Inference for travel time on transportation networks

Travel time is essential for making travel decisions in real-world transportation networks. Understanding its distribution can resolve many fundamental problems in transportation. Empirically, single-edge travel-time is well studied, but how to aggregate such information over many edges to arrive at the distribution of travel time over a route is still daunting. A range of statistical tools have been developed for network analysis; tools to study statistical behaviors of processes on dynamical networks are still lacking. This paper develops a novel statistical perspective to specific type of mixing ergodic processes (travel time), that mimic the behavior of travel time on real-world networks. Under general conditions on the single-edge speed (resistance) distribution, we show that travel time, normalized by distance, follows a Gaussian distribution with universal mean and variance parameters. We propose efficient inference methods for such parameters, and consequently asymptotic universal confidence and prediction intervals of travel time. We further develop path(route)-specific parameters that enable tighter Gaussian-based prediction intervals. We illustrate our methods with a real-world case study using mobile GPS data, where we show that the route-specific and universal intervals both achieve the 95\% theoretical coverage levels. Moreover, the route-specific prediction intervals result in tighter bounds that outperform competing models.

[1]  D. Blei.  Bayesian Nonparametrics I , 2016 .

[2]  T. Snijders,et al.  Modeling the Coevolution of Networks and Behavior , 2007 .

[3]  Hesham Rakha,et al.  Multistate Travel Time Reliability Models with Skewed Component Distributions , 2012 .

[4]  W. Wu.  Submitted to the Annals of Applied Probability RECURSIVE ESTIMATION OF TIME-AVERAGE VARIANCE CONSTANTS By , 2008 .

[5]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[6]  F. Hall,et al.  Approximation of conditional densities by smooth mixtures of regressions ∗ , 2009 .

[7]  P. Abbeel,et al.  Path and travel time inference from GPS probe vehicle data , 2009 .

[8]  Eric Horvitz,et al.  Predicting Travel Time Reliability using Mobile Phone GPS Data , 2017 .

[9]  Haris N. Koutsopoulos,et al.  Travel time estimation for urban road networks using low frequency probe vehicle data , 2013, Transportation Research Part B: Methodological.

[10]  Edward Chi-Fai Lo.  The Sum and Difference of Two Lognormal Random Variables , 2013, J. Appl. Math..

[11]  Klaus Nordhausen,et al.  Statistical Analysis of Network Data with R , 2015 .

[12]  J C Oppenlander,et al.  SAMPLE SIZE DETERMINATION FOR TRAVEL TIME AND DELAY STUDIES , 1976 .

[13]  Eric D. Kolaczyk,et al.  Models for Network Graphs , 2009 .

[14]  Hong Xu,et al.  Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables , 2009, Int. J. Math. Math. Sci..

[15]  Daniel Kifer,et al.  A simple baseline for travel time estimation using large-scale trip data , 2015, SIGSPATIAL/GIS.

[16]  I. Benjamini,et al.  Recurrence of Distributional Limits of Finite Planar Graphs , 2000, math/0011019.

[17]  H. Berbee,et al.  Convergence rates in the strong law for bounded mixing sequences , 1984 .

[18]  Jiguo Cao,et al.  Parameter estimation for differential equations: a generalized smoothing approach , 2007 .

[19]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[20]  Vlada Limic,et al.  Equidistribution, uniform distribution: a probabilist's perspective , 2016, 1610.02368.

[21]  Marco Pavone,et al.  On the Interaction between Autonomous Mobility-on-Demand and Public Transportation Systems , 2018, 2018 21st International Conference on Intelligent Transportation Systems (ITSC).

[22]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[23]  Huizhao Tu,et al.  Travel time unreliability on freeways: Why measures based on variance tell only half the story , 2008 .

[24]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[25]  R. C. Bradley.  Basic Properties of Strong Mixing Conditions , 1985 .

[26]  Ben Taskar,et al.  Learning Probabilistic Models of Relational Structure , 2001, ICML.

[27]  Armann Ingolfsson,et al.  Empirical Analysis of Ambulance Travel Times: The Case of Calgary Emergency Medical Services , 2010, Manag. Sci..

[28]  M. Rosenblatt.  A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION. , 1956, Proceedings of the National Academy of Sciences of the United States of America.

[29]  R. Suresh,et al.  Estimation of variance of partial sums of an associated sequence of random variables , 1995 .

[30]  Eric D. Kolaczyk,et al.  Statistical Analysis of Network Data with R , 2020, Use R!.

[31]  P. Billingsley,et al.  Ergodic theory and information , 1966 .

[32]  Shane G. Henderson,et al.  Travel time estimation for ambulances using Bayesian data augmentation , 2013, 1312.1873.

[33]  N. Herrndorf.  The invariance principle for ϕ-mixing sequences , 1983 .

[34]  D. Wilkinson,et al.  Bayesian Inference for Stochastic Kinetic Models Using a Diffusion Approximation , 2005, Biometrics.

[35]  P. O’Neill,et al.  Bayesian inference for stochastic epidemics in populations with random social structure , 2002 .

[36]  David Aldous,et al.  Applications of Random Walks on Finite Graphs , 1991 .

[37]  John Krumm,et al.  Hidden Markov map matching through noise and sparseness , 2009, GIS.

[38]  Mahmoud Mesbah,et al.  Estimation of trip travel time distribution using a generalized Markov chain approach , 2017 .

[39]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[40]  Wlodzimierz Bryc,et al.  Moment conditions for almost sure convergence of weakly correlated random variables , 1993 .

[41]  Fangfang Zheng,et al.  Urban link travel time estimation based on sparse probe vehicle data , 2013 .

[42]  Sebastian Schmutzhard,et al.  Universal series induced by approximate identities and some relevant applications , 2011, J. Approx. Theory.

[43]  L. Schmetterer.  Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete. , 1963 .

[44]  C. Caramanis.  What is ergodic theory , 1963 .

[45]  Magda Peligrad,et al.  On the asymptotic normality of sequences of weak dependent random variables , 1996 .

[46]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[47]  Christian Steglich,et al.  Beyond dyadic interdependence: Actor-oriented models for co-evolving social networks and individual behaviors , 2007 .

[48]  Alexandre M. Bayen,et al.  Large-Scale Estimation in Cyberphysical Systems Using Streaming Data: A Case Study With Arterial Traffic Estimation , 2013, IEEE Transactions on Automation Science and Engineering.