Short-Term Traffic Flow Prediction Using Variational LSTM Networks

S H O RT - T E R M T R A FFI C F L O W P R E D I C T I O N U S I N G V A R I A T I O NA L L S T M N E T W O R K S A P R E P R I N T Mehrdad Farahani Department of Computer Engineering Islamic Azad Univ ersity North T ehran Branch T ehran, Iran m3hrdadfi@gmail.com Marzieh Farahani Department of Computing Science Umeå Univ ersity Umeå, Sweden mafa2431@student.umu.se Mohammad Manthouri Department of Electrical and Electronic Engineering Shahed Univ erisity T ehran, Iran mmanthouri@shahed.ac.ir Okyay Kaynak Department of Electrical and Electronic Engineering Bogazici Uni versity Istanbul, T urke y okyay.kaynak@boun.edu.tr A B S T R A C T T raffic flo w characteristics are one of the most critical decision-making and traf fic policing factors in a region. A wareness of the predicted status of the traf fic flow has prime importance in traffic management and traf fic information di visions. The purpose of this research is to suggest a forecasting model for traffic flo w by using deep learning techniques based on historical data in the Intelligent T ransportation 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 V ariational Long Short-T erm Memory Encoder in brief VLSTM-E try to estimate the flow accurately in contrast to other conv entional methods. VLSTM-E can provide more reliable short-term traffic flow by considering the distribution and missing v alues. K eywords T raf fic Flow Prediction · Short-term Prediction · V ariational Encoder · Long Short-T erm Memory 1 Introduction Urban life has undergone many changes in the de velopment of local communities. This transport transformation and traf fic congestion lead to road-clogging, slo wer speeds, longer trip times, and increased v ehicular queuing in most of the urban and suburban passages in the world. This issue will be the trigger of abundant problems such as air pollution and noise pollution and in total, has a massi ve role in quality reductions. Therefore, gov ernors recognize intelligent traffic flo w control systems as a priority plan for their countries. The traffic flo w forecasting is a crucial step for obtaining time optimizers in the public traffic adapti ve control system. T raffic flo w prediction is a significant issue for both transport management from one side and drivers and ordinary people on the other side. These methods help managers to recognize hea vy traffics in the countrysides. Using some predefined paradigms and protocols can a void the incidence of long traf fic jams. On the other hand, dri vers and ordinary people can also make a better decision based on that prediction and contributing to decreasing traffic le vels. Therefore, predicting traf fic flo w characteristics in a geographical area is one of the most critical decision-making and polic ymakers that hav e a significant effect on urban traf fic management. Mainly traf fic flow prediction di vided into three categories [1]. • Short-term forecasting (the interv al is 5 minutes to 30 minutes) • Medium-term forecasting (a time interv al of 30 minutes to several hours) Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T • Long-term forecasting (ranges of one day to se veral days) The ultimate goal in this domain is to e valuate the traf fic flow prediction with the historical traffic data in a particular region before it happens. Ho wev er, unpredictable disturbances, including internal-ev ents in transportation ways (such as an accident, falling part of the route) and une xpected e xternal-events (such as a flood, storm) make long-term forecasting inaccurate 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 h ybrid deep learning method predicts the flow based on a complex generati ve model from the data, which can recognize the spatial and temporal correlation within the sequence of traffic flo ws in a particular range. Furthermore, in the following, the recommended model compares to other state-of-the-art models. The contribution of this paper can be summarized as follo ws: • Presenting a nov el hybrid deep learning model based on a V ariational Long Short-T erm Memory Encoder (VLSTM-E) • The proposed model is considering the distrib ution of data to forecast short-term traffic flo w • T ake into consideration the missing data, which occurred by sensors failure by the distrib uted data The paper is se gmented as follows; the next section gi ves a brief description of terminologies, challenges, and other methods of short-term traf fic forecasting research concerning se veral neural network techniques. In section 3, the background of the model is introduced. Then, In section 4, the suggested model is presented. The dataset is denoted in section 5, and the results, and performance ev aluation are presented in section 6. Finally , conclusions and future research are stated in section 7. 2 Related W orks T raffic flo w forecasting is one of the most useful tools in intelligent transportation systems (ITS). It allo ws the system to be in a control automatic operation state and anticipates the ev ents before they occur . It can be able to predict and assess the states and prepare itself for logical decision-making at the machine level, and based on human-made protocols can manage the condition [ 2 ]. Meanwhile, the short-term prediction of the traf fic flow is more critical than the other two before categories in the field of intelligent transportation systems, in which man y research and development are done in both academically and operationally [ 2 ]. A great deal of research on the short-term forecasting model can be classified into two main categories: • Parametric, Including methods such as state-space methods [ 3 ], Kalman filter methods [ 4 ], spectra analysis methods [ 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 has the most recepti ve 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 traf fic sensors around the country is one of the signs of technological advancement that brings up v aluable f acilities for the transportation systems of the country . The information provides a massiv e amount of patterns and paradigms of terrestrial transport in a geographic location. Moreo ver , the direct and indirect effects of that information present the foundation for the application of deep learning networks. Deep learning is a section of machine learning that grants short-term forecasts of traf fic flows to find latent dependence relationships in a set of patterns with high dimensions of explanatory v ariables. This model tries to detect extreme disturbances in the traffic flo w within a pool of latent relations providing by real-time sensors [ 27 , 28 ]. Nev ertheless, there is no clue that which types of deep learning models are the most appropriate model for forecasting traffic flows. All of these models are trying to find a part of these latent relations by presenting a different structure. For e xample, the Stacked Autoencoders model was introduced by considering time and space correlation, w as able to learn the general characteristics of the traffic flo w [ 29 ]. Another model that was able to achiev e better performance is the Long Short-T erm Memory (LSTM) and Gated Recurrent Unit (GR U) networks [ 30 ]. These models provided a solution for gaining better results with an increase in the length of the sequences of information. It is necessary to take into account the ef fects of time before, and after more on each day . The performance of these models is significantly downed due to the accumulation of errors. The LSTM+ model in [ 31 ] made it possible to achie ve better performance considering these effects. 2 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T In addition to predicting traffic flo w behavior , which is one the importance of the traf fic flow prediction, traffic sensors are usually controlling manually , so these collections of data from sensors accompany with various lengths, irregular sampling, and missing data. These dissonances make this prediction complicated. T o solve this challenge, the researcher proposed a model base on Long Short-T erm Memory in [ 32 ]. Also, Con volutional Neural Netw ork models, which showed their abilities to resolv e image issues, are used in this domain so that they could provide e xcellent results in prediction the traffic flo w [33]. 3 Background Since the central core of the proposed mode divided into two parts, v ariational and Long Short-T erm Memory (LSTM). In the following, each section introduced in detail. 3.1 Long Short-T erm Memory Long short-term memory (LSTM), as sho wn in Fig (1), proposed by [ 34 ], is a recursi ve neural network architecture that is capable of learning long-term dependencies. This model has been de veloped to deal with v anishing gradient problems and considered a deep neural netw ork architecture over time. The main component of the Long short-term memory 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 an output gate. The follo wing equations show step by step operation of a layer of memory cells for input time series as X = ( x 1 , x 2 , x 3 , ..., x n ) , hidden states memory cells H = ( h 1 , h 2 , h 3 , ..., h n ) . i t = σ  x t U i + h t − 1 W i  (1) f t = σ  x t U f + h t − 1 W f  (2) o t = σ  x t U o + h t − 1 W o  (3) ˜ C t = tanh  x t U g + h t − 1 W g  (4) C t = σ  f t ∗ C t − 1 + i t ∗ ˜ C t  (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 shown how the hidden layer calculated at a time h t . In the calculations above: • i, f , o are called the input, forget and output gates, respecti vely . • W i , W f , W o the weights connect the recurrence layer at t − 1 to the hidden layer at time t . • U i , U f , U o weights that connect the hidden layer at time t − 1 to the recursiv e layer at time t . At the end of the weighted non-linear calculation in the gates section, the output enters int a sigmoid acti vation function so that it can simulate the gating concept since the sigmoid activ ation function as shown in Eq (7) with a range from 0 to 1 can provide a gate way as an open or closed concept 3 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T σ x = 1 1 + e x (7) In Long Short-T erm Memory networks, the objectiv e function can be different depending on the structure of the problem, which cross-entropy , softmax, and l quadratic can be called accessible functions. 3.2 V ariational A utoencoders 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 by forcing an encoder network to the output in that case to a lo w dimensional representation z , which is then consumed by a decoder network to output the original data as sho wn in (2). Figure 2: Autoencoder model architecture. Howe ver , concerning the variational part [ 36 ], we must say that the goal is to achieve a model in which reproduction is not dependent only on data. V ariational Autoencoder tries to decode data from some kno wn probability distribution, in this case, Gaussian distribution that comes from encoding part to produce reasonable outputs e ven if they are not encoding actual data as shown in Fig (3). Suppose x = x (1) , x (2) , x (3) , ..., x ( N ) be a set of observed v ariables and z = z (1) , z (2) , z (3) , ..., z ( M ) be a set of hidden variables with joint distrib ution p ( Z, X ) . Label this distribution as p θ which parameterized by θ . T o generate a sample that looks like a real data point x ( i ) as shown in Fig (4). Then the inference issue is to calculate the conditional distrib ution of hidden variables gi ven the observ ations, 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 ) = Z p θ ( x | z ) p θ ( z ) d z Unfortunately , computing p θ ( x ) is quite dif ficult because it is very e xpensive to check all the possible v alues of z and sum them up. So, to solve this issue, approximate p θ ( z | x ) by another distib ution q φ ( z | x ) then can perform approximate inference of the intractable distrib ution. In order to ensure that q φ ( z | x ) and p θ ( z | x ) were similar to each other , we could minimize the KL div ergence between these two distrib utions, as shown in Eq (9). 4 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T Figure 3: V ariational Autoencoder model with the multiv ariate Gaussian assumption Figure 4: The graphical model of V ariational Autoencoder . Solid lines denote the generati ve distrib ution p θ ( z ) , and dashed lines denote the distribution q φ ( z | x ) to approximate the intractable posterior p θ ( z | x ) . D KL ( q φ ( z | x ) k p θ ( z | x )) (9) = Z q φ ( z | x ) log q φ ( z | x ) p θ ( z | x ) d z = Z 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 ) k 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. W e ha ve Eq (10); moreover , then the loss function would be as the variational lo wer bound, or evidence lo wer bound, as shown in Eq (11). log p θ ( x ) − D KL ( q φ ( z | x ) k p θ ( z | x )) (10) = E z ∼ q φ ( z | x ) log p θ ( x | z ) − D KL ( q φ ( z | x ) k p θ ( z )) 5 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T L V AE ( θ , φ ) = − log p θ ( x ) + D KL ( q φ ( z | x ) k p θ ( z | x )) (11) = − E z ∼ q φ ( z | x ) p θ ( x | z ) + D KL ( q φ ( z | x ) k p θ ( z )) θ ∗ , φ ∗ = arg min θ,φ L V AE Therefore by minimizing the loss, we are maximizing the lo wer bound of the probability of generating real data samples in Eq (12). − L V AE = log p θ ( x ) − D KL ( q φ ( z | x ) k p θ ( z | x )) ≤ log p θ ( x ) (12) 4 Proposed Method According to the previous approaches, the proposed model includes a V ariational Autoencoder, which uses LSTM as its encoder and decoder parts, as shown in Fig (5). Long Short-T erm Memory acts as an exploiter both the past and future information — finally , a multi-layer perceptron (MLP) netw ork, which is responsible for mapping the target with the samples 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 the distribution and feed into the Multilayer Perceptron model to estimate traf fic flow 6 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T 5 Experiments 5.1 Dataset Figure 6: The traffic flo w 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 of data by more than 39,000 indi vidual detectors across all major metropolitan areas of the state of California. Performance Measurement System pro vides a significant v ariety source of traffic data inte grated from Caltrans and other local agenc y systems. In this paper , the traf fic flo w dataset consists of sensors information in the California area, district se ven, between 2019-01-01 to 2019-05-30 in a fi ve minutes interv al detections. In the case of sensors f ailure, some records hav e no values (missing data). In this scenario, a combination of Spline-Interpolation and av erage over a 15 minutes interval, could help the model learn inner patterns desirably . Then the dataset prepared in preprocessing steps. In this particular case, the proposed model would be tested on the traffic flo ws of two points between station 716076 and 717060, as shown in Fig (6). Then for each record at time t , data related to time t 12 is selected as additional features. In other words, our data is picked up to 12 earlier records as a look back. Then the data is scaled into a Min-Max scaler . The data in 2019 between 2019-01-01 00:00:00 to 2019-03-31 23:59:00 chose as a training set others for testing, as shown in T able (1). Besides, typical daily traffic flo w charts are presented in Fig (7) for both training and testing parts regarding two stations. T able 1: Displays the dimensional division of data into training and testing Stations X T rain Y T rain X T est Y T est 716076 8628 x 12 x 1 5778 x 12 x 1 8628 x 1 5778 x 1 717060 8628 x 12 x 1 6187 x 12 x 1 8628 x 1 6187 x 1 5.2 Parametric Settings In terms of hardw are, the GPU we use is T esla k80 which provided by Google Colab[ 37 ]. The proposed VLSTM-E architecture and chosen networks were implemented on the T ensorFlo w platform (v1.14.0) [ 38 ]. The learning rate is 0.0001, and the batch size is 256, the sigmoid is used for both as the activ ation of the last layer . 7 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T (a) (b) Figure 7: T ypical daily traf fic flow pattern for two stations 716076 and 717060. (a) T raffic flo w from T uesday 1 January 2019 to Saturday 5 January 2019 as a training e xample. (b) T raffic flo w from Saturday 20 April 2019 to W ednesday 24 April 2019 as a testing example. 8 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T 5.3 Index of Perf ormance Four measurements introduced in this paper to e valuate the ef fectiveness of the proposed model, in the follo ws: e i = f i − b f i (13) M S E = 1 n n X t =1 e 2 i (14) RM S E = v u u t 1 n n X t =1 e 2 i (15) M AE = 1 n n X t =1 | e i | (16) M AP E = 100% n n X t =1     e i f i     (17) where n is the number of the test sample, f i is the real traffic flo w in sample i , and b f i denotes the predicted traffic flo w . 6 Results In the following, the results presented as e valuation results and forecasting the traffic flo w for VLSTM-E (T able (2), Fig (8)), LSTM (T able (3), Fig (9)), MCNNM (T able (4), Fig (10)), and SAEs (T able (5), Fig (11)), respectiv ely . T able 2: The e valuation results for the V ariational Long Short-T erm Memory Encoder (VLSTM-E) model. VLSTM-E Station ID MAPE [%] MAE MSE RMSE 716076 9.5954 0.0312 0.0018 0.0422 717060 8.8625 0.0276 0.0015 0.0381 T able 3: The e valuation results for the Long Short-T erm Memory (LSTM) model. LSTM Station ID MAPE [%] MAE MSE RMSE 716076 10.2718 0.0341 0.0024 0.0490 717060 10.8174 0.0366 0.0022 0.0464 9 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T (a) (b) Figure 8: T ypical daily traf fic flow forecasting for two stations 716076 and 717060 by VLSTM-E model between Saturday 20 April 2019 to W ednesday 24 April 2019. (a) T raffic flo w forecasting for 716076. (b) T raffic flo w forecasting for 717060. 10 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T (a) (b) Figure 9: T ypical daily traf fic flow forecasting for two stations 716076 and 717060 by LSTM model between Saturday 20 April 2019 to W ednesday 24 April 2019. (a) T raffic flo w forecasting for 716076. (b) T raffic flo w forecasting for 717060. 11 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T (a) (b) Figure 10: T ypical daily traf fic flow forecasting for two stations 716076 and 717060 by MCNNM model between Saturday 20 April 2019 to W ednesday 24 April 2019. (a) T raffic flo w forecasting for 716076. (b) T raffic flo w forecasting for 717060. 12 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T T able 4: The e valuation results for the Multiple Con volutional Neural Network for Multi variate (MCNNM) model. MCNNM Station ID MAPE [%] MAE MSE RMSE 716076 31.0840 0.0757 0.0129 0.1136 717060 24.0724 0.0603 0.0082 0.0905 T able 5: The e valuation results for the Stacked Autoencoders (SAEs) model. SAEs Station ID MAPE [%] MAE MSE RMSE 716076 9.9421 0.0326 0.0020 0.0449 717060 18.4939 0.0560 0.0040 0.0635 As the results show , the proposed model, VLSTM-E, has impro ved compared to other con ventional models like the Stacked Autoencoders, Long Short-T erm Memory , and Multiple Con volutional Neural Network, which introduced in 2015 [ 29 ], 2016 [ 30 ] and 2019 [ 33 ]. T o better understanding, this superiority , the a verage of the results according to the ev aluation criterion is presented in T able (6) which, shows the MSE score of the VLSTM-E is 0.0016. T able 6: A verage performance for all the models. A verage Models Station ID MAPE [%] MAE MSE RMSE VLSTM-E 9.2290 0.0294 0.0016 0.0402 LSTM [30] 10.5446 0.0353 0.0023 0.0477 MCNNM [33] 27.5782 0.0680 0.0106 0.1021 SAEs [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, April 20. As can be seen, in all stations, the VLSTM-E curve has a better estimation of the traf fic flow than other curves. In cases where the traf fic flow fluctuates in viewing a large amount of traf fic, the model can quickly con verge into that behavior . Also, in lo w volume v olatility , imitation shows a better response than the Long Short-T erm Memory model. Perhaps the reason for this impro vement can be found in the data structure; in some cases, the sensors in the stations can not detect the observ ation, or ev en this observation will not be highly accurate. In another word, these sensors might be failed in vehicle detection, so it caused missing v alues. Since the model related to the distribution of data, and the sample of this distribution feed into the network, it can be reduced the adv erse ef fects of these missing data in the learning process and lead to satisfactory results than the other models like Long Short-T erm Memory . 7 Conclusions This paper presents a Deep Learning approach with a V ariational Long Short-T erm Memory Encoder to predict the short-term traffic flow . In contrast to the pre vious approaches [ 30 ], this model considers the pattern of the data and provided a solution for missing data. So, it could achieve better results based on the four ev aluation criteria in contrast to the other models [ 29 , 30 , 33 ], which were introduced earlier . This model is implemented on the PeMS dataset. A suggestion for future work would be interesting if implemented on the other dataset that the stations and its sensors produce missing or lo w-value information. Also, on v arious distributions, such as Dirichlet distribution, can be useful in improving sample distrib ution in traffic flo w . 13 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T (a) (b) Figure 11: T ypical daily traf fic flow forecasting for two stations 716076 and 717060 by SAEs model between Saturday 20 April 2019 to W ednesday 24 April 2019. (a) T raffic flo w forecasting for 716076. (b) T raffic flo w forecasting for 717060. 14 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T Figure 12: Forecasting performance on V aritional Long Short-T erm Memory Encoder (VLSTME), Long Short-T erm Memory (LSTM), Multiple Con volutional Neural Network for Multi variate (MCNNM), and Stacked Autoencoders (SAEs) for 716076 station! 15 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T Figure 13: Forecasting performance on V aritional Long Short-T erm Memory Encoder (VLSTME), Long Short-T erm Memory (LSTM), Multiple Con volutional Neural Network for Multi variate (MCNNM), and Stacked Autoencoders (SAEs) for 717060 station! 16 Short-T erm T raffic Flo w Prediction Using V ariational LSTM Networks A P R E P R I N T References [1] Zhongsheng Hou and Xingyi Li. Repeatability and similarity of free way traf fic flow and long-term prediction under big data. 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