Computational Bayesian Statistics in Transportation Modeling: From Road Safety Analysis to Discrete Choice

In this paper, we review both the fundamentals and the expansion of computational Bayesian econometrics and statistics applied to transportation modeling problems in road safety analysis and travel behavior. Whereas for analyzing accident risk in transportation networks there has been a significant increase in the application of hierarchical Bayes methods, in transportation choice modeling, the use of Bayes estimators is rather scarce. We thus provide a general discussion of the benefits of using Bayesian Markov chain Monte Carlo methods to simulate answers to the problems of point and interval estimation and forecasting, including the use of the simulated posterior for building predictive distributions and constructing credible intervals for measures such as the value of time. Although there is the general idea that going Bayesian is just another way of finding an equivalent to frequentist results, in practice Bayes estimators have the potential of outperforming frequentist estimators and, at the same time, may offer more information. Additionally, Bayesian inference is particularly interesting for small samples and weakly identified models.     

Estimation of multinomial probit-kernel integrated choice and latent variable model: comparison on one sequential and two simultaneous approaches

Integrated choice and latent variable (ICLV) model incorporates latent factors into standard discrete choice model with aim to provide greater explanatory power. Using simulated datasets, this study makes a comparison among three estimation approaches corresponding to the sequential approach and two simultaneous approaches including the maximum simulated likelihood with GHK estimator and maximum approximate composite marginal likelihood (MACML) approach, to evaluate their abilities to recover the underlying parameters of multinomial probit-kernel ICLV model. The results show that both simultaneous approaches outperform the sequential approach in terms of estimates accuracy and efficiency irrespective of the sample sizes, and the MACML approach is the most preferable due to its best performance on recovering true values of parameters with relatively small standard errors, especially when the sample size is large enough.     

Bayesian inference for day-to-day dynamic traffic models

There is significant current interest in the development of models to describe the day-to-day evolution of traffic flows over a network. We consider the problem of statistical inference for such models based on daily observations of traffic counts on a subset of network links. Like other inference problems for network-based models, the critical difficulty lies in the underdetermined nature of the linear system of equations that relates link flows to the latent path flows. In particular, Bayesian inference implemented using Markov chain Monte Carlo methods requires that we sample from the set of route flows consistent with the observed link flows, but enumeration of this set is usually computationally infeasible.We show how two existing conditional route flow samplers can be adapted and extended for use with day-to-day dynamic traffic. The first sampler employs an iterative route-by-route acceptance–rejection algorithm for path flows, while the second employs a simple Markov model for traveller behaviour to generate candidate entire route flow patterns when the network has a tree structure. We illustrate the application of these methods for estimation of parameters that describe traveller behaviour based on daily link count data alone.     

A Bayesian approach for modeling origin-destination matrices

The majority of origin destination (OD) matrix estimation methods focus on situations where weak or partial information, derived from sample travel surveys, is available. Information derived from travel census studies, in contrast, covers the entire population of a specific study area of interest. In such cases where reliable historical data exist, statistical methodology may serve as a flexible alternative to traditional travel demand models by incorporating estimation of trip-generation, trip-attraction and trip-distribution in one model. In this research, a statistical Bayesian approach on OD matrix estimation is presented, where modeling of OD flows derived from census data, is related only to a set of general explanatory variables. A Poisson and a negative binomial model are formulated in detail, while emphasis is placed on the hierarchical Poisson-gamma structure of the latter. Problems related to the absence of closed-form expressions are bypassed with the use of a Markov Chain Monte Carlo method known as the Metropolis-Hastings algorithm. The methodology is tested on a realistic application area concerning the Belgian region of Flanders on the level of municipalities. Model comparison indicates that negative binomial likelihood is a more suitable distributional assumption than Poisson likelihood, due to the great degree of overdispersion present in OD flows. Finally, several predictive goodness-of-fit tests on the negative binomial model suggest a good overall fit to the data. In general, Bayesian methodology reduces the overall uncertainty of the estimates by delivering posterior distributions for the parameters of scientific interest as well as predictive distributions for future OD flows.     

Estimation of a constrained multinomial logit model

Identifying the set of available alternatives in a choice process after considering an individual’s bounds or thresholds is a complex process that, in practice, is commonly simplified by assuming exogenous rules in the choice set formation. The Constrained Multinomial Logit (CMNL) model incorporates thresholds in several attributes as a key endogenous process to define the alternatives choice/rejection mechanism. The model allows for the inclusion of multiple constraints and has a closed form. In this paper, we study the estimation of the CMNL model using the maximum likelihood function, develop a methodology to estimate the model overcoming identification problems by an endogenous partition of the sample, and test the model estimation with both synthetic and real data. The CMNL model appears to be suitable for general applications as it presents a significantly better fit than the MNL model under constrained behaviour and replicates the MNL estimates in the unconstrained case. Using mode choice real data, we found significant differences in the values of times and elasticities between compensatory MNL and semi-compensatory CMNL models, which increase as the thresholds on attributes become active.     

On finite sample performance of confidence intervals methods for willingness to pay measures

This paper systematically compares finite sample performances of methods to build confidence intervals for willingness to pay measures in a choice modeling context. It contributes to the field by also considering methods developed in other research fields. Various scenarios are evaluated under an extensive Monte Carlo study. Results show that the commonly used Delta method, producing symmetric intervals around the point estimate, often fails to account for skewness in the estimated willingness to pay distribution. Both the Fieller method and the likelihood ratio test inversion method produce more realistic confidence intervals for small samples. Some bootstrap methods also perform reasonably well, in terms of effective coverage. Finally, empirical data are used to illustrate an application of the methods considered.     

A new generalized heterogeneous data model (GHDM) to jointly model mixed types of dependent variables

This paper formulates a generalized heterogeneous data model (GHDM) that jointly handles mixed types of dependent variables—including multiple nominal outcomes, multiple ordinal variables, and multiple count variables, as well as multiple continuous variables—by representing the covariance relationships among them through a reduced number of latent factors. Sufficiency conditions for identification of the GHDM parameters are presented. The maximum approximate composite marginal likelihood (MACML) method is proposed to estimate this jointly mixed model system. This estimation method provides computational time advantages since the dimensionality of integration in the likelihood function is independent of the number of latent factors. The study undertakes a simulation experiment within the virtual context of integrating residential location choice and travel behavior to evaluate the ability of the MACML approach to recover parameters. The simulation results show that the MACML approach effectively recovers underlying parameters, and also that ignoring the multi-dimensional nature of the relationship among mixed types of dependent variables can lead not only to inconsistent parameter estimation, but also have important implications for policy analysis.     

Dynamic travel mode searching and switching analysis considering hidden model preference and behavioral decision processes

This paper proposes a conceptual framework to model the travel mode searching and switching dynamics. The proposed approach is structurally different from existing mode choice models in the way that a non-homogeneous hidden Markov model (HMM) has been constructed and estimated to model the dynamic mode srching process. In the proposed model, each hidden state represents the latent modal preference of each traveler. The empirical application suggests that the states can be interpreted as car loving and carpool/transit loving, respectively. At each time period, transitions between the states are functions of time-varying covariates such as travel time and travel cost of the habitual modes. The level-of-service (LOS) changes are believed to have an enduring impact by shifting travelers to a different state. While longitudinal data is not readily available, the paper develops an easy-to-implement memory-recall survey to collect required process data for the empirical estimation. Bayesian estimation and Markov chain Monte Carlo method have been applied to implement full Bayesian inference. As demonstrated in the paper, the estimated HMM is reasonably sensitive to mode-specific LOS changes and can capture individual and system dynamics. Once applied with travel demand and/or traffic simulation models, the proposed model can describe time-dependent multimodal behavior responses to various planning/policy stimuli.     

Novel Methods and the Maximum Likelihood Estimation Technique for Estimating Traffic Critical Gap

Most unsignalised intersection capacity calculation procedures are based on gap acceptance models. Accuracy of critical gap estimation affects accuracy of capacity and delay estimation. Several methods have been published to estimate drivers' sample mean critical gap, the Maximum Likelihood Estimation (MLE) technique regarded as the most accurate. This study assesses three novel methods; Average Central Gap (ACG) method, Strength Weighted Central Gap method (SWCG), and Mode Central Gap method (MCG), against MLE for their fidelity in rendering true sample mean critical gaps. A Monte Carlo event based simulation model was used to draw the maximum rejected gap and accepted gap for each of a sample of 300 drivers across 32 simulation runs. Simulation mean critical gap is varied between 3s and 8s, while offered gap rate is varied between 0.05veh/s and 0.55veh/s. This study affirms that MLE provides a close to perfect fit to simulation mean critical gaps across a broad range of conditions. The MCG method also provides an almost perfect fit and has superior computational simplicity and efficiency to the MLE. The SWCG method performs robustly under high flows; however, poorly under low to moderate flows. Further research is recommended using field traffic data, under a variety of minor stream and major stream flow conditions for a variety of minor stream movement types, to compare critical gap estimates using MLE against MCG. Should the MCG method prove as robust as MLE, serious consideration should be given to its adoption to estimate critical gap parameters in guidelines. Copyright © 2012 John Wiley & Sons, Ltd.     

Quasi-dynamic estimation of o–d flows from traffic counts: Formulation,statistical validation and performance analysis on real data

The paper proposes a “quasi-dynamic” framework for estimation of origin–destination (o–d) flow from traffic counts, under the assumption that o–d shares are constant across a reference period, whilst total flows leaving each origin vary for each sub-period within the reference period. The advantage of this approach over conventional within-day dynamic estimators is that of reducing drastically the number of unknowns given the same set of observed time-varying traffic counts. Obviously, the gain in accuracy depends on how realistic is the underlying assumption that total demand levels vary more rapidly over time than o–d shares. Firstly, the paper proposes a theoretical specification of the quasi-dynamic estimator. Subsequently, it proposes empirical and statistical tests to check the quasi-dynamic assumption and then compares the performances of the quasi-dynamic estimator of o–d flows with both classical off-line simultaneous dynamic estimators and on-line recursive Kalman filter-based estimators. Experiments are carried out on the real test site of A4–A23 motorways in North-Eastern Italy. Results confirm the acceptability of the assumption of quasi-dynamic o–d flows, even under the hypothesis of constant distribution shares for the whole day and show that the quasi-dynamic estimator outperforms significantly the simultaneous estimator. Data also suggest that using the quasi-dynamic estimates instead of the simultaneous estimates as historical o–d flows improves significantly the performances of the Kalman filter, which strongly depends of the quality of the seed o–d flows. In addition, it is shown that the aggregation of quasi-dynamic o–d estimates across subsequent time slices represents also the most effective way to obtain o–d estimates for larger time horizons (e.g. hourly estimates). Finally, a validation based on an hold-out sample of link flows (i.e. counts not used as inputs in the o–d estimation/updating process) revealed the quasi-dynamic estimator to be overall more robust and effective with respect to the other tested estimators.     

Interval estimation in the calibration of certain trip distribution models

This paper argues for interval, rather than point, estimation when calibrating some variants of the trip distribution “gravity” models. Analytic expressions are derived for the approximate asymptotic covariances of least squares and maximum likelihood estimates of the parameters in the impedance function under a variety of conditions. A comparative numerical example, and an application using migration flows, are also presented.     

Vehicle purchasing behaviors comparison in two-stage choice perspective before and after eco-car promotion policy in Japan

This paper compares the vehicle purchasing behaviors in Japan between before and after the eco-car (environmental friendly vehicle) promotion policy implemented. Consumer behaviors are modeled as a two-stage decision process: a consideration set formation stage and a choice-making stage. In the first stage, all available vehicle types are included in the choice set, and consumers are assumed to apply a conjunctive screening rule to construct consideration sets. In the second stage, consumers only evaluate the vehicles in the consideration set and choose the one with maximum utility. The applied Hierarchical Bayes model can avoid the issue of an indifferentiable and irregular likelihood surface caused by thresholds and discontinuities, and the data augmentation and Markov-Chain Monte Carlo estimation methods make it possible to estimate two stages simultaneously using only the information about the consumers’ actual choices. The estimations indicate that the change of consumer behavior during the formation of consideration sets after the policy implemented: more people preferred compact and hybrid vehicles because of their better fuel efficiency and more competitive prices under the tax reduction policy. The results show, however, that most of consumers who purchase hybrid vehicles after the policy implemented are only including hybrid vehicles in their consideration sets, and oil price and vehicle price still play important roles in the choice-making stage for these who consider both gasoline and hybrid vehicles.     

Critical assessment of five methods to correct for endogeneity in discrete-choice models

Endogeneity often arises in discrete-choice models, precluding the consistent estimation of the model parameters, but it is habitually neglected in practical applications. The purpose of this article is to contribute in closing that gap by assessing five methods to address endogeneity in this context: the use of Proxys (PR); the two steps Control-Function (CF) method; the simultaneous estimation of the CF method via Maximum-Likelihood (ML); the Multiple Indicator Solution (MIS); and the integration of Latent-Variables (LV). The assessment is first made qualitatively, in terms of the formulation, normalization and data needs of each method. Then, the evaluation is made quantitatively, by means of a Monte Carlo experiment to study the finite sample properties under a unified data generation process, and to analyze the impact of common flaws. The methods studied differ notably in the range of problems that they can address; their underlying assumptions; the difficulty of gathering proper auxiliary variables needed to apply them; and their practicality, both in terms of the need for coding and their computational burden. The analysis developed in this article shows that PR is formally inappropriate for many cases, but it is easy to apply, and often corrects in the right direction. CF is also easy to apply with canned software, but requires instrumental variables which may be hard to collect in various contexts. Since CF is estimated in two stages, it may also compromise efficiency and difficult the estimation of standard errors. ML guarantees efficiency and direct estimation of the standard errors, but at the cost of larger computational burden required for the estimation of a multifold integral, with potential difficulties in identification, and retaining the difficulty of gathering proper instrumental variables. The MIS method appears relatively easy to apply and requiring indicators that may be easier to obtain in various cases. Finally, the LV approach appears as the more versatile method, but at a high cost in computational burden, problems of identification and limitations in the capability of writing proper structural equations for the latent variable.     

Spatial Transferability and Updating Analysis of Mode Choice Models in Developing Countries

ABSTRACT

This article examines the spatial transferability of mode choice models in developing countries. An evaluation of the updating procedure and sample size are also included in the study. Because of the insufficiency of model coefficients in explaining differences in unmeasured modal attributes, naïvely transferring a model is not recommended. An understanding of the transport characteristics in both the estimation context and the application context is required, in order to justify whether a variable is transferable or not. Four updating procedures – updating alternative specific constants (ASCs), updating ASCs and scale parameter, the combined transfer estimator and Bayesian updating associated with three sets of small sample sizes – are applied to improve transferability. In general, the first three approaches produce significant improvements. It is also proposed that a minimum small sample size of 400 observations is necessary for updating purposes.     

Sampling of alternatives in Multivariate Extreme Value (MEV) models

We propose a methodology to achieve consistency, asymptotic normality and efficiency, while sampling alternatives in Multivariate Extreme Value (MEV) models, extending a previous result for Logit. We illustrate the methodology and study the finite sample properties of the estimators using Monte Carlo experimentation and real data on residential location choice from Lisbon, Portugal. Experiments show that the proposed methodology is practical, that it outperforms the uncorrected model, and that it yields acceptable results, even for relatively small samples of alternatives. The paper finishes with a synthesis and an analysis of the impact, limitations and potential extensions of this research.     

Calibrating aggregate travel demand models with traffic counts: Estimators and statistical performance

Traffic counts on network links constitute an information source on travel demand which is easy to collect, cheap and repeatable. Many models proposed in recent years deal with the use of traffic counts to estimate Origin/Destination (O/D) trip matrices under different assumptions on the type of "a-priori" information available on the demand (surveys, outdated estimates, models, etc.) and the type of network and assignment mapping (see Cascetta & Nguyen 1988). Less attention has been paid to the possibility of using traffic counts to estimate the parameters of demand models. In this case most of the proposed methods are relative to particular demand model structures (e.g. gravity-type) and the statistical analysis of estimator performance is not thoroughly carried out. In this paper a general statistical framework defining Maximum Likelihood, Non Linear Generalized Least Squares (NGLS) and Bayes estimators of aggregated demand model parameters combining counts-based information with other sources (sample or a priori estimates) is proposed first, thus extending and generalizing previous work by the authors (Cascetta & Russo 1992). Subsequently a solution algorithm of the projected-gradient type is proposed for the NGLS estimator given its convenient theoretical and computational properties. The algorithm is based on a combination of analytical/numerical derivates in order to make the estimator applicable to general demand models. Statistical performances of the proposed estimators are evaluated on a small test network through a Monte Carlo method by repeatedly sampling "starting estimates" of the (known) parameters of a generation/distribution/modal split/assignment system of models. Tests were carried out assuming different levels of "quality" of starting estimates and numbers of available counts. Finally NGLS estimator was applied to the calibration of the described model system on the network of a real medium-size Italian town using real counts with very satisfactory results in terms of both parameter values and counted flows reproduction.     

Introducing non-normality of latent psychological constructs in choice modeling with an application to bicyclist route choice

In the current paper, we propose the use of a multivariate skew-normal (MSN) distribution function for the latent psychological constructs within the context of an integrated choice and latent variable (ICLV) model system. The multivariate skew-normal (MSN) distribution that we use is tractable, parsimonious in parameters that regulate the distribution and its skewness, and includes the normal distribution as a special interior point case (this allows for testing with the traditional ICLV model). Our procedure to accommodate non-normality in the psychological constructs exploits the latent factor structure of the ICLV model, and is a flexible, yet very efficient approach (through dimension-reduction) to accommodate a multivariate non-normal structure across all indicator and outcome variables in a multivariate system through the specification of a much lower-dimensional multivariate skew-normal distribution for the structural errors. Taste variations (i.e., heterogeneity in sensitivity to response variables) can also be introduced efficiently and in a non-normal fashion through interactions of explanatory variables with the latent variables. The resulting model we develop is suitable for estimation using Bhat’s (2011) maximum approximate composite marginal likelihood (MACML) inference approach. The proposed model is applied to model bicyclists’ route choice behavior using a web-based survey of Texas bicyclists. The results reveal evidence for non-normality in the latent constructs. From a substantive point of view, the results suggest that the most unattractive features of a bicycle route are long travel times (for commuters), heavy motorized traffic volume, absence of a continuous bicycle facility, and high parking occupancy rates and long lengths of parking zones along the route.     

Hidden Markov Model-based population synthesis

Micro-simulation travel demand and land use models require a synthetic population, which consists of a set of agents characterized by demographic and socio-economic attributes. Two main families of population synthesis techniques can be distinguished: (a) fitting methods (iterative proportional fitting, updating) and (b) combinatorial optimization methods. During the last few years, a third outperforming family of population synthesis procedures has emerged, i.e., Markov process-based methods such as Monte Carlo Markov Chain (MCMC) simulations. In this paper, an extended Hidden Markov Model (HMM)-based approach is presented, which can serve as a better alternative than the existing methods. The approach is characterized by a great flexibility and efficiency in terms of data preparation and model training. The HMM is able to reproduce the structural configuration of a given population from an unlimited number of micro-samples and a marginal distribution. Only one marginal distribution of the considered population can be used as a boundary condition to “guide” the synthesis of the whole population. Model training and testing are performed using the Survey on the Workforce of 2013 and the Belgian National Household Travel Survey of 2010. Results indicate that the HMM method captures the complete heterogeneity of the micro-data contrary to standard fitting approaches. The method provides accurate results as it is able to reproduce the marginal distributions and their corresponding multivariate joint distributions with an acceptable error rate (i.e., SRSME=0.54 for 6 synthesized attributes). Furthermore, the HMM outperforms IPF for small sample sizes, even though the amount of input data is less than that for IPF. Finally, simulations show that the HMM can merge information provided by multiple data sources to allow good population estimates.     

The econometric estimation of airports’ cost function

The econometric estimation of cost functions has been proposed in the literature as a suitable approach in order to obtain estimations of marginal costs, efficiency levels and scale elasticities for transport industries. However, regarding the airport industry, no significant attention has been paid in developing an airport-specific estimation methodology rather than adapting the procedures applied to other industries. The lack of comparable airport data is one of the causes which could explain the scarcity of this literature in the past, as well as the use of very limited approaches to explain airport technology. This paper tries to overcome these limitations by developing an airport-specific methodology to estimate a multi-output long-run cost function using an unbalanced pooled database on 161 airports worldwide. The specification of hedonically-adjusted aircraft operations, domestic and international passengers, cargo and commercial revenues in the output vector, as well as the calculation of input prices are discussed. Both technical and allocative inefficiencies are specified in the model using a Stochastic Frontier method that has been estimated through Bayesian Inference and Markov Chain Monte Carlo methods.     

Choice of mode of transport for long-distance trips: Solving the problem of sparse data

Transport planning is usually based on models’ forecasts, but the reliability of their outputs depends so much on the quality of input-data they are fed with. Discrete-choice models are used to characterise travellers’ behaviour in choosing their transport mode. Their calibration process is usually based on data stemming from household survey campaigns. However, the modelling in multimodal and intermodal transport on an interurban level is far more complicated and costly than in the case of an urban area. An alternative way to reduce costs is achieved by designing a choice-based sampling strategy where household surveys are replaced by specific surveys for each transport mode. This strategy generates a non-random sample that has to be treated correctly during the estimation process. In principle, the sample does not represent population market quotas for each different transport option. Moreover, as a result of both physical and functional constraints, the survey period cannot cover all origin–destination pairs (O–D pairs) in an optimal way and, consequently, the above-mentioned bias also affects each different individual O–D pair or, at least, group of pairs. In order to overcome this problem, this study presents a new procedure derived from the introduction of maximum likelihood estimators. These estimators assume the original mode options in terms of population quotas and in terms of O–D groups of pairs. The procedure is based on the optimisation of an objective-function to correct the above-mentioned bias in a way similar to the estimators of samples based on different choice options. The method named DWELT estimates the parameters corresponding to each explanatory variable using mode shares for each O–D pair or group of pairs. DWELT has been successfully validated in the case study of the Madrid–Barcelona interurban corridor in Spain. This result allows to achieve a more flexible cheaper survey procedure for interurban transport planning activities. Therefore transport policy strategies could be better designed and tested with lower costs.