A car-following model considering asymmetric driving behavior based on long short-term memory neural networks

Transportation Research Part C 95 (2018) 346–362

Contents lists available at ScienceDirect

Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

A car-following model considering asymmetric driving behavior based on long short-term memory neural networks Xiuling Huang, Jie Sun, Jian Sun

T

⁎

Department of Traffic Engineering & Key Laboratory of Road and Traffic Engineering, Ministry of Education, Tongji University, Shanghai, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Traffic flow Car-following Asymmetric driving behavior Deep learning Long short-term memory

Asymmetric driving behavior is a critical characteristic of human driving behaviors and has a significant impact on traffic flow. In consideration of the asymmetric driving behavior, this paper proposes a long short-term memory (LSTM) neural networks (NN) based car-following (CF) model to capture realistic traffic flow characteristics by incorporating the driving memory. The NGSIM data are used to calibrate and validate the proposed CF model. Meanwhile, three characteristics closely related to the asymmetric driving behavior are investigated: hysteresis, discrete driving, and intensity difference. The simulation results show the good performance of the proposed CF model on reproducing realistic traffic flow features. Moreover, to further demonstrate the superiority of the proposed CF model, two other CF models including recurrent neural network based CF model and asymmetric full velocity difference model, are compared with LSTM-NN model. The results reveal that LSTM-NN model can capture the asymmetric driving behavior well and outperforms other models.

1. Introduction Over the past six decades, numerous car-following (CF) models have been developed to describe the longitudinal interactions of vehicles, including mathematical CF models (Chandler et al., 1958; Herman et al., 1959; Helly, 1959; Newell, 1961; Gipps, 1981; Bando et al., 1995; Helbing and Tilch, 1998; Treiber et al., 2000; Jiang et al., 2001; Gong et al., 2008, Tordeux et al., 2010; Xu et al, 2013; Liu et al., 2017) and data-driven CF models (Panwai and Dia, 2007; Khodayari et al., 2012; Wei and Liu, 2013; Zheng et al., 2013; Z. He et al., 2015; Wang et al., 2018). Mathematical CF models mathematically describe drivers’ behavior in the context of varying traffic conditions, while data-driven CF models precisely mimic the CF behavior based on a mass of empirical data without manual intervention. For both types of CF models, human behaviors (e.g. asymmetric driving behavior, distractive driving behavior) are critical in describing real traffic flow. However, only a few studies have incorporated human driving behaviors (e.g. asymmetric driving behavior) in CF modelling (Krauß et al., 1997; Zhang and Kim, 2005; Gong et al., 2008; Tordeux et al., 2010; Xu et al., 2013, 2015; Wei and Liu, 2013). The asymmetric driving behavior, which means that drivers are more attentive in deceleration than in acceleration, is closely related to the well-known traffic hysteresis phenomenon (Zhang, 1999). This study attempts to model CF behavior and investigate the related characteristics of asymmetric driving behavior: (1) hysteresis: drivers are used to keeping a larger headway when accelerating than decelerating given the same speed (Wei and Liu, 2013); (2) discrete driving: the acceleration and deceleration in CF are not consecutive (Yeo, 2008); (3) intensity difference: the response intensity of drivers to the positive and negative relative speed are

⁎

Corresponding author. E-mail address: [email protected] (J. Sun).

https://doi.org/10.1016/j.trc.2018.07.022 Received 6 February 2018; Received in revised form 20 July 2018; Accepted 23 July 2018 0968-090X/ © 2018 Elsevier Ltd. All rights reserved.

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

different though driving at the same condition, e.g., the same magnitudes of the relative speeds, the same speed, the same gap between the leading and following vehicles. (Zhang, 1999; Wei and Liu, 2013). In addition, Toledo (2007) pointed out that driver's decisions are always dependent on the historical driving behaviors and the past traffic states. In human driving process, the historical information usually have a fading effect which means that the memory is not always constant. Other researchers also demonstrated that CF models taking driving memory into account can describe the traffic flow characteristics better (Lee, 1966; Treiber and Helbing, 2003; Peng et al., 2016). Therefore, it is necessary to model the historical driving behaviors and fading memory effect. However, since it is difficult for mathematical CF models to consider a relative longperiod and varying memory, a data-driven model is thus needed for modeling the CF behavior. The Long Short-Term Memory (LSTM) neural networks (NN) proposed by Hochreiter and Schmidhuber (1997) is used for CF modeling in this study due to its advantages on considering massive historical information and fading memory effect. With NGSIM data (FHWA, 2008), the LSTM-NN model is optimized and analyzed. Meanwhile, the three characteristics of asymmetric driving behavior (i.e. hysteresis, discrete driving and intensity difference) are investigated to validate the efficiency of proposed CF model. In addition, one data-driven model (recurrent neural network (RNN) based CF model), and one mathematical model (asymmetric full velocity difference (AFVD) model (Gong et al., 2008)) are compared with the LSTM-NN model in terms of the simulated trajectories and asymmetric driving behavior characteristics. The remaining part of this paper is organized as follows: Section 2 reviews the studies about asymmetric driving behavior, and the development of data-driven CF models; Section 3 proposes the structure of LSTM-NN model; Section 4 optimizes the LSTM-NN model and other two CF models in terms of simulated trajectories; Section 5 discusses in detail the characteristics of the asymmetry driving behavior and the comparison with RNN model and AFDV model; Section 6 studies the ability of proposed CF model on reproducing stop-and-go wave and Section 7 summarizes main conclusions and discusses the future study. 2. Literature review 2.1. The asymmetric driving behavior Asymmetric driving behavior, which is a critical characteristic of human driving behaviors and has a significant impact on traffic flow, has been observed and verified by many studies. As early as 1960s, the asymmetry in acceleration and deceleration has been observed from the real traffic (Forbes, 1963; Foote, 1965; Edie, 1965; Newell, 1965). Among these studies, Newell (1965) is the first one who gave a theoretical explanation for the asymmetric phenomenon, believing that drivers tend to maintain different gaps from the leading vehicle in accelerating and decelerating and proposed two speed-gap curves with different jam spacings for acceleration and deceleration respectively. The spacing in accelerating is always larger than that in decelerating. Later, Treiterer and Myers (1974) found the traffic hysteresis phenomenon on macroscopic level. They concluded that the asymmetric driving behavior in acceleration and deceleration causes the hysteresis loops. Additionally, by using the CF model proposed by Gazis et al. (1961), they testified that drivers increase awareness on speed and space when decelerating than accelerating, which is consistent with Newell (1965). To testify the asymmetric reaction time in acceleration and deceleration, Ozaki (1993) proposed a piecewise function for reaction time and calibrated the model with observed data. Zhang (1999) is the first one that proposed a mathematical model for the hysteresis phenomenon from macroscopic level based on the analysis of three groups of observed data respectively representing different traffic conditions: acceleration, deceleration and equilibrium. Daganzo et al. (1999) proposed a possible explanation for the phase transitions considering the feature of asymmetric driving behavior. They also pointed out that the lane changing behavior contributes to counter-clock hysteresis loop. Recently, Yeo (2008) investigated the asymmetric driving behavior thoroughly and found three characteristics in asymmetric driving behavior: traffic hysteresis, intensity difference and discrete driving. Tordeux et al. (2010) found that driving behaviors in acceleration and deceleration are different by analyzing the NGSIM data. Khodayari et al. (2012) testified that the reaction time in acceleration is different with that in deceleration. Li and Chen (2017) pointed out that the right-skew feature of headway distributions which has been found by many researchers, results in asymmetric driving behavior. 2.2. CF models considering asymmetric driving behavior To further investigate the asymmetric driving behavior, many CF models incorporating asymmetric driving behavior have been proposed and are demonstrated to describe the traffic flow well. A summary of representative studies on these CF models are presented in Table 1. 2.3. Data-driven CF models For CF modeling, Adeli (2001) pointed out that data-driven model can capture the features from data (the examples or experiences) automatically, even though the input data has noise (error, incomplete, etc.). Due to these advantages, many researchers proposed data-driven CF models based on conventional Neural Networks (NN). e.g. Panwai and Dia (2007), Khodayari et al. (2012), and Zheng et al. (2013). All the studies have proved that the data-driven CF model outperform the mathematical models. Wei and Liu (2013) modeled the driving behavior based on Support Vector Machine (SVM) and pointed out that the performance of data-driven method is better than mathematical models in capturing the asymmetric driving behavior. However, the feature of intensity difference in Wei and Liu (2013) is inconsistent with the observed data. 347

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Table 1 Representative studies of CF models considering asymmetric driving behavior. Model

Researcher

Model consideration

Model validation

Mathematical models

Krauß et al. (1997)

Asymmetry in acceleration and deceleration Asymmetry in acceleration and deceleration Asymmetry in acceleration and deceleration Different time gap in accelerating and decelerating Hysteresis and intensity difference in acceleration and deceleration Asymmetry in acceleration and deceleration

Hysteresis loop on macroscopic level

Zhang and Kim (2005) AFVD model (Gong et al., 2008) Time-gap model (Tordeux et al., 2010) Xu et al. (2013)

Asymmetric Optimal Velocity model (Xu et al., 2015) Data-driven models

Artificial Neural Network based CF model (Khodayari et al., 2012) Support Vector Machine based CF model (Wei and Liu, 2013)

Hysteresis loop on macroscopic level Hysteresis loop in gap-speed diagram n/a Hysteresis loop in gap-speed diagram

Hysteresis loop in gap-speed diagram

Different reaction time in acceleration and deceleration

Location error comparisons between the simulated and the observed trajectories

Asymmetry in acceleration and deceleration

(1) hysteresis loop both on gap-speed diagram and densityflow diagram; (2) intensity difference with features: (a) there is a linear relationship between relative speed and acceleration. (b) It is not always true that the response intensity on deceleration is larger than on acceleration.

Recently, with the development of deep-learning method, some researchers proposed deep-learning based CF model. Zhou et al. (2017) developed CF model with RNN, which shows better performance than Intelligent Driver Model (IDM) (Treiber et al., 2000). Wang et al. (2018) also demonstrated that the Gate Recurrent Unit (GRU) NN (Cho et al., 2014) based CF model which includes 10 s of historical driving behavior outperforms the feedforward NN based CF model and IDM. However, one limitation of these studies is that they did not investigate the asymmetric driving behavior. From the previous studies, we can obtain some findings: (1) The asymmetric driving behavior generally exists in real traffic, which can impact the traffic flow significantly. (2) The incorporation of the asymmetric behavior or historical driving behavior can improve the accuracy of CF models. (3) Only considering hysteresis loop as the main indicator to assess the effectiveness of describing the asymmetric driving behavior is not sufficient. (4) Data-driven CF models have been proved to outperform the classic models. (5) The comparison between field data and the simulated trajectories regarding the asymmetric behavior is not fully studied yet. (6) The feature of discrete driving is not discussed in existing CF models. Therefore, in this paper, a data-driven CF model using LSTM-NN is proposed to depict the asymmetric driving behavior. The simulated trajectories are compared with the field data in detail for all the three characteristics of asymmetric driving behaviors. 3. LSTM-NN based CF model As stated before, to incorporate the asymmetric driving behavior into CF model, the historical driving information is of great importance, which is difficult to be combined in mathematical CF models. Therefore, an advanced data-driven method, deep learning, is adopted in this paper for several reasons: (1) it can thin the neural networks and simplify the computation via the input or/and hidden units dropout, the raw data filtering, the weight regularizing and replicating, saprse linear model estimation, etc. (Srivastava et al., 2014; Schmidhuber, 2015; Polson and Sokolov, 2017); (2) it can automatically extract and store information from the data within a long period (Hochreiter and Schmidhuber, 1997); (3) it has been applied in many domains successfully, such as speech recognition (Graves et al., 2013), visual object recognition (Donahue et al., 2015), video captioning (Yang et al., 2018) and obviously, transportation researches, including traffic speed prediction(Ma et al., 2015), driving distraction classification (Wollmer et al., 2011), traffic flow prediction (Polson and Sokolov, 2017; Wu et al., 2018), vehicle classification (Simoncini et al., 2018). Among numerous deep-learning algorithms, the LSTM-NN proposed by Hochreiter and Schmidhuber (1997) is used in this study for two main reasons: (1) to depict the driving behaviors more accurately, requires that more time-sequence historical information should be included. Although many machine-learning algorithms can deal with time-sequence data, LSTM-NN can consider more than 1000 time steps of historical data theoretically (Gers et al., 2002). (2) LSTM-NN contains fading memory function. As demonstrated by Gers et al. (2000), the LSTM-NN model with forget gate can not only obtain knowledge from the historical information, but also reset the memory block and remove the memory gradually when the historical data is out of date. This is quite consistent with human driving process, where drivers make decisions with varying short-term historical information. 3.1. Structure of LSTM-NN model In general, a LSTM-NN contains one input layer, several hidden layers and one output layer (Hochreiter and Schmidhuber, 1997; Ma et al., 2015). The input layer, the first one of the neural networks, initializes input data for subsequent layers. The output layer of 348

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Fig. 1. Memory block in LSTM-NN.

the neural networks is responsible for the feature classification or regression. The hidden layers are mainly used for the feature learning of the input data, where the main unit is memory block. (1) Memory block The memory block in the hidden layer contains at least five sub-units (as shown in Fig. 1) (Greff et al., 2017): a memory cell unit named Constant Error Carousel (CEC), three gates (a forget gate, an input gate and an output gate) with activation functions, and peephole connections (the blue dash lines in Fig. 1). The input gate is used to capture the information from input data flows ( x t ) at each time step. The output gate controls the hidden state output by telling the CEC when and what hidden state should output at each time step. The CEC is a recurrent self-connected unit, which makes the LSTM method capable of processing sequential data (Hochreiter and Schmidhuber, 1997). The forget gate solves the vanishing error problem by gradually releasing the information from internal data flow (ht − 1) if it is out of date when processing the continual sequential data to prevent the net breaking down (Gers et al., 2000). The peephole connections from CEC to gates help the cell inspect and control the gates, by which the precise timings can be learned more easily (Gers et al., 2002). Each sub-unit in Fig. 1 has been divided by red dot dash line. The ht − 1 and ht are the output of the memory block at time t −1 and time t , while the Ct − 1 and Ct are the hidden state of time t −1 and time t . Both the internal data flows ht and Ct return to all gates and are used for calculation at the next time step (Greff et al., 2017). To capture the information from the inputs, the relationship between the input flows ( x t ), internal flows (ht , ht − 1, Ct , Ct − 1) and the output flows of the gates in memory cell in Fig. 1 are given as following: Forget gate:

ft = σ (Wfx x t + Wfh ht − 1 + Wfc Ct − 1 + bf )

(1)

Input gate:

it = σ (Wix x t + Wih ht − 1 + Wic Ct − 1 + bi )

(

Ct = tanh Wc x x t + Wc h ht − 1 + bc

(2)

)

(3)

Memory cell unit:

Ct = ft ⊗ Ct − 1 + it ⊗ Ct

(4)

Output gate:

ot = σ (Wox x t + Woh ht − 1 + Woc Ct − 1 + bo)

(5)

ht = Who ot + bh

(6)

where x t is the input data which represent the historical driving information in CF model. Ct is the hidden state at time t . ⊗ represents the scalar product of two vectors, and σ (·) denotes the standard logistics sigmoid function as shown in Eq. (7), changing the inputs into range of (0, 1). tanh (·) is a hyperbolic tangent function as defined in Eq. (8), which transforms the input and output into range of [−1, 1]. W∗(∗ = fx , fh, fc, ix , ih, ic, ox , oh, oc, ho ) denote the weight matrices in each layer.b∗ (∗ = f , i, c , o, h ) denote 349

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

the bias of each gate.

σ (x ) =

1 1 + e−x

tanh (x ) =

(7)

exp(x )−exp(−x ) exp(x ) + exp(−x )

(8)

(2) Number of LSTM layer LeCun et al. (2015) has concluded that deep learning with 5–20 layers can capture the intricate feature from input data and be sensitive to minute details simultaneously. In Wang et al. (2018), the deep-learning based CF model containing 3 layers with 30, 10, 10 neurons in each layer showed the best performance. Since the number of the layers can significantly affect the ability of extracting feature information from input data, LSTM-NN models containing 3, 5, 8 LSTM layers with different number of neurons {32, 64, 128} in each hidden layer are constructed respectively to find the appropriate layer number. We finally set the number of hidden layers and neurons in each hidden layer as 8 and 32 respectively according to the comparison results, which is not shown here for the sake of concise. (3) Input and output variables In real traffic, drivers usually change speed according to the speed change of the leading vehicles by taking safety driving into account. Thus the speed of the subject vehicle at next time step is calculated based on the speed of the subject vehicle (vn ), its speed difference (Δvn = vn − 1−vn ) with the leading (n−1)th vehicle, the gap (Δx n ) between the two vehicles (the rear bumper of the (n−1)th vehicle to the front bumper of the n th vehicle) at time t. Other variables such as the speed of leading vehicle (vn − 1), time-to-collision (ttc = Δx n /Δvn ) are also usually incorporated in some CF models (Van Winsum, 1999; Naseri et al., 2015). However, the experiment results indicate that variables like vn − 1, ttc cannot improve the performance of the LSTM-NN. Therefore, only three variables including vn , Δvn , Δx n are adopted as the input variables of the LSTM-NN model. For CF models, the response of the following vehicle is usually represented by its acceleration or speed at the next time step. As per some previous studies (Zheng et al., 2013; Wang et al., 2018), in data-driven CF models, the speed variable can be a more intuitive way to describe the driving behavior. Thus, the speed of subject vehicle at the next time step is the output of this CF model, while the relationship of the stimulus and response in this study can be obtained as Eq. (9).

vn (t ) = f (vn (t −τ : t −Δt ), Δvn (t −τ : t −Δt ), Δx n (t −τ : t −Δt ))

(9)

where f (·) denotes the function of mapping relation between input and output variables; τ denotes the maximum historical time steps; and Δt denotes the updating time step in LSTM-NN model which is 0.1 s. (4) Historical time step As aforementioned, the decision in CF is closely dependent on the historical driving behaviors. However, the selections of historical time steps are diverse in the literature, such as 1 s in Chong et al. (2011) and 10 s in Wang et al. (2018). In order to study the effect of historical driving behaviors on the CF behavior, 1 s, 3 s, 5 s, 10 s of historical driving behaviors are incorporated in the LSTMNN model respectively to obtain the optimal one and the model with 5 s win out. In summary, a schematic structure of LSTM-NN model containing eight LSTM hidden layers and incorporating 5 s historical information are shown in Fig. 2. In the input layer, the size of input matrix is 3 × 50 at each time step, including three input variables (vn , Δvn , Δx n ), and 50 time steps, as one time step is 0.1 s. t = 0 s represents the current computing step, while t = 0.1s represents the next computing step. In the recurrent layer (hidden layer), all the eight internal flows hi0 (i = 1, 2⋯8) affect the feature extracting ability of each LSTM layer at the next time step, while the peephole connections that produced at time t = 0 s affect the feature extracting ability of the corresponding LSTM layer at the next time step. The superscript of h represents time step, while the subscript of h represents the index of LSTM layers. The blue dashed curves represent the peephole connections. The detail of the LSTM memory block is described in Section 3.1. The predicted results of vn′ and Δx n′ are then obtained in the output layer. 3.2. Configurations of LSTM-NN model After developing the structure of LTSM-NN, several configurations of LSTM-NN need to be specified to optimize the CF model, such as the optimization algorithm, activation function, loss function and epoch. (1) Optimization algorithm The optimization algorithm is a method for gradient descent (Hochreiter and Schmidhuber, 1997). In this paper, back propagation method and the stochastic gradient descent (SGD) are used, where the learning rate (lr ) and momentum (mom ) affect the performance of SGD significantly (Schaul et al., 2013). Therefore, the value of lr and mom should be determined cautiously. According to 350

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Fig. 2. Schematic structure of LSTM-NN model.

Schaul et al. (2013) and Simoncini et al. (2018), experiments of finding the best hyper-parameter from setting among lr ∈ {10−1, 10−2, 10−3, 10−4 , 10−5, 10−6} , mom ∈ [0.1, 0.9] with the a step of 0.1 are conducted. And the batch size is selected from the range of {50, 100, 200} with independent experiments. Finally, the lr , mom and batch size is determined as 10−3, 0.9 and 100 respectively. (2) Activation function The activation function can solve the drift problem of internal states at the beginning of learning, which can improve the ability of nonlinear modeling (Hochreiter and Schmidhuber, 1997). Similar to many studies (Bishop, 2006; Wang et al., 2018), the Hyperbolic tangent (tanh ) (see Eq. (8)) is used as activation function in the hidden layers. In the output layers, Parametric Rectified Linear Unit (PReLU) which can reduce the overfitting risk is adopted (K. He et al., 2015). The formulation of PRuLU is shown in Eq. (10). The value of α is set from 0.2 to 0.9 with a step size of 0.05 in the experiment of obtaining the appropriate α . Finally, the value of α is set as 0.9.

x if x i > 0 PRuLU (x i ) = ⎧ i ⎨ ⎩ αx i if x i ≤ 0

(10)

(3) Loss function The loss function, also known as objective function, calculates the error between the predicted and the realistic value. A biobjective loss function, which considers the speed and gap meanwhile, is adopted in this paper since the value of simulated gap may be abnormal when optimizing only one variable (Kesting and Treiber, 2008; Ossen, 2008). The Mean Square Error (MSE) with normalized data is used for calculating the error between the simulated and observed trajectories, as shown in Eq. (11).

f=

1 N

N

∑ (visim−viobs )2 + i=1

1 N

N

∑ (Δxisim−Δxiobs )2

(11)

i=1

where visim , Δx isim are the simulated speed and gap; viobs , Δx iobs are the observed speed and gap; N is the number of training samples in each time step. 351

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Table 2 Parameters for LSTM-NN model. Parameter

Value

Parameter

Value

Learning rate Momentum Batch size Activation function

0.001 0.9 100 tanh , PReLU (α = 0.9)

Loss function Number of neuron in hidden layers Number of neuron in output layer Optimizer

MSE of speed and gap 32 2 SGD

(4) Epoch The epoch means the number of iteration times in training. To get rid of over-fitting or under-fitting, 'Early Stopping' is introduced in the training. The model stops training when the results do not achieve improvement for five times. The epoch is selected from the range of {10, 15, 20, 50, 100, 150}. The LSTM-NN models are built using Keras running on the Theano backend (Chollet, 2015) and implemented in python. Based on the description and independent experiments in Sections 3.1 and 3.2, some parameters of LSTM-NN are first determined as shown in Table 2. 4. Model calibration and comparison In this section, we use the NGSIM data to calibrate and validate the LSTM-NN model. The simulated results are then compared with one data-driven model (RNN based CF model) and one mathematical model (AFVD model). 4.1. Data preparation Since deep learning is a data-driving method, massive data is necessary for applying the method. Therefore, the NGSIM datasets, which are collected from U.S. Highway 101 in Los Angeles, California, from 7:50 a.m. to 8:20 a.m. on June 15, 2005, are used in this paper. The temporal interval of the data is 0.1 s. In order to avoid the effect of lane changing and extract CF behavior feature accurately, the trajectories of vehicles that keep driving on the median three lanes (lane 2, 3, 4) without any lane-changing are collected. Finally, 799 vehicles’ trajectories are obtained, where 500 vehicles are used for calibrating and 299 vehicles are used for testing of the CF model. As pointed out by Wang et al. (2018), larger dataset does not necessarily lead to better results. Thus we conduct a sensitivity analysis in terms of the size of training dataset. 5 subsets are selected from the training datasets that contain 300, 350, 400, 450, 500 vehicles’ trajectories respectively. The dataset that contains 300 vehicles’ trajectories (214349 samples) is similar to the size of dataset used in Wang et al. (2018). For each training dataset, 70% of data are used for training and 30% are used for validation. Due to the low efficiency of calibrating mathematical CF models, only 100 vehicles selected from the training dataset (traindata100 for short) are used for the calibration of AFVD model. For the sake of consistency, traindata100 is also used for comparison of different trained LSTM-NN and RNN models in the following section. 100 vehicles from the testing dataset (testdata100 for short) are used for the comparison between different optimal CF models. In addition, since the original NGSIM data contain abnormal values of acceleration and deceleration (Punzo et al., 2011; Montanino and Punzo, 2013), we adopt Kalman filtering (Welch and Bishop, 2013) to reduce the measurement errors. 4.2. Model evaluation The correlation coefficient (R) of speed and the MSE of speed (mse v ) and gap (msegap ) are adopted as evaluation indicators for determining the parameter values of LSTM-NN based CF model. The indictor R shown in Eq. (12) can depict the uniformity of speed changing tendency between the simulated trajectories and field data. The indicators mse v and msegap illustrate the speed and gap error between the simulated trajectories and field data respectively, as shown in Eq. (13) and Eq. (14). Higher R and smaller mse v and msegap mean better model. N

R=

∑i = 1 (visim−v¯ sim)(viobs−v¯obs ) N

N

∑i = 1 (visim−v¯ sim)2 ·∑i = 1 (viobs−v¯obs )2

mse v =

1 N

msegap =

(12)

N

∑ (visim−viobs )2

(13)

i=1

1 N

N

∑ (Δxisim−Δxiobs )2

(14)

i=1

352

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Table 3 Results with different parameters. Index

Historical time step (s)

Training data (veh)

Epoch

R

msev (m/s )

msegap (m)

1 2 3 4 5 6 7 8

1

450 500 450 500 450 500 450 500

15 20 15 20 20 20 15 20

0.91 0.93 0.94 0.94 0.94 0.94 0.93 0.93

1.29 0.94 0.71 0.79 0.66 0.67 0.69 0.69

71.8 54.8 54.3 76.5 28.3 31.9 37.1 29.1

3 5 10

4.3. Calibration results of LSTM-NN model To obtain the optimal LSTM-NN based CF model, training and validation experiments are conducted with the configuration stated in Sections 3.1 and 3.2. The results show that: (1) All of the values of hidden layer number, epoch, train data size, and historical time step impact the performance of the LSTM-NN model. (2) The model would not always be better when the layer number increases. (3) The impact of layer number, epoch and train data size are interdependent. More specifically, the simulated results indicate that the R of the LSTM-NN model keeps stable despite using different training datasets. The msegap of CF models training with 450 and 500 vehicles is much smaller than that training with 300, 350, 400 vehicles, which means that we need to involve sufficient data to obtain fairly good results for this data-driven method. The epoch should not exceed 20, otherwise R will be less than 0.8. Therefore, for the sake of simplicity, with other parameters shown in Table 2, a sensitivity analysis in terms of the number of layers, considered historical time, size of training data, epoch are conducted. Finally, the simulation results of the LSTM-NN models with 8 hidden layers are shown in Table 3, since models with 8 hidden layers have the best results. From Table 3, we can see that the model with 5 s of historical time, 450 vehicles’ trajectories and 20 epoch has the best performance (R is 0.94, mse v is 0.66 m/s and msegap is 28.3 m). It is named as LSTM-5 s model, and used in the following section for comparison with field data regarding its manifestation of the asymmetric driving behavior. 4.4. Comparison with other CF models To verify the performance of the LSTM-NN model, the simulated trajectories are compared not only with the filed data, but also with the simulated trajectories reproduced by RNN model and AFVD model. 4.4.1. RNN based CF model Recurrent neural network (RNN) is a class of artificial neural network where connections between units form a directed graph along a sequence. This allows it to exhibit dynamic temporal behavior for a time sequence (Schmidhuber, 2015). RNN can use their internal state (memory) to process sequences of inputs, making them applicable to tasks such as handwriting recognition or speech recognition (Zhou et al., 2017). What’s more, the LSTM-NN is a specific variation of RNN. Therefore, the setup is similar to that of LSTM-NN while the parameter values in Table 2 are also used for RNN except the number of neuron in hidden layers. The rest parameters for RNN are decided by sensitivity analysis as shown in Table 4. The calibrated results are obtained when R reaches greatest (0.94) and msegap reaches least (42.8 m) with the traindata100 dataset. 4.4.2. AFVD model The AFVD model incorporates the asymmetry of acceleration and deceleration (Gong et al., 2008) into the full velocity difference model (Jiang et al., 2001). Xu et al. (2013) also validated the performance of the AFVD model in terms of the manifestation of the asymmetry driving behavior. It is formulated as follows:

an (t ) = k [V (sn (t ))−vn (t )]+λ1 H (−Δvn (t ))Δvn (t ) + λ2 H (Δvn (t ))Δvn (t )

(15)

where sn (t ) represents the spacing at time t . vn (t ) is the speed of the subject vehicle. Δvn (t ) = vn − 1 (t )−vn (t ) denotes the relative speed with the leading vehicle. H is the Heaviside function. V (sn (t )) is the optimal speed function in terms of the spacing (sn (t ) ) as Table 4 Parameters for RNN. Parameter

Range

Result

Neurons Epoch Historical time steps Hidden layer number Train data size

{10, 20, 30} {10, 15, 20, 25, 30, 35, 40, 45, 50, 100} with ‘early stopping’ {1 s, 3 s, 5 s} {3, 5, 8} {450, 500} vehicles

10 45 3s 5 450 vehicles

353

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Table 5 Calibration results of AFVD model. Parameter

Value

Parameter

Value

k V1 V2 C1

0.1 4.96 6.97 1.82

C2 λ1 λ2

12.35 0.53 0.9983

shown in Eq. (16). The express of λ1 and λ2 are given in Eqs. (17) and (18) respectively. k is the parameter that need to be calibrated. (16)

V (sn (t )) = V1 + V2 tanh[C1 sn (t )−C2] λ1 =

1 −[sn (t ) − s (vn (t ))]/ R′ e τ′

(17)

λ2 =

1 −|sn (t ) − s (vn (t ))|/ R‴ e τ‴

(18)

s (vn (t )) = s0 + Tvn (t )

(19)

where s(vn (t )) represents the safety gap. s0 denote the jam spacing. T denotes the safety time headway. V1, V2 , C1, C2 , R′, R‴, τ ′, τ ‴are parameters needed to be calibrated. Thus, there are 11 parameters need to be calibrated for the AFVD model. To simplify the calibration, similar to Xu et al. (2013), we fix the parameters λ1 and λ2 when calibrating. After calibrated with traindata100 and Genetic Algorithm, the optimal results are given in Table 5. 4.4.3. Comparison between different CF models To validate the performance of different CF models, all the optimal CF models are simulated with the testing dataset (testdata100). The statistics results of R, mse v and msegap for different models are shown in Table 6. In addition, the evaluation results for the training dataset (traindata100) are also presented. Generally, the comparison and concerning results are calculated from the testing dataset. The comparison between the training dataset and testing dataset can somehow reflect the ability of models on applying on “unknown” data. For testing results, the R values of CF models are close to each other, while both the msev and msegap of LSTM-NN model (0.71 m/s and 38.52 m) are much smaller than other models. The results indicate better performance of the proposed LSTM-NN model on mimicking trajectories. It is worth noting that although for all models the results from testing dataset are a little worse than that from the training dataset, the LSTM-NN and AFVD model have more stable performance than the RNN model for the two dataset. Furthermore, the gap profiles of two sample trajectories (vehicle 2572 and 1915) are illustrated for a comparison analysis in Fig. 3. It shows that all the models can follow the trend of observed vehicle well, whereas the simulated trajectories of LSTM-NN model are much close to the observed trajectories. The LSTM-NN model is thus proved outperforming the RNN model and AFVD model regarding the consistency of trajectories with empirical data. 5. Analysis of the asymmetry driving behavior The R and MSEs can only tell the error between simulated trajectories and the observed trajectories by considering one trajectory as a whole. However, different driving regimes exists in CF process, e.g. acceleration, deceleration, cruising, free flow, and stand still (Treiber and Kesting, 2013). The driving behaviors in different driving regimes are largely different (i.e. the asymmetric driving behavior) as several characteristics arises in traffic flow, such as hysteresis, discrete driving and intensity difference. Therefore, in this section, these characteristics are first analyzed with the empirical data. Moreover, they are compared with the simulation results of LSTM-NN model, RNN model and AFVD model. Table 6 Statistics results of the evaluation indicators. Model

Dataset

R

msev

msegap

Mean (SD)

Max

Min

Mean (SD)

Max

Min

Mean (SD)

Max

Min

LSTM-NN

Traindata100 Testdata100

0.94 (0.06) 0.94 (0.08)

0.99 0.99

0.62 0.50

0.66 (0.35) 0.71 (0.37)

2.21 2.33

0.26 0.21

28.36 (36.58) 38.52 (54.82)

212.88 317.78

1.89 0.79

RNN

Traindata100 Testdata100

0.94 (0.05) 0.93 (0.08)

0.99 0.99

0.69 0.47

0.89 (0.48) 1.11 (0.85)

3.15 4.85

0.35 0.27

42.78 (41.29) 61.08 (82.57)

228.88 433.49

2.81 2.62

AFVD

Traindata100 Testdata100

0.94 (0.06) 0.94 (0.07)

0.99 0.99

0.64 0.58

0.87 (0.42) 0.90 (0.45)

2.65 2.44

0.41 0.19

43.08 (47.86) 52.5 (80.64)

299.72 567.10

2.81 0.86

354

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

(a) vehicle 2572

(b) vehicle 1915

Fig. 3. The simulated and observed gaps of the sample trajectories.

5.1. Hysteresis 5.1.1. Hysteresis in real traffic Hysteresis is an important indicator of the asymmetric driving behavior (Gong et al., 2008; Yeo, 2008; Wei and Liu, 2013). One trajectory from testdata100 is plotted for gap-speed diagram in Fig. 4, which shows a hysteresis loop obviously. To explain this charateristic, Newell (1965) proposed the gap difference theory, which reveals that drivers tend to keep away from the leading vehicle farther when accelerating than decelerating at the given speed. As shown in Fig. 4, the gap at acceleration (7 m) is much greater than that at deceleration (2 m) when the speed is around 2 m/s . To quantify the magnitude of the hysteresis, Ahn et al. (2013) used the average gap difference (Δgap ) between acceleration and deceleration brims, while Liu et al. (2017) suggested that the size of hysteresis also impact the traffic flow. Therefore, in this paper, the average gap difference Δgap at the speed of 4 m/s , 6 m/s , 8 m/s and 10 m/s and their relative error (err ) are used to access the CF models’ ability of capturing hysteresis loop. The formulation of Δgap and err are shown in Eqs. (20) and (21) respectively.

Δgap = gapacc −gapdec

err =

(20)

Δgapsim −Δgapobs Δgapobs

(21)

where gapacc denotes the average gap at accelerating. gapdec denotes the average gap at decelerating. Δgapsim denotes the average gap difference of simulated trajectories. Δgapobs denotes the average gap difference of observed trajectories. The results of gap difference in field data is shown in Table 7.

Fig. 4. Gap-speed diagram of observed trajectories. 355

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Table 7 Results of gap difference. v (m/s)

Model

Δgap (m)

err

4

Observed LSTM-5 s AFVD RNN

1.75 1.81 1.76 2.54

3.43% 0.57% 45.14%

Observed LSTM-5 s AFVD RNN

1.43 2.05 2.3 3.33

43.36% 60.84% 132.87%

6

v (m/s)

Model

Δgap (m)

err

8

Observed LSTM-5 s AFVD RNN

1.41 2.2 2.55 3.28

56.03% 80.85% 132.62%

Observed LSTM-5 s AFVD RNN

1.43 2.04 1.84 3.07

42.66% 28.67% 114.69%

10

5.1.2. Hysteresis reproduced by CF models Three simulated trajectories, corresponding to the observed trajectory in Fig. 4, are respectively reproduced by the LSTM-NN model, RNN model and AFVD model as shown in Fig. 5(a)–(c). As shown in Fig. 5, all the models can reproduce the hysteresis loop. However, the shape of the hysteresis loop in Fig. 5(a) (LSTM-NN model) is much closer to the observed trajectory than that in Fig. 5(b) (RNN model) and Fig. 5(c) (AFVD model). To further compare the hysteresis magnitude (the internal space between the persistent acceleration curve and persistent deceleration curve) in quantification, the Δgap and err of the simulated trajectories reproduced by the three CF models at 4 m/s , 6 m/s , 8 m/s , 10 m/s respectively are calculated with Eqs. (20) and (21). The statistics summary results are presented in Table 7, from which significant difference between different models can been seen clearly. The comparison of errs indicate that both LSTM-NN model and AFVD model outperform RNN model, as all the errs of the LSTM-NN model and AFVD model are less than that of RNN model. The errs of AFVD model is less than LSTM-NN model when driving at 4 m/s and 10 m/s . However, the errs of AFVD model are greater than 60% when driving at 6 m/s and 8 m/s , while all the errs of LSTM-NN model are less than 60%. Moreover, the average gap difference Δgap of observed data is around 1.5 m, while that of the simulated trajectories of the LSTM-NN model, AFVD model and RNN model are 2.0 m, 2.1 m and 3.1 m, respectively. The results suggest the LSTM-NN model performs better than other CF models. 5.2. Discrete driving Yeo (2008) stated that both acceleration and deceleration action in CF behavior are discrete rather than completely continuous, which means that drivers would always paused the accelerating/deceleration for a moment to check the leading vehicle’s speed and gap when accelerating or decelerating. This feature is also clearly presented in Fig. 4, where the cruising (blue1 dots) exist between successive acceleration (green dots) or deceleration (red dots). From Fig. 5, we also can see that the simulated trajectories of LSTM-NN, AFVD and RNN models are not completely smooth, since there are few and small pauses (blue dots) during the accelerating or decelerating, which seems insufficient. However, there are some ripple waves in the Fig. 5(a) and (b) when decelerating or accelerating, which means a tendency of pausing. Yeo (2008) also suggested that fewer pauses in accelerating result in longer reaction time and larger gap. It is consistent with the simulated results in this study. 5.3. Intensity difference 5.3.1. Intensity difference in real traffic Another characteristic of the asymmetric driving behavior is intensity difference (Yeo, 2008; Wei and Liu, 2013). That is to say, the magnitudes of acceleration and deceleration is different while the gap between the pair vehicles (Δx ), speed of the following vehicle (v ) and the absolute magnitude of relative speed (Δv ) are constant. For instance, the acceleration a is −1.8 m/s2 and 2.2 m/s2 , when Δx = 8 m, v = 6 m/s and Δv is −4 m/s , 4 m/s respectively (Fig. 6(b)). To demonstrate this characteristic, the observed trajectories with the speed range from 3 m/s to 12 m/s when the gap equals to 4 m, 6 m, 8 m, 10 m, 12 m, 14 m respectively are collected from testdata100 set. Similar to Wei and Liu (2013), the acceleration a is hypothesized as a linear function of relative speed as shown in Eq. (22). Finally, the acceleration-relative speed fitting curve of the observed trajectories are obtained and shown in Fig. 6, where the goodness of fit (R-square) are greater than 0.8. (22)

a = k ·Δv + b From Fig. 6, we can conclude several findings about intensity difference exhibited in the observed trajectories:

(1) The solid lines that do not pass the point (0, 0) demonstrate the intensity difference characteristic and the linear relationship between relative speed and acceleration.

1

For interpretation of color in Fig. 4, the reader is referred to the web version of this article. 356

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

(a) LSTM-NN model

(b) RNN model

(c) AFVD model Fig. 5. Gap-speed diagrams of different models.

(2) However, this characteristic does not always exist. The dashed lines passing the point (0, 0) in Fig. 6 presents that there is no intensity difference characteristic. Taking the line of v = 5 m/s in Fig. 6(b) for example, the acceleration is about 2.4 m/s2 when Δv is 4 m/s , and about −2.4 m/s2 when Δv is −4 m/s . (3) The response intensity to the negative relative speed is not always higher than that of the positive relative speed as demonstrated by the line of v = 10 m/s in Fig. 6(a) and the line of v = 3 m/s in Fig. 6(b). 357

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

(a) Driving at

(b) Driving at

with different speeds

with different speeds

Fig. 6. Relationship between acceleration and relative speed of the observed data.

(4) The R-square of fitting curve of the observed data is less than 0.8, when gap = 12 m , gap = 14 m and gap = 4 m . It reveals that the linear relationship between acceleration and relative speed is only obvious when the gap is between 6 m and 10 m. (5) The linear relationship does not exist in all situations. For instance, there is no corresponding lines for gap = 8 m, v = 7, 8, 9, 10 m/s . (6) The slop of lines for different speed, when driving at the same gap, are different. 5.3.2. Intensity difference reproduced by CF models As shown by the solid lines which do not pass point (0, 0) in Fig. 7, the LSTM-NN model can capture the feature of intensity difference and the linear relationship between acceleration and relative speed when driving at the same condition (same gap and same speed). The dashed lines passing point (0, 0) in Fig. 7(b), reveals that the intensity difference is not always achieved. Additionally, there are not all the solid lines in Fig. 7 exhibiting higher response intensity to the negative relative speed than positive relative speed, such as the solid line of speed = 8 m/s in Fig. 7(b). Moreover, all the R-square values of the simulated trajectories reproduced by the LSTM-NN model are smaller than 0.8 when gap = 4 m , gap = 12 m and gap = 14 m , which is consistent with the fourth finding from the field data. Similar to the fifth finding from field data, the linear relation-ship between the relative-speed and acceleration is not always existing. And the slop of the lines for different speed in Fig. 7 are different. Thus, the LSTM-NN model can present the characteristic of intensity difference well. With respect to the RNN model, no dashed line existing in Fig. 8, which is inconsistent with the second finding from field data.

(a) Driving at

with different speeds

(b) Driving at

with different speeds

Fig. 7. Relationship between acceleration and relative speed with LSTM-NN model. 358

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

(a) Driving at

with different speeds

(b) Driving at

with different speeds

Fig. 8. Relationship between acceleration and relative speed with RNN model.

Moreover, when the gap is equal or greater than 8m , there always exists the linear relationship between the simulated relative speed and simulated acceleration for RNN model (R-square value is greater than 0.8), contradicting with the empirical finding as well. For the AFVD model, the linear relationship always exists as long as the data points exist (there is no data point when speed = 10 or 12 m/s and gap = 6 m ). This feature can be deduced by substituing Eqs. (16)–(19) into Eq. (15) with a result of Eq. (23), where the gap and following speed are both constant. From Eq. (23), we can see that no matter what the gap is, the linear relationship between relative speed and acceleration of simulated trajectories that reproduced by RNN always exists. However, this characteristic is inconsistent with the finding (4) and (5) that exhibited in the field data. Moreover, as shown in Fig. 9, all the lines have the same slope, as long as driving at the same gap, which is consistent with the finding in Wei and Liu (2013) but contradicts the feature of field data (Fig. 6).

an (t ) = p1 + p2 ·Δvn (t )

(23)

To sum up, the LSTM-NN model performs better than the AFVD and RNN model regarding the reproduction of intensity difference characteristic. 6. Reproducing of stop-and-go phenomenon Stop-and-go traffic is a common phenomenon in the congest traffic but nuisance (Laval and Leclercq, 2010; Treiber and Kesting,

(a) Driving at

with different speeds

(b) Driving at

Fig. 9. Relationship between acceleration and relative speed with AFVD model. 359

with different speeds

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

(a) Speed profiles of vehicles

(b) Location profiles of vehicles

Fig. 10. Stop-and-go phenomenon reproduced by LSTM-NN.

2017). A main objective of the CF models is to reproduce characteristics of human-driven traffic, such as capacity drop, traffic oscillation, and hysteresis. Previous analysis has proved the ability of proposed CF model on the asymmetric driving behavior including the hysteresis characteristic. To further validate its performance of reproducing stop-and-go traffic, a simulation experiment is designed where 13 vehicles driving on a single lane. The speed profile of the first vehicle is assigned with observed data (vehicle 301) as blue line in Fig. 10. The following vehicles are driving with the same LSTM-NN based CF model. As we can see from the simulated speed profiles of vehicles as shown in Fig. 10, the perturbation of the first vehicle becomes bigger and bigger when propagating upstream the platoon, while the following vehicles reduce speeds more intensely. The stop-and-go traffic thus forms in the platoon, which further demonstrate the ability of the proposed CF model of capturing characteristics of real traffic.

7. Conclusions The asymmetric driving behavior is one of the most significant human driving behaviors that affect the dynamics of traffic flow. In order to capture the three characteristics of asymmetric driving behavior (i.e. hysteresis, discrete driving and intensity difference), a LSTM-NN based CF model incorporating historical driving behaviors is proposed in this paper. The NGSIM data are used for model calibrating, validating, and simulation testing. In addition, the RNN model and AFVD model are also studied for the comparison with the LSTM-NN model. Several main findings are concluded from the comparison results between the field data and different models:

• The simulated trajectories reproduced by the LSTM-NN model are much close to the trajectories in real traffic, while the MSEs between the observed gap and simulated gap reproduced by the RNN and AFVD model are much greater. • The LSTM-NN model can reproduce the hysteresis loop in gap-speed diagram well, while the shape of the hysteresis loop and the hysteresis magnitude are closer to the field data. • Six findings about the intensity difference characteristic are obtained from the field data. The LSTM-NN model can capture all the findings while the simulated trajectories of the AFVD and RNN model are not always consistent with them. The • LSTM-NN model, AFVD model and RNN model show limited performance regarding the feature of discrete driving. • The proposed LSTM-NN model can regenerate the stop-and-go phenomenon well. In summary, the deep learning method of LSTM-NN can promote the performance of CF models on understanding the human driving behaviors in traffic flow. Especially, with the development of data collection and machine learning techniques, the datadriven CF models can play a more important role in traffic flow modeling and simulation. In addition, this study excludes the impact of lane-changing on the asymmetric behavior which could be a limitation. In this regard, LSTM-NN model can also be used for training an integrated model including the longitudinal and lateral driving behavior simultaneously and this work is ongoing.

Acknowledgements The authors would like to thank the anonymous reviewers’ constructive comments. This research is sponsored by the Natural Science Foundation of China (51422812, U1764261), the International Science & Technology Cooperation Program of Science and Technology Commission of Shanghai Municipality (16510711400), and the Fundamental Research Funds for the Central Universities. 360

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

References Adeli, H., 2001. Neural networks in civil engineering: 1989–2000. Comput.-Aided Civ. Infrastruct. Eng. 16, 126–142. Ahn, S., Vadlamani, S., Laval, J.A., 2013. A method to account for non-steady state conditions in measuring traffic hysteresis. Transp. Res. Part C 34, 138–147. Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y., 1995. Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035–1042. Bishop, C.M., 2006. Pattern Recognition and Machine Learning. Springer, Singapore. Chandler, R.E., Herman, R., Montrol, E.W., 1958. Traffic dynamics: studies in car-following. Oper. Res. 6, 165–184. Cho, K., Van Merriënboer, B., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., Bengio, Y., 2014. Learning phrase representations using RNN encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078. Chong, L., Abbas, M.M., Medina, A., 2011. Simulation of driver behavior with agent-based back-propagation neural network. Transp. Res. Rec. 2249, 44–51. Daganzo, C.F., Cassidy, M.J., Bertini, R.L., 1999. Possible explanations of phase transitions in highway traffic. Transp. Res. Part A 33, 365–379. Donahue, J., Anne Hendricks, L., Guadarrama, S., Rohrbach, M., Venugopalan, S., Saenko, K., Darrell, T., 2015. Long-term recurrent convolutional networks for visual recognition and description. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2625–2634. Edie, L.C., 1965. Discussion of traffic stream measurements and definitions, In: Proceedings of The Second International Symposium on The Theory of Road Traffic Flow, pp. 139–154. Federal Highway Administration (FHWA), 2008. The Next Generation Simulation (NGSIM) [online] < https://ops.fhwa.dotgov/trafficanalysistools/ngsim.htm > . Chollet, F., 2015. Keras Document < https://keras.io > . Foote, R.S., 1965. Single lane traffic flow control. In: Proceedings of The Second International Symposium on The Theory of Road Traffic Flow, pp. 84–103. Forbes, T.W., 1963. Human factor consideration in traffic flow theory. Highway Res. Rec. 15, 60–66. Gazis, D.C., Herman, R., Rothery, R.W., 1961. Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9, 545–567. Gers, F.A., Schmidhuber, J., Cummins, F., 2000. Learning to forget: continual prediction with LSTM. Neural Comput. 12, 2451–2471. Gers, F.A., Schraudolph, N.N., Schmidhuber, J., 2002. Learning precise timing with LSTM recurrent networks. J. Mach. Learn. Res. 3, 115–143. Gipps, P.G., 1981. A behavioural car-following model for computer simulation. Transp. Res. Part B 15, 105–111. Gong, H., Liu, H., Wang, B.H., 2008. An asymmetric full velocity difference car-following model. Phys. A: Statist. Mech. Appl. 387, 2595–2602. Graves, A., Mohamed, A., Hinton, G., 2013. Speech recognition with deep recurrent neural networks. In: IEEE International Conference, Speech and Signal Processing, pp. 6645–6649. Greff, K., Srivastava, R.K., Koutnik, J., Steunebrink, B.R., Schmidhuber, J., 2017. LSTM: a search space Odyssey. IEEE Trans. Neural Netw. Learn. Syst. 28, 2222–2232. He, K., Zhang, X., Ren, S., Sun, J., 2015a. Delving deep into rectifiers: surpassing human-level performance on Image-net classification. In: Proceedings of the IEEE International Conference on Computer Vision (ICCV), pp. 1026–1034. He, Z., Zheng, L., Guan, W., 2015. A simple nonparametric car-following model driven by field data. Transp. Res. Part B 80, 185–201. Helbing, D., Tilch, B., 1998. Generalized force model of traffic dynamics. Phys. Rev. 58, 133–138. Helly, W., 1959. Simulation of bottlenecks in single lane traffic flow. In: Proceedings of the Symposium on Theory of Traffic Flow, pp. 207–238. Herman, R., Montroll, E.W., Potts, R.B., Rothery, R.W., 1959. Traffic dynamics: analysis of stability in car following. Oper. Res. 7, 86–106. Hochreiter, S., Schmidhuber, J., 1997. Long short-term memory. Neural Comput. 9, 1735–1780. Jiang, R., Wu, Q., Zhu, Z., 2001. Full velocity difference model for a car-following theory. Phys. Rev. E 64, 017101. Kesting, A., Treiber, M., 2008. Calibrating car-following models using trajectory data: methodological study. Transp. Res. Rec. 2088, 148–156. Khodayari, A., Ghaffari, A., Kazemi, R., Braunstingl, R., 2012. A modified car-following model based on a neural network model of the human driver effects. IEEE Trans. Syst., Man, Cybernet. - Part A: Syst. Hum. 42, 1440–1449. Krauß, S., Wagner, P., Gawron, C., 1997. Metastable states in a microscopic model of traffic flow. Phys. Rev. E 55, 5597–5602. Laval, J.A., Leclercq, L., 2010. A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic. Philos. Trans. R. Soc. A: Math., Phys. Eng. Sci. 368, 4519–4541. LeCun, Y., Bengio, Y., Hinton, G., 2015. Deep learning. Nature 521, 436–444. Lee, G., 1966. A generalization of linear car-following theory. Oper. Res. 14, 595–606. Li, L., Chen, X., 2017. Vehicle headway modeling and its inferences in macroscopic/microscopic traffic flow theory: a survey. Transp. Res. Part C 76, 170–188. Liu, D.W., Shi, Z.K., Ai, W.H., 2017. Enhanced stability of car-following model upon incorporation of short-term driving memory. Commun. Nonlin. Sci. Numer. Simul. 47, 139–150. Ma, X., Tao, Z., Wang, Y., Yu, H., Wang, Y., 2015. Long short-term memory neural network for traffic speed prediction using remote microwave sensor data. Transp. Res. Part C 54, 187–197. Montanino, M., Punzo, V., 2013. Making NGSIM data usable for studies on traffic flow theory: multistep method for vehicle trajectory reconstruction. Transp. Res. Rec. 99–111. Naseri, H., Nahvi, A., Karan, F.S.N., 2015. A new psychological methodology for modeling real-time car following maneuvers. Travel Behav. Soc. 2, 124–130. Newell, G.F., 1961. Nonlinear effects in the dynamics of car following. Oper. Res. 9, 209–229. Newell, G.F., 1965. Instability in dense highway traffic, a review. In: Proceedings of The Second International Symposium on The Theory of Road Traffic Flow, pp. 73–83. Ossen, S.J.L., 2008. Longitudinal Driving Behavior: Theory and Empirics. TRAIL Research School, Netherlands. Ozaki, H., 1993. Reaction and anticipation in the car-following behavior. Transport. Traffic Theory 12, 349–366. Panwai, S., Dia, H., 2007. Neural agent car-following models. IEEE Trans. Intell. Transp. Syst. 8, 60–70. Peng, G., Lu, W., He, H., Gu, Z., 2016. Nonlinear analysis of a new car-following model accounting for the optimal velocity changes with memory. Commun. Nonlin. Sci. Numer. Simul. 40, 197–205. Polson, N.G., Sokolov, V.O., 2017. Deep learning for short-term traffic flow prediction. Transport. Res. Part C 79, 1–17. Punzo, V., Borzacchiello, M.T., Ciuffo, B., 2011. On the assessment of vehicle trajectory data accuracy and application to the Next Generation SIMulation (NGSIM) program data. Transp. Res. Part C 19, 1243–1262. Schaul, T., Zhang, S., LeCun, Y., 2013. No more pesky learning rates. In: International Conference on Machine Learning, pp. 343–351. Schmidhuber, J., 2015. Deep learning in neural networks: an overview. Neural Netw. 61, 85–117. Simoncini, M., Taccari, L., Sambo, F., Bravi, L., Salti, S., Lori, A., 2018. Vehicle classification from low-frequency GPS data with recurrent neural networks. Transp. Res. Part C 91, 176–191. Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., Salakhutdinov, R., 2014. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 15, 1929–1958. Toledo, T., 2007. Driving Behavior: Models and Challenges. Transp. Rev. 27, 65–84. Tordeux, A., Lassarre, S., Roussignol, M., 2010. An adaptive time gap car-following model. Transp. Res. Part B 44, 1115–1131. Treiber, M., Helbing, D., 2003. Memory effects in microscopic traffic models and wide scattering in flow-density data. Phys. Rev. E 68, 046119. Treiber, M., Kesting, A., 2013. Microscopic calibration and validation of car-following models- a systematic approach. Proc.-Soc. Behav. Sci. 80, 922–939. Treiber, M., Kesting, A., 2017. The intelligent driver model with stochasticity-new insights into traffic flow oscillations. Transp. Res. Proc. 23, 174–187. Treiber, M., Kesting, A., Helbing, D., 2000. Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E 62, 1805–1824. Treiterer, J., Myers, J., 1974. The hysteresis phenomenon in traffic flow. Transport. Traffic Theory 6, 13–38. Van Winsum, W., 1999. The human element in car following models. Transport. Res. Part F 2, 207–211. Wang, X., Jiang, R., Li, L., Lin, Y., Zheng, X., Wang, F.Y., 2018. Capturing car-following behaviors by deep learning. IEEE Trans. Intell. Transport. Syst. 19, 910–920. Wei, D., Liu, H., 2013. Analysis of asymmetric driving behavior using a self-learning approach. Transp. Res. Part B 47, 1–14. Welch, G., Bishop, G., 2013. An Introduction to the Kalman Filter, vol. 8. University of North Carolina at Chapel Hill, pp. 127–132.

361

Transportation Research Part C 95 (2018) 346–362

X. Huang et al.

Wollmer, M., Blaschke, C., Schindl, T., Schuller, B., Farber, B., Mayer, S., Trefflich, B., 2011. Online driver distraction detection using long short-term memory. IEEE Trans. Intell. Transp. Syst. 12, 574–582. Wu, Y., Tan, H., Qin, L., Ran, B., Jiang, Z., 2018. A hybrid deep learning based traffic flow prediction method and its understanding. Transport. Res. Part C 90, 166–180. Xu, H., Liu, H., Gong, H., 2013. Modeling the asymmetry in traffic flow (a): microscopic approach. Appl. Math. Model. 37, 9431–9440. Xu, X., Pang, J., Monterola, C., 2015. Asymmetric optimal-velocity car-following model. Phys. A: Statist. Mech. Appl. 436, 565–571. Yang, Y., Zhou, J., Ai, J., Bin, Y., Hanjalic, A., Shen, H.T., Ji, Y., 2018. Video captioning by adversarial LSTM. IEEE Trans. Image Process. 1–12. Yeo, H., 2008. Asymmetric Microscopic Driving Behavior Theory. University of California Transportation Center PhD thesis. Zhang, H.M., 1999. A mathematical theory of traffic hysteresis. Transp. Res. Part B 33, 1–23. Zhang, H.M., Kim, T., 2005. A car-following theory for multiphase traffic flow. Transp. Res. Part B 39, 386–399. Zheng, J., Suzuki, K., Fujita, M., 2013. Car-following behavior with instantaneous driver–vehicle reaction delay: a neural-network-based methodology. Transport. Res. Part C 36, 339–351. Zhou, M., Qu, X., Li, X., 2017. A recurrent neural network based microscopic car following model to predict traffic oscillation. Transport. Res. Part C 84, 245–264.

362