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ORF 510: Directed Research II
“Analyzing travel time distributions using GPS data”
Santiago Arroyo
Advisor: Prof. Alain L. Kornhauser
January 13th, 2004
Objectives
Obtain travel time distributions between “monuments” on the US road network in order to:• Find more realistic shortest paths (according to travel time, not
distance)
• Obtain travel time patterns according to categories:• Day of week (week/weekend)
• Time of day
• Road type
• Forecast travel times in real time
Concepts
Road network US (Copilot®): • Different levels:
• Level 0: 30 x 106 arcs
• Level 1: 500.000 arcs
• Level 2: 10.000 arcs
• Level 3: 1.000 arcs
• Nodes: 5 million approx.
Monuments:• Midpoints of some arcs on level 1 (used originally to build network)
• Number: 280.000
Data
Copilot® :• Generated on
Nov 14th, 2003• Every 3 seconds• Matched to link
using position and heading
• Biased geographically and by users
• Identified by vehicle, position, heading, speed, date and time
Data
1.500.000 monument to monument (m2m) timesMax # of observations for m2m pair: 1202Total # of m2m pairs: 34879Only 1730 (4.96%) pairs have 100 observations or moreBiased around Princeton area
0 200 400 600 800 1000 1200
05
00
01
00
00
15
00
0
Observations
Nu
mb
er
of
M2
M p
air
s
Time vs. Speed
Time: • Absolute (don’t care about what happens in between measures)
• Readily extractable from data
Speed:•Calculate estimated time using speed measurements
•More difficult to take into account changes in speed between points (intersections, left turns, stops)
•Significant errors around intersections due to matching
Case Study
Mean: 75.55
Median: 73
St Dev: 30.23
Mean: 119.41
Median: 117
St Dev: 19.52
100 200 300 400 500 600 700
02
00
40
06
00
M1 to M2 (seconds)
M2M travel times
100 200 300 400 500
01
00
30
05
00
M2 to M3 (seconds)
Case Study
Mean: 74.09
Median: 73
St Dev: 7.91
Mean: 118.25
Median: 117
St Dev: 9.06
60 80 100 120 140 160 180
05
01
00
15
02
00
M1 to M2 (seconds)
M2M travel times (without high values)
100 120 140 160 180
02
04
06
08
01
00
M2 to M3 (seconds)
Case Study
Empirical Cumulative Distribution M1 to M2
Travel Time (seconds)
CD
F
0 200 400 600
0.0
0.2
0.4
0.6
0.8
1.0
Empirical Cumulative Distribution M2 to M3
Travel Time (seconds)
CD
F100 200 300 400 500 600
0.0
0.2
0.4
0.6
0.8
1.0
Case StudyM2M time passing through another M
M2M time
De
nsi
ty
0 200 400 600
0.0
0.0
05
0.0
10
0.0
15
M1 to M2M2 to M3M1 to M3
Problem
Stochastic Shortest Path: • Travel time is a function of departure time
• Discrete time of day intervals?
• Continuous function relating travel time and departure time
• Edge weights are random variables• What are their distributions?
• Are they independent?
• How do we “add” distributions on a path?
• Memory limitations• Can’t store all possible paths
• Which paths should we store?
Travel time as a function of departure time
Schrader & Kornhauser (2003): • Ten-parameter function fit to data from the Milwaukee Highway System:
Travel time as a function of departure time
Schrader & Kornhauser (2003): • Milwaukee Highway System (Weekday travel time):
),(),(),()( 333222111 CCCKtfTT
where:
TT = travel timet = departure time
22 2/)(22
1),(
te
Travel time as a function of departure time
Case study: Travel time as function of departure time
Seconds from midnight
Tra
vel T
ime
(se
con
ds)
0 20000 40000 60000 80000
10
02
00
30
04
00
50
06
00
Edge weights are random variables
Extension of deterministic algorithms: • Dijkstra’s, Bellman-Ford, A* and variations
• Use expected value as edge weight:• Need to know travel time distribution
Stochastic Shortest Path algorithms:• Priority First Search (PFS) with dominance pruning (Wellman et al.,
1995)
• Adaptive Path Planning (Wellman et al., 1995)
• Vertex-Potential Model (Cooper et al., 1997)
• Path Optimality Indexes (Sigal et al., 1980)
Edge weights are random variables
PFS with dominance pruning: • Stochastically consistent network:
cij(x)= time dependent travel time from i to j
For all i, j, s<=t, and z: Pr{s+ cij(s)<=z}>=Pr{t+ cij(t)<=z}
i.e. the probability of arriving by any given time z cannot be increased by leaving later
Edge weights are random variables
PFS with dominance pruning:
• Stochastic dominance:• Arrival time distribution at a node dominates another iff cumulative
probability function is uniformly greater or equal to that of the other
Edge weights are random variables
PFS with dominance pruning: • Utility is nonincreasing with respect to arrival time• How do we find arrival time distributions from a path with two edges or more?
(+) = ?
A B C
• Priority queue: only keep stochastically undominated paths• Apply variations like A*
Further Research
Establish m2m time travel distribution approximations (normal, lognormal, exponential, etc.)“Addition” of distributions on a path (stochastic model)Investigate on independence of distributions in a path (maybe as a function of edges in the path)Categorize by time of day, day of week, road typeDevelop travel time as a function of departure time using Copilot® dataConstruct network with expected travel times instead of distances (PTNM)Use travel times from Copilot® data to forecast travel times, incorporating real time data
BibliographyC. Cooper, A. Frieze, K. Mehlhorn and V. Priebe, Average-case of shortest-paths problems in the vertex-potential model, International Workshop RANDOM’97, Bologna, Italy
A.M. Frieze and G.R. Grimmett, The Shortest-Path Problem for Graphs with Random Arc-Lengths, Discrete Applied Mathematics, 10 (1985) 57-77.
S. Pallottino and M. Scutella, Shortest Path Algorithms in Transportation Models: Classical and Innovative Aspects. In Marcotte, P., Nguyen, S., eds.: Equilibrium and Advanced Transportation Modelling. Kluwer, Amsterdam (1998) 245-281.
C. Schrader and Alain L. Kornhauser, Using Historical Information in Forecasting in Travel Times, BSE Thesis, Princeton University, 2003
C. Elliot Sigal, A. Alan B. Pritsker and James J. Solberg, The Stochastic Shortest Route Problem, Operations Research, (1980) 1122-1129.
Michael P. Wellman, Kenneth Larson, Matthew Ford and Peter R. Wurman, Path Planning Under Time-Dependent Uncertainty, Eleventh Conference on Uncertainty in Artificial Intelligence, 28-5 (1995) 532-539.
Fastest Path Problems in Dynamic Transportation Networks, in www.husdal.com
