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New Approaches for Traffic State Estimation:

Calibrating Heterogeneous Car-Following Behavior using Vehicle Trajectory Data

Dr. Xuesong Zhou & Jeffrey Taylor, Univ. of Utah

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Outline Background on Dynamic Time Warping (DTW) Application to Newell’s Simplified CFM Calibration Results Important Considerations

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Motivations: I Real-time Traffic Management

Automatic Vehicle IdentificationAutomatic Vehicle Location

Loop Detector

Video Image Processing

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Motivation 2: Self-driving Cars as Mobile Sensor Controlled , coordinated movements

Proactive approach

Applications Automated cars Unmanned aerial vehicles

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Motivation 3: Detecting Distracted/Risky Drivers

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Underlying Theory:Cross-resolution Traffic Modeling

Reaction distance/spacing δ Reaction time lag τW = δ/ τ

Time

Space

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How to Estimate Driver-specific Car-following Parameters? Input and output

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Intro to Dynamic Time Warping (DTW)

Time

Position

1 2 3 4 5 6 7 8

(A) Vehicle Trajectories with DTW Match Solution

X: Leader

Y: Follower

Match Solution

Time

Velocity

X: LeaderY: Follower

1 2 3 4 5 6 7 8

(B) Vehicle Velocity Time Series

• Matches points by measure of similarity

Euclidean Vs Dynamic Time Warping

Euclidean DistanceSequences are aligned “one to one”.

“Warped” Time AxisNonlinear alignments are possible.

Reference: Eamonn Keogh

Computer Science & Engineering DepartmentUniversity of California - Riverside

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Construct Cost Matrix for Traffic Trajectory Matching

1 2 3 4 5 6 7 81 0 0 0 0 20 18 18 52 0 0 0 0 20 18 18 53 0 0 0 0 20 18 18 54 20 20 20 20 3 3 3 255 20 20 20 20 3 3 3 256 5 5 5 5 23 23 23 07 5 5 5 5 23 23 23 08 5 5 5 5 23 23 23 0

Time

Velocity Y: Follower

TimeVelocity

X: Leader

(C) Cost Matrix (for Velocity)

wij

jxixrdjxixijYXYXC FL

FLjiji

)()()()(),(

Time

Position

1 2 3 4 5 6 7 8

(A) Vehicle Trajectories with DTW Match Solution

X: Leader

Y: Follower

Match Solution

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Cumulative Cost Matrix Dynamic programming

Calculate the least cost for matching a pair of points

Warp path Least cost matching

points from end to beginning

1 2 3 4 5 6 7 81 0 0 0 0 20 38 56 62 0 0 0 0 20 38 56 613 0 0 0 0 20 38 56 614 20 20 20 20 3 6 9 345 40 40 40 40 6 6 9 346 45 45 45 45 29 29 29 97 50 50 50 50 52 52 52 98 55 55 55 55 75 75 75 9

(D) Cumulative Cost Matrix with Highlighted Warp Path

τ = TF – TL

d = XL(tL) - XF(tF)

Follower:

Lea

der

Singularity

Time

Position

1 2 3 4 5 6 7 8

(A) Vehicle Trajectories with DTW Match Solution

X: Leader

Y: Follower

Match Solution

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Application to Newell’s Model Follower separated

by leader by reaction time and critical jam spacing

Algorithm finds optimal τn (time lag) for best velocity match Calculate dn for all time

steps along the trajectory

Sn

Sn’

τn

dn

Xn(t)

Xn-1(t)

Time, tD

ista

nce

, X

nnnn dtxtx )()( 1

Calibrated Parameters: Car 1737

1 25 49 73 97 1211451691932172412652893133373613854094334574815055295535770

5

10

15

20

25

0

0.5

1

1.5

2

2.5

3

3.5

Critical Jam Spacing Backward Wave Speed Reaction Time

Spacin

g (

m)

& W

ave S

peed (

km

/h)

Reacti

on T

ime (

seconds)

Reaction Time Lag (sec)

Critical Spacing (m)

Backward Wave Speed (km/h)

Avg 2.62 13.39 18.46

St. Dev 0.41 2.08 1.05

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NGSIM Data: I-80 Lane 4

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NGSIM Data: I-80 Lane 4: Reaction Time Distribution

-1 0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5x 10

4

Reaction Time (seconds)

Fre

quen

cy

Mean = 1.48 seconds

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NGSIM Data: I-80 Lane 4Critical Spacing Distribution

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5x 10

4

Spacing (meters)

Fre

quen

cy

Mean = 8.06 meters

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NGSIM Data: I-80 Lane 4Wave Speed Distribution

0 5 10 15 20 25 30 35 400

2000

4000

6000

8000

10000

12000

14000

16000

Wave Speed (km/h)

Fre

quen

cy

Mean = 20.55 km/h

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Current Issues in DTW Application Singularities

Locations with more than one match solution Data reduction algorithms

Parameter estimates differ with available methods

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Singularities

1 2 3 4 5 6 7 81 0 0 0 0 20 38 56 62 0 0 0 0 20 38 56 613 0 0 0 0 20 38 56 614 20 20 20 20 3 6 9 345 40 40 40 40 6 6 9 346 45 45 45 45 29 29 29 97 50 50 50 50 52 52 52 98 55 55 55 55 75 75 75 9

(D) Cumulative Cost Matrix with Highlighted Warp Path

τ = TF – TL

d = XL(tL) - XF(tF)

Follower:Le

ader

5160 5180 5200 5220 5240 5260 5280 5300 5320300

350

400

450

500

550

600

650

Time (1/10 sec)

Pos

ition

(ft

)

DTW Trajectory Match (Based on Acceleration Data)

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Singularity Implications 1st Interpretation: Many

responses to 1 stimulus 2nd Interpretation: 1

response to many stimuli

3rd Interpretation: Algorithm drawback Increases uncertainty in

parameter estimates LCSS force 1-to-1 match

LCSS : Longest Common Subsequence

5160 5180 5200 5220 5240 5260 5280 5300 5320300

350

400

450

500

550

600

650

Time (1/10 sec)

Pos

ition

(ft

)

DTW Trajectory Match (Based on Acceleration Data)

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Singularities

Without Prior Information

With Prior Information

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Data Reduction Algorithms

• Piecewise Linear Approximation/Regression– Somewhat subjective in application, needs dynamic

parameters

– Difficulties creating new points application with Newell’s model

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Potential Applications Analyze intradriver heterogeneity Markov Chain Monte Carlo method for reaction

time/critical jam spacing Analyze relationships between parameters

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Markov Chain Transition Matrix

Reaction Time + + + 0 0 0 - - -Acceleration + 0 - + 0 - + 0 -

+ 0.1 0.050 0.001 0.048 0.036 0.199 0.102 0.058 0.043 0.0430.0 0.950 0.949 0.952 0.918 0.801 0.898 0.942 0.858 0.751

- 0.1 0.001 0.049 0.000 0.046 0.000 0.000 0.000 0.099 0.207Sum 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Reaction Time + + + 0 0 0 - - -

Acceleration + 0 - + 0 - + 0 -

0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.010 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98

-0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

Sum 100.0%####

#100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

Hypothetical case:

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Trajectory Prediction (MCMC)

0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

1600

Original Trajectory

Predicted Trajectory

~ 5% MAPE