Bierlaire, Michel

Frejinger, Emma

Infoscience - École polytechnique fédérale de Lausanne

Importance sampling of alternatives for routechoice modelsEmma Frejinger and Michel BierlaireTransport and Mobility Laboratory, EPFL,transp-or.epfl.chImportance sampling of alternatives for route choice models – p.1/25Outline• Introduction to route choice modeling• Modeling framework• Estimation• Issues• Stochastic path enumeration approach• Sampling of alternatives• Preliminary numerical resultsImportance sampling of alternatives for route choice models – p.2/25Route choice modelingGiven a transportation network composed of nodes, links,origin and destinations.For a given transportation mode and origin-destinationpair, which is the chosen route?Importance sampling of alternatives for route choice models – p.3/25Route choice modeling• Deterministic approach: Travelers use the shortest(with regard to any arbitrary generalized cost) routeamong all• Behaviorally unrealistic• Random utility models (discrete choice models)Importance sampling of alternatives for route choice models – p.4/25Framework• Utility maximization• An individual n associates a utility Ujn with each path jin his/her choice set Cn and chooses the alternativewith the highest utilityImportance sampling of alternatives for route choice models – p.5/25Random Utility ModelsUjn = Vjn + εjnVjn: Deterministic part Vjn = βTXjnβ: vector of parameters to be estimatedXjn: attributesεjn: random termMultinomial Logit modelP (i|Cn) = eVin∑j∈CneVjnImportance sampling of alternatives for route choice models – p.6/25Estimation• Maximum likelihood estimationL∗(βˆ1, ..., βˆK) = maxβ∈RL(β) =N∑n=1lnPn(β)• BIOGEME: estimation softwareBierlaire’s Optimization Toolbox for GEV ModelEstimationImportance sampling of alternatives for route choice models – p.7/25Problem characteristics• Universal choice set very large• Individual specific choice set unknown• Correlated alternatives due to overlapping paths• Data issuesImportance sampling of alternatives for route choice models – p.8/25Path Enumeration• Many heuristics are proposed in the literature• Deterministic and stochasticExamples: link elimination (Azevedo et al., 1993),labeled paths (Ben-Akiva et al., 1984), simulation(Ramming, 2001) and doubly stochastic (Bovy andFiorenzo-Catalano, 2006)• These approaches assume that generated choicesets include all alternatives considered by thetravelersImportance sampling of alternatives for route choice models – p.9/25Importance Sampling Approach• All paths belong to the true choice set• Objective: define choice set allowing for unbiasedestimation and prediction results• We view stochastic path enumeration algorithms asimportance sampling of alternatives• In order to obtain unbiased results, path utilities mustbe corrected• We propose a stochastic path enumeration algorithmthat allows the computation of sampling correctionImportance sampling of alternatives for route choice models – p.10/25Stochastic Path Enumeration• We choose to include in the choice set a link ℓ or asequence of links in a stochastic way based on itsdistance to the shortest path• Paths can be generated using different algorithms• Kumaraswamy distribution, cumulative distributionfunction F (xℓ|a, b) = 1− (1− xℓa)b for xℓ ∈ [0, 1].xℓ =SP (o, d)SP (o, i) + C(ℓ) + SP (j, d)Importance sampling of alternatives for route choice models – p.11/25Stochastic Path Enumeration0 1.001.0xℓF(xℓ|a,b)1 2 5 10 30b = 1a = 1, 2, 5, 10, 30Importance sampling of alternatives for route choice models – p.12/25Stochastic Path Enumeration• Biased random walk algorithmq(j) =∏ℓ∈Γjq(ℓ|Ev)• Γj: set of all links in j• v: source node of j• Ev: set of all outgoing links from v• q(ℓ|Ev) is distributed Kumaraswamy• Issue: the set of all paths U is unbounded but weassume∑j∈U q(j) ≈ 1 and treat it as boundedImportance sampling of alternatives for route choice models – p.13/25Sampling of Alternatives• Multinomial Logit model: Probability of i conditional onthe choice set Cn defined by the analyst (e.g.Ben-Akiva and Lerman, 1985)P (i|Cn) = q(Cn|i)P (i)∑j∈Cnq(Cn|j)P (j)=eVin+ln q(Cn|i)∑j∈CneVjn+ln q(Cn|j)q(Cn|j): probability of sampling Cn given that j is thechosen alternativeImportance sampling of alternatives for route choice models – p.14/25Sampling of Alternatives• Sampling protocol: a set C˜n is generated by drawing Rpaths with replacement from the universal set of pathsU and adding the chosen path to itOutcome of sampling: (k˜1, k˜2, . . . , k˜J) and∑j∈U k˜j = RP (k˜1, k˜2, . . . , k˜J) =R!∏j∈U k˜j!∏j∈Uq(j)ekj• Alternative j appears kj = k˜j + δcj in C˜nImportance sampling of alternatives for route choice models – p.15/25Sampling of Alternatives• Let Cn = {j ∈ U | kj > 0}• Following Ben-Akiva (1993)q(C˜n|i) = R!(ki − 1)!∏j∈Cnj 6=ikj!q(i)ki−1∏j∈Cnj 6=iq(j)kj = KCnkiq(i)KCn =R!Qj∈Cnkj !∏j∈Cnq(j)kjP (i|C˜n) = eVin+ln( kiq(i))∑j∈CneVjn+ln“kjq(j)”Importance sampling of alternatives for route choice models – p.16/25Preliminary Numerical Results• Estimation of models based on synthetic datagenerated with postulated models• Non-correlated paths• Correlated paths in a “grid-like” network• True parameter values are compared to estimatesImportance sampling of alternatives for route choice models – p.17/25Preliminary Numerical Results• True model: multinomial logitUj = βL lengthj + βSB nbspeedbumpsj + εjβL = −0.6 and βSB = −0.3εj is distributed Gumbel with location parameter 0 andscale 1• 500 observations• Biased random walk using 40 draws with a = 2 andb = 1Generated choice sets include at least 7, maximum 18and on average 11.9 pathsImportance sampling of alternatives for route choice models – p.18/25Preliminary Numerical ResultsMNL MNLSampling correction without withbβL -0.203 -0.286Scaled estimate -0.600 -0.600Robust std. 0.0193 0.019Robust t-test -10.53 -15.01bβSB -0.0194 -0.143Scaled estimate -0.0573 -0.300Robust std. 0.0662 0.0661Robust t-test -0.29 -2.17Null log-likelihood -1069.453 -1633.501Final log-likelihood -788.42 -759.848Adjusted ρ¯2 0.261 0.288BIOGEME has been used for all model estimations.Importance sampling of alternatives for route choice models – p.19/25Preliminary Numerical ResultsODSB SBSB SBSB SBSB SBImportance sampling of alternatives for route choice models – p.20/25Preliminary Numerical Results• True model: probit (Burrell, 1968)Uℓ = βL lengthℓ + βSB nbspeedbumpsℓ + σ√LℓνℓβL = −0.6 and βSB = −0.4νℓ is distributed standard NormalLink utility variance assumed proportional to lengthwith parameter σ = 0.8• Path utilities are link additive• 382 observations are generated after 500 realizations ofthe link utilitiesImportance sampling of alternatives for route choice models – p.21/25Preliminary Numerical Results• Biased random walk using 30 draws with a = 2 andb = 1Generated choice sets include at least 7, maximum 19and on average 13.5 pathsImportance sampling of alternatives for route choice models – p.22/25Preliminary Numerical ResultsMNL MNL PSL PSLSampling correction without with without withbβL -0.627 -0.978 -0.619 -0.969Scaled estimate -0.600 -0.600 -0.600 -0.600Robust std. 0.0397 0.032 0.0407 0.0358Robust t-test -15.79 -30.57 -15.22 -27.04bβSB -0.0822 -0.0801 -0.347 -0.461Scaled estimate -0.0787 -0.0491 -0.336 -0.285Robust std. 0.052 0.0559 0.182 0.158Robust t-test -1.58 -1.43 -1.90 -2.92bβPS 1.17 1.74Scaled estimate 1.13 1.08Robust std. 0.788 0.705Robust t-test 1.49 2.47Importance sampling of alternatives for route choice models – p.23/25Preliminary Numerical ResultsMNL MNL PSL PSLSampling correction without with without withNull log-likelihood -988.63 -2769.959 -988.63 -2769.959Final log-likelihood -676.111 -653.396 -674.481 -649.268Adjusted ρ¯2 0.314 0.337 0.315 0.340BIOGEME has been used for all model estimations.Importance sampling of alternatives for route choice models – p.24/25Conclusions and Future Work• Ongoing research• Modeling path enumeration as importance sampling ofalternatives is promising however some work remain• Implications of ∑j∈U q(j) ≈ 1• Empirical results on real data• Correction in predictionImportance sampling of alternatives for route choice models – p.25/25

Importance sampling of alternatives for route choice models

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Importance sampling of alternatives for route choice models

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Infoscience - École polytechnique fédérale de Lausanne