Research ArticleVehicle Trajectory Reconstruction for Signalized Intersectionswith Low-Frequency Floating Car Data

Hua Wang 1 Changlong Gu 1 and Washington Yotto Ochieng2

1School of Transportation Science and Engineering Harbin Institute of Technology No 73 Huanghe Road Nangang DistrictHarbin 150090 China2Department of Civil and Environmental Engineering Imperial College London London SW7 2AZ UK

Correspondence should be addressed to Hua Wang wanghuahiteducn

Received 28 November 2018 Revised 31 March 2019 Accepted 7 May 2019 Published 20 May 2019

Academic Editor Jaeyoung Lee

Copyright copy 2019 Hua Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Floating car data are beneficial in estimating traffic conditions in wide areas and are playing an increasing role in traffic surveillanceHowever widespread application is limited by low-sample frequency which makes it hard to get a complete picture of a vehiclersquosmotion An accurate and reliable reconstruction of a vehiclersquos trajectory could effectively result in a higher sampling frequencyenabling a more accurate estimation of road traffic parameters Existing methods require additional information such as nearbyvehicles signal timing strategies and queue patterns which are not always available To address this problem this paper presents amethod used with low-sample frequency data to reconstruct vehicle trajectories through intersections without the need for extrainformation Furthermore the additional parameters for the speed-time curve distributions for deceleration rate and accelerationrate are generatedA piecewise deceleration and accelerationmodel is developed to calculate the acceleration rate for different travelmodes in the trajectory The distribution parameters of the acceleration data for each travel mode are then estimated using a newExpectation Maximization (EM) algorithm The acceleration statistics are then used to reconstruct the corresponding parts of thetrajectory Compared to the reference trajectories (truth) the test results show that the method developed in this paper achievesimprovement in accuracy ranging from 16 to 67 over the commonly used linear interpolation method In addition the proposedmethod is not very sensitive to the sampling interval of the floating car data unlike the linear interpolationmethod where the errorgrows rapidly with increasing sampling interval

1 Introduction

Thepotential for the contribution of floating car data to trafficstate monitoring has attracted significant research in the pastdecade [1ndash3] However constrained by the penetration rateof equipped vehicles and cost of data storage and transmis-sion the sampling interval of floating car data can be lowrestricting their application Liu et al evaluated the sensitivityof delay measurement to sampling frequency of floating cardata showing that delays estimated from data sampled at a10-second interval are consistent with those from a 5-secondinterval for 74 of the cases However when the samplinginterval is 60 seconds the consistency drops to 37 [4] Patireet al investigated the question of how much GPS data areneeded for a traffic information system capable of providingaccurate speed (and thus travel time) information Theypoint

out that this question must address issues of data quality interms of sample rate and penetration rate [5] Bucknell etal investigated the relationship between penetration rate andsampling rate concluding that in general increasing samplefrequency is more beneficial when the current penetrationrate is low [6] High-resolution trajectory data are ideal fortraffic state monitoring However the majority of trajectorydata (especially those for commercial use) today are collectedat a relatively low sampling frequency Hence it is necessaryto develop methods to reconstruct vehicle trajectories fromlow-frequency floating car data Researchers have conductedvarious studies on vehicle trajectory reconstruction classifiedinto two categories One category aims at eliminating thetrajectory noise caused by GPS positioning error while theother attempts are to reconstruct the vehiclersquos trajectory fromlow-frequency floating car data

HindawiJournal of Advanced TransportationVolume 2019 Article ID 9417471 14 pageshttpsdoiorg10115520199417471

2 Journal of Advanced Transportation

For trajectory error correction Huang et al propose anapproach for predicting vehicle trajectory by using infor-mation from nearby vehicles to construct the local drivingenvironment [7] Fard et al present a simple two-stepmethod based on wavelet analysis for filtering errors andreconstructing trajectories Wavelet transform is employedto identify and modify the outliers [8] Marcello et aldevelop an approach with amultistep procedure based on theinformation of traffic kinematics and vehicle dynamics [9]Since high-frequency data cannot be collected at a large scaleit is impossible to provide high-accuracy trafficmonitoring ofthe road network

To reconstruct the trajectory between spares probes Sunet al apply the Variation Formulation (VF) method to obtainthe complete picture of traffic flow using sample vehicletrajectories However to properly apply the VF method theshockwave boundaries have to be known which is estimatedfrom the fixed sensors data [10] Wan et al propose Expec-tation Maximization (EM) and Maximum Likelihood (ML)methods to reconstruct trajectories between two consecutivetransit bus GPS updates The path is divided into shortsegments and the segment travel time is allocated iterativelywith the EM method The maximum likelihood trajectory isgenerated based on travel time statistics [11] In the methodit is assumed that the traffic signal timing is known Hao etal propose a model investigating all possible driving modesequences between data points Detailed trajectories arereconstructed based on the optimal driving mode sequences[12] Hao et al develop a modal activity-based stochasticmodel for reconstructing the trajectories from sparse data[13] However in their approach the distribution parametersof four types of modal activity are estimated with the second-by-second trajectory in the NGSIM Lankershim dataset TheNGSIM data consist of detailed vehicle trajectory and arecollected using digital video camera The precise location ofeach vehicle on a 05 to 10 kilometer section of roadway isrecorded every one-tenth of a secondThe applicability of thismethod to different locations and traffic conditions is still tobe proven Wang et al proposed an approach for trajectoryreconstruction with sparse probe data for estimating controldelayThemethod describes the deceleration and accelerationas a piecewise constant deceleration process and accelerationprocess respectively This method has the limitation of poorstability [14]

The problem for trajectory reconstruction is that onlya few discrete sampling points are available Furthermorethe correlation between sampling points is weak as aresult of low-frequency sampling Therefore in the currentmethods additional information is required for reconstruc-tion For example Huang et al exploit information fromproximate vehicles Wan et al use signal timing strate-gies and queue patterns to assist in the trajectory recon-struction In the approach proposed by Hao et al theparameters for deceleration and acceleration are estimatedfrom second-by-second vehicle trajectory data Howeverin reality the extra data are not always available Hencethis paper proposes a trajectory reconstruction methodfor signalized intersections with low-frequency floating cardata

In this paper it is assumed that the vehicle driving modethrough an intersection evolves in a given pattern that iscruise (1) ndash deceleration (2) ndash idle (3) ndash acceleration (4) ndashcruise (5) More precisely a deceleration (acceleration) rate-based piecewise process is used to model the driving modeBased on this assumption (ie the 5 driving modes) a modelis developed to calculate deceleration and acceleration ratefrom selected historical data An Expectation Maximization(EM) algorithm is specified to estimate the correspondingdistribution parameters For given low-frequency samplepoints when the distribution parameters of deceleration andacceleration are available the time period and distance foreach of the fivemodes are calculated by solving a ConstrainedQuadratic Programming (CQP) problem

In this paper unlike the current methods only low-frequency floating car data are used with no additionalor extra information Furthermore in addition to position-time curve speed-time curve and distribution parametersof acceleration (deceleration) rate are obtained By recon-structing the trajectory of the low-frequency trajectory moreaccurate and valuable information could be extracted fromtrajectories enabling floating car data to play a more impor-tant role in traffic state monitoring

The paper is structured as follows The next sectionpresents the methodology followed by the results and theiranalysis and conclusions

2 Methodology

Different from the current methods the method developedin this paper reconstructs the trajectory in two parts deceler-ation process and acceleration process (ie not the trajectorybetween two consecutive updates) Traffic signal timing andhistorical queue length data are not needed Themain factorsthat may influence trajectory reconstruction include trafficconditions number of sample points and a vehiclersquos stopposition relative to the intersection To better reconstruct thelow-frequency trajectory the numbers of sample points andstop positions are investigated in the first instance It shouldbe noted that in our statistics even if there are many samplepoints with zero speed the number of points is recorded as1 For an arterial street with signalized control we generatethe number of sample points and stop positions of floatingcars for different periods of the day over a timespan of threeweeks The results are shown in Figure 1

As shown in Figures 1(a) and 1(b) whether in the peakhour (700-900 am) or off-peak hour (1000-1200 am) thedistribution is skewed to the left This means that for thefloating car through the intersection there are more casesgenerating 3 4 and 5 sample points than 6 7 and 8 pointswith the most likely being 4 Figures 1(c) and 1(d) showthe vehicle stop position histogram exhibiting a bimodaldistribution There is no obvious difference between the peakhour and off-peak hour However in both figures the vehiclesrarely stopped between 100 and 300m The data used for theanalysis here were collected by taxis Hence the gap in thestop position histograms is attributed to the traffic volumeWhen the traffic volume was low the taxi drivers preferredto stop close to the intersection by changing lane However

Journal of Advanced Transportation 3

120

100

80

60

40

20

0

Cou

nts

3 4 5 6 7 8

Number of GPS sample points

(a) 700-900 AM

100

80

60

40

20

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Cou

nts

3 4 5 6 7 8

Number of GPS sample points

(b) 1000-1200 AM

Cou

nts

25

20

15

10

5

0

0 100 200 300 400

Positions (m)

(c) 700-900 AM

Cou

nts

25

20

15

10

5

0

30

0 100 200 300 400

Positions (m)

(d) 1000-1200 AM

Figure 1 Histogram of GPS sample points and stop position

when the traffic volume was high all the lanes were mostlyoccupied with other vehicles so that the taxis had to queueHence in most cases the taxis stopped relatively far from theintersection This led to the vehicle stopping either close toor far from the intersection From the histogram it could beinferred that the maximum queue length of the intersectionwas about 400m

A vehiclersquos motion is disrupted at the point where thespeed is zero Hence the trajectory reconstruction is dividedinto two processes deceleration and acceleration

The schematic of this research is shown in Figure 2 forreconstructing the maximum likelihood trajectory from allpossible trajectory sets 1 2 3 with sparse sample pointsA B and C Figure 3 presents the reconstruction steps withthe main elements of estimation of the distribution para-meters for different modes and maximum likelihood trajec-tory reconstruction

21 Acceleration Rate Estimation for Different Modes Themodel for calculating the acceleration and deceleration rateswith the low-frequency floating car data originates fromprevious work by the authors on the estimation of vehiclecontrol delay [13] The model is developed on the basis of apiecewise constant deceleration model and a Simple PlatoonAdvancement (SPA) model As illustrated in Figure 4 decel-eration and acceleration are both divided into three parts Forthe deceleration process the first stage represents a vehicletravelling at the free-flow speed The second and third stagesrepresent vehicle deceleration rates 1198861 and 1198862 in ms2 Simi-larly for the acceleration process the first and second stagesare represented by acceleration rates 1198863 and 1198864 The thirdstage represents a vehicle travelling at the free-flow speedHence the vehicle driving pattern through the intersectionis as follows cruise (1) ndash deceleration1 (2) ndash deceleration2 (3)ndash idle (4) ndash acceleration1 (5) ndash acceleration2 (6) ndash cruise (7)

4 Journal of Advanced Transportation

Time

loca

tion

1

2

3

A

B

C

Figure 2 Schematic of trajectory reconstruction with low-frequency probe data

Based on the model and driving pattern for a low-frequency trajectory deceleration rates a2 for mode2 (decel-eration1) and a3 for mode3 (deceleration2) and accelerationrates a5 for mode5 (acceleration1) and a6 for mode6 (acceler-ation2) can be calculated by solving a constrained nonlinearprogramming problem as follows

For a typical low-frequency trajectory of the probe vehicleshown in Figure 5 the vehicle travels at the free-flow speedat Point 1 and then decelerates between the point in thedotted box and Point 2 At Point 3 the vehicle has stoppedand is waiting for the traffic light to turn green From Point4 to Point 5 the vehicle accelerates to reach the free-flowspeed Points 1-5 cover the entire motion process of thevehicle through the intersection The deceleration onset andacceleration end points correspond to the points in the dottedboxes

The deceleration rates 1198862 and 1198863 are calculated with thedeveloped model

Let 119905dy be the time interval between 1199051 and 11990510158401 and let 119905aybe the time interval between 11990510158404 and 1199055 The time and distancebetween Point 1 and Point 2 are expressed by the followingequations

119905119889119910 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (1)

V1 lowast 119905119889119910 + V21 minus V211989821198862 + V2119898 minus V2221198863 = 11987112 (2)

0 le 119905119889119910 le 30 0 le 1198862 le 9 0 le 1198863 le 9 (3)

where T is the sampling interval of the probe vehicle dataand is also the time travelled between Point 1 and Point 2 V1and V2 are the instantaneous velocities at Point 1 and Point 2respectively In this paper the sample interval is 30 seconds11987112 is the distance between Point 1 and Point 2 V119898 = (V1 +V2)2 In (1) and (2) there are three unknown parametersThe only solution cannot be obtained from this nonlinearequation group Therefore to obtain 1198862 and 1198863 expressions

1 and 2 are converted into a nonlinear programming problemwith the objective function as follows

119911 = min[(119905d119910 + V1 minus V1198981198862 + V119898 minus V21198863 minus 30)2

+ (V1 lowast 119905119889119910 + V21 minus V211989821198862 + V2119898 minus V2221198863 minus 11987112)2](4)

subject to

0 le 119905119889119910 le 300 le 1198862 le 90 le 1198863 le 9

(5)

By solving the nonlinear programming problem the valuesof 1198862 and 1198863 could be calculated For the acceleration modelthe acceleration rates of 1198865 and 1198866 are calculated as follows

119911 = min[(119905119886119910 + V119899 minus V41198865 + V5 minus Vn1198866 minus 30)2

+ (V5 lowast 119905119886119910 + V2119899 minus V2421198865 + V25 minus V211989921198866 minus 11987145)2](6)

subject to

0 le 119905a119910 le 300 le 1198865 le 90 le 1198866 le 9

(7)

where v4 and v5 are the instantaneous speeds at Point 4 andPoint 5 respectively Vn=(v4+v5)2 and L45 is the distancebetween Point 4 and Point 5

With a2 a3 a5 and a6 the trajectory can be recon-structed However the model only works for the scenarioswhere there is a sample point before deceleration and indeceleration For the acceleration process the requirement isthat there is a sample point in acceleration and after accel-eration Nevertheless probe vehicle updates normally occurat random positions and times so there may be no samplepoint in or before deceleration in a trajectory To addressthese issues we propose an approach to reconstruct themaximum likelihood trajectory to alleviate the effect causedby the randomness of sampling The essential parameters ofour method are distribution parameters of deceleration rateand acceleration rate for different modes

22 Distribution Parameters Estimation of Acceleration Ratefor Different Modes In this section the valid trajectories areselected from the historical trajectory data A valid trajectoryhas a sample point before deceleration and in decelerationand in acceleration and after acceleration By modeling andsolving the nonlinear programming problem (see (4) and(6)) the acceleration rate for modes 2 3 5 and 6 for each

Journal of Advanced Transportation 5

Selected historical low-frequencytrajectory

Decelerationtrajectory

Accelerationtrajectory

Assumed travelpattern and

model

Decelerationrate of

different mode

Accelerationrate of

different mode

EMalgorithm

Distributionparameters for

deceleration rate ofdifferent mode

Distributionparameters for

deceleration rate ofdifferent mode

A new low-frequencytrajectory

A newdeceleration

trajectory

A newaccelerationtrajectory

Maximum likelihooddeceleration

trajectory

Maximum likelihoodaccelerationtrajectory

Whole trajectory

Figure 3 Trajectory reconstruction process

V

T0

６1

６2

６Ｇ

(a)T

V

0６1

６2

６Ｇ

(b)

Figure 4 Relationship between speed and time in deceleration model (a) and accelerationmodel (b)

6 Journal of Advanced Transportation

1 2 4 53

Ｎ1 Ｎ2 Ｎ3 Ｎ4 Ｎ5Ｎ1

Ｎ4

Figure 5 Low-frequency vehicle trajectory at intersection

valid trajectory is obtained To avoid errors by using thesedirectly to calculate the distribution parameters of accel-eration rate for different modes we adopt an ExpectationMaximization method to iteratively calculate the accelerationrate for each mode It is assumed that the deceleration rateand acceleration rate are independent from each other Forthe idle and cruise modes the mode time mainly depends onthe signal plan traffic condition and intersection placementIn this paper it is assumed that the idle and cruisemode traveltimes t are uniformly distributed

119905119894 sim U (119887119894 119888119894) (8)

where i = 1 4 7 correspond to cruise (1) idle (4) and cruise(7)modesFor the deceleration and accelerationmodes it is assumed

that the acceleration rate follows a Gaussian distribution Thetravel time t that a vehicle spends under the decelerationand acceleration modes is the product of speed variation Δvdivided by corresponding time period

119886119895 = ΔV119895119905119895 sim 119873(120583119895 1205902119895 ) (9)

where j = 2356 correspond to deceleration1 (mode2) decel-eration2 (mode3) acceleration1 (mode5) and acceleration2(mode6) Each valid trajectory provides an observation ofdeceleration rate and acceleration rate for different modesAssume that there are M floating cars passing through theintersection Suppose that k⩽M is the trajectory sequencenumber The accelerate rate for each mode is calculated

E Step For mode j (j=2 3 5 6) we calculate the mean 120583119895and standard deviation 120590119895 of the corresponding accelerationUnder the assumption that the acceleration rate follows aGaussian distribution the probability density function for 119886119896119895is determined as

119901 (119886119896119895 | 120583119895 120590119895) = 1radic2120587120590119895 119890minus(119886119896119895minus120583119895)

221205902119895 (10)

where a119896119895 is the acceleration rate for mode i of kth trajectoryj=2356 For ease of calculation the log-likelihood functionof a119896119895 is used

log [119901 (119886119896119895 | 120583119895 120590119895)] = minus(119886119896119895 minus 120583119895)221205902119895 minus log (radic2120587120590119895) (11)

M Step In this step for each valid trajectory decelerationrates and acceleration rates for different modes a119896119895 are recalcu-lated such that the likelihood function is maximized For the

deceleration process of each trajectory its likelihood functioncan be represented as

argmax( sum119894=1119895=23

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=23

(119886119896119895 minus 120583119895)221205902119895 )

= argmin( sum119895=23

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(12)

subject to the equality

1199051198961 + sum119895=23

ΔV119896119895119886119896119895 = 119879 (13)

1199051198961 lowast V1198961 + (V1198961)2 minus (V119896119898)22 lowast 1198861198962 + (V119896119898)2 minus (V1198962)22 lowast 1198861198963 = 11987111989612 (14)

where 1199051198961 is travel time of mode1 for the kth trajectory V1198961and V1198962 are the speeds at the first sample point and secondsample point for kth trajectory respectively V119896119898 = (V1198961 + V1198962)2ΔV1198962 = V1198961 minus V119896119898 ΔV1198963 = V119896119898 minus V1198962 and 11987111989612 is the distancebetween the first sample point and the second sample pointIn the M step 120583119895 and 120590119895 are considered to be constants and119901(119905119896119894 | 119880119894) is fixed Therefore the terms containing them canbe ignored when maximizing the likelihood function This isa Constrained Quadratic Programming (CQP) problemThesolution of this CQP problem is the travel time of mode1 andthe distribution parameters of acceleration rate for mode 2and mode 3

In a similar way the likelihood function for the accelera-tion process is formulated as

argmax( sum119894=7119895=56

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=56

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(15)

subject to the equality

1199051198967 + sum119895=56

ΔV119896119895119886119896119895 = 119879 (16)

1199051198967 lowast V1198965 + (V1198965)2 minus (V119896119899)22 lowast 1198861198966 + (V119896119899)2 minus (V1198964)22 lowast 1198861198965 = 11987111989645 (17)

where 1199051198967 is the travel time of mode7 for the kth trajectoryV1198964 and V1198965 are the speeds at the fourth and the fifth samplepoints for the kth trajectory V119896n = (V1198964 + V1198965)2 ΔV1198965 = V119896n minus V1198964

Journal of Advanced Transportation 7

ΔV1198966 = V1198965 minus V119896n and 11987111989645 is the distance between the fourthand the fifth sample points It should be noted that the Mstep is run separately for each trajectory When the M stepends the method goes back to the E step to update the meanand variance based on all trajectories When the differencebetween consecutive iterations is below a threshold theiteration process stops

23 Maximum Likelihood Trajectory Estimation With thedistribution parameters of acceleration for each mode beingknown the maximum likelihood trajectory of a given low-frequency trajectory is estimated As a result of low-frequencysampling the sample points may be located at any position ofthe intersection Besides the number of sample points in atrajectory is uncertain To better capture the difference wedefine scenarios for the deceleration stage of a low-frequencytrajectory each with different numbers and reconstruct themaximum likelihood trajectory for each

Scenario 1 1 sample point before deceleration none in thedeceleration stage

Scenario 2 1 sample point before deceleration 1 in decelera-tion

Scenario 3 Multiple sample points before deceleration nonein the deceleration stage

The scenario definitions for the acceleration stage aresimilar to those for deceleration

Scenario 1 1 sample point after acceleration none in theacceleration stage

Scenario 2 1 sample point after acceleration 1 in acceleration

Scenario 3 Multiple sample points after acceleration

Here the sample points refer to the points where thespeed is above zero Different trajectories contain different

numbers of sample points and different combinations ofscenarios The trajectory reconstruction is divided into twoparts deceleration process and acceleration process Thereconstruction method is the same for both processes henceonly the deceleration process reconstruction with differentsample points is illustrated here

(1) Scenario 1 for Deceleration Trajectory It is assumed thatP1 is the sample point where the speed is above zero P2 isthe sample point where the speed is zero The distributionparameters of deceleration rate for mode 2 and mode 3 are(1205832 1205902) and (1205833 1205903) In order to reconstruct the maximumlikelihood trajectory of deceleration process the following isformulated and solved

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(18)

subject to

1199051 + V12 lowast 1198862 + V12 lowast 1198863 lt 119879 (19)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987112 (20)

In scenario 1 v2=0 v1-Vm=Vm-v2=v12 The solution tothis problem generates the travel time for mode1 anddeceleration rates of mode2 and mode3 119905m=t1+v1(2a2)11990521015840=t1+v1(2a2)+v1(2a3) This information is used to con-struct the speed-time curve and the position-time curve thatis trajectory of the vehicle v is the speed of the vehicle and yis the position of the vehicle at time t According to themodelthe expressions for v and y are as follows

V =

V1 0 lt t lt 1199051V1 + 1198862 lowast (119905 minus 1199051) 1199051 lt t lt 1199051 + V121198862V12 + 1198863 lowast (119905 minus 1199051 minus V121198862) 1199051 + V121198862 lt t lt 1199051 + V121198862 + V121198863

(21)

119910 =

V1 lowast 119905 0 lt t lt 1199051V1 lowast 1199051 + V1 lowast (119905 minus 1199051) + 11988622 lowast (119905 minus 1199051)2 1199051 lt 119905 lt 1199051 + V121198862V1 lowast 1199051 + 3V2181198862 + V12 lowast (119905 minus 1199051 minus V121198862) + 11988632 lowast (119905 minus 1199051 minus V121198862)

2 1199051 + V121198862 lt 119905 lt 1199051 + V121198862 + V121198863(22)

(2) Scenario 2 for Deceleration Trajectory As shownin Figure 6(b) there is a sample point P1 before

deceleration and another P2 in the deceleration stageof a trajectory In this scenario the trajectory between

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

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Journal of Advanced Transportation 3

120

100

80

60

40

20

0

Cou

nts

3 4 5 6 7 8

Number of GPS sample points

(a) 700-900 AM

100

80

60

40

20

0

Cou

nts

3 4 5 6 7 8

Number of GPS sample points

(b) 1000-1200 AM

Cou

nts

25

20

15

10

5

0

0 100 200 300 400

Positions (m)

(c) 700-900 AM

Cou

nts

25

20

15

10

5

0

30

0 100 200 300 400

Positions (m)

(d) 1000-1200 AM

Figure 1 Histogram of GPS sample points and stop position

when the traffic volume was high all the lanes were mostlyoccupied with other vehicles so that the taxis had to queueHence in most cases the taxis stopped relatively far from theintersection This led to the vehicle stopping either close toor far from the intersection From the histogram it could beinferred that the maximum queue length of the intersectionwas about 400m

A vehiclersquos motion is disrupted at the point where thespeed is zero Hence the trajectory reconstruction is dividedinto two processes deceleration and acceleration

The schematic of this research is shown in Figure 2 forreconstructing the maximum likelihood trajectory from allpossible trajectory sets 1 2 3 with sparse sample pointsA B and C Figure 3 presents the reconstruction steps withthe main elements of estimation of the distribution para-meters for different modes and maximum likelihood trajec-tory reconstruction

21 Acceleration Rate Estimation for Different Modes Themodel for calculating the acceleration and deceleration rateswith the low-frequency floating car data originates fromprevious work by the authors on the estimation of vehiclecontrol delay [13] The model is developed on the basis of apiecewise constant deceleration model and a Simple PlatoonAdvancement (SPA) model As illustrated in Figure 4 decel-eration and acceleration are both divided into three parts Forthe deceleration process the first stage represents a vehicletravelling at the free-flow speed The second and third stagesrepresent vehicle deceleration rates 1198861 and 1198862 in ms2 Simi-larly for the acceleration process the first and second stagesare represented by acceleration rates 1198863 and 1198864 The thirdstage represents a vehicle travelling at the free-flow speedHence the vehicle driving pattern through the intersectionis as follows cruise (1) ndash deceleration1 (2) ndash deceleration2 (3)ndash idle (4) ndash acceleration1 (5) ndash acceleration2 (6) ndash cruise (7)

4 Journal of Advanced Transportation

Time

loca

tion

1

2

3

A

B

C

Figure 2 Schematic of trajectory reconstruction with low-frequency probe data

Based on the model and driving pattern for a low-frequency trajectory deceleration rates a2 for mode2 (decel-eration1) and a3 for mode3 (deceleration2) and accelerationrates a5 for mode5 (acceleration1) and a6 for mode6 (acceler-ation2) can be calculated by solving a constrained nonlinearprogramming problem as follows

For a typical low-frequency trajectory of the probe vehicleshown in Figure 5 the vehicle travels at the free-flow speedat Point 1 and then decelerates between the point in thedotted box and Point 2 At Point 3 the vehicle has stoppedand is waiting for the traffic light to turn green From Point4 to Point 5 the vehicle accelerates to reach the free-flowspeed Points 1-5 cover the entire motion process of thevehicle through the intersection The deceleration onset andacceleration end points correspond to the points in the dottedboxes

The deceleration rates 1198862 and 1198863 are calculated with thedeveloped model

Let 119905dy be the time interval between 1199051 and 11990510158401 and let 119905aybe the time interval between 11990510158404 and 1199055 The time and distancebetween Point 1 and Point 2 are expressed by the followingequations

119905119889119910 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (1)

V1 lowast 119905119889119910 + V21 minus V211989821198862 + V2119898 minus V2221198863 = 11987112 (2)

0 le 119905119889119910 le 30 0 le 1198862 le 9 0 le 1198863 le 9 (3)

where T is the sampling interval of the probe vehicle dataand is also the time travelled between Point 1 and Point 2 V1and V2 are the instantaneous velocities at Point 1 and Point 2respectively In this paper the sample interval is 30 seconds11987112 is the distance between Point 1 and Point 2 V119898 = (V1 +V2)2 In (1) and (2) there are three unknown parametersThe only solution cannot be obtained from this nonlinearequation group Therefore to obtain 1198862 and 1198863 expressions

1 and 2 are converted into a nonlinear programming problemwith the objective function as follows

119911 = min[(119905d119910 + V1 minus V1198981198862 + V119898 minus V21198863 minus 30)2

+ (V1 lowast 119905119889119910 + V21 minus V211989821198862 + V2119898 minus V2221198863 minus 11987112)2](4)

subject to

0 le 119905119889119910 le 300 le 1198862 le 90 le 1198863 le 9

(5)

By solving the nonlinear programming problem the valuesof 1198862 and 1198863 could be calculated For the acceleration modelthe acceleration rates of 1198865 and 1198866 are calculated as follows

119911 = min[(119905119886119910 + V119899 minus V41198865 + V5 minus Vn1198866 minus 30)2

+ (V5 lowast 119905119886119910 + V2119899 minus V2421198865 + V25 minus V211989921198866 minus 11987145)2](6)

subject to

0 le 119905a119910 le 300 le 1198865 le 90 le 1198866 le 9

(7)

where v4 and v5 are the instantaneous speeds at Point 4 andPoint 5 respectively Vn=(v4+v5)2 and L45 is the distancebetween Point 4 and Point 5

With a2 a3 a5 and a6 the trajectory can be recon-structed However the model only works for the scenarioswhere there is a sample point before deceleration and indeceleration For the acceleration process the requirement isthat there is a sample point in acceleration and after accel-eration Nevertheless probe vehicle updates normally occurat random positions and times so there may be no samplepoint in or before deceleration in a trajectory To addressthese issues we propose an approach to reconstruct themaximum likelihood trajectory to alleviate the effect causedby the randomness of sampling The essential parameters ofour method are distribution parameters of deceleration rateand acceleration rate for different modes

22 Distribution Parameters Estimation of Acceleration Ratefor Different Modes In this section the valid trajectories areselected from the historical trajectory data A valid trajectoryhas a sample point before deceleration and in decelerationand in acceleration and after acceleration By modeling andsolving the nonlinear programming problem (see (4) and(6)) the acceleration rate for modes 2 3 5 and 6 for each

Journal of Advanced Transportation 5

Selected historical low-frequencytrajectory

Decelerationtrajectory

Accelerationtrajectory

Assumed travelpattern and

model

Decelerationrate of

different mode

Accelerationrate of

different mode

EMalgorithm

Distributionparameters for

deceleration rate ofdifferent mode

Distributionparameters for

deceleration rate ofdifferent mode

A new low-frequencytrajectory

A newdeceleration

trajectory

A newaccelerationtrajectory

Maximum likelihooddeceleration

trajectory

Maximum likelihoodaccelerationtrajectory

Whole trajectory

Figure 3 Trajectory reconstruction process

V

T0

６1

６2

６Ｇ

(a)T

V

0６1

６2

６Ｇ

(b)

Figure 4 Relationship between speed and time in deceleration model (a) and accelerationmodel (b)

6 Journal of Advanced Transportation

1 2 4 53

Ｎ1 Ｎ2 Ｎ3 Ｎ4 Ｎ5Ｎ1

Ｎ4

Figure 5 Low-frequency vehicle trajectory at intersection

valid trajectory is obtained To avoid errors by using thesedirectly to calculate the distribution parameters of accel-eration rate for different modes we adopt an ExpectationMaximization method to iteratively calculate the accelerationrate for each mode It is assumed that the deceleration rateand acceleration rate are independent from each other Forthe idle and cruise modes the mode time mainly depends onthe signal plan traffic condition and intersection placementIn this paper it is assumed that the idle and cruisemode traveltimes t are uniformly distributed

119905119894 sim U (119887119894 119888119894) (8)

where i = 1 4 7 correspond to cruise (1) idle (4) and cruise(7)modesFor the deceleration and accelerationmodes it is assumed

that the acceleration rate follows a Gaussian distribution Thetravel time t that a vehicle spends under the decelerationand acceleration modes is the product of speed variation Δvdivided by corresponding time period

119886119895 = ΔV119895119905119895 sim 119873(120583119895 1205902119895 ) (9)

where j = 2356 correspond to deceleration1 (mode2) decel-eration2 (mode3) acceleration1 (mode5) and acceleration2(mode6) Each valid trajectory provides an observation ofdeceleration rate and acceleration rate for different modesAssume that there are M floating cars passing through theintersection Suppose that k⩽M is the trajectory sequencenumber The accelerate rate for each mode is calculated

E Step For mode j (j=2 3 5 6) we calculate the mean 120583119895and standard deviation 120590119895 of the corresponding accelerationUnder the assumption that the acceleration rate follows aGaussian distribution the probability density function for 119886119896119895is determined as

119901 (119886119896119895 | 120583119895 120590119895) = 1radic2120587120590119895 119890minus(119886119896119895minus120583119895)

221205902119895 (10)

where a119896119895 is the acceleration rate for mode i of kth trajectoryj=2356 For ease of calculation the log-likelihood functionof a119896119895 is used

log [119901 (119886119896119895 | 120583119895 120590119895)] = minus(119886119896119895 minus 120583119895)221205902119895 minus log (radic2120587120590119895) (11)

M Step In this step for each valid trajectory decelerationrates and acceleration rates for different modes a119896119895 are recalcu-lated such that the likelihood function is maximized For the

deceleration process of each trajectory its likelihood functioncan be represented as

argmax( sum119894=1119895=23

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=23

(119886119896119895 minus 120583119895)221205902119895 )

= argmin( sum119895=23

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(12)

subject to the equality

1199051198961 + sum119895=23

ΔV119896119895119886119896119895 = 119879 (13)

1199051198961 lowast V1198961 + (V1198961)2 minus (V119896119898)22 lowast 1198861198962 + (V119896119898)2 minus (V1198962)22 lowast 1198861198963 = 11987111989612 (14)

where 1199051198961 is travel time of mode1 for the kth trajectory V1198961and V1198962 are the speeds at the first sample point and secondsample point for kth trajectory respectively V119896119898 = (V1198961 + V1198962)2ΔV1198962 = V1198961 minus V119896119898 ΔV1198963 = V119896119898 minus V1198962 and 11987111989612 is the distancebetween the first sample point and the second sample pointIn the M step 120583119895 and 120590119895 are considered to be constants and119901(119905119896119894 | 119880119894) is fixed Therefore the terms containing them canbe ignored when maximizing the likelihood function This isa Constrained Quadratic Programming (CQP) problemThesolution of this CQP problem is the travel time of mode1 andthe distribution parameters of acceleration rate for mode 2and mode 3

In a similar way the likelihood function for the accelera-tion process is formulated as

argmax( sum119894=7119895=56

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=56

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(15)

subject to the equality

1199051198967 + sum119895=56

ΔV119896119895119886119896119895 = 119879 (16)

1199051198967 lowast V1198965 + (V1198965)2 minus (V119896119899)22 lowast 1198861198966 + (V119896119899)2 minus (V1198964)22 lowast 1198861198965 = 11987111989645 (17)

where 1199051198967 is the travel time of mode7 for the kth trajectoryV1198964 and V1198965 are the speeds at the fourth and the fifth samplepoints for the kth trajectory V119896n = (V1198964 + V1198965)2 ΔV1198965 = V119896n minus V1198964

Journal of Advanced Transportation 7

ΔV1198966 = V1198965 minus V119896n and 11987111989645 is the distance between the fourthand the fifth sample points It should be noted that the Mstep is run separately for each trajectory When the M stepends the method goes back to the E step to update the meanand variance based on all trajectories When the differencebetween consecutive iterations is below a threshold theiteration process stops

23 Maximum Likelihood Trajectory Estimation With thedistribution parameters of acceleration for each mode beingknown the maximum likelihood trajectory of a given low-frequency trajectory is estimated As a result of low-frequencysampling the sample points may be located at any position ofthe intersection Besides the number of sample points in atrajectory is uncertain To better capture the difference wedefine scenarios for the deceleration stage of a low-frequencytrajectory each with different numbers and reconstruct themaximum likelihood trajectory for each

Scenario 1 1 sample point before deceleration none in thedeceleration stage

Scenario 2 1 sample point before deceleration 1 in decelera-tion

Scenario 3 Multiple sample points before deceleration nonein the deceleration stage

The scenario definitions for the acceleration stage aresimilar to those for deceleration

Scenario 1 1 sample point after acceleration none in theacceleration stage

Scenario 2 1 sample point after acceleration 1 in acceleration

Scenario 3 Multiple sample points after acceleration

Here the sample points refer to the points where thespeed is above zero Different trajectories contain different

numbers of sample points and different combinations ofscenarios The trajectory reconstruction is divided into twoparts deceleration process and acceleration process Thereconstruction method is the same for both processes henceonly the deceleration process reconstruction with differentsample points is illustrated here

(1) Scenario 1 for Deceleration Trajectory It is assumed thatP1 is the sample point where the speed is above zero P2 isthe sample point where the speed is zero The distributionparameters of deceleration rate for mode 2 and mode 3 are(1205832 1205902) and (1205833 1205903) In order to reconstruct the maximumlikelihood trajectory of deceleration process the following isformulated and solved

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(18)

subject to

1199051 + V12 lowast 1198862 + V12 lowast 1198863 lt 119879 (19)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987112 (20)

In scenario 1 v2=0 v1-Vm=Vm-v2=v12 The solution tothis problem generates the travel time for mode1 anddeceleration rates of mode2 and mode3 119905m=t1+v1(2a2)11990521015840=t1+v1(2a2)+v1(2a3) This information is used to con-struct the speed-time curve and the position-time curve thatis trajectory of the vehicle v is the speed of the vehicle and yis the position of the vehicle at time t According to themodelthe expressions for v and y are as follows

V =

V1 0 lt t lt 1199051V1 + 1198862 lowast (119905 minus 1199051) 1199051 lt t lt 1199051 + V121198862V12 + 1198863 lowast (119905 minus 1199051 minus V121198862) 1199051 + V121198862 lt t lt 1199051 + V121198862 + V121198863

(21)

119910 =

V1 lowast 119905 0 lt t lt 1199051V1 lowast 1199051 + V1 lowast (119905 minus 1199051) + 11988622 lowast (119905 minus 1199051)2 1199051 lt 119905 lt 1199051 + V121198862V1 lowast 1199051 + 3V2181198862 + V12 lowast (119905 minus 1199051 minus V121198862) + 11988632 lowast (119905 minus 1199051 minus V121198862)

2 1199051 + V121198862 lt 119905 lt 1199051 + V121198862 + V121198863(22)

(2) Scenario 2 for Deceleration Trajectory As shownin Figure 6(b) there is a sample point P1 before

deceleration and another P2 in the deceleration stageof a trajectory In this scenario the trajectory between

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

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4 Journal of Advanced Transportation

Time

loca

tion

1

2

3

A

B

C

Figure 2 Schematic of trajectory reconstruction with low-frequency probe data

Based on the model and driving pattern for a low-frequency trajectory deceleration rates a2 for mode2 (decel-eration1) and a3 for mode3 (deceleration2) and accelerationrates a5 for mode5 (acceleration1) and a6 for mode6 (acceler-ation2) can be calculated by solving a constrained nonlinearprogramming problem as follows

For a typical low-frequency trajectory of the probe vehicleshown in Figure 5 the vehicle travels at the free-flow speedat Point 1 and then decelerates between the point in thedotted box and Point 2 At Point 3 the vehicle has stoppedand is waiting for the traffic light to turn green From Point4 to Point 5 the vehicle accelerates to reach the free-flowspeed Points 1-5 cover the entire motion process of thevehicle through the intersection The deceleration onset andacceleration end points correspond to the points in the dottedboxes

The deceleration rates 1198862 and 1198863 are calculated with thedeveloped model

Let 119905dy be the time interval between 1199051 and 11990510158401 and let 119905aybe the time interval between 11990510158404 and 1199055 The time and distancebetween Point 1 and Point 2 are expressed by the followingequations

119905119889119910 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (1)

V1 lowast 119905119889119910 + V21 minus V211989821198862 + V2119898 minus V2221198863 = 11987112 (2)

0 le 119905119889119910 le 30 0 le 1198862 le 9 0 le 1198863 le 9 (3)

where T is the sampling interval of the probe vehicle dataand is also the time travelled between Point 1 and Point 2 V1and V2 are the instantaneous velocities at Point 1 and Point 2respectively In this paper the sample interval is 30 seconds11987112 is the distance between Point 1 and Point 2 V119898 = (V1 +V2)2 In (1) and (2) there are three unknown parametersThe only solution cannot be obtained from this nonlinearequation group Therefore to obtain 1198862 and 1198863 expressions

1 and 2 are converted into a nonlinear programming problemwith the objective function as follows

119911 = min[(119905d119910 + V1 minus V1198981198862 + V119898 minus V21198863 minus 30)2

+ (V1 lowast 119905119889119910 + V21 minus V211989821198862 + V2119898 minus V2221198863 minus 11987112)2](4)

subject to

0 le 119905119889119910 le 300 le 1198862 le 90 le 1198863 le 9

(5)

By solving the nonlinear programming problem the valuesof 1198862 and 1198863 could be calculated For the acceleration modelthe acceleration rates of 1198865 and 1198866 are calculated as follows

119911 = min[(119905119886119910 + V119899 minus V41198865 + V5 minus Vn1198866 minus 30)2

+ (V5 lowast 119905119886119910 + V2119899 minus V2421198865 + V25 minus V211989921198866 minus 11987145)2](6)

subject to

0 le 119905a119910 le 300 le 1198865 le 90 le 1198866 le 9

(7)

where v4 and v5 are the instantaneous speeds at Point 4 andPoint 5 respectively Vn=(v4+v5)2 and L45 is the distancebetween Point 4 and Point 5

With a2 a3 a5 and a6 the trajectory can be recon-structed However the model only works for the scenarioswhere there is a sample point before deceleration and indeceleration For the acceleration process the requirement isthat there is a sample point in acceleration and after accel-eration Nevertheless probe vehicle updates normally occurat random positions and times so there may be no samplepoint in or before deceleration in a trajectory To addressthese issues we propose an approach to reconstruct themaximum likelihood trajectory to alleviate the effect causedby the randomness of sampling The essential parameters ofour method are distribution parameters of deceleration rateand acceleration rate for different modes

22 Distribution Parameters Estimation of Acceleration Ratefor Different Modes In this section the valid trajectories areselected from the historical trajectory data A valid trajectoryhas a sample point before deceleration and in decelerationand in acceleration and after acceleration By modeling andsolving the nonlinear programming problem (see (4) and(6)) the acceleration rate for modes 2 3 5 and 6 for each

Journal of Advanced Transportation 5

Selected historical low-frequencytrajectory

Decelerationtrajectory

Accelerationtrajectory

Assumed travelpattern and

model

Decelerationrate of

different mode

Accelerationrate of

different mode

EMalgorithm

Distributionparameters for

deceleration rate ofdifferent mode

Distributionparameters for

deceleration rate ofdifferent mode

A new low-frequencytrajectory

A newdeceleration

trajectory

A newaccelerationtrajectory

Maximum likelihooddeceleration

trajectory

Maximum likelihoodaccelerationtrajectory

Whole trajectory

Figure 3 Trajectory reconstruction process

V

T0

６1

６2

６Ｇ

(a)T

V

0６1

６2

６Ｇ

(b)

Figure 4 Relationship between speed and time in deceleration model (a) and accelerationmodel (b)

6 Journal of Advanced Transportation

1 2 4 53

Ｎ1 Ｎ2 Ｎ3 Ｎ4 Ｎ5Ｎ1

Ｎ4

Figure 5 Low-frequency vehicle trajectory at intersection

valid trajectory is obtained To avoid errors by using thesedirectly to calculate the distribution parameters of accel-eration rate for different modes we adopt an ExpectationMaximization method to iteratively calculate the accelerationrate for each mode It is assumed that the deceleration rateand acceleration rate are independent from each other Forthe idle and cruise modes the mode time mainly depends onthe signal plan traffic condition and intersection placementIn this paper it is assumed that the idle and cruisemode traveltimes t are uniformly distributed

119905119894 sim U (119887119894 119888119894) (8)

where i = 1 4 7 correspond to cruise (1) idle (4) and cruise(7)modesFor the deceleration and accelerationmodes it is assumed

that the acceleration rate follows a Gaussian distribution Thetravel time t that a vehicle spends under the decelerationand acceleration modes is the product of speed variation Δvdivided by corresponding time period

119886119895 = ΔV119895119905119895 sim 119873(120583119895 1205902119895 ) (9)

where j = 2356 correspond to deceleration1 (mode2) decel-eration2 (mode3) acceleration1 (mode5) and acceleration2(mode6) Each valid trajectory provides an observation ofdeceleration rate and acceleration rate for different modesAssume that there are M floating cars passing through theintersection Suppose that k⩽M is the trajectory sequencenumber The accelerate rate for each mode is calculated

E Step For mode j (j=2 3 5 6) we calculate the mean 120583119895and standard deviation 120590119895 of the corresponding accelerationUnder the assumption that the acceleration rate follows aGaussian distribution the probability density function for 119886119896119895is determined as

119901 (119886119896119895 | 120583119895 120590119895) = 1radic2120587120590119895 119890minus(119886119896119895minus120583119895)

221205902119895 (10)

where a119896119895 is the acceleration rate for mode i of kth trajectoryj=2356 For ease of calculation the log-likelihood functionof a119896119895 is used

log [119901 (119886119896119895 | 120583119895 120590119895)] = minus(119886119896119895 minus 120583119895)221205902119895 minus log (radic2120587120590119895) (11)

M Step In this step for each valid trajectory decelerationrates and acceleration rates for different modes a119896119895 are recalcu-lated such that the likelihood function is maximized For the

deceleration process of each trajectory its likelihood functioncan be represented as

argmax( sum119894=1119895=23

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=23

(119886119896119895 minus 120583119895)221205902119895 )

= argmin( sum119895=23

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(12)

subject to the equality

1199051198961 + sum119895=23

ΔV119896119895119886119896119895 = 119879 (13)

1199051198961 lowast V1198961 + (V1198961)2 minus (V119896119898)22 lowast 1198861198962 + (V119896119898)2 minus (V1198962)22 lowast 1198861198963 = 11987111989612 (14)

where 1199051198961 is travel time of mode1 for the kth trajectory V1198961and V1198962 are the speeds at the first sample point and secondsample point for kth trajectory respectively V119896119898 = (V1198961 + V1198962)2ΔV1198962 = V1198961 minus V119896119898 ΔV1198963 = V119896119898 minus V1198962 and 11987111989612 is the distancebetween the first sample point and the second sample pointIn the M step 120583119895 and 120590119895 are considered to be constants and119901(119905119896119894 | 119880119894) is fixed Therefore the terms containing them canbe ignored when maximizing the likelihood function This isa Constrained Quadratic Programming (CQP) problemThesolution of this CQP problem is the travel time of mode1 andthe distribution parameters of acceleration rate for mode 2and mode 3

In a similar way the likelihood function for the accelera-tion process is formulated as

argmax( sum119894=7119895=56

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=56

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(15)

subject to the equality

1199051198967 + sum119895=56

ΔV119896119895119886119896119895 = 119879 (16)

1199051198967 lowast V1198965 + (V1198965)2 minus (V119896119899)22 lowast 1198861198966 + (V119896119899)2 minus (V1198964)22 lowast 1198861198965 = 11987111989645 (17)

where 1199051198967 is the travel time of mode7 for the kth trajectoryV1198964 and V1198965 are the speeds at the fourth and the fifth samplepoints for the kth trajectory V119896n = (V1198964 + V1198965)2 ΔV1198965 = V119896n minus V1198964

Journal of Advanced Transportation 7

ΔV1198966 = V1198965 minus V119896n and 11987111989645 is the distance between the fourthand the fifth sample points It should be noted that the Mstep is run separately for each trajectory When the M stepends the method goes back to the E step to update the meanand variance based on all trajectories When the differencebetween consecutive iterations is below a threshold theiteration process stops

23 Maximum Likelihood Trajectory Estimation With thedistribution parameters of acceleration for each mode beingknown the maximum likelihood trajectory of a given low-frequency trajectory is estimated As a result of low-frequencysampling the sample points may be located at any position ofthe intersection Besides the number of sample points in atrajectory is uncertain To better capture the difference wedefine scenarios for the deceleration stage of a low-frequencytrajectory each with different numbers and reconstruct themaximum likelihood trajectory for each

Scenario 1 1 sample point before deceleration none in thedeceleration stage

Scenario 2 1 sample point before deceleration 1 in decelera-tion

Scenario 3 Multiple sample points before deceleration nonein the deceleration stage

The scenario definitions for the acceleration stage aresimilar to those for deceleration

Scenario 1 1 sample point after acceleration none in theacceleration stage

Scenario 2 1 sample point after acceleration 1 in acceleration

Scenario 3 Multiple sample points after acceleration

Here the sample points refer to the points where thespeed is above zero Different trajectories contain different

numbers of sample points and different combinations ofscenarios The trajectory reconstruction is divided into twoparts deceleration process and acceleration process Thereconstruction method is the same for both processes henceonly the deceleration process reconstruction with differentsample points is illustrated here

(1) Scenario 1 for Deceleration Trajectory It is assumed thatP1 is the sample point where the speed is above zero P2 isthe sample point where the speed is zero The distributionparameters of deceleration rate for mode 2 and mode 3 are(1205832 1205902) and (1205833 1205903) In order to reconstruct the maximumlikelihood trajectory of deceleration process the following isformulated and solved

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(18)

subject to

1199051 + V12 lowast 1198862 + V12 lowast 1198863 lt 119879 (19)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987112 (20)

In scenario 1 v2=0 v1-Vm=Vm-v2=v12 The solution tothis problem generates the travel time for mode1 anddeceleration rates of mode2 and mode3 119905m=t1+v1(2a2)11990521015840=t1+v1(2a2)+v1(2a3) This information is used to con-struct the speed-time curve and the position-time curve thatis trajectory of the vehicle v is the speed of the vehicle and yis the position of the vehicle at time t According to themodelthe expressions for v and y are as follows

V =

V1 0 lt t lt 1199051V1 + 1198862 lowast (119905 minus 1199051) 1199051 lt t lt 1199051 + V121198862V12 + 1198863 lowast (119905 minus 1199051 minus V121198862) 1199051 + V121198862 lt t lt 1199051 + V121198862 + V121198863

(21)

119910 =

V1 lowast 119905 0 lt t lt 1199051V1 lowast 1199051 + V1 lowast (119905 minus 1199051) + 11988622 lowast (119905 minus 1199051)2 1199051 lt 119905 lt 1199051 + V121198862V1 lowast 1199051 + 3V2181198862 + V12 lowast (119905 minus 1199051 minus V121198862) + 11988632 lowast (119905 minus 1199051 minus V121198862)

2 1199051 + V121198862 lt 119905 lt 1199051 + V121198862 + V121198863(22)

(2) Scenario 2 for Deceleration Trajectory As shownin Figure 6(b) there is a sample point P1 before

deceleration and another P2 in the deceleration stageof a trajectory In this scenario the trajectory between

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

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Journal of Advanced Transportation 5

Selected historical low-frequencytrajectory

Decelerationtrajectory

Accelerationtrajectory

Assumed travelpattern and

model

Decelerationrate of

different mode

Accelerationrate of

different mode

EMalgorithm

Distributionparameters for

deceleration rate ofdifferent mode

Distributionparameters for

deceleration rate ofdifferent mode

A new low-frequencytrajectory

A newdeceleration

trajectory

A newaccelerationtrajectory

Maximum likelihooddeceleration

trajectory

Maximum likelihoodaccelerationtrajectory

Whole trajectory

Figure 3 Trajectory reconstruction process

V

T0

６1

６2

６Ｇ

(a)T

V

0６1

６2

６Ｇ

(b)

Figure 4 Relationship between speed and time in deceleration model (a) and accelerationmodel (b)

6 Journal of Advanced Transportation

1 2 4 53

Ｎ1 Ｎ2 Ｎ3 Ｎ4 Ｎ5Ｎ1

Ｎ4

Figure 5 Low-frequency vehicle trajectory at intersection

valid trajectory is obtained To avoid errors by using thesedirectly to calculate the distribution parameters of accel-eration rate for different modes we adopt an ExpectationMaximization method to iteratively calculate the accelerationrate for each mode It is assumed that the deceleration rateand acceleration rate are independent from each other Forthe idle and cruise modes the mode time mainly depends onthe signal plan traffic condition and intersection placementIn this paper it is assumed that the idle and cruisemode traveltimes t are uniformly distributed

119905119894 sim U (119887119894 119888119894) (8)

where i = 1 4 7 correspond to cruise (1) idle (4) and cruise(7)modesFor the deceleration and accelerationmodes it is assumed

that the acceleration rate follows a Gaussian distribution Thetravel time t that a vehicle spends under the decelerationand acceleration modes is the product of speed variation Δvdivided by corresponding time period

119886119895 = ΔV119895119905119895 sim 119873(120583119895 1205902119895 ) (9)

where j = 2356 correspond to deceleration1 (mode2) decel-eration2 (mode3) acceleration1 (mode5) and acceleration2(mode6) Each valid trajectory provides an observation ofdeceleration rate and acceleration rate for different modesAssume that there are M floating cars passing through theintersection Suppose that k⩽M is the trajectory sequencenumber The accelerate rate for each mode is calculated

E Step For mode j (j=2 3 5 6) we calculate the mean 120583119895and standard deviation 120590119895 of the corresponding accelerationUnder the assumption that the acceleration rate follows aGaussian distribution the probability density function for 119886119896119895is determined as

119901 (119886119896119895 | 120583119895 120590119895) = 1radic2120587120590119895 119890minus(119886119896119895minus120583119895)

221205902119895 (10)

where a119896119895 is the acceleration rate for mode i of kth trajectoryj=2356 For ease of calculation the log-likelihood functionof a119896119895 is used

log [119901 (119886119896119895 | 120583119895 120590119895)] = minus(119886119896119895 minus 120583119895)221205902119895 minus log (radic2120587120590119895) (11)

M Step In this step for each valid trajectory decelerationrates and acceleration rates for different modes a119896119895 are recalcu-lated such that the likelihood function is maximized For the

deceleration process of each trajectory its likelihood functioncan be represented as

argmax( sum119894=1119895=23

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=23

(119886119896119895 minus 120583119895)221205902119895 )

= argmin( sum119895=23

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(12)

subject to the equality

1199051198961 + sum119895=23

ΔV119896119895119886119896119895 = 119879 (13)

1199051198961 lowast V1198961 + (V1198961)2 minus (V119896119898)22 lowast 1198861198962 + (V119896119898)2 minus (V1198962)22 lowast 1198861198963 = 11987111989612 (14)

where 1199051198961 is travel time of mode1 for the kth trajectory V1198961and V1198962 are the speeds at the first sample point and secondsample point for kth trajectory respectively V119896119898 = (V1198961 + V1198962)2ΔV1198962 = V1198961 minus V119896119898 ΔV1198963 = V119896119898 minus V1198962 and 11987111989612 is the distancebetween the first sample point and the second sample pointIn the M step 120583119895 and 120590119895 are considered to be constants and119901(119905119896119894 | 119880119894) is fixed Therefore the terms containing them canbe ignored when maximizing the likelihood function This isa Constrained Quadratic Programming (CQP) problemThesolution of this CQP problem is the travel time of mode1 andthe distribution parameters of acceleration rate for mode 2and mode 3

In a similar way the likelihood function for the accelera-tion process is formulated as

argmax( sum119894=7119895=56

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=56

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(15)

subject to the equality

1199051198967 + sum119895=56

ΔV119896119895119886119896119895 = 119879 (16)

1199051198967 lowast V1198965 + (V1198965)2 minus (V119896119899)22 lowast 1198861198966 + (V119896119899)2 minus (V1198964)22 lowast 1198861198965 = 11987111989645 (17)

where 1199051198967 is the travel time of mode7 for the kth trajectoryV1198964 and V1198965 are the speeds at the fourth and the fifth samplepoints for the kth trajectory V119896n = (V1198964 + V1198965)2 ΔV1198965 = V119896n minus V1198964

Journal of Advanced Transportation 7

ΔV1198966 = V1198965 minus V119896n and 11987111989645 is the distance between the fourthand the fifth sample points It should be noted that the Mstep is run separately for each trajectory When the M stepends the method goes back to the E step to update the meanand variance based on all trajectories When the differencebetween consecutive iterations is below a threshold theiteration process stops

23 Maximum Likelihood Trajectory Estimation With thedistribution parameters of acceleration for each mode beingknown the maximum likelihood trajectory of a given low-frequency trajectory is estimated As a result of low-frequencysampling the sample points may be located at any position ofthe intersection Besides the number of sample points in atrajectory is uncertain To better capture the difference wedefine scenarios for the deceleration stage of a low-frequencytrajectory each with different numbers and reconstruct themaximum likelihood trajectory for each

Scenario 1 1 sample point before deceleration none in thedeceleration stage

Scenario 2 1 sample point before deceleration 1 in decelera-tion

Scenario 3 Multiple sample points before deceleration nonein the deceleration stage

The scenario definitions for the acceleration stage aresimilar to those for deceleration

Scenario 1 1 sample point after acceleration none in theacceleration stage

Scenario 2 1 sample point after acceleration 1 in acceleration

Scenario 3 Multiple sample points after acceleration

Here the sample points refer to the points where thespeed is above zero Different trajectories contain different

numbers of sample points and different combinations ofscenarios The trajectory reconstruction is divided into twoparts deceleration process and acceleration process Thereconstruction method is the same for both processes henceonly the deceleration process reconstruction with differentsample points is illustrated here

(1) Scenario 1 for Deceleration Trajectory It is assumed thatP1 is the sample point where the speed is above zero P2 isthe sample point where the speed is zero The distributionparameters of deceleration rate for mode 2 and mode 3 are(1205832 1205902) and (1205833 1205903) In order to reconstruct the maximumlikelihood trajectory of deceleration process the following isformulated and solved

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(18)

subject to

1199051 + V12 lowast 1198862 + V12 lowast 1198863 lt 119879 (19)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987112 (20)

In scenario 1 v2=0 v1-Vm=Vm-v2=v12 The solution tothis problem generates the travel time for mode1 anddeceleration rates of mode2 and mode3 119905m=t1+v1(2a2)11990521015840=t1+v1(2a2)+v1(2a3) This information is used to con-struct the speed-time curve and the position-time curve thatis trajectory of the vehicle v is the speed of the vehicle and yis the position of the vehicle at time t According to themodelthe expressions for v and y are as follows

V =

V1 0 lt t lt 1199051V1 + 1198862 lowast (119905 minus 1199051) 1199051 lt t lt 1199051 + V121198862V12 + 1198863 lowast (119905 minus 1199051 minus V121198862) 1199051 + V121198862 lt t lt 1199051 + V121198862 + V121198863

(21)

119910 =

V1 lowast 119905 0 lt t lt 1199051V1 lowast 1199051 + V1 lowast (119905 minus 1199051) + 11988622 lowast (119905 minus 1199051)2 1199051 lt 119905 lt 1199051 + V121198862V1 lowast 1199051 + 3V2181198862 + V12 lowast (119905 minus 1199051 minus V121198862) + 11988632 lowast (119905 minus 1199051 minus V121198862)

2 1199051 + V121198862 lt 119905 lt 1199051 + V121198862 + V121198863(22)

(2) Scenario 2 for Deceleration Trajectory As shownin Figure 6(b) there is a sample point P1 before

deceleration and another P2 in the deceleration stageof a trajectory In this scenario the trajectory between

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

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6 Journal of Advanced Transportation

1 2 4 53

Ｎ1 Ｎ2 Ｎ3 Ｎ4 Ｎ5Ｎ1

Ｎ4

Figure 5 Low-frequency vehicle trajectory at intersection

valid trajectory is obtained To avoid errors by using thesedirectly to calculate the distribution parameters of accel-eration rate for different modes we adopt an ExpectationMaximization method to iteratively calculate the accelerationrate for each mode It is assumed that the deceleration rateand acceleration rate are independent from each other Forthe idle and cruise modes the mode time mainly depends onthe signal plan traffic condition and intersection placementIn this paper it is assumed that the idle and cruisemode traveltimes t are uniformly distributed

119905119894 sim U (119887119894 119888119894) (8)

where i = 1 4 7 correspond to cruise (1) idle (4) and cruise(7)modesFor the deceleration and accelerationmodes it is assumed

that the acceleration rate follows a Gaussian distribution Thetravel time t that a vehicle spends under the decelerationand acceleration modes is the product of speed variation Δvdivided by corresponding time period

119886119895 = ΔV119895119905119895 sim 119873(120583119895 1205902119895 ) (9)

where j = 2356 correspond to deceleration1 (mode2) decel-eration2 (mode3) acceleration1 (mode5) and acceleration2(mode6) Each valid trajectory provides an observation ofdeceleration rate and acceleration rate for different modesAssume that there are M floating cars passing through theintersection Suppose that k⩽M is the trajectory sequencenumber The accelerate rate for each mode is calculated

E Step For mode j (j=2 3 5 6) we calculate the mean 120583119895and standard deviation 120590119895 of the corresponding accelerationUnder the assumption that the acceleration rate follows aGaussian distribution the probability density function for 119886119896119895is determined as

119901 (119886119896119895 | 120583119895 120590119895) = 1radic2120587120590119895 119890minus(119886119896119895minus120583119895)

221205902119895 (10)

where a119896119895 is the acceleration rate for mode i of kth trajectoryj=2356 For ease of calculation the log-likelihood functionof a119896119895 is used

log [119901 (119886119896119895 | 120583119895 120590119895)] = minus(119886119896119895 minus 120583119895)221205902119895 minus log (radic2120587120590119895) (11)

M Step In this step for each valid trajectory decelerationrates and acceleration rates for different modes a119896119895 are recalcu-lated such that the likelihood function is maximized For the

deceleration process of each trajectory its likelihood functioncan be represented as

argmax( sum119894=1119895=23

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=23

(119886119896119895 minus 120583119895)221205902119895 )

= argmin( sum119895=23

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(12)

subject to the equality

1199051198961 + sum119895=23

ΔV119896119895119886119896119895 = 119879 (13)

1199051198961 lowast V1198961 + (V1198961)2 minus (V119896119898)22 lowast 1198861198962 + (V119896119898)2 minus (V1198962)22 lowast 1198861198963 = 11987111989612 (14)

where 1199051198961 is travel time of mode1 for the kth trajectory V1198961and V1198962 are the speeds at the first sample point and secondsample point for kth trajectory respectively V119896119898 = (V1198961 + V1198962)2ΔV1198962 = V1198961 minus V119896119898 ΔV1198963 = V119896119898 minus V1198962 and 11987111989612 is the distancebetween the first sample point and the second sample pointIn the M step 120583119895 and 120590119895 are considered to be constants and119901(119905119896119894 | 119880119894) is fixed Therefore the terms containing them canbe ignored when maximizing the likelihood function This isa Constrained Quadratic Programming (CQP) problemThesolution of this CQP problem is the travel time of mode1 andthe distribution parameters of acceleration rate for mode 2and mode 3

In a similar way the likelihood function for the accelera-tion process is formulated as

argmax( sum119894=7119895=56

log [119901 (119886119896119895 | 120583119895 120590119895) + 119901 (119905119896119894 | 119880119894)])

= argmin( sum119895=56

(119886119896119895 221205902119895 minus1198861198961198951205831198951205902119895 ))

(15)

subject to the equality

1199051198967 + sum119895=56

ΔV119896119895119886119896119895 = 119879 (16)

1199051198967 lowast V1198965 + (V1198965)2 minus (V119896119899)22 lowast 1198861198966 + (V119896119899)2 minus (V1198964)22 lowast 1198861198965 = 11987111989645 (17)

where 1199051198967 is the travel time of mode7 for the kth trajectoryV1198964 and V1198965 are the speeds at the fourth and the fifth samplepoints for the kth trajectory V119896n = (V1198964 + V1198965)2 ΔV1198965 = V119896n minus V1198964

Journal of Advanced Transportation 7

ΔV1198966 = V1198965 minus V119896n and 11987111989645 is the distance between the fourthand the fifth sample points It should be noted that the Mstep is run separately for each trajectory When the M stepends the method goes back to the E step to update the meanand variance based on all trajectories When the differencebetween consecutive iterations is below a threshold theiteration process stops

23 Maximum Likelihood Trajectory Estimation With thedistribution parameters of acceleration for each mode beingknown the maximum likelihood trajectory of a given low-frequency trajectory is estimated As a result of low-frequencysampling the sample points may be located at any position ofthe intersection Besides the number of sample points in atrajectory is uncertain To better capture the difference wedefine scenarios for the deceleration stage of a low-frequencytrajectory each with different numbers and reconstruct themaximum likelihood trajectory for each

Scenario 1 1 sample point before deceleration none in thedeceleration stage

Scenario 2 1 sample point before deceleration 1 in decelera-tion

Scenario 3 Multiple sample points before deceleration nonein the deceleration stage

The scenario definitions for the acceleration stage aresimilar to those for deceleration

Scenario 1 1 sample point after acceleration none in theacceleration stage

Scenario 2 1 sample point after acceleration 1 in acceleration

Scenario 3 Multiple sample points after acceleration

Here the sample points refer to the points where thespeed is above zero Different trajectories contain different

numbers of sample points and different combinations ofscenarios The trajectory reconstruction is divided into twoparts deceleration process and acceleration process Thereconstruction method is the same for both processes henceonly the deceleration process reconstruction with differentsample points is illustrated here

(1) Scenario 1 for Deceleration Trajectory It is assumed thatP1 is the sample point where the speed is above zero P2 isthe sample point where the speed is zero The distributionparameters of deceleration rate for mode 2 and mode 3 are(1205832 1205902) and (1205833 1205903) In order to reconstruct the maximumlikelihood trajectory of deceleration process the following isformulated and solved

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(18)

subject to

1199051 + V12 lowast 1198862 + V12 lowast 1198863 lt 119879 (19)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987112 (20)

In scenario 1 v2=0 v1-Vm=Vm-v2=v12 The solution tothis problem generates the travel time for mode1 anddeceleration rates of mode2 and mode3 119905m=t1+v1(2a2)11990521015840=t1+v1(2a2)+v1(2a3) This information is used to con-struct the speed-time curve and the position-time curve thatis trajectory of the vehicle v is the speed of the vehicle and yis the position of the vehicle at time t According to themodelthe expressions for v and y are as follows

V =

V1 0 lt t lt 1199051V1 + 1198862 lowast (119905 minus 1199051) 1199051 lt t lt 1199051 + V121198862V12 + 1198863 lowast (119905 minus 1199051 minus V121198862) 1199051 + V121198862 lt t lt 1199051 + V121198862 + V121198863

(21)

119910 =

V1 lowast 119905 0 lt t lt 1199051V1 lowast 1199051 + V1 lowast (119905 minus 1199051) + 11988622 lowast (119905 minus 1199051)2 1199051 lt 119905 lt 1199051 + V121198862V1 lowast 1199051 + 3V2181198862 + V12 lowast (119905 minus 1199051 minus V121198862) + 11988632 lowast (119905 minus 1199051 minus V121198862)

2 1199051 + V121198862 lt 119905 lt 1199051 + V121198862 + V121198863(22)

(2) Scenario 2 for Deceleration Trajectory As shownin Figure 6(b) there is a sample point P1 before

deceleration and another P2 in the deceleration stageof a trajectory In this scenario the trajectory between

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

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Journal of Advanced Transportation 7

ΔV1198966 = V1198965 minus V119896n and 11987111989645 is the distance between the fourthand the fifth sample points It should be noted that the Mstep is run separately for each trajectory When the M stepends the method goes back to the E step to update the meanand variance based on all trajectories When the differencebetween consecutive iterations is below a threshold theiteration process stops

23 Maximum Likelihood Trajectory Estimation With thedistribution parameters of acceleration for each mode beingknown the maximum likelihood trajectory of a given low-frequency trajectory is estimated As a result of low-frequencysampling the sample points may be located at any position ofthe intersection Besides the number of sample points in atrajectory is uncertain To better capture the difference wedefine scenarios for the deceleration stage of a low-frequencytrajectory each with different numbers and reconstruct themaximum likelihood trajectory for each

Scenario 1 1 sample point before deceleration none in thedeceleration stage

Scenario 2 1 sample point before deceleration 1 in decelera-tion

Scenario 3 Multiple sample points before deceleration nonein the deceleration stage

The scenario definitions for the acceleration stage aresimilar to those for deceleration

Scenario 1 1 sample point after acceleration none in theacceleration stage

Scenario 2 1 sample point after acceleration 1 in acceleration

Scenario 3 Multiple sample points after acceleration

Here the sample points refer to the points where thespeed is above zero Different trajectories contain different

numbers of sample points and different combinations ofscenarios The trajectory reconstruction is divided into twoparts deceleration process and acceleration process Thereconstruction method is the same for both processes henceonly the deceleration process reconstruction with differentsample points is illustrated here

(1) Scenario 1 for Deceleration Trajectory It is assumed thatP1 is the sample point where the speed is above zero P2 isthe sample point where the speed is zero The distributionparameters of deceleration rate for mode 2 and mode 3 are(1205832 1205902) and (1205833 1205903) In order to reconstruct the maximumlikelihood trajectory of deceleration process the following isformulated and solved

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(18)

subject to

1199051 + V12 lowast 1198862 + V12 lowast 1198863 lt 119879 (19)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987112 (20)

In scenario 1 v2=0 v1-Vm=Vm-v2=v12 The solution tothis problem generates the travel time for mode1 anddeceleration rates of mode2 and mode3 119905m=t1+v1(2a2)11990521015840=t1+v1(2a2)+v1(2a3) This information is used to con-struct the speed-time curve and the position-time curve thatis trajectory of the vehicle v is the speed of the vehicle and yis the position of the vehicle at time t According to themodelthe expressions for v and y are as follows

V =

V1 0 lt t lt 1199051V1 + 1198862 lowast (119905 minus 1199051) 1199051 lt t lt 1199051 + V121198862V12 + 1198863 lowast (119905 minus 1199051 minus V121198862) 1199051 + V121198862 lt t lt 1199051 + V121198862 + V121198863

(21)

119910 =

V1 lowast 119905 0 lt t lt 1199051V1 lowast 1199051 + V1 lowast (119905 minus 1199051) + 11988622 lowast (119905 minus 1199051)2 1199051 lt 119905 lt 1199051 + V121198862V1 lowast 1199051 + 3V2181198862 + V12 lowast (119905 minus 1199051 minus V121198862) + 11988632 lowast (119905 minus 1199051 minus V121198862)

2 1199051 + V121198862 lt 119905 lt 1199051 + V121198862 + V121198863(22)

(2) Scenario 2 for Deceleration Trajectory As shownin Figure 6(b) there is a sample point P1 before

deceleration and another P2 in the deceleration stageof a trajectory In this scenario the trajectory between

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

8 Journal of Advanced Transportation

P1 and P2 is reconstructed with the following expres-sions

argmax( log (119901 (1198862 | 1205832 1205902)+ log (119901 (1198863 | 1205833 1205903) + 119901 (1199051 | 1198801))

= argmin( sum119895=23

(119886119896119895 221205902119895 +1198861198961198951205831198951205902119895 ))

(23)

subject to

1199051 + V1 minus V1198981198862 + V119898 minus V21198863 = 119879 (24)

1199051 lowast V1 + (V1)2 minus (V119898)22 lowast 1198862 + (V119898)2 minus (V2)22 lowast 1198863 = 11987112 (25)

However a part of the deceleration is unclear To reconstructthe entire deceleration process it is assumed that a vehicledecelerates at rate 1198863 from Point P2 to Point P21015840 as shownin Figure 6(b) The time and position when the vehicle isstationary can be calculated and this estimated point isregarded as a virtual sample point P21015840 If the position ofP3 is different from that of point P21015840 this indicates thatthe estimated trajectory has an obvious error However theestimated trajectory could be corrected and the estimationerror reduced Keep the obtained acceleration rates 1198862 and 1198863unchanged and assume that a vehicle travels at speed V10158401

1199051 lowast V10158401 + (V10158401)2 minus (V119898)22 lowast 1198862 + (V119898)22 lowast 1198863 = 11987113 (26)

1199051 + V10158401 minus V1198981198862 + V1198981198863 = 119905121015840 (27)

The solution of this problem generates the travel time ofmode1 and acceleration rate of mode2 and mode3The speedand position at time t of the vehicle are obtained as inscenario 1The trajectory reconstruction for scenario 2 is thencompleted

(3) Scenario 3 for Deceleration Trajectory For scenario 3 asshown in Figure 6(c) there is no sample point in decelerationand there are two sample points before deceleration Thetrajectory reconstruction between P2 and P3 is the same asscenario 1 However for the trajectory between P1 and P2the assumption that a vehicle travels at a constant speedmay not hold Because the probe data are based on afixed time frequency more sample points represent a longertime at the intersection Under this circumstance a vehiclemay experience fluctuation in speed before decelerationespecially in congested traffic Hence we propose a piecewiseconstant acceleration rate method to construct the trajectorybetween P1 and P2 This method adapts to various trafficconditions and the constant speed motion is a special caseFirstly another virtual point P is assumed with the corre-sponding speed V12 which is the average speed between P1and P2 It is assumed that a vehiclersquos speed at point P1 is

V1 the vehiclersquos speed reaches V12 at time 11990512 at a constantacceleration rate Then the vehiclersquos speed becomes V2 at thesame acceleration rate at timestampTThe speed and positionare then determined by the following expressions1003816100381610038161003816V1 minus V12

10038161003816100381610038161003816100381610038161003816V2 minus V121003816100381610038161003816 = 11990512119879 minus 11990512 (28)

(V12 + V1) lowast 119905122 + (V12 + V2) lowast (119879 minus 11990512)2 = 11987112 (29)

T is the sample interval After solving the problem thespeed and position of the vehicle between P1 and P2 areobtained Combined with the trajectory between P2 and P3the trajectory of the deceleration process is constructed

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

3 Field Experiments

In this paper the site for experiment site was a portion ofthe south bound carriageway on Songshan Road spanningthe intersection of Huanghe and Songsha Roads in theNangang District of the city of Harbin in China (Figure 7)To estimate the distribution parameters of deceleration rateand acceleration rate accumulated historical data collectedby taxi from January to June 2017 are used There are about66700 taxis in Harbin in 2017 proving a good coverageof the road network The typical sample frequency of thetaxi is 130 Hz As the type of the floating car in thispaper is a taxi to guarantee the statistical significance ofthe results of the distribution parameter estimation thetrajectories that included pick-up and drop-off points wereexcluded The low-frequency trajectory passing through thetargeted intersection for scenario 2 is selected to estimate thedistribution parameters of deceleration rate and accelerationrate

To test the accuracy of the proposed method a fieldexperiment was conducted in the experiment site Four probevehicles were equipped with GPS receivers to collect high-frequency GPS data at 1Hz The vehicles were driven inthe north-south direction (red arrow direction in Figure 7)traversing straight through the intersection repeatedly duringthe periods of 0600-1100 and 1330-1900 on a normalweekday capturing the morning and evening peaks and off-peaks respectively To obtain the possible trajectories of thevehicles through the intersection the vehicles joined theflow at different locations The process lasted for six hoursgenerating 189 valid high-frequency GPS tracks

The statistical results for acceleration are shown inTable 1As can be seen in Table 1 in the deceleration process the

vehiclersquos acceleration in the first stage is lower than that inthe second stage However in the acceleration process thevehiclersquos acceleration in the first stage is higher than that inthe second stage The standard deviation for mode4 is thesmallest The standard deviation is a function amongst othersof differences in driving behavior and habits of the driver

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 9

0

P1

v

t

P2

t1 tＧ t2

(a)

v

t0

P1

P2

P3

t1 tＧ t2

P2

(b)

v

t0

P1 P2

P3

P

(c)

Figure 6 Different sample scenarios for deceleration process

Table 1 Parameters for each mode

Modes Mean (ms2) Standard Deviation (ms2)Deceleration (Mode2) 06916 0203Deceleration (Mode3) 0894 0202Acceleration (Mode5) 0961 0239Acceleration (Mode6) 0688 0141

Figure 7 The experiment site

4 Result Analysis and Discussion

41 Trajectory Reconstruction for Different Sample Pointswith the Proposed Method With the distribution parametersof acceleration rate for each mode the trajectory fromthe low-frequency floating car data can be reconstructedwith the proposed method To test the proposed methodrsquosperformance the estimated trajectories are compared withthe observed trajectories achieved in a field experimentFigure 8 shows this comparison for three typical low-frequency trajectories eachwith a different number of samplepoints and combinations of sample scenarios Figure 8(a)represents the trajectory with 4 sample points scenario 1for deceleration trajectory and scenario 3 for accelerationtrajectory Figure 8(b) illustrates trajectories with 5 samplepoints scenario 2 for deceleration trajectory and scenario 3

for acceleration trajectory Figure 8(c) illustrates trajectorieswith 7 sample points scenario 3 for deceleration trajectoryand scenario 3 for acceleration trajectory

From Figure 8 it can be seen that in general the errorbetween the estimated trajectory and the observed trajectoryis small for the most part of the trajectory which indicates theeffectiveness of our method For the trajectory in Figure 8(b)the performance of the method is the best For the trajectoryin Figure 8(a) the trajectory error between the third andfourth sample points is relatively large while for the trajectoryin Figure 8(c) the large estimation error occurs in thefirst and third sample points To explain this phenomenonFigure 9 shows the estimated and observed speed-time curveof the three trajectories It can be seen that for the tra-jectories in Figures 8(b) and 8(c) the vehicle experiencesanother deceleration after accelerating For the trajectoryin Figure 8(c) the vehicle even experiences another stop(inconsistent with our assumption) and experiences speedfluctuation before decelerating resulting in the relativelylarge trajectory estimation error In general the proposedmethod provides a good reconstruction of the low-frequencytrajectory in all the cases investigated

To further demonstrate the significance of the proposedmethod the estimated trajectories were compared withtrajectories that were created with the linear interpolationmethod In the linear method it is assumed that the distancethe vehicle travels between any consecutive updates is fixedAs illustrated in Figure 10 the trajectories estimated bythe proposed method have a better performance than thetrajectories created by the linear interpolation especially forthe estimation of the stop period In addition the proposedmethod captures the change in the trajectory caused bythe change in speed while the linear-interpolation methodtreats all the trajectories in a similar way From Figure 10it can be seen that the proposed methodrsquos performance

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved Trajectory

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

0 20 40 60 80

Time (s)

400

300

200

120100

Estimated TrajectoryObserved Trajectory

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

Time (s)

400

300

200

100

0

0

50 100 150 200

Estimated TrajectoryObserved Trajectory

(c) Estimated and observed trajectory for trajectory with 7samplepoints

Figure 8 Estimated and observed trajectories for different types of trajectory

is obviously better than the linear-interpolation methodespecially for the deceleration and acceleration processesof each trajectory which indicates the effectiveness andpracticality of our method Note that even in the cases wherethe selected trajectory is relatively complex with fluctuationsin acceleration and deceleration the estimated trajectory isclose to the observed trajectory for the most part

To quantitatively analyze the estimation results the MeanAbsolute Error (MAE) between the estimated and groundtruth trajectories is calculatedTheMAE is defined as follows

MAE = 1119899119899sum119894=1

10038161003816100381610038161198901198941003816100381610038161003816 = 1119899119899sum119894=1

1003816100381610038161003816119909119894 minus 1199091198941003816100381610038161003816 (30)

where119909119894 is the estimated position and 119909119894 is the actual positionat time step i Hence the estimation error of a group oftrajectories MMAE is defined as

MMAE = 1m

119898sum119894=1

1003816100381610038161003816MAE1198941003816100381610038161003816 (31)

where MAEi is the estimation error of the ith trajectoryand m is the number of trajectories The MAE betweenthe estimated and ground truth trajectories is calculatedBesides the stop period is also an important evaluation

indicator TAE is defined as the absolute error of the estimatedtime period and the actual time period for each trajectoryHence MTAE is defined as the mean of the TAE of agroup of trajectories To test the ability of the proposedmethod to reconstruct trajectories generated in differenttraffic conditions and stop positions the trajectories wereclassified and counted according to the speed of decelerationonset point and stop position TheMMAE and MTAE valuesfor each category of trajectory are listed in Table 2

Compared with the linear interpolation method theproposed method generally provides better estimation interms of both MMAE and MTAE For the proposed methodwhen the speed at the deceleration onset point ranges from0 to 15 kmh which indicates that the traffic is congestedthe MMAE and MTAE are obviously larger than those forother speed ranges However for the linear-interpolationmethod there is no obvious difference between each speedrange For the proposed method the MMAE ranges from347 to 1053 with MTAE from 38 to 99 For the linear-interpolation method the MMAE range is 935-139 andthe MTAE range is 97-131 When the speed range at thedeceleration onset point is 45-60 kmh the proposedmethodachieves smaller MMAE and MTAE than the other speedintervals This may be because when a vehicle travels ata relatively high speed the travel pattern is in line with

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 11

0 20 40 60 80

Time (s)

Estimated speedObserved Speed

25

20

15

10

5

0

Spee

d (k

mh

)

(a) Estimated and observed speed for trajectory with 4 samplepoints

0 20 40 60 80

Time (s)120100

Estimated speedObserved Speed

Spee

d (k

mh

)

50

40

30

20

10

0

(b) Estimated and observed speed for trajectory with 5 samplepoints

Time (s)0 50 100 150 200

Estimated speedObserved Speed25

20

15

10

5

0

Spee

d (k

mh

)

(c) Estimated and observed speed for trajectory with 7 samplepoints

Figure 9 Estimated and observed speed for different types of trajectory

Table 2 Trajectory estimation error for proposed method and benchmarkmethod

Deceleration onset Stopped position(m) Number of Proposed method Linear InterpolationMethod

Speed (Kmh) trajectories MMAE (m) MTAE (s) MMAE (m) MTAE (s)[0-15) [0 100) 5 1053 66 113 97[0-15) [100 200) 7 934 89 948 131[0-15) [200 300) 7 986 95 1251 127[0-15) [300 400) 16 1026 99 139 116[15 30) [0 100) 14 823 63 111 103[15 30) [100 200) 10 611 74 995 105[15 30) [200 300) 9 757 60 1066 112[15 30) [300 400) 7 835 62 1138 99[30 45) [0 100) 20 739 65 1017 104[30 45) [100 200) 16 646 58 935 117[30 45) [200 300) 13 632 63 1136 109[30 45) [300 400) 10 749 61 977 98[45 60] [0 100) 23 649 46 959 105[45 60] [100 200) 15 593 57 984 99[45 60] [200 300) 17 347 38 1097 107[45 60] [300 400) - - - - -

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Journal of Advanced Transportation

300

250

200

150

100

Posit

ion

(m)

0 20 40 60 80

Time (s)

Estimated TrajectoryObserved TrajectoryBench Method

(a) Estimated and observed trajectory for trajectory with 4 samplepoints

Posit

ion

(m)

400

300

200

0 20 40 60 80

Time (s)120100

Estimated TrajectoryObserved TrajectoryBench Method

(b) Estimated and observed trajectory for trajectory with 5 samplepoints

Posit

ion

(m)

400

300

200

100

0

Time (s)0 50 100 150 200

Estimated TrajectoryObserved TrajectoryBench Method

(c) Estimated and observed trajectory for trajectory with 7 samplepoints

Figure 10 Comparison of estimated trajectory with the observed and linear method based trajectories

the modal structure in our proposed method Converselywhen the vehicle travels at a low speed more sample pointsare generated and the vehicle may experience stop and goand even pass through the intersection in two or moresignal cycles This circumstance is not consistent with ourmodelrsquos assumption This could explain the differences inperformance of the proposed method under different travelspeed ranges It should be noted that the worst performanceappears when the vehicle travels at low speed and stops at thefurthest distance from the intersection This is the case forboth the proposed and linear interpolation methods

In this paper the number of sample points of thetrajectory is an important factor for trajectory reconstructionTo givemore insight into the proposedmethodrsquos performancewith different number of sample points the MMAE fordifferent number of sample points and travel speed ranges canbe visualized in Figure 11

Figure 11 shows that for a specific stop position rangewhen the number of sample points is below 6 the MMAE isnot affected by the number of sample points However whenthe number is above 6 as the number increases the MMAEincreasesThe further the stop position from the intersectionthe larger the error When the number of sample points is

Stop

ped

posit

ion

rang

e

[0100)

[100200)

[200300)

[300400]

3 4 5 6 7 8

Number of sample points

12

10

8

6

4

75 61 65 83 12 13

55 63 42 65 11 12

67 53 65 74 97 11

49 53 5 81 91 10

Figure 11 MMAE for different number of sample points and stopposition ranges

below 7 the stop position range has no effect on the MMAEHowever when the number is above 6 the MMAE increaseswith the distance from the stop position to the intersec-tion

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 13

200

175

150

125

100

75

50

MM

AE

(m)

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

(a) Boxplot of MMAE

30-P 30-B 40-P 40-B 50-P 50-B 60-P 60-BSample frequency (s)

35

30

25

20

15

10

5

MTA

E (m

)(b) Boxplot of MTAE

Figure 12 Sensitivity analysis with different sample interval

In this paper the low-frequency floating car data issampled at 30s The effect of higher sampling intervals isinvestigated through a sensitivity analysis employing both theproposed and linear interpolation methods

In Figure 12 ldquoPrdquo represents the proposed method andldquoBrdquo the linear-interpolation method with the number beforethem being the sample interval in seconds The point insideeach box represents the mean value In Figure 12(a) it isshown that the linear-interpolation method is more sensitiveto the sample interval By contrast the proposed method hasa better performance in stability For the sample intervals30 40 50 and 60 the corresponding MMAE values fromthe proposed method are 72 90 105 and 117m Thecorresponding MMAE values for the linear-interpolationmethod are 110 135 154 and 170m It can be seen thatthe difference in performance between the proposed andlinear-interpolation methods increases with the length of thesample interval Figure 12(b) presents the MTAE for differentsample intervals for the proposed method and the linear-interpolation method As the length of the sample intervalincreases the MTAE for the linear-interpolation methodshows greater growth and includes frequent abnormal valuesHence the estimation results from the proposed method aremore reliable than the linear-interpolation method

5 Conclusions

In this paper a novel method is proposed to reconstructa vehiclersquos trajectory from low-frequency floating car dataUnder the assumption that the vehicle travels in a certainpattern a model is developed to estimate the distributionparameters of deceleration rate and acceleration rate withselected historical data With the distribution parameters as

prior information trajectories for different scenarios are con-structed Compared to a reference trajectory (truth) the pro-posed method achieves encouraging accuracy with MMAEranging from 347m to1053m higher than the commonlyused linear-interpolation method which that MMAE rangingfrom935m to 1251m In ourmethod extra information is notrequired and in addition the speed curve and distributionparameters of the deceleration rate and acceleration rate areobtained The proposed method therefore paves the way fora widespread application of floating car data in transport

The presented work has established a framework forvehicle trajectory reconstruction that could be modified orsolve other traffic problems in the future To improve andextend the application of our model future work will includethe following cases

(1) Vehicles experiencing stop-and-go behavior when thequeue at an intersection is long In this case the vehi-cle movement may deviate from our assumed drivingpattern increasing trajectory estimation error Wewill extend our model to account for the limitationsof our assumptions to improve its robustness

(2) Queue length at an intersection is long such thatvehicles pass through the intersection in two or moresignal cycles In this case we do not recommendthe application of our method We mine multicyclevehicle trajectories to capture the vehicle movementpatterns and use them to improve our model

(3) Queue profile at the intersection Combined withthe traffic shockwave theory the queue length andcontrol delay of the intersection could be effectivelyestimated We will update our model to incorporatetraffic shockwave models

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

14 Journal of Advanced Transportation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

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

References

[1] S Pan B Jiang N Zou and L Jia ldquoAverage travel speed esti-mation using multi-type floating car datardquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 192ndash197 IEEE June 2011

[2] M Rahmani E Jenelius and H N Koutsopoulos ldquoNon-parametric estimation of route travel time distributions fromlow-frequency floating car datardquo Transportation Research PartC Emerging Technologies vol 58 pp 343ndash362 2015

[3] S Y R Rompis M Cetin and F Habtemichael ldquoProbe vehiclelane identification for queue length estimation at intersectionsrdquoJournal of Intelligent Transportation Systems Technology Plan-ning and Operations vol 22 no 1 pp 10ndash25 2018

[4] K Liu T Yamamoto and T Morikawa ldquoEstimating delay timeat signalized intersections by probe vehiclesrdquo in Proceedings ofthe 5th International Conference on Traffic and TransportationStudies (ICTTS rsquo06) pp 644ndash655 August 2006

[5] A D PatireMWright B Prodhomme andAM Bayen ldquoHowmuch GPS data do we needrdquo Transportation Research Part CEmerging Technologies vol 58 pp 325ndash342 2015

[6] C Bucknell and J C Herrera ldquoA trade-off analysis betweenpenetration rate and sampling frequency of mobile sensors intraffic state estimationrdquo Transportation Research Part C Emerg-ing Technologies vol 46 pp 132ndash150 2014

[7] J Huang and H-S Tan ldquoVehicle future trajectory predictionwith a DGPSINS-based positioning systemrdquo in Proceedings ofthe American Control Conference pp 5831ndash5836 IEEE June2006

[8] M Rafati Fard A Shariat Mohaymany and M Shahri ldquoAnew methodology for vehicle trajectory reconstruction basedon wavelet analysisrdquo Transportation Research Part C EmergingTechnologies vol 74 pp 150ndash167 2017

[9] M Montanino and V Punzo ldquoTrajectory data reconstructionand simulation-based validation against macroscopic trafficpatternsrdquo Transportation Research Part B Methodological vol80 pp 82ndash106 2015

[10] Z Sun and X J Ban ldquoVehicle trajectory reconstruction forsignalized intersections using mobile traffic sensorsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp2680-2283 2013

[11] N Wan A Vahidi and A Luckow ldquoReconstructing maximumlikelihood trajectory of probe vehicles between sparse updatesrdquoTransportation Research Part C Emerging Technologies vol 65pp 16ndash30 2016

[12] P Hao K Boriboonsomsin GWu andM Barth ldquoProbabilisticmodel for estimating vehicle trajectories using sparse mobile

sensor datardquo in Proceedings of the 17th IEEE InternationalConference on Intelligent Transportation Systems (ITSC rsquo14) pp1363ndash1368 October 2014

[13] P Hao K Boriboonsomsin G Wu and M J Barth ldquoModalactivity-based stochastic model for estimating vehicle trajec-tories from sparse mobile sensor datardquo IEEE Transactions onIntelligent Transportation Systems vol 18 no 3 pp 701ndash711 2017

[14] Z Zhang Y Wang G Zhang and H Wang ldquoEstimatingcontrol delays at signalised intersections using low-resolutiontransit bus-based global positioning systemdatardquo IET IntelligentTransport Systems vol 10 no 2 pp 73ndash78 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom