Three challenges in route choice modeling

Transcription

1 Three challenges in route choice modeling Michel Bierlaire and Emma Frejinger transp-or.epfl.ch Transport and Mobility Laboratory, EPFL Three challenges in route choice modeling p.1/61

2 Route choice modeling Given a transportation network composed of nodes, links, origin and destinations. For a given transportation mode and origin-destination pair, which is the chosen route? Three challenges in route choice modeling p.2/61

3 Applications Intelligent transportation systems GPS navigation Transportation planning Three challenges in route choice modeling p.3/61

4 Challenges Alternatives are often highly correlated due to overlapping paths Data collection Large size of the choice set Three challenges in route choice modeling p.4/61

5 Dealing with correlation Frejinger, E. and Bierlaire, M. (2007). Capturing correlation with subnetworks in route choice models, Transportation Research Part B: Methodological 41(3): Three challenges in route choice modeling p.5/61

6 Existing Approaches Few models explicitly capturing correlation have been used on large-scale route choice problems C-Logit (Cascetta et al., 1996) Path Size Logit (Ben-Akiva and Bierlaire, 1999) Link-Nested Logit (Vovsha and Bekhor, 1998) Logit Kernel model adapted to route choice situation (Bekhor et al., 2002) Probit model (Daganzo, 1977) permits an arbitrary covariance structure specification but cannot be applied in a large-scale route choice context Three challenges in route choice modeling p.6/61

7 Existing Approaches Link based path-multilevel logit model (Marzano and Papola, 2005) Illustrated on simple examples and not estimated on real data Three challenges in route choice modeling p.7/61

8 Subnetworks How can we explicitly capture the most important correlation structure without considerably increasing the model complexity? Three challenges in route choice modeling p.8/61

9 Subnetworks How can we explicitly capture the most important correlation structure without considerably increasing the model complexity? Which are the behaviorally important decisions? Three challenges in route choice modeling p.8/61

10 Subnetworks How can we explicitly capture the most important correlation structure without considerably increasing the model complexity? Which are the behaviorally important decisions? Our hypothesis: choice of specific parts of the network (e.g. main roads, city center) Concept: subnetwork Three challenges in route choice modeling p.8/61

11 Subnetworks Subnetwork approach designed to be behaviorally realistic and convenient for the analyst Subnetwork component is a set of links corresponding to a part of the network which can be easily labeled Paths sharing a subnetwork component are assumed to be correlated even if they are not physically overlapping Three challenges in route choice modeling p.9/61

12 Subnetworks - Example D S a O Path 3 Path 2 Path 1 S b Three challenges in route choice modeling p.10/61

13 Subnetworks - Methodology Factor analytic specification of an error component model (based on model presented in Bekhor et al., 2002) U n = β T X n + F n Tζ n + ν n F n (JxQ) : factor loadings matrix (f n ) iq = l niq T (QxQ) = diag (σ 1,σ 2,...,σ Q ) ζ n (Qx1) : vector of i.i.d. N(0,1) variates ν (Jx1) : vector of i.i.d. Extreme Value distributed variates Three challenges in route choice modeling p.11/61

14 Subnetworks - Example U 1 = β T X 1 + p l 1a σ a ζ a + p l 1b σ b ζ b + ν 1 U 2 = β T X 2 + p l 2a σ a ζ a + ν 2 D U 3 = β T X 3 + p l 3b σ b ζ b + ν 3 S a O Path 3 Path 2 Path 1 S b FTT T F T = 2 l 1a σa 2 + l 1b σ 2 b l1a l2a σ 2 3 a l1b l3b σb l1a l2a σa 2 l 2a σa l3b l1b σb 2 0 l 3b σb 2 Three challenges in route choice modeling p.12/61

15 Empirical Results The approach has been tested on three datasets: Boston (Ramming, 2001), Switzerland, and Borlänge Deterministic choice set generation Link elimination GPS data from 24 individuals 2978 observations, 2179 origin-destination pairs Borlänge network 3077 nodes and 7459 links BIOGEME (biogeme.epfl.ch, Bierlaire, 2003) has been used for all model estimations Three challenges in route choice modeling p.13/61

16 Borlänge Road Network Three challenges in route choice modeling p.14/61

17 Model Specifications Six different models: MNL, PSL, EC 1, EC 1, EC 2 and EC 2 EC 1 and EC 1 have a simplified correlation structure EC 1 and EC 2 do not include a Path Size attribute Deterministic part of the utility V i = β PS ln(ps i ) + β EstimatedTime EstimatedTime i + β NbSpeedBumps NbSpeedBumps i + β NbLeftTurns NbLeftTurns i + β AvgLinkLength AvgLinkLength i Three challenges in route choice modeling p.15/61

18 Estimation Results Parameter estimates for explanatory variables are stable across the different models Path size parameter estimates Parameter PSL EC 1 EC 2 Path Size Scaled estimate Rob. T-test All covariance parameters estimates in the different models are significant except the one associated with R.50 S Three challenges in route choice modeling p.16/61

19 Estimation Results Model Nb. σ Nb. Estimated Final Adjusted Estimates Parameters L-L Rho-Square MNL PSL EC 1 (with PS) EC EC 2 (with PS) EC pseudo-random draws for Maximum Simulated Likelihood estimation 2978 observations Null log likelihood: BIOGEME (biogeme.epfl.ch) has been used for all model estimations. Three challenges in route choice modeling p.17/61

20 Forecasting Results Comparison of the different models in terms of their performance of predicting choice probabilities Five subsamples of the dataset Observations corresponding to 80% of the origin destination pairs (randomly chosen) are used for estimating the models The models are applied on the observations corresponding to the other 20% of the origin destination pairs Comparison of final log-likelihood values Three challenges in route choice modeling p.18/61

21 Forecasting Results Same specification of deterministic utility function for all models Same interpretation of these models as for those estimated on the complete dataset Coefficient and covariance parameter values are stable across models Three challenges in route choice modeling p.19/61

22 Forecasting Results Three challenges in route choice modeling p.20/61

23 Conclusion - Subnetworks Models based on subnetworks are designed for route choice modeling of realistic size Correlation on subnetwork is explicitly captured within a factor analytic specification of an Error Component model Estimation and prediction results clearly shows the superiority of the Error Component models compared to PSL and MNL The subnetwork approach is flexible and the model complexity can be controlled by the analyst Three challenges in route choice modeling p.21/61

24 Network-free data Bierlaire, M., Frejinger, E., and Stojanovic, J. (2006). A latent route choice model in Switzerland. Proceedings of the European Transport Conference (ETC) September 18-20, Bierlaire, M., and Frejinger, E. (2007). Route choice modeling with network-free data. Technical report TRANSP-OR Transport and Mobility Laboratory, ENAC, EPFL. Three challenges in route choice modeling p.22/61

25 Data collection and processing Link-by-link descriptions of chosen routes necessary for route choice modeling but never directly available Data processing in order to obtain network compliant paths Map matching of GPS points Reconstruction of reported paths Difficult to verify and may introduce bias and errors Three challenges in route choice modeling p.23/61

26 Modeling with network-free data An observation i is a sequence of individual pieces of data related to an itinerary. Examples: sequence of GPS points or reported locations For each piece of data we define a Domain of Data Relevance (DDR) that is the physical area where it is relevant The DDRs bridge the gap between the network-free data and the network model Three challenges in route choice modeling p.24/61

27 Example - GPS data Three challenges in route choice modeling p.25/61

28 Example - Reported trip Intersection 5 Main St and Cross St 4 City center Mall 1 Home 3 Three challenges in route choice modeling p.26/61

29 Domain of Data Relevance For each piece of data d we generate a list of relevant network elements e (links and nodes) We define an indicator function 1 if e is related to the DDR of d δ(d,e) = 0 otherwise Three challenges in route choice modeling p.27/61

30 Model estimation We aim at estimating the parameters β of route choice model P(p C n (s);β) We have a set S i of relevant od pairs The probability of reproducing observation i of traveler n, given S i is defined as P n (i S i ) = P n (s S i ) P n (i p)p n (p C n (s);β) s S i p C n (s) Three challenges in route choice modeling p.28/61

31 Model estimation Measurement equation P n (i p) Reported trips P n (i p) = 1 if i corresponds to p 0 otherwise GPS data P n (i p) = 0 if i does not correspond to p If i corresponds to p then P n (i p) is a function of the distance between i and p Three challenges in route choice modeling p.29/61

32 Model estimation Measurement equation P n (i p) for GPS data Distance between i and a the closest point on a link l is D(d,p) = min l Apd (d,l) 4 (4, 6) (2, 4) (4, 5) (d 4, (2, 4)) d 4 (d 4, (4, 5)) (d 4, (4, 6)) Three challenges in route choice modeling p.30/61

33 Model estimation P n (i S i ) = s S i P n (s S i ) p C n (s) P n (i p)p n (p C n (s);β) P(i s) = P(i p 1 )P(p 1 C(s);β) + P(i p 2 )P(p 2 C(s);β) Three challenges in route choice modeling p.31/61

34 Empirical Results Simplified Swiss network (39411 links and nodes) RP data collection through telephone interviews Long distance car travel The chosen routes are described with the origin and destination cities as well as 1 to 3 cities or locations that the route pass by 940 observations available after data cleaning and verification Three challenges in route choice modeling p.32/61

35 Empirical Results Three challenges in route choice modeling p.33/61

36 Empirical Results No information available on the exact origin destination pairs P(s i) = 1 S i s S i P(r i) is modeled with a binary variable 1 if r corresponds to i δ ri = 0 otherwise Three challenges in route choice modeling p.34/61

37 Empirical Results Two origin-destination pairs are randomly chosen for each observation 46 routes per choice set are generated with a choice set generation algorithm After choice set generation 780 observations are available 160 observations were removed because either all or none of the generated routes crossed the observed zones Three challenges in route choice modeling p.35/61

38 Empirical Results Probability of an aggregate observation i P(i) = s S i 1 S i r C s δ ri P(r C s ) We estimate Path Size Logit (Ben-Akiva and Bierlaire, 1999) and Subnetwork (Frejinger and Bierlaire, 2007) models BIOGEME (biogeme.epfl.ch) used for all model estimations Three challenges in route choice modeling p.36/61

39 Empirical Results - Subnetwork Subnetwork: main motorways in Switzerland Correlation among routes is explicitly modeled on the subnetwork Combined with a Path Size attribute Linear-in-parameters utility specifications Three challenges in route choice modeling p.37/61

40 Empirical Results - Subnetwork Three challenges in route choice modeling p.38/61

41 Parameter PSL Subnetwork ln(path size) based on free-flow time 1.04 (0.134) (0.141) 7.78 Scaled Estimate Freeway free-flow time 0-30 min (0.877) (0.984) Scaled Estimate Freeway free-flow time 30min - 1 hour (0.875) (1.03) Scaled Estimate Freeway free-flow time 1 hour (0.772) (1.00) Scaled Estimate CN free-flow time 0-30 min (0.882) (0.975) Scaled Estimate CN free-flow time 30 min (0.331) (0.384) Scaled Estimate Main free-flow travel time 10 min (0.502) (0.624) Scaled Estimate Small free-flow travel time (0.685) (0.804) Scaled Estimate Proportion of time on freeways -2.2 (0.812) (0.865) Scaled Estimate Proportion of time on CN 0 fixed 0 fixed Proportion of time on main (0.752) (0.800) Scaled Estimate Proportion of time on small (0.992) (1.03) Scaled Estimate Covariance parameter (0.0543) 4.00 Scaled Estimate 0.205

42 Empirical Results PSL Subnetwork Covariance parameter (Rob. Std. Error) Rob. T-test (0.0543) 4.00 Number of simulation draws Number of parameters Final log-likelihood Adjusted rho square Sample size: 780, Null log-likelihood: Three challenges in route choice modeling p.39/61

43 Empirical Results All parameters have their expected signs and are significantly different from zero The values and significance level are stable across the two models The subnetwork model is significantly better than the Path Size Logit (PSL) model Three challenges in route choice modeling p.40/61

44 Concluding remarks Network-free data are more reliable Data processing may bias the result We prefer to model explicitly the relationship between the data and the model Three challenges in route choice modeling p.41/61

45 Choice set generation Frejinger, E. and Bierlaire, M. (2007). Stochastic Path Generation Algorithm for Route Choice Models. Proceedings of the Sixth Triennial Symposium on Transportation Analysis (TRISTAN) June 10-15, Three challenges in route choice modeling p.42/61

46 Introduction Choice sets need to be defined prior to the route choice modeling Path enumeration algorithms are used for this purpose, many heuristics have been proposed, for example: Deterministic approaches: link elimination (Azevedo et al., 1993), labeled paths (Ben-Akiva et al., 1984) Stochastic approaches: simulation (Ramming, 2001) and doubly stochastic (Bovy and Fiorenzo-Catalano, 2006) Three challenges in route choice modeling p.43/61

47 Introduction Underlying assumption: the actual choice set is generated Empirical results suggest that this is not always true Our approach: True choice set = universal set Too large Sampling of alternatives Three challenges in route choice modeling p.44/61

48 Sampling of Alternatives Multinomial logit model (e.g. Ben-Akiva and Lerman, 1985): P(i C n ) = q(c n i)p(i) j C n q(c n j)p(j) = ev in+ln q(c n i) e Vjn+ln q(cn j) j C n C n : set of sampled alternatives q(c n j): probability of sampling C n given that j is the chosen alternative Three challenges in route choice modeling p.45/61

49 Importance Sampling of Alternatives Attractive paths have higher probability of being sampled than unattractive paths Path utilities must be corrected in order to obtain unbiased estimation results Three challenges in route choice modeling p.46/61

50 Stochastic Path Enumeration Flexible approach that can be combined with various algorithms, here a biased random walk approach The probability of a link l with source node v and sink node w is modeled in a stochastic way based on its distance to the shortest path Kumaraswamy distribution, cumulative distribution function F(x l a,b) = 1 (1 x a l ) b for x l [0, 1]. x l = SP(v,d) C(l) + SP(w,d) Three challenges in route choice modeling p.47/61

51 Stochastic Path Enumeration 1.0 a = 1, 2, 5, 10, 30 b = 1 F(xl a,b) x l Three challenges in route choice modeling p.48/61

52 Stochastic Path Enumeration Probability for path j to be sampled q(j) = q((v,w) E v ) l=(v,w) Γ j Γ j : ordered set of all links in j v: source node of j E v : set of all outgoing links from v Issue: in theory, the set of all paths U is unbounded. We treat it as bounded with size J. Three challenges in route choice modeling p.49/61

53 Sampling of Alternatives Following Ben-Akiva (1993) Sampling protocol 1. A set C n is generated by drawing R paths with replacement from the universal set of paths U 2. Add chosen path to C n Outcome of sampling: ( k 1, k 2,..., k J ) and J j=1 k j = R P( k 1, k 2,..., k J ) = R! k j U j! j U q(j) e k j Alternative j appears k j = k j + δ cj in C n Three challenges in route choice modeling p.50/61

54 Sampling of Alternatives Let C n = {j U k j > 0} q(c n i) = q( C n i) = R! K Cn = Q j Cn k j! R! (k i 1)! j C n j i j C n q(j) k j k j! q(i)k i 1 j C n j i q(j) k j = K Cn k i q(i) P(i C n ) = in+ln( q(i)) ki ev j C n e Vjn+ln kj q(j) Three challenges in route choice modeling p.51/61

55 Preliminary Numerical Results Estimation of models based on synthetic data generated with postulated models Non-correlated paths Postulated model same as estimated model (multinomial logit) Correlated paths in a grid-like network Postulated model is probit and estimated models are multinomial logit and path size logit True parameter values are compared to estimates Three challenges in route choice modeling p.52/61

56 Preliminary Numerical Results SB 1 = 3, L 1 = 2 SB 2 = 4, L 2 = 4 SB 3 = 1, L 3 = 6 O SB j, L j D SB 39 = 0, L 39 = 78 SB 40 = 0, L 40 = 80 Three challenges in route choice modeling p.53/61

57 Preliminary Numerical Results True model: multinomial logit U j = β L length j + β SB nbspeedbumps j + ε j β L = 0.6 and β SB = 0.3 ε j is distributed Extreme Value with location parameter 0 and scale observations, therefore 500 choice sets are sampled Biased random walk using 40 draws with a = 2 and b = 1 Generated choice sets include at least 7, maximum 18 and on average 11.9 paths Three challenges in route choice modeling p.54/61

58 Preliminary Numerical Results MNL MNL Sampling correction without with bβ L (-0.6) Scaled estimate Robust std Robust t-test bβ SB (-0.3) Scaled estimate Robust std Robust t-test Null log-likelihood Final log-likelihood Adjusted ρ BIOGEME has been used for all model estimations. Three challenges in route choice modeling p.55/61

59 Preliminary Numerical Results SB SB D SB SB SB SB O SB SB Three challenges in route choice modeling p.56/61

60 Preliminary Numerical Results True model: probit (Burrell, 1968) U l = β L length l + β SB nbspeedbumps l + σ L l ν l β L = 0.6 and β SB = 0.4 ν l is distributed standard Normal Link utility variance assumed proportional to length with parameter σ = 0.8 Path utilities are link additive 382 observations are generated after 500 realizations of the link utilities Three challenges in route choice modeling p.57/61

61 Preliminary Numerical Results Biased random walk using 30 draws with a = 2 and b = 1 (382 choice sets) Generated choice sets include at least 7, maximum 19 and on average 13.5 paths Three challenges in route choice modeling p.58/61

62 Preliminary Numerical Results MNL MNL PSL PSL Sampling correction without with without with bβ L (-0.6) Scaled estimate Robust std Robust t-test bβ SB (-0.4) Scaled estimate Robust std Robust t-test bβ PS Scaled estimate Robust std Robust t-test Three challenges in route choice modeling p.59/61

63 Preliminary Numerical Results MNL MNL PSL PSL Sampling correction without with without with Null log-likelihood Final log-likelihood Adjusted ρ BIOGEME has been used for all model estimations. Three challenges in route choice modeling p.60/61

64 Conclusions and Future Work Stochastic path enumeration algorithms are viewed as an approach for importance sampling of alternatives We propose an algorithm that allows for computation of path selection probabilities and correction for sampling Ongoing research, further work will be dedicated, for example, to empirical results on real data and correction in prediction Three challenges in route choice modeling p.61/61