A sampling of alternatives approach for route choice models

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1 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 1/34 A sampling of alternatives approach for route choice models Emma Frejinger Centre for Transport Studies, KTH, Stockholm Joint work with Michel Bierlaire and Moshe Ben-Akiva

2 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 2/34 Short about CTS at KTH A collaboration between KTH (Transport and Location Analysis, Transport and Logistics) VTI Transport Economics SIKA, WSP, JIBS 10 year financing VINNOVA, Rail adm., Road adm., Rikstrafiken The partners Model development, appraisal methodology, applied appraisal and analysis

3 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 3/34 Outline Introduction to route choice modeling and choice set generation Sampling of alternatives and derivation of sampling correction Stochastic path generation Expanded path size attribute Numerical results Conclusions and future work

4 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 4/34 Introduction Given a uni-modal transportation network an origin-destination pair link and path attributes socio-economic characteristics of a traveler identify the route a (given) traveler would select Important problem in e.g. intelligent transport systems, GPS navigation and transportation planning

5 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 5/34 Introduction Most simple route choice model: travelers use the shortest path with respect to any generalized cost function Behaviorally unrealistic Random utility models Here we consider estimation of random utility models based on disaggregate revealed preferences data

6 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 6/34 Introduction Network Trips

7 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 6/34 Introduction Network Trips Path Observations

8 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 6/34 Introduction Network Trips Choice sets Path Observations

9 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 6/34 Introduction Network Trips Choice sets Path Observations Description of correlation Route choice model

10 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 7/34 Introduction Set of all paths U from o to d Path generation Deterministic M U Stochastic M n U Choice set formation Deterministic Probabilistic Route choice model P(i C n ) P(i) = X C n G n P(i C n )P(C n )

11 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 8/34 Introduction Underlying assumption in existing approaches: the actual choice set is generated Empirical results suggest that this is not always true In order to avoid bias in the econometric model we propose True choice set = universal set U Too large Sampling of alternatives

12 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 9/34 Sampling of Alternatives Multinomial Logit model can be consistently estimated on a subset of alternatives (McFadden, 1978) P(i C n ) = q(c n i)p(i) j C n q(c n j)p(j) = eµv inln q(c n i) e µvjnln 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

13 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 10/34 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

14 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 11/34 MNL Route Choice Models Path Size Logit (Ben-Akiva and Ramming, 1998 and Ben-Akiva and Bierlaire, 1999) and C-Logit (Cascetta et al. 1996) Additional attribute in the deterministic utilities capturing correlation among alternatives These attributes should reflect the true correlation structure Here we use the Path Size Logit with an Expanded Path Size attribute

15 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 12/34 Sampling Correction Based on Ben-Akiva (1993) Sampling protocol 1. A set C n is generated by drawing R n paths with replacement from the universal set of paths U 2. Add chosen path to C n Outcome of sampling: ( k 1n, k 2n,..., k Jn ) and j U k jn = R n P( k 1n, k 2n,..., k Jn ) = R n! k j U jn! q(j) e k jn j U

16 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 13/34 Sampling Correction Alternative j appears k jn = k jn δ jc in C n 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 ) = inln( q(i)) ki eµv j C n e µvjnln kj q(j)

17 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 14/34 Sampling Correction General methodology valid for any definition of U if it exists an algorithm generating any path in U with non zero probability path sampling probabilities can be computed

18 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 15/34 Stochastic Path Generation Biased (towards shortest path) random walk algorithm A measure of distance x l [0, 1] to the shortest path for link l = (v,w) x l = SP(v,s d ) C(l) SP(w,s d ) Parametrized function mapping [0, 1] into itself (inspired from the Kumaraswamy distribution) ω(l b 1,b 2 ) = 1 ( 1 x b 1 l ) b2

19 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 16/34 Stochastic Path Generation 1.0 b 1 = 1, 2, 5, 10, 30 b 2 = 1 ω(l b1, b2) x l

20 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 17/34 Stochastic Path Generation Initialize v = s o, Γ = Loop While v s d perform the following Weights For each link l = (v,w) E v, where E v is the set of outgoing links from v, we compute the weights ω(l b 1,b 2 ) Probability For each link l = (v,w) E v, we compute q(l E v,b 1,b 2 ) = ω(l b 1,b 2 ) m E v ω(m b 1,b 2 ) Draw Randomly select a link (v,w ) in E v based on the above probability distribution Update path Γ = Γ (v,w ) Next node v = w.

21 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 18/34 Stochastic Path Generation Probability q(j) of generating path j is q(j) = l Γ j q(l E v,b 1,b 2 )

22 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 19/34 Expanded Path Size Path size attribute should reflect the true correlation structure Original formulation PS C in = a Γ i L a L i 1 M an, M an = j C n δ aj Expanded formulation EPS in = L a 1, M L i M EPS an EPS a Γ an i = j C n δ aj Φ jn

23 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 20/34 Expanded Path Size Expansion factor Φ jn = 1 if δ jc = 1 or q(j)r n 1 1 q(j)r n otherwise Chosen alternative always included Duplicates are ignored Asymptotically valid if R n then q(j)r n 1 j U and M EPS an j U δ aj

24 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 21/34 Numerical Results Evaluation of the impact on estimation results of the sampling correction, the definition of the PS attribute, the biased random walk algorithm parameters and the definition of the postulated model used for generating the data

25 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 22/34 Numerical Results Data SB D SB SB SB SB SB O

26 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 23/34 Numerical Results Data Two sets of observations with two postulated Path Size Logit models U in = β PS ln PS U i β L Length i β SB NbSB i ε in β PS = 1, β SB = 0.1, ε in i.i.d. EV(0,1) β L = 0.3 and β L = 1 Path size PS U i = L a 1 L i a Γ i δ aj j U 3000 observations per dataset

27 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 24/34 Numerical Results Probability Path number β L = 0.3 β L = 1

28 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 25/34 Numerical Results Models ) M NoCorr PS(C) V in = µ (β PS ln PS Cin β L Length i β SB NbSB i ) M Corr PS(C) V in = µ (β PS ln PS Cin β L Length i β SB NbSB i M NoCorr PS(U) V i = µ M Corr PS(U) V in = µ (β PS ln PS Ui β L Length i β SB NbSB i ) (β PS ln PS Ui β L Length i β SB NbSB i ) ln( k in q(i) ) ln( k in q(i) ) M Corr EPS V in = µ (β PS ln EPS in β L Length i β SB NbSB i ) ln( k in q(i) )

29 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 26/34 Numerical Results Sampling Number of draws: 10, 20, 40, 80, 120, 170, 250 Parameters of the distribution: b 1 = 1, 2, 3, 5, 10, 15, 20 with b 2 always fixed to one

30 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 27/34 Numerical Results Average choice set size Number of draws b 1 = 1 b 1 = 2 b 1 = 3

31 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 28/34 Numerical Results Estimation Over 300 models have been estimated Two datasets, five model specifications, different settings of the sampling algorithm Detailed results for one setting: β L = 1 dataset using 40 draws and b 1 = 1 Average size of sampled choice sets: BIOGEME (Bierlaire, 2003, and Bierlaire, 2007)

32 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 29/34 Numerical Results M NoCorr P S(U) M Corr P S(U) M NoCorr P S(C) M Corr P S(C) M Corr EPS bβ PS Rob. std Rob. t-test bβ SB Rob. std Rob. t-test bµ Rob. std Rob. t-test Final L-L Adj. rho bar sq

33 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 30/34 Numerical Results Speed Bump Coef. β b SB b 1 = 1 b 1 = 2 b 1 = β L = 1, MEPS Corr β L = 1, M Corr P S(C) Speed Bump Coef. β b SB Scale Param. bµ Scale Param. bµ PS Coef. b β PS PS Coef. β b PS

34 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 31/34 Numerical Results Speed Bump Coef. β b SB b 1 = 1 b 1 = 2 b 1 = β L = 0.3, MEPS Corr β L = 0.3, M Corr P S(C) Speed Bump Coef. β b SB Scale Param. bµ Scale Param. bµ PS Coef. b β PS PS Coef. β b PS

35 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 32/34 Conclusions Network Trips Sampling of paths Expanded Path Size Description of correlation Choice sets Sampling correction Path Observations Route choice model

36 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 33/34 Conclusions New paradigm for choice set generation and route choice model estimation Aim: avoid bias in the econometric model Path generation is importance sampling of alternatives Sampling correction of path utilities and Path Size attribute Numerical results show that true parameter values can be retrieved

37 Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models p. 34/34 Future Work Results based on real data Forecasting

Route choice models: Introduction and recent developments

Route choice models: Introduction and recent developments Michel Bierlaire transp-or.epfl.ch Transport and Mobility Laboratory, EPFL Route choice models: Introduction and recent developments p.1/40 Route