Arterial Lane Selection Model

Moshe Ben-Akiva,

Charisma Choudhury, Varun Ramanujam,

Tomer Toledo

ITS ProgramJanuary 21, 2007

2

Outline

1. Introduction and dataset2. Estimation

2.1 Lane choice at intersection2.2 Lane changing within section

3. Validation3.1 Methodology3.2 Data3.2 Results

3

1. Introduction and dataset

4

Introduction

• Objective – Develop a lane changing model for urban arterials

• Tasks– Specify lane changing model and estimate using

NGSIM trajectory data – Implement and verify in MITSIMLab– Calibrate and validate using aggregate data

5

Dataset• Site

– Lankershim Blvd. near Universal Studios CA

• Coverage– Four signalized intersections

• Only destinations available for the fourth intersection

1

2

3

4

6

Dataset (cont.)

1 2 3 4

7

Summary statistics*

* Source: NGSIM Lankershim Data Analysis (2006)

• Number of Vehicles: 2443 – 703 turning vehicles

• 32 mins of data at 1/10 th second resolution• Number of Observations: 1,607,321• Number of Lane-changes: 2284 • Average Speed: 12.4 m/sec• Average Travel Time

– Northbound: 67.3 sec– Southbound: 60.3 sec

8

Models

• Lane choice at intersection– Target lane choice– Immediate lane choice

• Lane changing within section– Target lane choice– Gap acceptance– Execution of lane change

9

2.1 Lane choice at intersection

10

? 1? 2? 3? 4

Lane choice at intersection

11

2.1.1 Summary statistics

• 703 turning vehicles– 269 turn to the naturally

connecting lane

– 435 later change to different lanes within

the section

• 29 OD pairs

• Sections traversed– 1 section: 139

– 2 sections: 171

– 3 sections: 393

0%

25%

50%

75%

100%

1 2 3

Left

Right

Straight

Turning Directions of Vehicles in Lankershim, CA

1 2 3 Section traversed

12

2.1.2 Model structure

1 2 3 4

1 2 3 4

Target Lane

Immediate Lane

( | , )nP l

( | , )nP i l

( | , ) ( | , ) ( | , )n

n n nl C

P i P i l P l

13

• Probability of driver n selecting lane l as target lane

2.1.2 Model structure (cont.)

exp( ( ) )( | , )

exp( ( ) )

T

T

n

l lnt n n

n j jnt n n

j C

XP l

X

,

lnt n

n

n

l

where

X attributes of lane l for vehicle n at time t, can be function of

individual specific plan-ahead distance

cofficients

individual specific aggressiveness

coefficient of individual s

pecific random effect for lane l

14

????

????

Driver classes

Myopic driver

(considers immediate section only)

Driver who plans-ahead

(considers more than one section ahead)

15

• Probability of driver n selecting lane i as immediate lane given that the target lane is l

2.1.2 Model structure (cont.)

exp( )( | , )

exp( )

T

T

n

i int n

n k knt n

k C

XP i l

X

,int

i

where

X attributes of lane i for vehicle n at time t

coefficient of individual specific random effect for lane i

16

2.1.2 Model structure (cont.)• Probability of driver n selecting lane i

• Modeling unobserved aggressiveness and planning capability (‘plan-ahead’)

• Probability of class membership estimated with other model parameters

• Aggressiveness– Individual specific random term

( | , ) ( | , ) ( | , )n

n n nl C

P i P i l P l

( ) ( | , ) ( ) ( )n nP i P i p f d

1 1or (1 )

n ~N(0,1)

1

11n

myopic drivers (consider immediate section only) p

drivers who plan more than one section ahead

17

2.1.3 Variables: Target lane choice

• Path plan variables– Distance to intended downstream turn

– Number of lane changes • Lane attributes

– Queue length and queue discharge rate

– Average speeds

• Driving style and capabilities– Individual driver/vehicle characteristics

• planning capability (‘plan-ahead’)

• aggressiveness

18

2.1.3 Variables: Immediate lane choice

• Current position of the driver– Departure from the naturally connecting lane

• Neighborhood variables– Presence of other vehicles and their actions

• Driving style and capabilities– Individual driver/vehicle characteristics

• aggressiveness

• performance capabilities of the vehicle

(e.g. required turning radius)

19

2.1.4 Estimation resultsInitial Log-likelihood -2797.9Final Log-likelihood -2115.8Number of Parameters 20Number of Observations 703Variable Parameter valuet-Stat

Lane 2 constant -0.837 -3.64Lane 3 constant 1.30 7.62Lane 4 constant 3.25 8.16Anticipated delay (second) -0.477 -0.56Lanes away from turning lane (myopic) coefficient-myopic drivers -0.0240 -0.63

constant-myopic drivers 1.43 0.83heterogenity coefficient -myopic drivers 1.53 0.75

Lanes away from turning lane (with plan-ahead) coefficient-drivers who plan-ahead -4.08 -1.98constant-drivers who plan-ahead 2.05 3.01

heterogenity coefficient -drivers who plan-ahead 0.466 0.74Expected maximum utility from immediate lane* 0.915 7.22

Target Lane

1 2* E( ( ))

,

ln

ln

n

i Iln ln ln ln

iln

Expected maximum utility derived from lane l

utility of immediate lane i for individual n given target lane is l

C 1,2,...,i,...,I

EMU max U ,U ,...,U ,...,U

where EMU

U

20

2.1.4 Estimation results (cont.)

Lanes away from connecting lane : coefficient -1.01 -1.19constant 0.691 1.94

heterogenity coefficient 1.96 3.48Target lane dummy 3.16 4.54Lanes away from target lane : coefficient -4.42 -3.00

constant 2.12 2.14heterogenity coefficient 0.0904 0.36

Lateral conflict dummy -1.76 -9.63

Driver population with >1 section plan-ahead (%) 18.3 2.07Driver Class

Immediate Lane

21

2.1.4 Estimation results (cont.)

• Utility associated with target lane l for vehicle n

1

2

1

l

l

n

l

n

n

where

constant for lane l

q anticipated delay function in lane l for myopic drivers

q anticipated delay function in lane l for drivers who plan-ahead

if the driver plans-ahead beyond immedi

1

2

ln

ln

l

ate section

e lanes away from turning lane for myopic drivers

e lanes away from turning lane for drivers who plan-ahead (consider path-plan beyond current section)

EMU expected maximum utility der

ived from selecting lane l as immediate lane

1 2 1

2

0.0240.477 ( )(1 ) ( )( ) ( )(1 )

1.43 1.53

4.08( ) 0.915( )

2.05 0.466

l ll l ln n n n nn n

n

l ln n

n

U q q ev

e EMUv

22

2.1.4 Estimation results (cont.)

• Utility associated with immediate lane i for vehicle n given his target lane is l

1.01( ) 3.16( 0)

0.691 1.96

4.42( ) 1.76

2.12 .0904

1

i i in n n

n

i in n

n

in

in n

in

U c lv

lv

where

c lanes away from connecting lane

l lanes away from target lane l, l C

if maneuver to lane i is obstructed by

adjacent vehicle

23

1 1 1 2 1 2 2

11,2

1 exp( )

/ , /

kl

nt klnt

l l l l l l lnt nt n nt nt nt n

klnt

kint

kint

q kq

q d r q q d r

where

q anticipated delay in lane l considering k sections ahead

d queue length in lane i in section k at time t (vehicles)

r

average queue discharge rate of lane i in section k (vehicles/sec)

Anticipated Delay

• Delay associated with lane l for driver class k:

Effect of anticipated delay on the utility of lane

0.4

0.8

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Anticipated delay (sec)

Uti

lity

of

lan

e

2.1.5 Interpretation of results

24

2.1.5 Interpretation of results (cont.)

• Next section- Lane 4 does not continue

1234

0

0.25

0.5

0.75

1 2 3 4

Lane

Pro

bab

ilit

y

Plan-ahead = 1 section Plan-ahead > 1 section

Effect of path-plan in target lane choice

25

2.1.5 Interpretation of results (cont.)

Heterogeneity in Immediate Lane Choice(CL=2,TL=1)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4

Immediate Lane

Pro

ba

bili

ty

Normal Driver

Aggressive Driver

• Aggressive Driver-Lower inertia to stay in current lane

26

2.1.6 Model selection

Statistic Base

Model

New

Model

Log likelihood value -2120.4 -2115.8

Number of parameters (K) 19 20

Akaike information criteria (AIC)* -2139.3 -2135.8

Bayesian information criteria (BIC)* -2182.5 -2181.4

*AIC=LL-K, BIC=LL -K/2*ln*(N)

• Compared with a single level lane-choice model

• Significant improvement in goodness-of-fit

27

2.2 Lane-changing within section

28

2.2.1 Summary statistics

– No. of vehicles in sampled dataset : 401 • NB: 160, SB: 241

• 150 vehicles turning vehicles

• 125 vehicles entering through intermediate intersections

– Total no. of lane changes in sample : 249 • 0.621 per vehicle

– Lane changes by turning vehicles:104• 0.693 per vehicle

• Majority (80.8%) of these lane changes occur in the last section before turn

29

2.2.2 Model structure

Lane 2

Lane 1

Target Lane

Execution

Gap Acceptance

Reject Gap

No Change

AcceptGap

Change Right

No Change

No Change

RejectGap

No Change

AcceptGap

Change Left

No Change

RejectGap

No Change

Accept Gap

Change Left

No Change

Lane 2 Lane 3 Lane 4

Lane 1

Lane 2

Lane 3

Lane 4

30

2.2.2 Model structure: Target lane choice

• Probability of driver n selecting target lane i at time t

• The desired lane shift (left, current or right) is implied by the target lane choice

exp( )( | )

exp( )

T

T

TL i int n

nt TL j jnt n

availablelanes j

XP i

X

,int

TL

n

i

where

X attributes of lane i for vehicle n at time t

cofficients

individual specific aggressiveness

coefficient of individual specific random effect for lane i

31

2.2.2 Model structure: Gap acceptance

• Evaluate adjacent lead and lag gaps • Accept gap if

Available gap >= Critical gap

Subjectvehicle

Leadvehicle

Lagvehicle

Adjacent gap

Lead gapLag gap

Traffic direction

32

2.2.2 Model structure: Gap acceptance (cont.)

• Critical gaps in the adjacent lane in the direction of target lane i

• Assuming is normally distributed:

, exp( )

,

{ , }

Tig cr g ig ig ignt nt n nt

ignt

G X

where

X attributes of gap g in the adjacent lane in the direction of target lane i

g lead lag

ignt

, ,

2

, , ,

ln ln

~ (0, ), ~ (

T T

i lead i lead cr i lag i lag crnt n nt nt n nt nt n

iLead Lead iLead iLead iLag Lag iLag iLagnt nt n nt nt n

Lead Lag

iLead iLagnt Lead nt

P d i P G G i P G G i

G X G X =

N N

20, )Lag

33

2.2.3 Model structure: Execution of lane change

• Critical gaps unacceptable– No execution

• Critical gaps acceptable– Decision to change in current time step or not

– Modeled as a binary logit

nt n l l lnt n

lnt

l

l

1P ( l|d ,i, )

1+exp(- X - )

where,

X var iables associated with the lane changing by n at time t

cofficients

coefficient of individual specific random effect for lane i

34

2.2.2 Model structure (cont.)

• Probability of driver n executing lane action l

• Modeling unobserved aggressiveness

• Aggressiveness– Individual specific random term

( | ) ( | , ) ( | , , )nt

nt n nt nt nti C

P P i P d i P l d i

l

( ) ( | ) ( )nt ntP P f d

l l

n ~N(0,1)

35

2.2.3 Variables: Target lane choice

• Path plan variables– Distance to intended downstream turn– Number of lane changes

• Neighborhood variables– Immediate surroundings (e.g. leaders)

• Lane attributes– Speed, density, queue lengths

• Driving effort– Lanes away from current lane, inertia

36

2.2.3 Variables: Gap acceptance

• Critical gaps may depend on– Lead and lag vehicle relative speeds and

spacing

• May differ from freeways– Different speed ranges – Impact of traffic lights – Impatience

37

2.2.3 Variables: Execution of lane change

• May depend on– Speed of the subject vehicle– Remaining distance to the turn– Density

38

2.2.4 Estimation results: Target lane Initial Log-Likelihood -2094.9Final Log-Likelihood -1004Number of Observations 16696Number of Vehicles 400Number of Parameters 21Variable Name Parameter t-stat

Current Lane Dummy 3.02 3.96

Path Plan impact : No. of lane changes to exit lane -1.28 -4.54

Path plan impact : No. of lane changes to exit interacted with distance from exit

-1.36 -3.49

Exponent of dist. to exit in no. of lanes to exit- dist. to exit interaction

0.707 3.35

No. of lane changes from current lane = 1 -0.815 -1.10

No. of lane changes from current lane >= 2 -5.68 -0.27

Queue length ahead in lane (m) -0.349 -4.19

Front vehicle rel. speed negative, interacted with front veh. Gap (m/s per m)

0.0284 0.57

Extra Bay Dummy 0.148 0.54

αCL -0.418 -2.11

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Variable Name Parameter t-stat

Lead gap constant 2.38 59.25Relative lead speed (m/s) -0.0221 -3.43σlead 0.00741 0.23αlead -1.75 -42.0

Lag gap constant 1.44 28.6Relative lag speed (m/s) 0.263 18.4σlag 0.00851 0.24αlag -1.86 -36.3

Lead Critical Gap

Lag Critical Gap

2.2.4 Estimation results: Gap acceptance

lead,TLntV

lag,TLntV

40

2.2.4 Estimation results: Execution of lane change

ntV

Variable Name Parameter t-statIntercept -3.7 -3.89αlead 0.357 0.32Subject speed 0.647 3.97

41

Utility associated with lane i for vehicle n at time t :i ic i i exit -exp(0.707) i

nt nt nt nt nt nt

i i ic i i int nt n nt nt nt nt

ic cnt nt

V = 3.024 d - 1.28 e - 1.36 e (d ) -0.815 (k = 1)

- 5.68 ( k >= 2) k - 0.418 u d - 0.349 (q )(q <= 3) - 1.05 (q >= 4)

+(d )[0.0284 * min(0, fr ) /(1+exp cnt (fs ))]

2.2.4 Estimation resultsTarget lane choice

| |)

icnt

int

int

int nt

current lane dummy, 1 if lane i is current lane, 0 ow

q queue ahead in lane i

v average speed ahead in lane i

k number of lane changes required from current lane to lane i ( i j

nt

int

exitnt

cnt

n

where j current lane of individual n at time t

e number of lane changes reqd. to take the desired exit/turn from lane i

d remaining distance to exit/turn

fr = front vehicle rel. speed

fs

ct = front vehicle spacing

42

2.2.4 Estimation resultsGap acceptance

lead TL,cr lead,TL leadnt nt n nt

lead 2nt

G = exp(2.38 - 0.022 V - 1.75 v + )

~ N(0,0.0074 )

,gi gi

nt nt nt

nt

gint

where

V V -V

V speed of subject vehicle n at time t

V speed of vehicle associated with gap g of subject n at time t in direction of target lane i

Critical Lead Gap

Critical Lag Gap

)lag TL,cr lag,TL lagnt nt n nt

lag 2nt

G = exp(1.44 + 0.263 V - 1.86 +

~ N(0,0.0085 )

43

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CL=1 CL=2 CL=3 CL=4

Pro

ba

bil

ity

Lane 1

Lane 2

Lane 3

Lane 4

2.2.5 Interpretation of resultsInertia and path plan effect

Trade-off between current lane inertia and path plan effectDistance to exit = 410 m ( 2 sections), Turning/exit lane = Lane 4*

*Lane 4 is the rightmost lane, Lane 1 is the leftmost lane

44

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CL=1 CL=2 CL=3 CL=4

Pro

ba

bili

ty

Lane 1

Lane 2

Lane 3Lane 4

Trade-off between current lane inertia and path plan effectDistance to exit = 150 m (1 section), Turning/exit lane = Lane 4*

2.2.5 Interpretation of resultsInertia and path plan effect (cont.)

45

2.2.5 Interpretation of resultsInertia and path plan effect (cont.)

Trade-off between current lane inertia and path plan effectDistance to exit = 75 m, Turning/exit lane = Lane 4*

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CL=1 CL=2 CL=3 CL=4

Pro

ba

bili

ty

Lane 1

Lane 2

Lane 3

Lane 4

46

2.2.5 Interpretation of resultsHeterogeneity in critical gap

Critical Lag Gap Variation

-4

-3

-2

-1

0

1

2

3

4

5

6

7

relv = -10 relv = 0 relv = 10

Rel. Lag Speed(m/s)

La

g C

riti

ca

l G

ap

(ln

(m

))

niu = 1

niu = 0

niu = -1

47

2.2.6 Model selection

Statistic Base

Model

New

Model

Log likelihood value -1186.9 -1004.1

Number of parameters (K) 17 22

Akaike information criteria (AIC)* -1203.9 -1126.1

Bayesian information criteria (BIC)* -1271.53 -897.14*AIC=LL-K, BIC=LL –(K/2)*ln(N)

• Compared with simplified model

• Significant improvement in goodness-of-fit

RIGHTLEFT CURRENTLane shift

Gapacceptance

CHANGERIGHT

CHANGELEFT

NOCHANGE

NOCHANGE

NOCHANGE

48

3. Validation

49

Overall Model Development Process

Estimation of merging model

Data collection

Aggregate calibration of simulation model

Disaggregate testing

Model refinement

Dis

aggr

egat

e

data

Agg

rega

te

data

Aggregate validation

Calibrated and validated simulation

model

Open sourced MITSIMLab

Tra

ject

ory

data

Sen

sor

Dat

a

Implementation and verification

Operational Validation

Conceptual Validation

50

3.1 Data

• ‘Synthetic’ sensor data generated from trajectory data– Three sensor stations per section

• Total available data: 8:30 -9:00 am– Calibration: 8:30-8:50 am– Validation: 8:50-9:00 am

‘Synthetic’ Sensor Location

51

3.2 Model comparison• Compared against performance of simplified models

– Lane selection at intersection: rule based model

– Lane selection within section: re-estimated lane-shift model

• Measures of performance– Lane specific speeds

– Lane distributions

RIGHTLEFT CURRENTLane shift

Gapacceptance

CHANGERIGHT

CHANGELEFT

NOCHANGE

NOCHANGE

NOCHANGE

52

3.2 Model comparison (cont.)

Base NewRMSE 11.91 10.64RMSPE 1.82 3.78ME 6.89 2.30MPE 0.47 0.36

Comparison of lane specific speeds

22

1 1

1 1

1 1

1 1

sim obsN Nsim obs n nn n obs

n n n

sim obsN Nsim obs n nn n obs

n n n

obsn

simn

M MRMSE M M ,RMSPE

N N M

M MME M M ,MPE

N N M

M observed speeds / 5min

M simulated speeds / 5min

53

3.2 Model comparison (cont.)

Comparison of lane distributions

Lane Distributions

0

10

20

30

40

1 2 3 4

Lane

%

Observed

Base

New

54

Summary• Intersection lane choice

– Effect of path-plan and ‘plan-ahead’

– Driver heterogeneity• latent driver classes• aggressiveness

• Within section lane change– Effect of path-plan

– Modeling execution of lane change

• Significant improvement in goodness-of-fit compared to simplified models estimated with same data• Improved simulation capability

• Measures of performance– Lane specific speeds

– Lane distributions