Estimating Travel Time Reliability: Can we Safely Ignore Correlation?

1. Estimating Travel Time Reliability: Can we Safely Ignore Correlation? Professor Alan Nicholson University of Canterbury Christchurch, New Zealand

2. Introduction (I) Travel time reliability has been of interest for at least 60 years: • Turner & Wardrop (1951) found trip times in Central London were not Normally distributed; • Herman & Lam (1974) found the same for trips to/from work in Detroit; • Richardson & Taylor (1978) found the Lognormal distribution fitted trip times for peak-period trips to/from central Melbourne better than the Normal: • analysed time data for 19 sections of trip:

3. Introduction (II) • found 13.9% & 5.6% of correlations for adjacent sections were statistically significantly different from zero at the 5% & 1% levels; • concluded “reasonable to accept the hypothesis that route section unit travel times (i.e. the travel time per unit distance) were independent”. NB. Congestion levels varied a lot between sections, so driver behaviour could have reduced the level of correlation (e.g. drivers delayed in one section going faster in next section).

4. Introduction (III) The MVA Consultancy (1996) noted that both positive & negative correlations of travel times on parts of trips are possible: • positive when have common influences (e.g. bad weather) or congestion ~ trip time variance more than sum of variances for parts of trip; • negative when there is a bottle-neck ~ trip time variance less than sum of variances for parts of trip. • found correlations between adjacent parts were generally positive & often as high as 0.4

5. Introduction (IV) Subsequent studies by Mott MacDonald (2000a, 2000b, 2003) allowed for correlations of 0.2 to 0.4, between adjacent links only: • aim was a practical method for including travel time variability in traffic assignment; • economic appraisal of projects to reduce travel time variability might not produce accurate result; • flows used in economic appraisal might well differ greatly from actual flows if drivers consider travel time reliability when choosing their routes.

6. NZ Economic Appraisal Method (I) Allows for variations in times for trips undertaken at the same time every day: • does not account for major incidents. Estimates variation in mean travel times: • less than variation in travel times of individuals. Standard deviation (SD) of travel time (s) used as the measure of link travel time variability: • assumed to depend on volume/capacity ratio (v/c). Decrease in SD is 0.8 & 1.2 times the value of same decrease in travel time (cars & trucks, respectively).

7. NZ Economic Appraisal Method (II)

8. NZ Economic Appraisal Method (III) Assumes variance of total trip time is Where have correlation, the total variance is That is, the NZ method uses the ‘variance term’ only, and ignores the ‘covariance term’. Is it safe to ignore covariance? In particular, what correlations exist, and what is the effect of ignoring them?

9. North-bound carriageway, from the Horikiri junction to Kasai junction (11.9 km) Divided into 39 links, lengths generally 300 ± 50 m, and flows rates about 26 thousand veh/day/lane. Collected flow rates and speeds on each link for each minute on 93 days. East Central Circular Route Case Study Area

10. Links & Link Lengths

11. Links, Segments & Sections

12. Instantaneous Method (IM)

13. Trip-Based Method (TBM) Same TTs for links 1-3, but different TTs for links 4 &5

14. Stage 1 Analysis (I) Had flow rate & speed for 39 links for one-minute intervals for 93 days. Data were grouped: • weekdays (incl. Saturdays) ~ n = 60; • Sundays & National Holidays ~ n = 20; • all days ~ n = 93. Potentially 741 [=(392-39)/2] correlation coefficients for each minute: • 1,067,040 (i.e. 741×1440) correlation coefficients.

15. Stage 2 Analysis (II) Calculated correlation of link travel times for all pairs of links, for trips along whole length of expressway and commencing during: • interval 6.00am-6.01am (‘peak’); • interval 1.00pm-1.01pm (‘off-peak’). Calculated for each interval: • standard deviation of travel times for each link; • ‘total variance’, ‘variance term’ and ‘correlation term’ for trips commencing during each interval.

16. Correlation Coefficients (Peak): IM

17. Correlation Coefficients (Peak): TBM

18. Correlation Coefficients (Off-Peak): IM

19. Correlation Coefficients (Off-Peak): TBM

20. Variance Components (Peak): IM

21. Variance Components (Peak): TBM

22. Variance Components (Off-Peak): IM

23. Variance Components (Off-Peak): TBM

24. Discussion of Stage 1 Results (I) The results for the instantaneous and trip-based methods are fairly similar: • probably because link travel speeds fairly stable for typical time to travel along the expressway; • this is unlikely to apply elsewhere all the time; • trip-based method better, despite involving more effort, because it is more robust. Main results: • correlations ranged from -0.7 to almost +1.0;

25. Discussion of Stage 1 Results (II) • many statistically significantly different from zero; • evidence of ‘bottle-necks’; • ‘off-peak’ travel time variance about 10 times ‘peak’ travel time variance; • as expected (viz. decision to compare 6.00am-6.01am and 1.00pm-1.01pm intervals); • ‘correlation term’ about 10 times the ‘variance term’; • ‘total variance’ >> ‘variance term’.

26. Stage 2 Analysis (I) The Stage 1 analysis looked at the variations and correlations for trips commencing in two specific short periods; • the Stage 2 analysis examined the general variations and correlations, for trips commencing at any time; • a ‘bootstrap’ analysis of the complete set of data was undertaken; • estimated correlations using a ‘bootstrap’ sample;

27. Stage 2 Analysis (II) • i.e. a set of data points, each chosen randomly & independently from the observed data set, with all values being equally likely. A ‘block bootstrap analysis’ analysis was done: • involves re-sampling data blocks, to capture any dependencies between data points in close temporal or spatial proximity; • a ‘block bootstrap’ sample combines several blocks of minute-by-minute link travel time data points;

28. Stage 2 Analysis (III) • each data block includes potentially correlated data points; • for this study, data blocks covered four hours and all links of the expressway; • wanted to estimate correlations between all pairs of links; • each ‘block bootstrap’ sample contained several blocks of 250 minute-by-minute travel time data points for each expressway link.

29. Stage 2 Analysis (IV) Correlations coefficients were estimated for: • all pairs of links; • selected pairs of links ~ adjoining, moderately separated and distant. The samples were used to get estimates of the ‘variance term’ and ‘covariance term’ for trips of varying length along the expressway; • to assess effect of spatial aggregation.

30. Correlation Coefficients (I)

31. Correlation Coefficients (II)

32. Correlation Coefficients (III)

33. Correlation Coefficients (IV)

34. Covariance & Variance Terms (I)

35. Covariance & Variance Terms (II)

36. Covariance & Variance Terms (III)

37. Covariance & Variance Terms (IV)

38. Discussion of Stage 2 Results Results of Stage 2 analysis confirm results of Stage 1 analysis: • have statistically significant correlations; • correlations less extreme (highly positive or highly negative); • correlations reduce with separation; • covariance term about 10 times the variance term, for wide range of trip lengths.

39. Conclusion • statistically significant correlations possible and should be considered; • likely to be larger and/or more common as networks & corridors become more congested; • if ignore correlation, can get large errors in estimate of trip time variance and benefits of projects; • good level of agreement on value of reductions in travel time variability, but not on estimation of changes in trip time variability; • how to include trip time variability in route choice (traffic assignment) is a major unresolved issue.

40. Acknowledgements • Tokyo Metropolitan Expressway Authority ~ for allowing the collection and use of their data. • Keiko Munakata (MSc student) ~ for data collation and Stage 1 analysis. • Dr Kenneth Kuhn ~ for Stage 2 (bootstrap method) analysis.

41. References (I) Herman, R. & Lam, T. (1974). Trip Characteristics of Journeys To and From Work. Proc. 6th Int. Sym. on Transportation & Traffic Flow; in “Transportation and Traffic Theory” (ed. Buckley, D.J), A.H. & A.W. Reed, Sydney. MVA Consultancy (1996), “The Benefits of Reduced Travel Time Variability”, Report 02/C/2923, Department for Transport, London, UK: 100pp. Richardson, A.J. & Taylor, M.A.P. (1978). Travel Time Variability on Commuter Journeys. High Speed Ground Transportation, 12(1): 77-99. Turner, J.K. & Wardrop, J.G. (1951). The Variation of Journey Times in Central London. Road Note RN/1511/JKT.JGW, Road Research Laboratory, Crowthorne, UK.

42. References (II) Mott MacDonald (2000a). Travel Time Variability Follow-On Research 1.1. Report for UK Department of the Environment, Transport and the Regions, London. Mott MacDonald (2000b). Travel Time Variability Follow-On Research 1.2. Report for UK Department of the Environment, Transport and the Regions, London. Mott MacDonald (2003). Travel Time Variability Follow-On Research 1.3. Report for UK Department for Transport, London.