ORF 510: Directed Research II “Analyzing travel time distributions using GPS data” Santiago Arroyo Advisor: Prof. Alain L. Kornhauser January 13 th, 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 10 6 arcs Level 1: arcs Level 2: arcs Level 3: arcs Nodes: 5 million approx. Monuments: Midpoints of some arcs on level 1 (used originally to build network) Number:

Data Copilot ® : Generated on Nov 14 th, 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 monument to monument (m2m) times Max # of observations for m2m pair: 1202 Total # of m2m pairs: Only 1730 (4.96%) pairs have 100 observations or more Biased around Princeton area

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 Monument 1 Monument 2 Monument 3

Case Study Mean: Median: 73 St Dev: Mean: Median: 117 St Dev: 19.52

Case Study Mean: Median: 73 St Dev: 7.91 Mean: Median: 117 St Dev: 9.06

Case Study

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): where: TT = travel time t = departure time

Travel time as a function of departure time Case study:

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 : c ij (x)= time dependent travel time from i to j For all i, j, s =Pr{t+ c ij (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? (+)= ? ABC 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 type Develop travel time as a function of departure time using Copilot ® data Construct network with expected travel times instead of distances (PTNM) Use travel times from Copilot ® data to forecast travel times, incorporating real time data

Bibliography C. 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) 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) 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) 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) Fastest Path Problems in Dynamic Transportation Networks, in