Estimating Acceleration and Lane-Changing Dynamics from Next Generation Simulation Trajectory Data

Transcription

1 Etimating Acceleration and Lane-Changing Dynamic from Next Generation Simulation Trajectory Data Chritian Thiemann, Martin Treiber, and Arne Keting The Next Generation Simulation (NGSIM) trajectory data et provide longitudinal and lateral poitional information for all vehicle in certain patiotemporal region. Velocity and acceleration information cannot be extracted directly becaue the noie in the NGSIM poitional information i greatly increaed by the neceary numerical differentiation. A moothing algorithm i propoed for poition, velocitie, and acceleration that can alo be applied near the boundarie. The moothing time interval i etimated on the bai of velocity time erie and the variance of the proceed acceleration time erie. The velocity information obtained in thi way i then applied to calculate the denity function of the two-dimenional ditribution of velocity and invere ditance and the denity of the ditribution correponding to the microcopic fundamental diagram. It i alo ued to calculate the ditribution of time gap and time to colliion, conditioned to everal range of velocitie and velocity difference. By imulating virtual tationary detector, it i hown that the probability for critical value of the time to colliion i greatly underetimated when etimated from ingle-vehicle data of tationary detector. Finally, the lane-changing proce i invetigated, and a quantitative criterion i formulated for the duration of lane change that i baed on the trajectory denity in normalized coordinate. There i a noiy but ignificant velocity advantage in favor of the targeted lane that decreae immediately before the change due to anticipatory acceleration. FHWA originated Next Generation Simulation (NGSIM) to improve the quality and performance of imulation tool, promote the ue of imulation for reearch and application, and achieve wider acceptance of validated imulation reult (1). A part of the program, a firt data et wa collected at the Berkeley Highway Laboratory (BHL) in Emeryville, California, by Cambridge Sytematic and the California Center for Innovative Tranportation at the Univerity of California, Berkeley. BHL i part of I-80 on the eat coat of the San Francico Bay. Six camera mounted on top of the 97-m Pacific Park Plaza tower recorded 4,733 vehicle on a road ection approximately 900 m long in a 30-min period in December 003. The reult wa publihed a a prototype data et. A part of the California Partner for Advanced Highway and Tranit Program, the Intitute of Tran- C. Thiemann, Department for Nonlinear Dynamic, Max Planck Intitute for Dynamic and Self-Organization, Bunentraße 10, D Göttingen, Germany. M. Treiber and A. Keting, Intitute for Tranport and Economic, Techniche Univerität Dreden, Andrea-Schubert-Straße 3, D-0106 Dreden, Germany. Correponding author: A. Keting, keting@vwi.tu-dreden.de. Tranportation Reearch Record: Journal of the Tranportation Reearch Board, No. 088, Tranportation Reearch Board of the National Academie, Wahington, D.C., 008, pp DOI: / portation Studie at Univerity of California at Berkeley further enhanced the data-collection procedure (), and in April 005, another trajectory data et wa recorded at the ame location by uing even camera and capturing a total of 5,648 vehicle trajectorie in three 15-min interval on a road ection of approximately 500 m. Thi wa later publihed a the I-80 data et. In June 005, a data collection wa made by uing eight camera on top of the 154-m 10 Univeral City Plaza building, next to the Hollywood Freeway, US-101. On a road ection of 640 m, 6,101 vehicle trajectorie were recorded in three conecutive 15-min interval. Thi data et wa publihed a the US-101 data et. All data et are freely available for download from the NGSIM webite ( Thi amount of trajectory data i unique in the hitory of traffic reearch and provide a better bai for the validation and calibration of microcopic traffic model. Lu and Skabardoni examined the backward propagation peed of traffic hock wave by uing the two later data et (3). However, mot recent attention focue on the invetigation of lane change: Roe and Ulerio ued the two later data et to tudy ome trend and enitivitie in weaving ection (4), epecially lane change. Zhang and Kovvali (5) and Gowami and Bham (6) invetigated the gap acceptance behavior in lane-changing ituation on freeway. By uing the prototype and I-80 data et, Toledo and Zohar invetigated the duration of lane change (7). Choudhury et al. calibrated a lane-changing model by uing the I-80 data et and validated the model by uing virtual loop detector placed into the US-101 data (8). Leclercq et al. calibrated a model of the headway relaxation phenomenon oberved in lane-changing ituation by uing the I-80 data et (9). Other tudie have alo ued the NGSIM data (10 13). In all thee tudie, the longitudinal and lateral poition information of the trajectory data wa ued directly. There have been few invetigation of the data regarding topic for which velocitie and acceleration play a ignificant role, uch a teting or calibrating car-following model (14) or lane-changing model or etimating fuel conumption (15). Since velocitie and acceleration are derived quantitie, the noie in the NGSIM poitional information i greatly increaed and a direct application i not poible. Thi work propoe and motivate a moothing method that allow the NGSIM data to be ued for data analyi by uing velocity or acceleration information. The velocitie are ued to calculate the denity function of the two-dimenional ditribution of velocity and invere ditance and the denity of the ditribution correponding to a microcopic fundamental diagram. The data alo are ued to calculate the ditribution of time gap and time to colliion, conditioned to everal range of velocitie and velocity difference. The meaurement of patial quantitie by virtual loop detector are compared to the real value determined from the trajectory data. A 90

2 Thiemann, Treiber, and Keting 91 method i propoed to determine the lane-change duration from the NGSIM data. EXTRACTING VELOCITY AND ACCELERATION INFORMATION Trajectory data available for download appear to be unfiltered and alo exhibit ome noie artifact. All data et include velocity and acceleration. However, they appear to have been numerically derived from the tracked vehicle poition without any proceing. Figure 1 viualize the problem of the data. In the prototype data et, two-third of all acceleration are beyond ±3 m/ (which are then reported a ±3 m/ in the data file). The example trajectory how that the driver i allegedly changing between hard acceleration and hard deceleration everal time a econd, which i unrealitic. In the I-80 and US-101 data et, the acceleration ditribution are more realitic although approximately 10% are beyond ±3 m/. However, in the later data et, the velocity ditribution are very piky, that i, velocitie tend to nap to certain value. The velocitie of an example trajectory exhibit an unrealitic behavior: taken a real, they would mean that driver do not moothly brake or accelerate but rather ue the ga and brake pedal only occaionally although pre hard, to quickly change between preferred velocitie. Alo, to produce the pike in the velocity ditribution, all driver mut happen to like the ame velocitie. Thi i unrealitic and therefore the velocity pike mut be an artifact of the meaurement method. One may credit the velocity pike to dicretization error (time and pace are dicretized; thu velocity can take only certain dicrete value a well), but two obervation object to that. Firt, the pike are not delta peak; other velocitie till appear. Second, given the time dicretization dt = 1 10 and the approximate ditance between the velocity pike dv = 0.7 m/, thi would mean that the patial accuracy of the meaurement method i jut 7 m, which i obviouly not the cae. It i upected therefore that the velocity pike are introduced by ome data potproceing. To correct thoe artifact, a ymmetric exponential moving average filter (EMA) i applied to all trajectorie before any further data analyi. Let x α (t i ) denote the meaured poition of vehicle α at time t i, where i = 1... N α and N α denote the number of data point of the trajectory. The moothing kernel i given by g(t) = exp( t / T), where T i the moothing width. Since the data point are equiditant in time with interval dt, the moothing operation can be formulated by uing data point indice intead of time. The moothed poition x (t i ) are given by x α where Z = i+ D 1 i k Δ ( ti)= xα( tk) e Z k= i D i+ D e i k Δ k= i D The moothing width Δ i given by T dt and tranparently handle the different time interval in the data et. (The prototype data et ue dt = 1 15, wherea the later two ue dt = 1 10 ). The ame real-time moothing width T can be ued for all data et, and T Δ= dt will be the correponding moothing width meaured in data point for the pecific data et. The moothing window width D = min{3δ, i 1, N α i} i choen to be three time the moothing kernel width for any data point that i not cloer than D data point to either trajectory boundary. For the point near the boundarie, the moothing width i decreaed to enure that the moothing width i alway ymmetric. () 1 Prob. Denity [/m] Prob. Denity [ /m] (c) (d) FIGURE 1 Problem of original, unmoothed data: acceleration ditribution, example trajectory excerpt, (c) velocity ditribution of 7:50 8:05 a.m. data file of US-101 data et, and (d) example trajectory (velocity).

3 9 Tranportation Reearch Record 088 It may be objected that other filter would work a well or even better for example, the Kalman filter or a imple moving average. A moving average filter, which would correpond to Equation 1 with the exponential removed, ha noncontinuou filter boundarie, that i, with moving, the filter data point uddenly lip into the moothing window with full weight or uddenly drop out. Thi can caue moothing artifact that are prevented by uing a weighted moving average, where the weight decreae with increaing ditance from the moothing window center. Thi way, data point will be moothly incorporated into the moothing window and fall out moothly a well. It wa found that an exponential weight function lead to better reult than a Gauian filter, and o EMA wa choen. The Kalman filter need a imple traffic model and thu introduce ome ignificant aumption into the moothing proce. Alo, the Kalman filter ha more parameter, wherea the EMA method ha only one parameter, T, and doe not introduce complicated aumption. Another poible filter would be not to ue ome moving kernel filter but rather to increae the tep ize from dt to n dt in calculation of the velocitie and acceleration, that i, = ( ( + ) ( )) ( i i ) vt xt n idt xt n idt n dt It can be hown that thi filter i equal to a imple moving average for the velocitie and a compoition of two moving average for the acceleration (which implifie to a triangular moving average when boundary region are neglected). Thi filter i a fater but omewhat wore alternative to the propoed method. Having defined the fundamental moothing mechanim, there are till two open quetion. Firt, the order of differentiation and moothing operation need to be defined. Second, a moothing width T mut be found. There are three poible anwer to the firt quetion: mooth poition, then differentiate to velocitie and acceleration; firt differentiate to velocitie and acceleration and then mooth all three variable; and (c) mooth poition, differentiate to velocitie, mooth velocitie, differentiate to acceleration, and mooth acceleration. For D + i N α D 1, the moothing (1) commute with the differentiation, and all thee method are equivalent. In view of the hort trajectorie, however, the point cloer to the boundary cannot be neglected. The firt method i problematic, a can be een by the following reaoning. Conider an artificial trajectory with contant acceleration: ( i) = 1 x t at i Any ymmetric moothing kernel will overetimate the poition and produce a trajectory x (t i ) > x(t i ). Sufficiently far away from the boundarie, the moothing window width D i contant and the moothed trajectory = + 1 ha a contant error proportional to the variance σ g of the moothing kernel. Near the boundarie, however, D and σ g will become maller and vanih for i = 1 and i = N, which reult in x (t 1 ) = x(t 1 ) and x (t N ) = x(t N ). Thu, the offet of the moothed poition become maller when approaching the boundarie, which of coure induce a bia to the velocity. Moreover, if the moothing kernel doe not completely vanih at the moothing window border, the tranition between contant offet and decreaing offet will not be continux t x t aσ i i g ouly differentiable, inducing a jump in velocity and thu an even larger jump in the acceleration. Therefore, ue of thi moothing method i dicouraged. To chooe the econd or third moothing method, artificial benchmark trajectorie have been generated and white noie ha been added to the poition. The econd method firt the differentiation to velocitie and acceleration and then the moothing of the three variable turned out to better reproduce the original trajectorie, and o thi method wa ued. Thi left the difficult quetion of which moothing width T to ue. There i no generic recipe, but ome hint help make thi deciion not completely arbitrary. Firt, the mot vivid trajectorie thoe with a large velocity range were extracted from each data et, and the variance of the acceleration, σ a, were compared for different moothing width (Figure 1a). For T, the acceleration variance of the moothed trajectory would vanih, but the variance that i caued by the noie vanihe much fater than the one caued by the real acceleration data. Thu, with finite T the noie i moothed out very quickly, leading to a fat drop in σ a(t) at mall T. For larger T, σ a (T) appear to be nearly contant. Keeping in mind that the real acceleration data are moothed a bit a well, the plot ugget a moothing width of about 4. However, thi value i a uggetion for the acceleration moothing width only. It i not neceary to ue uch large moothing width for the poition and velocitie. Let X α (t i ) be a random variable decribing the poition of vehicle α with expectation value x α(t i ) and variance σ x(t i ). The meaured trajectory x α (t i ) i a realization of X α (t i ) and, auming unbiaed noie, the real trajectory i equal to x α(t i ). Two new random variable decribe the velocitie and acceleration of vehicle α in term of ymmetric difference quotient: Xα ti + dt Xα ti dt Vα ( ti ) = dt Xα ti + dt Xα ti Xα ti dt Aα ( ti ) = () 3 dt Since thi i a linear combination of random variable, the expectation value of V α (t i ) and A α (t i ) will be the firt and econd derivative of x α(t i ), repectively (the real velocitie and the real acceleration). Auming uncorrelated noie, the variance of V α (t i ) and A α (t i ) are given by σ and σ V A ( t ) = i σ X ti dt 6σ X ti ( ti)= 4 dt ( ) + ( ) ( ) ( 4) Thu, the noie will be trongly amplified by the differentiation, and therefore the velocitie mut be weaker moothed than the acceleration and the poition weaker than the velocitie. Figure b and c, the lateral poition and longitudinal velocitie and acceleration of a ample trajectory of the US-101 data et are plotted a original data and for different moothing width. The poition moothing width T x i critical, becaue the lane change duration i quite enitive to it. A viible in the plot, a large T x will ignificantly mear out the trajectory leading to larger lane-change duration. To reolve the iue with the preferred velocitie, the moothing of the velocitie hould be trong enough o that the moothed velocitie

4 Thiemann, Treiber, and Keting 93 Acc. Variance [m / 4 ] Smoothing Kernel Width [] Acceleration [m/ ] Prob. Denity [/m] Prob. Denity [ /m] Lateral Poition [m] (d) (e) Acceleration [m/ ] Velocity [m/] Velocity [m/] (c) Time [] (f) FIGURE Effect of applied trajectory moothing: dependency of acceleration variance on moothing kernel width, acceleration ditribution in prototype data et, and (c) velocity ditribution in 7:50 8:05 a.m. data file of US-101 data et. Sample trajectory of US-101 data et with different applied moothing kernel width: (d) lateral poition, (e) longitudinal velocity, and ( f ) acceleration. no longer follow the trend of thi emiquantization. However, the moothing hould be a weak a poible becaue the velocity moothing width alo quantitatively influence ome reult. The moothing time T x = 05. T v = 1 and T a = 4 ( 5) were choen. The effect of thi moothing on the acceleration ditribution of the prototype data et and the velocity ditribution of the two later data et can be een in Figure d through f. RESULTS Mot empirical traffic tate data are gathered by tationary loop detector that can meaure quantitie at different time but at a ingle location only. Thee meaurement device are therefore capable of meauring temporal quantitie but not patial quantitie. However, ince both patial and temporal quantitie are important in traffic cience, it i common practice to derive the patial quantitie from temporal meaurement by uing ome conervation aumption (e.g., contant vehicle velocitie within a certain period). Modern trajectory data like the NGSIM recording provide enough data to enable a validation of thee practice. The following decribe the analyi proce to obtain patial information from temporal data, and vice vera, and it accuracy i checked for three example: the microcopic fundamental diagram and the ditribution of the time gap and time to colliion. Later, lane change in the NGSIM data are invetigated. All analyi ue the moothed data et obtained by the moothing

5 94 Tranportation Reearch Record 088 method introduced and motivated earlier; all reference to any NGSIM data et are to be undertood a reference to the moothed data et. Spatial and Temporal Quantitie from Momentary and Stationary Meaurement The two meaurement type to be compared are the traditional tationary loop detector, which i ingular in pace but continuou in time, and an aerial photograph, which i continuou in pace but ingular in time. The baic idea of the analyi i to place virtual loop detector into the trajectory data. Thee would correpond to line parallel to the time axi in a pace time plot, wherea line parallel to the pace axi correpond to momentary naphot (virtual photograph) of the meaurement area (Figure 3). Wherever thoe line interect, both tationary and momentary meaurement are available for comparion. To maximize the amount of data available for comparion, the following algorithm wa applied to the data: For every 10th data point of each trajectory, the patial leader and the temporal leader are determined. The patial leader α 1 i the vehicle currently driving ahead of the vehicle α, and the temporal leader i the vehicle that mot recently paed the actual poition of vehicle α. (For implicity, the temporal leader i denoted by α 1 a well.) The firt information i available only to momentary meaurement, wherea the econd i available only to tationary meaurement. Auming double loop detector for the tationary meaurement, the paage time t α and t α 1 of vehicle α and α 1 and their velocitie at the time of paing the detector are available: v α (t α ), v α 1 (t α 1 ). Furthermore, the length of the leading vehicle l α 1 i known, a are the poition (front bumper) at the time of paing the detector: x α (t α ) = x α 1 (t α 1 ). From the momentary meaurement at time t α one obtain the poition of the two vehicle, x α (t α ) and x α 1 (t α ), a well a the length of the leading vehicle l α 1. Auming that two conecutive photograph are taken, one can alo determine the velocitie v α (t α ) and v α 1 (t α ). From thi momentary meaurement, the following patial quantitie can be calculated: = patial gap t x t x t l α α α 1 α α α α 1 6 () approaching rate Δv t v t v t Auming contant velocitie within the time interval Δt α = t α t α 1, one can etimate the ame quantitie from data collected by a tationary detector: et ( t ) = v ( t ) i Δ t l () α α α 1 α 1 α α 1 8 Δv t v t v t Furthermore, the time gap T defined by the gap related to the actual velocity, v, i a crucial quantity for the afety and capacity of traffic flow. From the time interval between two vehicle paing the tationary detector, Δt α = t α t α 1, one can etimate the time gap while paing the detector: et, pt T t Δt Thi definition aume contant velocity of the leading vehicle in the time interval Δt α. The real time gap, however, would be obtained by meauring the time where the rear bumper of the leading vehicle paed the detector: Tα( tα) = tα t with t uch that x t l x t Both quantitie are illutrated in Figure 3b. Alternatively, one can etimate the time gap T α et,mom from data collected by a momentary detector, again auming contant velocitie of the vehicle: T et = α α α α α 1 α 1 9 α α α α 1 α 1 α α 11 et, mom α = = ( tα) = v lα 1 v t tα t α α 1( α) = α α α α α 1 α 7 α 1 α 1 () ( 10) ( 1) Stationary Detector t α α 1 Time [] Momentary Detector T α et α T α et Longitudinal Poition [m] X FIGURE 3 Virtual detector in pace time plot; tationary detector (loop detector) correpond to line parallel to time axi and momentary detector (aerial photograph) correpond to line parallel to the pace axi. Illutration et,pt et,pt of time gap T according to Equation 10, auming contant velocitie and real time gap T. (T i etimate from et,mom tationary meaurement, wherea T a defined in Equation 1 i etimate from momentary meaurement.)

6 Thiemann, Treiber, and Keting 95 Data Preparation In total, 184,171 data point in the prototype data et and 7,904 in the two other data et were invetigated. Data point that were too cloe to the downtream boundary needed to be dicarded ince no patial leader could be identified. Furthermore, data point that were cloer than 3 to a lane-changing event were ignored, leaving 146,13 data point from the prototype data et and 675,660 from the I-80 and US-101 data et. Becaue of tracking or vehicle dimenion detection error, ome patial and time gap are negative or very mall. A mall patial gap α lead to a very large invere time to colliion τ α, which would dominate any higher-order moment of the τ α ditribution. Thu, the data were filtered uch that α 1 m and T α 0.1 hold for every data point. Thi filter removed a further 3,070 data point (.1%) from the prototype data et extract and 11,755 (1.7%) from the extract of the two later data et. Microcopic Fundamental Diagram and Stopped Traffic From the decribed patiotemporal meaurement, one can derive the invere of the pace headway, (Δx α ) 1 = (x α 1 x α ) 1, and the invere of the time headway, (Δt α ) 1. Thee quantitie are more intuitively decribed a microcopic denity and microcopic flow, repectively, and are referred to by thee name throughout thi ection. For the prototype data et and the combined other two data et, the ditribution of velocity and microcopic denity are plotted in Figure 4a and 4b. The prototype data et mainly feature free traffic and ome bound traffic, wherea the two later data et feature only bound and jammed traffic. By plotting microcopic flow veru microcopic denity for all three data et, the fundamental diagram wa plotted (Figure 4c). The free flow part of the diagram i completely provided by the prototype data et, and the bound and jammed part i almot completely provided by the I-80 and US-101 data et. Notice that in contrat to the prototype data et, the later two et exhibit tripe correponding to the preferred velocitie a een in Figure 1, which are much more prominent when applying the ame procedure to the original, unmoothed data. From the rich amount of data in the jammed traffic regime, it i alo poible to determine the average headway of tanding vehicle. All data point with velocitie v α < 0.05 m/ were extracted, and the ditribution of Δx α i plotted in Figure 4f. The mode i at approximately 7 m for car and 8 m for truck (with a maller econd peak at 14 m). However, the ditribution i right-kewed, o that the mean value are a little higher: 8.3 m for car and 9.7 m for truck. Note that for principal reaon, thi ditribution cannot be obtained from tationary detector data. Time Gap Ditribution Conider now the time gap a defined in Equation 10, 11, and 1. Figure 5 plot the time gap ditribution in three different traffic regime: free traffic (v >. m/), jammed traffic (v < 15 m/), and bound traffic (intermediate velocitie). Furthermore, in every plot, the real time gap T α defined by Equation 11 a obtained from the trajectorie i compared to the etimated time gap from momentary meaurement T α et,mom (Equation 1). Firt note the remarkable Microcopic Vehicle Flow [veh/h] Probability Denity [km*h/veh ] Velocity [km/h] Probability Denity [km /veh/h] Microcopic Vehicle Denity [veh/km] Microcopic Vehicle Denity [veh/km] Microcopic Vehicle Flow [veh/h] Probability Denity [km*h/veh ] Probability Denity [1/m] Microcopic Vehicle Denity [veh/km] (c) Headway [m] (d) FIGURE 4 Probability denity of two-dimenional ditribution of microcopic denity (x 1 x ) 1 veru velocity v in prototype data et (upper left) and two later I-80 and US-101 data et. Probability denity of (c) microcopic denity veru microcopic flow T 1 and of (d) ditribution of headway x 1 x in topped traffic (v < 0.05 m/); mean value i 8.3 m for car and 0.0 m for truck, repectively.

7 96 Tranportation Reearch Record 088 Probability Denity [1/] Probability Denity [1/m] Time Gap [] Spatial Gap [m] (d) Conditional Probability Denity p(t v) [1/] Probability Denity [1/] Approaching Rate Δv [m/] Probability Denity [1/] Velocity v [m/] Time Gap [] Time Gap T [] (e) Conditional Probability Denity for Fixed Δv [1/] Time Gap [] (c) Time Gap [] (f) FIGURE 5 Ditribution: time gap T in jammed traffic, time gap T in bound traffic, (c) time gap T in free traffic, (d) patial gap in jammed traffic, (e) ditribution of time gap for different given velocitie v, and ( f ) ditribution of time gap for different given approaching rate v. White line how the mean value for each row of plot, that i, mean of time gap for different value of v or v. indifference of the ditribution to the meaurement method. For comparion, the patial gap ditribution in jammed traffic wa plotted (Figure 5b), which the tationary meaurement hift to larger value. In the other two traffic regime the patial gap ditribution agree very well. The mode of the time gap ditribution hift from approximately 1.5 in jammed traffic to 1 in free traffic. Thi effect i hown in Figure 5d. The mean time gap i.6 in jammed traffic, 1.9 in bound traffic, and.0 in free traffic. Figure 5f how another dependency of the time gap: although data become pare toward larger value, there i a ignificant tendency toward larger time gap if the velocity difference to the leading vehicle i large (regardle of whether approaching the vehicle or falling behind). In addition to comparing time gap meaured by tationary detector to time gap meaured by momentary detector, there are other way to determine the time gap with a tationary detector. The real time gap i the time between the leader rear bumper and the following front bumper paing the detector (Equation 11). However, if detector produce only paage time and vehicle length and velocitie, one need to etimate the time gap from the paage by auming contant velocity of the leader vehicle while paing the detector (Equation 10). Thi error i very mall in mot cae: only 10% of the ample data point had an error in the etimate from paage time T α et,pt that exceeded 10% of the real time gap T α. Time to Colliion Another relevant quantity i the time to colliion (TTC), which erve a a afety meaure for traffic ituation becaue it tate the time

8 Thiemann, Treiber, and Keting 97 1 FIGURE 6 Ditribution of invere time to colliion : in prototype data et and in two later data et et compared with etimated time to colliion ( ) 1 obtained from tationary meaurement. Upper figure how 1 et both ditribution, and lower figure how ditribution of meaurement error ( ) 1 1. left until the vehicle will crah into it leader unle at leat one of the driver change peed (16, 17). The TTC a a patial quantity i defined by Equation 6 and 7 a = α α τ α t α Δvα tα The TTC can alo be etimated from tationary (temporal) meaurement in Equation 8 and 9: et et α α τ α ( t α )= vα tα vα 1 tα 1 The impact of the contant-velocity aumption ued to derive the TTC τ α et from tationary meaurement i invetigated next. Since the TTC diverge for Δv α = 0, it i more convenient to dicu the TTC in term of it invere, τ α = 1 Δv α α ( t ) ( t ) ( 13) ( 14) Figure 6a and 6c plot the ditribution of the invere TTC in the prototype data et, and Figure 6b and 6d plot the ditribution in the two later data et. In contrat to the patial and time gap ditribution, the invere TTC ditribution differ ignificantly between the two meaurement method. The invere TTC i enitive to error in the patial gap, epecially when the gap i mall. Therefore, invere TTC value with abolute value larger than 1 were ignored in computing tatitical propertie of the ditribution. In thi way, 0.59% of all data point were ignored. The mean of the abolute error Δτ α 1 : = (τ α et ) 1 τ α 1 i in the prototype data et and in the two later data et. The ame can be oberved when plitting the data from all data et into traffic regime. The mean error i in jammed traffic, in bound traffic, and 0.01 in free traffic. The variance of the error i tronget in jammed traffic (0.036), wherea it i in bound traffic and i in free traffic. Statitical propertie of the invere TTC ditribution have been collected into Table 1. In the mean, variance, and kewne column, the top value i obtained from momentary meaurement (the real value), wherea the bottom TABLE 1 Statitical Propertie of Invere TTC Ditribution Data Set Mean Variance Skewne Sign Change (%) Prototype I-80 US Jammed traffic Bound traffic Free traffic

9 98 Tranportation Reearch Record 088 value i obtained from tationary meaurement (the etimated value). In the ign change column, the top value tate the amount of data point for which the tationary meaurement determine a poitive time-to-colliion wherea the momentary meaurement determine a negative value. The bottom value give the amount of data point for which the ign change i the other way around. The jammed, bound, and free traffic data et are combined from the prototype and the two later NGSIM data et. A data point wa aigned to jammed traffic if the vehicle velocity wa below 15 m/, to free traffic if v α >. m/, and to bound traffic otherwie. The kewne i conitently hifted toward higher value by the tationary meaurement. Thi i viible in the plot a well. In view of the application of the TTC a a afety meaure, it i critical that tationary meaurement conitently decreae the probability of meauring a large poitive invere-time-to-colliion value that correpond to a mall poitive τ α, indicating a dangerou traffic ituation. For example, in free traffic (Figure 7), the fraction of poitive TTC value below 5 (0.8% of the data point) that i conidered critical (16, 17) i underetimated by the tationary meaurement by about a factor of two. Thu, tationary meaurement tend to euphemize the danger of colliion. Lane Change In addition to the ability to compare tationary and momentary meaurement, the NGSIM trajectory data et provide a good bai to invetigate lane change. To determine the lane change duration, all lane change in the NGSIM data were collected. However, the proceed video data upplied with the NGSIM data et how that ometime the tracking algorithm accidentally miplaced a vehicle acro the lane boundary and back after a few time tep. Alo, ometime driver aborted a lane change or quickly croed two lane. To examine only real and normal ingle-lane lane change, all lane change were filtered out that were cloer than a certain threhold τ th to another lane change, choen to be τ th = 5. Alo orted out were all lane change that did not involve one of the four leftmot lane, to reduce the effect of the on and off ramp on the lane change analyi. With λ α (t) denoting the lane ued by vehicle α at time t, a lanechanging event occur at time t 1c if λ α (t 1c ) λ α (t 1c + t) (where t i the time interval between two conecutive data point of a trajectory). For each lane-changing event, a 0- environment of the trajectory with time, longitudinal, and lateral poition relative to the lane-changing event wa extracted: relative time τ= t t ( 15) lc = ( + ) relative longitudinal poition ξα τ xα τ tl c xα t lc ( 16) relative lateral poition ηα( τ)= yα( τ+ tlc ) yα t lc ( 17) Thu a plot can be produced of the conditional probability denity p(η τ) that a vehicle i at a relative lateral poition η at a certain time τ relative to the lane-changing event time (Figure 8a). From thi, the lane-change duration can be roughly etimated at approximately 5 to 6 by looking at the curvature of the two mode value η +(τ) = argmax η>0 {p(η τ)} and η (τ) = argmax η<0 {p(η τ)}. Thi procedure i imilar to the approach by Toledo and Zohar (7), in which the lane-change tart and end time of each trajectory were determined by looking at the curvature of the lateral poition y α (t). However, finding the correct point in the curvature may be omewhat (c) 1 FIGURE 7 Ditribution of invere time to colliion of all data et compared to etimated time to colliion ( et ) 1 obtained from tationary meaurement: jammed traffic, bound traffic, and (c) free traffic.

10 Thiemann, Treiber, and Keting 99 Rel. Lateral Poition η [m] Conditional Probability Denity p(η τ) [1/m] Relative Time τ [] Conditional Probability Denity [/m] Probability Denity Velocity Advantage [m/] Lane Change Duration [] Relative Time τ [] (c) FIGURE 8 Lane change: conditional probability p( ) of finding vehicle on lateral poition relative to lane boundary at given time relative to lane-changing event time, ditribution of lane-change duration T lc, according to Equation 0; mean lane-change duration i T lc ( ), and (c) conditional probability denity of velocity difference between leader on detination lane and leader on ource lane for fixed time relative to lanechanging event (white line how mean value). arbitrary, and thu the following conider a more well-defined way to meaure a lower bound of the lane-change duration. The NGSIM vehicle detection algorithm detect not only the vehicle poition but alo it length l α and width w α. Since the lane aignment algorithm work uch that each data point i placed into the lane in which it midpoint front-bumper poition (x α, y α ) lie, it i poible to determine the time at which a lane-changing vehicle firt intruded into the detination lane and the time at which it completely left the ource lane. Given the lane-changing event time t 1c and the relative time and poition a defined in Equation 15 through 17, the relative tart time τ and end time τ e of the lane change may be defined a follow (a higher lane index λ α correpond to a larger lateral poition η α ): τ τ e ττ< 0 and ηα( τ) + wα < 0 max = if λα ( tlc ) < λα tlc + Δt max{ ττ< 0 and ηα( τ) wα > 0 otherwie ττ> 0 and ηα( τ) wα > 0 min = if λα ( tlc ) < λα tlc + Δt min{ ττ> 0 and ηα( τ) + wα > 0 otherwie Then the lane change duration i obtained trivially from Tlc = e ( 18) ( 19) τ τ ( 0) In total, 1,31 lane change were invetigated, 1,105 of which were uitable to calculate T lc according to Equation 0. In the remaining 16 cae, either τ or τ e were undefined becaue the correponding condition wa not fulfilled for any τ [ 10, 10] within the 0- environment around the lane change. Thi can be attributed to vehicle dimenion detection error or vehicle tracking error, both leading to a trajectory where the vehicle drive on the lane boundary for ome time. Figure 8b how the ditribution of the lane-change duration of the examined lane change. The figure how that mot lane change take about 3 (mode value of the ditribution), a value found valid for German highway in 1978 (18), but which i ubtantially different from the one obtained by rule of thumb from the conditional probability denity p(η τ). The mean and tandard variation of the ditribution are T lc = 4. 01±. 31 ( 1) However, one hould be aware that Equation 19 meaure the time pan where the vehicle occupie two lane, which only can be taken a a lower bound of the real lane-change duration. Including the preparation and poible potproceing of a lane change, a value of 5 to 6 may appear realitic. Since the real beginning of a lane change the deciion for making the lane change i impoible to meaure, and the phyical beginning the moment at which the driver tart to turn the wheel i very difficult if not impoible to meaure, the propoed definition i a good etimator for the lane-change

11 100 Tranportation Reearch Record 088 duration, becaue it ue well-defined and eaily meaurable quantitie. Figure 8c how the conditional probability denity of the velocity difference between the leader on the detination lane and the leader on the ource lane for different fixed time relative to the lane-changing event. A indicated by the white line, the mean value rie before the lane change by approximately 1 m/. Thi indicate that driver perceive a velocity advantage on the detination lane before performing the lane-changing maneuver and take anticipatory action. DISCUSSION AND FUTURE RESEARCH The availability of the NGSIM data et purred coniderable reearch activity, particularly for lane changing, where larger-cale empirical invetigation are poible for the firt time. Mot reearcher have ued only the poitional information, which allow, for example, invetigation of the lane-changing rate, the duration of lane change, the gap acceptance behavior, or the propagation velocity of longitudinal denity wave. The full potential of the data, that i, uing the poitional information together with that for velocity and acceleration, ha not been tapped. Thi may be becaue the velocity and acceleration information cannot be ued directly ince the noie of the poitional information i greatly increaed by the neceary numerical differentiation. Thi paper developed a filter to extract more realitic velocity and acceleration information from the poitional data. Since the trajectorie are comparatively hort, the boundary region were included in the filtered output by reducing the width of the neceary moothing operation near the boundary. Thi implie determining the mot efficient order of the moothing and differentiation operation of the filter ince they no longer commute, and a wrong order may even lead to a ytematic bia. It i inherently difficult to determine the optimal filter parameter that eliminate mot of the noie while retaining the real information. Thi i particularly crucial for mean-reverting quantitie uch a the acceleration, where large moothing time interval will eventually uppre the whole information. Clearly, further reearch i neceary to develop more ophiticated, poibly nonlinear, filter. The velocity and acceleration information of the trajectorie can be ued in many way. Thi work invetigated the ytematic error in determining patial quantitie from temporal information, and vice vera. The background i that patial quantitie, uch a the gap to the leading vehicle, the denity, or the time to colliion, uually are etimated by ingle-vehicle data from tationary detector, that i, by uing temporal information. Ue of virtual tationary detector that are fed with the trajectory data, and imulation of the etimation procedure, allowed quantitative determination of the reulting etimation error. In addition to the well-known underetimation of the real denity of congeted traffic, the percentage of critical value of time-to-colliion wa found to be underetimated by a factor of and more when etimated from ingle-vehicle data. Thi i relevant for afety-related application. Another application field i empirical tet and parameter calibration for car-following and lane-changing model. Thi work howed that before a dicretionary lane change, there i a noiy and mall, but ignificant, velocity difference in favor of the target lane. From thi it can be concluded that lane-changing deciion are baed not only on gap and velocitie but alo on velocity difference and, poibly, on acceleration (19). More generally, the trajectory data allow one to empirically invetigate the trategical and tactical action for preparing or facilitating a lane change (0). Apart from the action of the lane-changing driver, thi include the action of the other driver involved, uch a cooperative action of the follower on the target lane to allow zip-like merging. Thi i relevant for microcopic imulation oftware ince it turned out to be notoriouly difficult to model realitic lane change, particularly in the cae of mandatory change in congeted traffic. The acceleration information in the data can be ued to invetigate to what extent the local traffic environment (coniting, e.g., of the next-nearet and further leading vehicle) influence longitudinal driving behavior (0). For example, it ha been propoed that driving tyle i influenced by local velocity variance a determined from few leading vehicle (1). Finally, the velocity and acceleration information can be ued to determine the influence of traffic congetion on the fuel conumption and emiion (15). Since reliable characteritic map are available for the intantaneou fuel conumption and emiion rate of variou pollutant a a function of velocity and acceleration, thee quantitie can now be etimated, for real ituation, with unprecedented accuracy. ACKNOWLEDGMENT The author thank FHWA for providing the NGSIM trajectory data ued in thi tudy. REFERENCES 1. NGSIM: Next Generation Simulation. FHWA, U.S. Department of Tranportation. Acceed May 5, Skabardoni, A. Etimating and Validating Model of Microcopic Driver Behavior with Video Data. Technical report. California Partner for Advanced Tranit and Highway, Lu, X.-Y., and A. Skabardoni. Freeway Traffic Shockwave Analyi: Exploring NGSIM Trajectory Data. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Roe, R. P., and J. M. Ulerio. Analyi of Four Weaving Section: Implication for Modeling. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Zhang, L., and V. G. Kovvali. Freeway Gap Acceptance Behavior Baed on Vehicle Trajectory Analyi. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Gowami, V., and G. H. Bham. Gap Acceptance Behavior in Mandatory Lane Change Under Congeted and Uncongeted Traffic on a Multilane Freeway. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Toledo, T., and D. Zohar. Modeling Duration of Lane Change. In Tranportation Reearch Record: Journal of the Tranportation Reearch Board, No. 1999, Tranportation Reearch Board of the National Academie, Wahington, D.C., 007, pp Choudhury, C. F., M. E. Ben-Akiva, T. Toledo, G. Lee, and A. Rao. Modeling Cooperative Lane Changing and Forced Merging Behavior. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Leclercq, L., N. Chiabaut, J. A. Laval, and C. Buion. Relaxation Phenomenon After Changing Lane: Experimental Validation with NGSIM Dataet. In Tranportation Reearch Record: Journal of the Tranportation Reearch Board, No. 1999, Tranportation Reearch Board of the National Academie, Wahington, D.C., 007, pp Vu, T. T., R. P. Roe, J. M. Ulerio, and E. S. Praa. Simulation of a Weaving Section. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Jin, W.-L., and L. Li. A Study of Firt-In Firt-Out Violation in Freeway Traffic. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., 007.

12 Thiemann, Treiber, and Keting Webter, N. A., T. Suzuki, E. Chung, and M. Kuwahara. Tactical Driver Lane Change Model Uing Forward Search. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Alecandru, C., and S. Ihak. Accounting for Random Driving Behavior and Nonlinearity of Backward Wave Speed in the Cell Tranmiion Model. Preented at 86th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Keting, A., and M. Treiber. Calibrating Car-Following Model Uing Trajectory Data: Methodological Study. In Tranportation Reearch Record: Journal of the Tranportation Reearch Board, No. 088, Tranportation Reearch Board of the National Academie, Wahington, D.C., 008, pp Treiber, M., A. Keting, and C. Thiemann. How Much Doe Traffic Congetion Increae Fuel Conumption and Emiion? Applying a Fuel Conumption Model to NGSIM Trajectory Data. Preented at 87th Annual Meeting of the Tranportation Reearch Board, Wahington, D.C., Hirt, S., and R. Graham. The Format and Preentation of Colliion Warning. In Ergonomic and Safety of Intelligent Driver Interface (Y. Noy, ed.), Lawrence Erlbaum Aociate, Mahwah, N.J., Minderhoud, M. M., and P. H. L. Bovy. Extended Time-to-Colliion Meaure for Road Traffic Safety Aement. Accident Analyi and Prevention, Vol. 33, 001, pp Sparmann, U. Spurwechelvorgänge auf zweipurigen BAB-Richtungfahrbahnen. Forchung Straßenbau und Straßenverkehrtechnik, Vol. 63, Treiber, M., A. Keting, and D. Helbing. Delay, Inaccuracie and Anticipation in Microcopic Traffic Model. Phyica A, Vol. 360, 006, pp Keting, A., M. Treiber, and D. Helbing. General Lane-Changing Model MOBIL for Car-Following Model. In Tranportation Reearch Record: Journal of the Tranportation Reearch Board, No. 1999, Tranportation Reearch Board of the National Academie, Wahington, D.C., 007, pp Treiber, M., A. Keting, and D. Helbing. Undertanding Widely Scattered Traffic Flow, the Capacity Drop, and Platoon a Effect of Variance- Driven Time Gap. Phyical Review E, Vol. 74, 006, p The Traffic Flow Theory and Characteritic Committee ponored publication of thi paper.