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Floating car data are beneficial in estimating traffic conditions in wide areas and are playing an increasing role in traffic surveillance. However, widespread application is limited by low-sample frequency which makes it hard to get a complete picture of a vehicle’s motion. An accurate and reliable reconstruction of a vehicle’s trajectory could effectively result in a higher sampling frequency enabling a more accurate estimation of road traffic parameters. Existing methods require additional information such as nearby vehicles, signal timing strategies, and queue patterns which are not always available. To address this problem, this paper presents a method used with low-sample frequency data to reconstruct vehicle trajectories through intersections, without the need for extra information. Furthermore, the additional parameters for the speed-time curve distributions for deceleration rate and acceleration rate are generated. A piecewise deceleration and acceleration model is developed to calculate the acceleration rate for different travel modes in the trajectory. The distribution parameters of the acceleration data for each travel mode are then estimated using a new Expectation Maximization (EM) algorithm. The acceleration statistics are then used to reconstruct the corresponding parts of the trajectory. Compared to the reference trajectories (truth), the test results show that the method developed in this paper achieves improvement in accuracy ranging from 16 to 67% over the commonly used linear interpolation method. In addition, the proposed method is not very sensitive to the sampling interval of the floating car data, unlike the linear interpolation method where the error grows rapidly with increasing sampling interval.

The potential for the contribution of floating car data to traffic state monitoring has attracted significant research in the past decade [

For trajectory error correction, Huang

To reconstruct the trajectory between spares probes, Sun

The problem for trajectory reconstruction is that only a few discrete sampling points are available. Furthermore, the correlation between sampling points is weak as a result of low-frequency sampling. Therefore, in the current methods, additional information is required for reconstruction. For example, Huang

In this paper, it is assumed that the vehicle driving mode through an intersection evolves in a given pattern, that is, cruise

In this paper, unlike the current methods, only low-frequency floating car data are used, with no additional or extra information. Furthermore, in addition to position-time curve, speed-time curve, and distribution parameters of acceleration (deceleration) rate are obtained. By reconstructing the trajectory of the low-frequency trajectory, more accurate and valuable information could be extracted from trajectories, enabling floating car data to play a more important role in traffic state monitoring.

The paper is structured as follows. The next section presents the methodology followed by the results and their analysis and conclusions.

Different from the current methods, the method developed in this paper reconstructs the trajectory in two parts, deceleration process and acceleration process (i.e., not the trajectory between two consecutive updates). Traffic signal timing and historical queue length data are not needed. The main factors that may influence trajectory reconstruction include traffic conditions, number of sample points, and a vehicle’s stop position relative to the intersection. To better reconstruct the low-frequency trajectory, the numbers of sample points and stop positions are investigated in the first instance. It should be noted that in our statistics even if there are many sample points with zero speed, the number of points is recorded as 1. For an arterial street with signalized control, we generate the number of sample points and stop positions of floating cars for different periods of the day over a timespan of three weeks. The results are shown in Figure

Histogram of GPS sample points and stop position.

7:00-9:00 AM

10:00-12:00 AM

7:00-9:00 AM

10:00-12:00 AM

As shown in Figures

A vehicle’s motion is disrupted at the point where the speed is zero. Hence, the trajectory reconstruction is divided into two processes: deceleration and acceleration.

The schematic of this research is shown in Figure

Schematic of trajectory reconstruction with low-frequency probe data.

Trajectory reconstruction process.

The model for calculating the acceleration and deceleration rates with the low-frequency floating car data originates from previous work by the authors on the estimation of vehicle control delay [^{2}. Similarly, for the acceleration process, the first and second stages are represented by acceleration rates

Relationship between speed and time in deceleration model (a) and acceleration model (b).

Based on the model and driving pattern for a low-frequency trajectory, deceleration rates_{2} for mode2 (deceleration1) and_{3} for mode3 (deceleration2) and acceleration rates_{5} for mode5 (acceleration1) and_{6} for mode6 (acceleration2) can be calculated by solving a constrained nonlinear programming problem as follows.

For a typical low-frequency trajectory of the probe vehicle shown in Figure

Low-frequency vehicle trajectory at intersection.

The deceleration rates

Let _{4} and_{5} are the instantaneous speeds at Point 4 and Point 5, respectively, _{4}+_{5})/2, and_{45} is the distance between Point 4 and Point 5.

With_{2},_{3},_{5}, and_{6}, the trajectory can be reconstructed. However, the model only works for the scenarios where there is a sample point before deceleration and in deceleration. For the acceleration process, the requirement is that there is a sample point in acceleration and after acceleration. Nevertheless, probe vehicle updates normally occur at random positions and times, so there may be no sample point in or before deceleration in a trajectory. To address these issues, we propose an approach to reconstruct the maximum likelihood trajectory to alleviate the effect caused by the randomness of sampling. The essential parameters of our method are distribution parameters of deceleration rate and acceleration rate for different modes.

In this section, the valid trajectories are selected from the historical trajectory data. A valid trajectory has a sample point before deceleration and in deceleration and in acceleration and after acceleration. By modeling and solving the nonlinear programming problem (see (

For the deceleration and acceleration modes, it is assumed that the acceleration rate follows a Gaussian distribution. The travel time

In a similar way, the likelihood function for the acceleration process is formulated as

With the distribution parameters of acceleration for each mode being known, the maximum likelihood trajectory of a given low-frequency trajectory is estimated. As a result of low-frequency sampling, the sample points may be located at any position of the intersection. Besides, the number of sample points in a trajectory is uncertain. To better capture the difference, we define scenarios for the deceleration stage of a low-frequency trajectory each with different numbers and reconstruct the maximum likelihood trajectory for each.

1 sample point before deceleration, none in the deceleration stage.

1 sample point before deceleration, 1 in deceleration.

Multiple sample points before deceleration, none in the deceleration stage.

The scenario definitions for the acceleration stage are similar to those for deceleration.

1 sample point after acceleration, none in the acceleration stage.

1 sample point after acceleration, 1 in acceleration.

Multiple sample points after acceleration.

Here the sample points refer to the points where the speed is above zero. Different trajectories contain different numbers of sample points and different combinations of scenarios. The trajectory reconstruction is divided into two parts: deceleration process and acceleration process. The reconstruction method is the same for both processes; hence, only the deceleration process reconstruction with different sample points is illustrated here.

_{2}=0,_{1}-_{2}=_{1}/2. The solution to this problem generates the travel time for mode1 and deceleration rates of mode2 and mode3. _{1}+_{1}/(2_{2}); _{1}+_{1}/(2_{2})+_{1}/(2_{3}). This information is used to construct the speed-time curve and the position-time curve, that is, trajectory of the vehicle.

Different sample scenarios for deceleration process.

The trajectory reconstruction of the acceleration process is similar to that for deceleration. For a trajectory, when deceleration and acceleration reconstructions are completed, the stop position and duration can be calculated. The low-frequency trajectory reconstruction is then completed.

In this paper, the site for experiment site was a portion of the south bound carriageway on Songshan Road, spanning the intersection of Huanghe and Songsha Roads in the Nangang District of the city of Harbin in China (Figure

The experiment site.

To test the accuracy of the proposed method, a field experiment was conducted in the experiment site. Four probe vehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz. The vehicles were driven in the north-south direction (red arrow direction in Figure

The statistical results for acceleration are shown in Table

Parameters for each mode.

Modes | Mean (m/s^{2}) | Standard Deviation (m/s^{2}) |

| ||

| 0.6916 | 0.203 |

| 0.894 | 0.202 |

| 0.961 | 0.239 |

| 0.688 | 0.141 |

As can be seen in Table

With the distribution parameters of acceleration rate for each mode, the trajectory from the low-frequency floating car data can be reconstructed with the proposed method. To test the proposed method’s performance, the estimated trajectories are compared with the observed trajectories achieved in a field experiment. Figure

Estimated and observed trajectories for different types of trajectory.

Estimated and observed trajectory for trajectory with 4 sample points

Estimated and observed trajectory for trajectory with 5 sample points

Estimated and observed trajectory for trajectory with 7sample points

From Figure

Estimated and observed speed for different types of trajectory.

Estimated and observed speed for trajectory with 4 sample points

Estimated and observed speed for trajectory with 5 sample points

Estimated and observed speed for trajectory with 7 sample points

To further demonstrate the significance of the proposed method, the estimated trajectories were compared with trajectories that were created with the linear interpolation method. In the linear method, it is assumed that the distance the vehicle travels between any consecutive updates is fixed. As illustrated in Figure

Comparison of estimated trajectory with the observed and linear method based trajectories.

Estimated and observed trajectory for trajectory with 4 sample points

Estimated and observed trajectory for trajectory with 5 sample points

Estimated and observed trajectory for trajectory with 7 sample points

To quantitatively analyze the estimation results, the Mean Absolute Error (MAE) between the estimated and ground truth trajectories is calculated. The MAE is defined as follows:_{i} is the estimation error of the i

Trajectory estimation error for proposed method and benchmark method.

Deceleration onset | Stopped position (m) | Number of | Proposed method | Linear Interpolation Method | ||

Speed (Km/h) | trajectories | MMAE (m) | MTAE (s) | MMAE (m) | MTAE (s) | |

| ||||||

| | 5 | 10.53 | 6.6 | 11.3 | 9.7 |

| | 7 | 9.34 | 8.9 | 9.48 | 13.1 |

| | 7 | 9.86 | 9.5 | 12.51 | 12.7 |

| | 16 | 10.26 | 9.9 | 13.9 | 11.6 |

| | 14 | 8.23 | 6.3 | 11.1 | 10.3 |

| | 10 | 6.11 | 7.4 | 9.95 | 10.5 |

| | 9 | 7.57 | 6.0 | 10.66 | 11.2 |

| | 7 | 8.35 | 6.2 | 11.38 | 9.9 |

| | 20 | 7.39 | 6.5 | 10.17 | 10.4 |

| | 16 | 6.46 | 5.8 | 9.35 | 11.7 |

| | 13 | 6.32 | 6.3 | 11.36 | 10.9 |

| | 10 | 7.49 | 6.1 | 9.77 | 9.8 |

| | 23 | 6.49 | 4.6 | 9.59 | 10.5 |

| | 15 | 5.93 | 5.7 | 9.84 | 9.9 |

| | 17 | 3.47 | 3.8 | 10.97 | 10.7 |

| | - | - | - | - | - |

Compared with the linear interpolation method, the proposed method generally provides better estimation in terms of both MMAE and MTAE. For the proposed method, when the speed at the deceleration onset point ranges from 0 to 15 km/h, which indicates that the traffic is congested, the MMAE and MTAE are obviously larger than those for other speed ranges. However, for the linear-interpolation method, there is no obvious difference between each speed range. For the proposed method, the MMAE ranges from 3.47 to 10.53 with MTAE from 3.8 to 9.9. For the linear-interpolation method, the MMAE range is 9.35-13.9 and the MTAE range is 9.7-13.1. When the speed range at the deceleration onset point is 45-60 km/h, the proposed method achieves smaller MMAE and MTAE than the other speed intervals. This may be because when a vehicle travels at a relatively high speed, the travel pattern is in line with the modal structure in our proposed method. Conversely, when the vehicle travels at a low speed, more sample points are generated and the vehicle may experience stop and go and even pass through the intersection in two or more signal cycles. This circumstance is not consistent with our model’s assumption. This could explain the differences in performance of the proposed method under different travel speed ranges. It should be noted that the worst performance appears when the vehicle travels at low speed and stops at the furthest distance from the intersection. This is the case for both the proposed and linear interpolation methods.

In this paper, the number of sample points of the trajectory is an important factor for trajectory reconstruction. To give more insight into the proposed method’s performance with different number of sample points, the MMAE for different number of sample points and travel speed ranges can be visualized in Figure

MMAE for different number of sample points and stop position ranges.

Figure

In this paper, the low-frequency floating car data is sampled at 30s. The effect of higher sampling intervals is investigated through a sensitivity analysis employing both the proposed and linear interpolation methods.

In Figure

Sensitivity analysis with different sample interval.

Boxplot of MMAE

Boxplot of MTAE

In this paper, a novel method is proposed to reconstruct a vehicle’s trajectory from low-frequency floating car data. Under the assumption that the vehicle travels in a certain pattern, a model is developed to estimate the distribution parameters of deceleration rate and acceleration rate with selected historical data. With the distribution parameters as prior information, trajectories for different scenarios are constructed. Compared to a reference trajectory (truth), the proposed method achieves encouraging accuracy, with MMAE ranging from 3.47m to10.53m, higher than the commonly used linear-interpolation method which that MMAE ranging from 9.35m to 12.51m. In our method, extra information is not required and, in addition, the speed curve and distribution parameters of the deceleration rate and acceleration rate are obtained. The proposed method, therefore, paves the way for a widespread application of floating car data in transport.

The presented work has established a framework for vehicle trajectory reconstruction that could be modified or solve other traffic problems in the future. To improve and extend the application of our model, future work will include the following cases:

Vehicles experiencing stop-and-go behavior when the queue at an intersection is long. In this case, the vehicle movement may deviate from our assumed driving pattern, increasing trajectory estimation error. We will extend our model to account for the limitations of our assumptions to improve its robustness.

Queue length at an intersection is long such that vehicles pass through the intersection in two or more signal cycles. In this case, we do not recommend the application of our method. We mine multicycle vehicle trajectories to capture the vehicle movement patterns and use them to improve our model.

Queue profile at the intersection. Combined with the traffic shockwave theory, the queue length and control delay of the intersection could be effectively estimated. We will update our model to incorporate traffic shockwave models.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The work is supported by the National Natural Science Foundation of China (NSFC) (Grant no. 51578198).