z (1) , z (2) , z (3) , ..., z ( M ) be a set of hiddenvariables with joint distribution p ( Z, X ) . Label this distribution as p θ which parameterized by θ . To generate a samplethat looks like a real data point x ( i ) as shown in Fig (4).Then the inference issue is to calculate the conditional distribution of hidden variables given the observations, that is, p θ ( z | x ) which can write as shown in Eq (8). p θ ( z | x ) = p θ ( z , x ) p θ ( x ) (8) p θ ( x ) = (cid:90) p θ ( x | z ) p θ ( z ) d z Unfortunately, computing p θ ( x ) is quite difficult because it is very expensive to check all the possible values of z andsum them up. So, to solve this issue, approximate p θ ( z | x ) by another distibution q φ ( z | x ) then can perform approximateinference of the intractable distribution. In order to ensure that q φ ( z | x ) and p θ ( z | x ) were similar to each other, wecould minimize the KL divergence between these two distributions, as shown in Eq (9).4hort-Term Traffic Flow Prediction Using Variational LSTM Networks A P
REPRINT
Figure 3: Variational Autoencoder model with the multivariate Gaussian assumptionFigure 4: The graphical model of Variational Autoencoder. Solid lines denote the generative distribution p θ ( z ) , anddashed lines denote the distribution q φ ( z | x ) to approximate the intractable posterior p θ ( z | x ) . D KL ( q φ ( z | x ) (cid:107) p θ ( z | x )) (9) = (cid:90) q φ ( z | x ) log q φ ( z | x ) p θ ( z | x ) d z = (cid:90) q φ ( z | x ) log q φ ( z | x ) p θ ( x ) p θ ( z , x ) d z = log p θ ( x ) + D KL ( q φ ( z | x ) (cid:107) p θ ( z )) − E z ∼ q φ ( z | x ) log p θ ( x | z ) Then rearrange the left and right-hand side of the equation. We have Eq (10); moreover, then the loss function would beas the variational lower bound, or evidence lower bound, as shown in Eq (11). log p θ ( x ) − D KL ( q φ ( z | x ) (cid:107) p θ ( z | x )) (10) = E z ∼ q φ ( z | x ) log p θ ( x | z ) − D KL ( q φ ( z | x ) (cid:107) p θ ( z )) A P
REPRINT L VAE ( θ, φ ) = − log p θ ( x ) + D KL ( q φ ( z | x ) (cid:107) p θ ( z | x )) (11) = − E z ∼ q φ ( z | x ) p θ ( x | z ) + D KL ( q φ ( z | x ) (cid:107) p θ ( z )) θ ∗ , φ ∗ = arg min θ,φ L VAE
Therefore by minimizing the loss, we are maximizing the lower bound of the probability of generating real data samplesin Eq (12). − L VAE = log p θ ( x ) − D KL ( q φ ( z | x ) (cid:107) p θ ( z | x )) ≤ log p θ ( x ) (12) According to the previous approaches, the proposed model includes a Variational Autoencoder, which uses LSTM as itsencoder and decoder parts, as shown in Fig (5). Long Short-Term Memory acts as an exploiter both the past and futureinformation — finally, a multi-layer perceptron (MLP) network, which is responsible for mapping the target with thesamples of distribution, which learned by the VLSTM-E.Figure 5: Illustration of the proposed model architecture.In this proposed approach, the network simultaneously learns the distribution of z and transmits samplings from thedistribution and feed into the Multilayer Perceptron model to estimate traffic flow6hort-Term Traffic Flow Prediction Using Variational LSTM Networks A P
REPRINT
Figure 6: The traffic flow between two station in the San Bernardino Fwy.Caltrans Performance Measurement System (PeMS) used as a public dataset. It was collected in the real-time form ofdata by more than 39,000 individual detectors across all major metropolitan areas of the state of California. PerformanceMeasurement System provides a significant variety source of traffic data integrated from Caltrans and other local agencysystems.In this paper, the traffic flow dataset consists of sensors information in the California area, district seven, between2019-01-01 to 2019-05-30 in a five minutes interval detections. In the case of sensors failure, some records have novalues (missing data). In this scenario, a combination of Spline-Interpolation and average over a 15 minutes interval,could help the model learn inner patterns desirably. Then the dataset prepared in preprocessing steps. In this particularcase, the proposed model would be tested on the traffic flows of two points between station 716076 and 717060, asshown in Fig (6).Then for each record at time t , data related to time t is selected as additional features. In other words, our data ispicked up to 12 earlier records as a look back. Then the data is scaled into a Min-Max scaler. The data in 2019 between2019-01-01 00:00:00 to 2019-03-31 23:59:00 chose as a training set others for testing, as shown in Table (1). Besides,typical daily traffic flow charts are presented in Fig (7) for both training and testing parts regarding two stations. Table 1: Displays the dimensional division of data into training and testing
Stations X Train Y Train X Test Y Test716076 8628 x 12 x 1 5778 x 12 x 1 8628 x 1 5778 x 1717060 8628 x 12 x 1 6187 x 12 x 1 8628 x 1 6187 x 1
In terms of hardware, the GPU we use is Tesla k80 which provided by Google Colab[37]. The proposed VLSTM-Earchitecture and chosen networks were implemented on the TensorFlow platform (v1.14.0) [38]. The learning rate is0.0001, and the batch size is 256, the sigmoid is used for both as the activation of the last layer.7hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT (a)(b)
Figure 7: Typical daily traffic flow pattern for two stations 716076 and 717060. (a) Traffic flow from Tuesday 1 January2019 to Saturday 5 January 2019 as a training example. (b) Traffic flow from Saturday 20 April 2019 to Wednesday 24April 2019 as a testing example.8hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT
Four measurements introduced in this paper to evaluate the effectiveness of the proposed model, in the follows: e i = f i − (cid:98) f i (13) M SE = 1 n n (cid:88) t =1 e i (14) RM SE = (cid:118)(cid:117)(cid:117)(cid:116) n n (cid:88) t =1 e i (15) M AE = 1 n n (cid:88) t =1 | e i | (16) M AP E = 100% n n (cid:88) t =1 (cid:12)(cid:12)(cid:12)(cid:12) e i f i (cid:12)(cid:12)(cid:12)(cid:12) (17)where n is the number of the test sample, f i is the real traffic flow in sample i , and (cid:98) f i denotes the predicted traffic flow. In the following, the results presented as evaluation results and forecasting the traffic flow for VLSTM-E (Table (2), Fig(8)), LSTM (Table (3), Fig (9)), MCNNM (Table (4), Fig (10)), and SAEs (Table (5), Fig (11)), respectively.Table 2: The evaluation results for the Variational Long Short-Term Memory Encoder (VLSTM-E) model.
VLSTM-EStation ID MAPE [%] MAE MSE RMSE716076 9.5954 0.0312 0.0018 0.0422717060 8.8625 0.0276 0.0015 0.0381
Table 3: The evaluation results for the Long Short-Term Memory (LSTM) model.
LSTMStation ID MAPE [%] MAE MSE RMSE716076 10.2718 0.0341 0.0024 0.0490717060 10.8174 0.0366 0.0022 0.0464
A P
REPRINT (a)(b)
Figure 8: Typical daily traffic flow forecasting for two stations 716076 and 717060 by VLSTM-E model betweenSaturday 20 April 2019 to Wednesday 24 April 2019. (a) Traffic flow forecasting for 716076. (b) Traffic flowforecasting for 717060.10hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT (a)(b)
Figure 9: Typical daily traffic flow forecasting for two stations 716076 and 717060 by LSTM model between Saturday20 April 2019 to Wednesday 24 April 2019. (a) Traffic flow forecasting for 716076. (b) Traffic flow forecasting for717060.11hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT (a)(b)
Figure 10: Typical daily traffic flow forecasting for two stations 716076 and 717060 by MCNNM model betweenSaturday 20 April 2019 to Wednesday 24 April 2019. (a) Traffic flow forecasting for 716076. (b) Traffic flowforecasting for 717060.12hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT
Table 4: The evaluation results for the Multiple Convolutional Neural Network for Multivariate (MCNNM) model.
MCNNMStation ID MAPE [%] MAE MSE RMSE716076 31.0840 0.0757 0.0129 0.1136717060 24.0724 0.0603 0.0082 0.0905
Table 5: The evaluation results for the Stacked Autoencoders (SAEs) model.
SAEsStation ID MAPE [%] MAE MSE RMSE716076 9.9421 0.0326 0.0020 0.0449717060 18.4939 0.0560 0.0040 0.0635
As the results show, the proposed model, VLSTM-E, has improved compared to other conventional models like theStacked Autoencoders, Long Short-Term Memory, and Multiple Convolutional Neural Network, which introduced in2015 [29], 2016 [30] and 2019 [33]. To better understanding, this superiority, the average of the results according to theevaluation criterion is presented in Table (6) which, shows the MSE score of the VLSTM-E is 0.0016.Table 6: Average performance for all the models.
Average ModelsStation ID MAPE [%] MAE MSE RMSEVLSTM-E 9.2290 0.0294 0.0016 0.0402LSTM [30] 10.5446 0.0353 0.0023 0.0477MCNNM [33] 27.5782 0.0680 0.0106 0.1021SAEs [29] 14.2180 0.0443 0.0030 0.0542
Figures (12, 13) shows the prediction results for the two stations 716076, and 717060 for the test dataset on 2019, April20. As can be seen, in all stations, the VLSTM-E curve has a better estimation of the traffic flow than other curves. Incases where the traffic flow fluctuates in viewing a large amount of traffic, the model can quickly converge into thatbehavior. Also, in low volume volatility, imitation shows a better response than the Long Short-Term Memory model.Perhaps the reason for this improvement can be found in the data structure; in some cases, the sensors in the stationscan not detect the observation, or even this observation will not be highly accurate. In another word, these sensorsmight be failed in vehicle detection, so it caused missing values. Since the model related to the distribution of data, andthe sample of this distribution feed into the network, it can be reduced the adverse effects of these missing data in thelearning process and lead to satisfactory results than the other models like Long Short-Term Memory.
This paper presents a Deep Learning approach with a Variational Long Short-Term Memory Encoder to predict theshort-term traffic flow. In contrast to the previous approaches [30], this model considers the pattern of the data andprovided a solution for missing data. So, it could achieve better results based on the four evaluation criteria in contrastto the other models [29, 30, 33], which were introduced earlier. This model is implemented on the PeMS dataset. Asuggestion for future work would be interesting if implemented on the other dataset that the stations and its sensorsproduce missing or low-value information. Also, on various distributions, such as Dirichlet distribution, can be usefulin improving sample distribution in traffic flow. 13hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT (a)(b)
Figure 11: Typical daily traffic flow forecasting for two stations 716076 and 717060 by SAEs model between Saturday20 April 2019 to Wednesday 24 April 2019. (a) Traffic flow forecasting for 716076. (b) Traffic flow forecasting for717060.14hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT
Figure 12: Forecasting performance on Varitional Long Short-Term Memory Encoder (VLSTME), Long Short-TermMemory (LSTM), Multiple Convolutional Neural Network for Multivariate (MCNNM), and Stacked Autoencoders(SAEs) for 716076 station!15hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT
Figure 13: Forecasting performance on Varitional Long Short-Term Memory Encoder (VLSTME), Long Short-TermMemory (LSTM), Multiple Convolutional Neural Network for Multivariate (MCNNM), and Stacked Autoencoders(SAEs) for 717060 station!16hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT
References [1] Zhongsheng Hou and Xingyi Li. Repeatability and similarity of freeway traffic flow and long-term prediction underbig data.
IEEE Transactions on Intelligent Transportation Systems , 17:1786–1796, 2016.[2] Se do Oh, Young jin Kim, and Ji sun Hong. Urban traffic flow prediction system using a multifactor patternrecognition model.
IEEE Transactions on Intelligent Transportation Systems , 16:2744–2755, 2015.[3] Anthony Stathopoulos and Matthew G. Karlaftis. A multivariate state space approach for urban traffic flow modelingand prediction.
Transportation Research Part C: Emerging Technologies , 11(2):121–135, April 2003.[4] Teng Zhou, Dazhi Jiang, Zhizhe Lin, Guoqiang Han, Xuemiao Xu, and Jing Qin. Hybrid dual kalman filteringmodel for short-term traffic flow forecasting.
IET Intelligent Transport Systems , 13(6):1023–1032, June 2019.[5] Yanru Zhang, Yunlong Zhang, and Ali Haghani. A hybrid short-term traffic flow forecasting method based onspectral analysis and statistical volatility model.
Transportation Research Part C: Emerging Technologies , 43:65–78,June 2014.[6] Milan Krbálek, Jiˇrí Apeltauer, and František Šeba. Traffic flow merging – statistical and numerical modeling ofmicrostructure.
Journal of Computational Science , 32:99–105, March 2019.[7] Xianglong Luo, Liyao Niu, and Shengrui Zhang. An algorithm for traffic flow prediction based on improved sarimaand ga.
KSCE Journal of Civil Engineering , 22(10):4107–4115, Oct 2018.[8] Qinzhong Hou, Junqiang Leng, Guosheng Ma, Weiyi Liu, and Yuxing Cheng. An adaptive hybrid model forshort-term urban traffic flow prediction.
Physica A: Statistical Mechanics and its Applications , 527:121065, August2019.[9] Chukwutoo C. Ihueze and Uchendu O. Onwurah. Road traffic accidents prediction modelling: An analysis ofanambra state, nigeria.
Accident Analysis & Prevention , 112:21–29, March 2018.[10] Guangyu Zhu, Kang Song, Peng Zhang, and Li Wang. A traffic flow state transition model for urban road networkbased on hidden markov model.
Neurocomputing , 214:567–574, November 2016.[11] Liguo Zhang and Christophe Prieur. Stochastic stability of markov jump hyperbolic systems with application totraffic flow control.
Automatica , 86:29–37, December 2017.[12] Darong Huang and Xing rong Bai. A wavelet neural network optimal control model for traffic-flow predictionin intelligent transport systems. In
Advanced Intelligent Computing Theories and Applications. With Aspects ofArtificial Intelligence , pages 1233–1244. Springer Berlin Heidelberg, 2007.[13] Shaurya Agarwal, Pushkin Kachroo, and Emma Regentova. A hybrid model using logistic regression and wavelettransformation to detect traffic incidents.
IATSS Research , 40(1):56–63, July 2016.[14] Dick Apronti, Khaled Ksaibati, Kenneth Gerow, and Jaime Jo Hepner. Estimating traffic volume on wyominglow volume roads using linear and logistic regression methods.
Journal of Traffic and Transportation Engineering(English Edition) , 3(6):493–506, December 2016.[15] Pinlong Cai, Yunpeng Wang, Guangquan Lu, Peng Chen, Chuan Ding, and Jianping Sun. A spatiotemporalcorrelative k-nearest neighbor model for short-term traffic multistep forecasting.
Transportation Research Part C:Emerging Technologies , 62:21–34, January 2016.[16] A. Sharma, R. Vijay, G. L. Bodhe, and L. G. Malik. An adaptive neuro-fuzzy interface system model for trafficclassification and noise prediction.
Soft Computing , 22(6):1891–1902, November 2016.[17] Jianhua Guo, Zhao Liu, Wei Huang, Yun Wei, and Jinde Cao. Short-term traffic flow prediction using fuzzyinformation granulation approach under different time intervals.
IET Intelligent Transport Systems , 12(2):143–150,March 2018.[18] Weihong Chen, Jiyao An, Renfa Li, Li Fu, Guoqi Xie, Md Zakirul Alam Bhuiyan, and Keqin Li. A novel fuzzydeep-learning approach to traffic flow prediction with uncertain spatial–temporal data features.
Future GenerationComputer Systems , 89:78–88, December 2018.[19] Carl Goves, Robin North, Ryan Johnston, and Graham Fletcher. Short term traffic prediction on the UK motorwaynetwork using neural networks.
Transportation Research Procedia , 13:184–195, 2016.[20] Jithin Raj, Hareesh Bahuleyan, and Lelitha Devi Vanajakshi. Application of data mining techniques for trafficdensity estimation and prediction.
Transportation Research Procedia , 17:321–330, 2016.[21] Kui-Lin Li, Chun-Jie Zhai, and Jian-Min Xu. Short-term traffic flow prediction using a methodology based onARIMA and RBF-ANN. In . IEEE, October 2017.17hort-Term Traffic Flow Prediction Using Variational LSTM Networks
A P
REPRINT [22] Bharti Sharma, Sachin Kumar, Prayag Tiwari, Pranay Yadav, and Marina I. Nezhurina. ANN based short-termtraffic flow forecasting in undivided two lane highway.
Journal of Big Data , 5(1), December 2018.[23] Jingyuan Wang, Yukun Cao, Ye Du, and Li Li. DST: A deep urban traffic flow prediction framework basedon spatial-temporal features. In
Knowledge Science, Engineering and Management , pages 417–427. SpringerInternational Publishing, 2019.[24] Anyu Cheng, Xiao Jiang, Yongfu Li, Chao Zhang, and Hao Zhu. Multiple sources and multiple measuresbased traffic flow prediction using the chaos theory and support vector regression method.
Physica A: StatisticalMechanics and its Applications , 466:422–434, January 2017.[25] Yuxing Sun, Biao Leng, and Wei Guan. A novel wavelet-SVM short-time passenger flow prediction in beijingsubway system.
Neurocomputing , 166:109–121, October 2015.[26] Jianli Xiao, Chao Wei, and Yuncai Liu. Speed estimation of traffic flow using multiple kernel support vectorregression.
Physica A: Statistical Mechanics and its Applications , 509:989–997, November 2018.[27] Nicholas G. Polson and Vadim O. Sokolov. Deep learning for short-term traffic flow prediction.
TransportationResearch Part C: Emerging Technologies , 79:1–17, June 2017.[28] Yuankai Wu, Huachun Tan, Lingqiao Qin, Bin Ran, and Zhuxi Jiang. A hybrid deep learning based traffic flowprediction method and its understanding.
Transportation Research Part C: Emerging Technologies , 90:166–180,May 2018.[29] Yisheng Lv, Yanjie Duan, Wenwen Kang, Zhengxi Li, and Fei-Yue Wang. Traffic flow prediction with big data: Adeep learning approach.
IEEE Transactions on Intelligent Transportation Systems , pages 1–9, 2014.[30] Rui Fu, Zuo Zhang, and Li Li. Using LSTM and GRU neural network methods for traffic flow prediction. In . IEEE, November 2016.[31] Bailin Yang, Shulin Sun, Jianyuan Li, Xianxuan Lin, and Yan Tian. Traffic flow prediction using LSTM withfeature enhancement.
Neurocomputing , 332:320–327, March 2019.[32] Yan Tian, Kaili Zhang, Jianyuan Li, Xianxuan Lin, and Bailin Yang. LSTM-based traffic flow prediction withmissing data.
Neurocomputing , 318:297–305, November 2018.[33] Kang Wang, Kenli Li, Liqian Zhou, Yikun Hu, Zhongyao Cheng, Jing Liu, and Cen Chen. Multiple convolutionalneural networks for multivariate time series prediction.
Neurocomputing , May 2019.[34] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory.
Neural Computation , 9(8):1735–1780,November 1997.[35] Jürgen Schmidhuber. Deep learning in neural networks: An overview.
Neural networks : the official journal ofthe International Neural Network Society , 61:85–117, 2015.[36] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes.