Structural estimation for a route choice model with uncertain measurement

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1 Structural estimation for a route choice model with uncertain measurement Yuki Oyama* & Eiji Hato *PhD student Behavior in Networks studies unit Department of Urban Engineering School of Engineering, The University of Tokyo September 25, 2016 Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Sep. 25, 2016

2 Outline Outline Outline 1. Introduction 2. Link-based route measurement model 3. Structural estimation method 4. Numerical examples 5. Conclusions Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

3 Outline Outline Outline 1. Introduction 2. Link-based route measurement model 3. Structural estimation method 4. Numerical examples 5. Conclusions Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

4 Motivation Route choice analysis Introduction Parameter estimation results largely depend on accuracy of route measurement GPS data Route measurement model Route choice data Route choice model Estimated parameters Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

5 Motivation Route choice analysis Introduction Parameter estimation results largely depend on accuracy of route measurement GPS data Route measurement model Route choice data Route choice model Estimated parameters Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

Motivation Introduction Pedestrian route

Dense and high-resolution network N Dense

errors ü Along river ü Wide street ü Narrow

6 Motivation Introduction Pedestrian route choice analysis Measurement uncertainty; Dense and high-resolution network N Dense network Spatial dependence of Measurement errors ü Along river ü Wide street ü Narrow street ü With arcade (m) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

7 Literature review Introduction Route measurement models (1) Sequential approach infers the true location at each data in chorological order : GPS data Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

8 Literature review Introduction Route measurement models (1) Sequential approach can output meaningless paths Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

9 Literature review Introduction Route measurement models (2) Path-based probabilistic approach evaluates path likelihood regarding all GPS data included in a trip Pyo et al. (2001); Bierlaire et al. (2013) Domain of Data Relevance Likelihood 10% 40% 50% Path 1 Path 2 Path 3 Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

10 Literature review Introduction Route measurement models (2) Path-based probabilistic approach suffers with trade-off between computational efficiency and measurement accuracy Pyo et al. (2001); Bierlaire et al. (2013) Large Small Path candidate set Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

σ : Large σ : Small Measurement probability 0.00 0.01 0.02 0.03 0.04 0.

11 Literature review Introduction Route measurement models (2) Assuming error variance constant on network distort measurement probabilities PDF of GPS measurement error: p( ˆx j x j ;σ ) = ˆx j x j σ 2 " exp$ ˆx j x j $ 2σ 2 # 2 % ' ' & σ : Large σ : Small Measurement probability ˆx t j x j = d σ =10 σ = 50 σ = Distance from the true location Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

Literature review Introduction Route measurement models (3) Bayesian approach incorporates behavioral models into measurement models Chen et al. (2013); Danalet et al.

12 Literature review Introduction Route measurement models (3) Bayesian approach incorporates behavioral models into measurement models Chen et al. (2013); Danalet et al. (2014) Prior (Route choice model) 25% 15% 60% Measurement 10% 40% 50% Posterior 16% 6% 78% Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

13 Literature review Introduction Route measurement models (3) Bayesian approach has a problem regarding parameter inconsistency Given parameters Arbitrary Previous study Other data GPS data Route measurement model Route choice model Prob. Route choice data Route choice model Estimated parameters Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

14 Literature review Introduction Route measurement models Challenges: Disconnected path Not suitable for route choice models Setting of the measurement parameter Possible to miss the true path Ignorance of spatial difference distorts path likelihood Parameter inconsistency of route choice model Estimated parameter includes biases regarding initial parameter Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

15 Framework Introduction Framework 1. Link-based route measurement model Matching each decomposed trip data to a link Estimating a measurement parameter for each link Incorporating a link-based route choice model as prior 2. Structural estimation method Parameters at convergence satisfy the parameter consistency of the route choice model Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

16 Outline Outline Outline 1. Introduction 2. Link-based route measurement model 3. Structural estimation method 4. Numerical examples 5. Conclusions Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

17 Problem & Notation Problem & Notation Link-based route measurement model Matching row GPS data to ˆm the transportation network G GPS data ˆm = ( ˆx, ˆ τ ) Pair of coordinateswith ˆx = ( ˆx lat, ˆx lon ) error variance Timestamp τˆ A given trip ˆm = ( ˆm 1,..., ˆm n,..., ˆm N ) Network G = (V, A) Node v V : the horizontal position x v = {x lat, x lon } Link: a = (v u, v d ) A the vector of spatial attributes Network connection δ(a' a) : 1/0 σ y a Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

18 Flow Link-based route measurement model Link-based route measurement Matching all data observed within a period to the same link ˆm 1 ˆm 2 ˆm n ˆm N t = T ˆm ˆm 1 J ˆm 1 ˆm J GPS data of a trip ˆm = ( ˆm 1,..., ˆm n,..., ˆm N ) ˆm n = ( ˆx n, ˆn) Data decompopsition ˆm = ( ˆm 1,..., ˆm t,..., ˆm T ) ˆm t = ( ˆm t 1,..., ˆm t j,..., ˆm t J ) Estimation: a 1 a 2 a T at p(a t ˆm t, a t 1 ) p( ˆm t a t ; a t )p(a t a t 1 ; ) Link probabilities a 1 Path inference: a 2 a T = [a 1,..., a t,..., a T ] p( ˆm t a t ) : Measurement probability of ˆm t given a t ; measurement equation p(a t a t 1 ) : Prior probability of a t given a t 1 ; system equation Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

19 Link probability a t Link probability Link-based route measurement model p(a t ˆm t, a t 1 ) The probability of given measurements and state Candidate set: A(a t 1 ) = {a t δ(a t a t 1 ) =1} Calculate link probabilities for all a t A(a t 1 ) ˆm t a t 1 ˆm 1 t 1 ˆm 3 t a t 1 ˆm 2 t 2 3 : GPS data ˆm t = { ˆm 1 t, ˆm 2 t, ˆm 3 t } Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

20 Measurement equation Measurement equation The probability of measurements ˆm t given a t Link-based route measurement model p( ˆm t a t ;σ at ) Assumption: τˆ Timestamp has no measurement error; p( ˆm t a t ) = p( ˆx t a t ) Measurement probability of data is independent from each other Traveler moves at the constant speed on the same link p( ˆx 1 t,..., ˆx J t a t ;σ at ) = p( ˆx j t J j=1 J x j a t a t ;σ at ) = p( ˆx j t j=1 x t j, a t ;σ at ) p(x j a t )dx j PDF of GPS measurement error: Rayleigh distribution (van Diggelen, 2007) p( ˆx j t x t j, a t ;σ at ) = ˆx t j x " j exp ˆx t 2 $ j x j 2 2 σ at $ 2σ # at % ' ' & Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

Measurement equation Link-based route measurement model Estimation of measurement parameter Link-based map matching can regard error variance as a link peculiar variable σ at Measurement likelihood

21 Measurement equation Link-based route measurement model Estimation of measurement parameter Link-based map matching can regard error variance as a link peculiar variable σ at Measurement likelihood maximization Where, p( ˆm t a t ;σ at ) = J j=1 σ at x j a t = argmax σ ) + *, + p( ˆm t a t ;σ ) ˆx t j x " j exp ˆx t 2 $ j x j 2 2 σ at $ 2σ # at % - ' ' p(x a ) + j t. dx j & / + Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

22 System equation The prior probability of given a state a t System equation Link-based route measurement model p(a t a t 1 ;θ) a t 1 Link-based route choice model Utility function: A(a t 1 ) u(a t a t 1 ) = v(a t a t 1 )+ε(a t ) =θy at a t 1 +ε(a t ) 1 a t 1 2 v( ) : Deterministic component of utility ε( ) : Probabilistic component of utility (i.i.d. gumbel distribution) 3 y at a t 1 θ : Vector of explanatory variables : Vector of parameters Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

23 System equation The prior probability of given a state a t System equation Link-based route measurement model p(a t a t 1 ;θ) a t 1 Link-based route choice model 1 A(a t 1 ) Choice probability option: Markov model p(a t a t 1 ) = a t A(a t 1 ) e v(a t a t 1 ) e v(a t a t 1 ) a t 1 2 Recursive logit model Fosgerau et al. (2013) 3 p(a t a t 1 ) = And others: e.g., a t A(a t 1 ) e v(a t a t 1 )+V d (a t ) e v(a t a t 1 )+V d (a t ) Mai et al. (2015); Mai (2016); Oyama et al. (2016) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

24 Link inference Link inference Link-based route measurement model Link (posterior) probability: a t The probability of given measurements and a state ˆm t a t 1 p(a t ˆm t, a t 1 ) p( ˆm t a t ;σ at )p(a t a t 1 ;θ) Link inference: Link likelihood maximization subject to switching condition a t = argmax a t A(a t 1 ) p(a t ˆm t, a t 1 ) s.t., max p( ˆm t+1 a;σ a ) > γ a A(a t ) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

25 Outline Outline Outline 1. Introduction 2. Link-based map matching algorithm 3. Structural estimation method 4. Numerical examples 5. Conclusions Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

26 A fixed point problem Methodology Structural estimation method Need to solve a fixed point problem of route measurement and estimation Route measurement Route choice probability p(a t a t 1 ;! θ ) Path ψ Measurement probability p( ˆm t a t ;σ at ) Where, # LL(θ;ψ) = ln% $ Route choice model θ = argmax θ s.t., i δ at T i LL(θ;ψ) θ! θ < ζ t=2 " = 1, if a i,t ψ i # $ 0, otherwise p(a t a t 1 ;θ) δ i at & ( ' Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

27 Methodology Structural estimation method Structural estimation A method for parameter estimation of models with fixed point problem NFXP (Nested Fixed Point) Rust (1987) NPL (Nested Pseudo Likelihood) Aguirregabiria and Mira (2002) MPEC (Mathematical Programming with Equilibrium Constraint) Su and Judd (2012) Structural estimation for route choice model with uncertain data Solving a fixed problem regarding parameter of route choice model Inner problem: Route measurement model Outer problem: Parameter estimation of route choice model Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

28 Structural estimation Methodology Structural estimation method A estimation method for solving a fixed point problem of route choice parameter h = 1, (h) = GPS data ˆm Measurement model p i (a t ˆm i t, a t 1 ) = a t a i,t = argmax a t p( ˆm i a t ; a t )p(a t a t 1 ; p( ˆm t i a t ; a t )p(a t a t 1 ; ˆm i = ( ˆm i 1,..., ˆm i t,..., ˆm i T ) p i (a t ˆm t i, a t 1 ; (h) i = [a i,1,..., a i,t,..., a i,t ] (h) ) (h) ) (h) ) (h) Behavior model max LL( ) = log I i t=2 (h) a i,t = 1, if a (h) i,t i 0, otherwise (h+1) = argmax LL( ; T (h ) p i (a t a t 1 ; ) a i,t (h) ) (h) h := h+1 No (h+1) (h) < Yes = (h), = (h) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

29 Outline Outline Outline 1. Introduction 2. Link-based map matching algorithm 3. Structural estimation method 4. Numerical examples 5. Conclusions Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

30 Twins experiment Numerical examples Twins experiments Simulation y y = 90 (2/0/5) (2/0/5) (2/0/5) d D Settings y = 60 y = 30 O o (2/1/5) (2/1/5) (2/1/5) (3/1/5) (1/0/20) (1/0/20) (1/0/20) (3/1/5) (2/0/5) (2/1/5) (3/1/5) (2/1/10) (2/1/10) (2/1/10) (3/1/5) (3/1/5) (3/1/5) (2/1/5) (2/1/5) (1/1/5) (2/1/5) x = 40 x = 80 x = 120 *(continuous cost: CC a / discrete cost: xdc a / variance: ) v(a k) =θ 1 TT a +θ 2 CC a +θ 3 DC a +θ 4 UT a k True parameter: Period interval: Data generation: a! θ = [ 0.1, 2, 1.5, 4] t = 30s ˆ τ j ˆ τ j 1 =10s : True path : Measurement y y y x x x (a) (b) (c) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

31 Twins experiment Numerical examples Twins experiments Measurement results Which model improves the route measurement accuracy? Table: Measurement accuracy and the difference of the parameter from the true value Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

32 Twins experiment Numerical examples Twins experiments Estimation results Does structural estimation method improve the parameter estimation results? Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

33 Real data Real data Numerical examples Matsuyama Probe Person data in 2007, 30 pedestrians, 729 locations N (m) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

34 Real data Numerical examples Real data Model specification Route choice model: (static) Markov model Target: Pedestrian trip in city center Utility function: v(a k) =θ 1 TT a +θ 2 CU a +θ 3 DU a +θ 4 UT a k TT : Travel time (min.) CU : DU : Sidewalk width (m) Arcade dummy variable UT : U-turn dummy variable Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

35 Real data Numerical examples Real data Parameter estimation results Travel time ( θ 1 ) seems to be significant from the result of one-way model, however, Structural estimation results show that links with arcade ( θ 3 ) are the most likely to be passed by pedestrians; travel time ( θ 1 ) is not significant Other t-values and rho-square ( ρ 2 ) indicate that the structural estimation improves parameter estimation results Travel time (min.) Sidewalk width (m) With arcade U-turn Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

Real data Numerical examples Real data Estimation results of error variance Estimated measurement error variance is dependent on each link Estimation result

36 Real data Numerical examples Real data Estimation results of error variance Estimated measurement error variance is dependent on each link Estimation result of σ ~ 20 (or no measurement) 20 ~ ~ ~ ~ Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

37 Outline Outline Outline 1. Introduction 2. Link-based map matching algorithm 3. Structural estimation method 4. Numerical examples 5. Conclusions Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

38 Conclusions Conclusions Conclusions and Future work Conclusions A link-based measurement model with route choice model Estimation of measurement parameter for each link Structural estimation method for solving a fixed point problem regarding route choice parameters Future work Comparison of computational efficiency with previous measurement models Alternatives and utility of pedestrian link choice Characteristics of the fixed point problem Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

39 Thank you for attention! Questions? Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

40 References Reference Reference 1. Bierlaire, M., Chen, J., Newman, J., A probabilistic map matching method for smartphone GPS data. Transportation Research Part C: Emerging Technologies 26, Bierlaire, M., Frejinger, E., Route choice modeling with network-free data. Transportation Research Part C: Emerging Technologies 16(2), Chen, J., Bierlaire, M., Probabilistic multimodal map matching with rich smartphone data. Journal of Intelligent Transportation System 19(2), Danalet, A., Farooq, B., Bierlaire, M., A bayesian approach to detect pedestrian destination-sequences from WiFi signatures. Transportation Research Part C: Emerging Technologies 44, Fuse, T., Nakanishi, W., A study on multiple human tracking by integrating pedestrian behavior model (in Japanese). Journal of JSCE Series D3: Infrastructure Planning and Management 68(2), Fosgerau, M., Frejinger, E. and Karlstrm A., A link-based network route choice model with unrestricted choice set. Transportation Research Part B, 56, pp Hunter, T., Abbeel, P., Bayen, A., The path inference filter: model-based low-latency map matching of probe vehicle data. IEEE Transactions on Intelligent Transportation Systems 15(2), Pyo, J.S., Shin, D.H., Sung, T.K., Development of a map matching method using the multiple hypothesis technique. IEEE Proceedings on Intelligent Transportation Systems, Quddus, M.A., Noland, R.B., Ochieng, W.Y., Validation of map matching algorithms using high precision positioning with GPS. The Journal of Navigation 58(2), Quddus, M.A., Ochieng, W.Y., Noland, R.B., Current map-matching algorithms for transport applications: Stateof-the art and future research directions. Transportation Research Part C: Emerging Technologies 15(5), Quddus, M.A., Ochieng, W.Y., Zhao, L., Noland, R.B., A general mao matching algorithm for transport telematics applications. GPS Solutions 7(3), Quddus, M.A., Washington, S., Shortest path and vehicle trajectory aided mapmatching for low frequency GPS data. Transportation Research Part C: Emerging Technologies 55, van Diggelen, F., GNSS accuracy: lies, damn lies, and statistics. GPS World 18(1), Velaga, N.R., Quddus, M.A., Bistow, A.L., Developing an enhanced weight-based topological map-matching algorithm for intelligent transport systems. Transportation Research Part C: Emerging Technologies 17, White, C.E., Bernstein, D., Kornhauser, A.L., Some map matching algorithms for personal navigation assistants. Transportation Research Part C: Emerging Technologies 8, Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

41 Appendix Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

42 Literature review Introduction Previous route measurement models Geometric White et al. (2000) - Point-to-point - Point-to-curve - Curve-to-curve Topological Greenfield (2002) - Adjacency - Connectivity - Vehicle heading Probabilistic Ouchieng et al. (2004) Quddus et al. (2006) - Error region - Fuzzy logic Path-based - MHT Pyo et al. (2006) Bielraire et al. (2013) - Measurement equation 30% 70% Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

43 Methodology Link switching errors Link-based route measurement model Difficulties regarding link connectivity because of myopic optimization ˆm ˆm 2 1 ˆm 3 1 ˆm ˆm 3 2 ˆm ˆm ˆm : Estimated path ˆm 3 3 : More likely path Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

44 Link switching Link switching algorithm Link-based route measurement model STEP1: Calculating probabilities p( ˆm t p i (a t ˆm t i, a t 1 ) = i a t ; a t )p(a t a t 1 ; ) p( ˆm t i a t ; a t )p(a t a t 1 ; ) a t A(a t 1 ) STEP2: Sorting candidates by probabilities [a t,1,..., a t,r,..., a t, A(at 1 ) ] p(a t,1 ) p(a t,r ) p(a t, A(at 1 ) ) r = 1 STEP3: Calculating measurement equation at (t+1) where, a t+1 = argmax a ( ) LLm r = log p( ˆm t+1 a t+1 ; a t+1 ) p(a ˆm t+1, a t,r ) r := r+1 No LLm r J No < r = A(a t 1 ) Yes a t = a t,r r := 1 Yes Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

45 Real data Numerical examples Twins experiments iterations 100 Ave Link accuracy (%) Estimation error: a Sum #iteration Estimation error: : input : input : input = [ 1.5, 0.1, 2, 10] (wrong parameters) = [0,0,0,0] (no information) = [ 0.1, 2, 1.5, 4] (true parameters) Oyama, Y. (The University of Tokyo) 15 th Behavior Modeling summer course Jury Sep. 29, 25, 2016

A path choice approach to ac,vity modeling with a pedestrian case study

A path choice approach to ac,vity modeling with a pedestrian case study Antonin Danalet Sta,s,que, transport et ac,vités, Grenoble November 5, 2013 Presenta,on outline Motivation: Why pedestrian activities?