Estimation of origin-destination matrices from link counts and sporadic routing data

Estimation of origin-destination (OD) matrices from link count data is a challenging problem because of the highly indeterminate relationship between the observations and the latent route flows. Conversely, estimation is straightforward if we observe the path taken by each vehicle. We consider an intermediate problem of increasing practical importance, in which link count data is supplemented by routing information for a fraction of vehicles on the network. We develop a statistical model for these combined data sources and derive some tractable normal approximations thereof. We examine likelihood-based inference for these normal models under the assumption that the probability of vehicle tracking is known. We show that the likelihood theory can be non-standard because of boundary effects, and provide conditions under which such irregular behaviour will be observed in practice. For regular cases we outline connections with existing generalised least squares methods. We then consider estimation of OD matrices under estimated and/or misspecified models for the probability of vehicle tracking. Theoretical developments are complemented by simulation experiments and an illustrative example using a section of road network from the English city of Leicester.     

Dynamic user optimal traffic assignment on congested multidestination networks

An equivalent continuous time optimal control problem is formulated to predict the temporal evolution of traffic flow pattern on a congested multiple origin-destination network, corresponding to a dynamic generalization of Wardropian user equilibrium. Optimality conditions are derived using the Pontryagin minimum principle and given economic interpretations, which are generalizations of similar results previously reported for single-destination networks. Analyses of sufficient conditions for optimality and of singular controls are also given. Under the steady-state assumptions, the model is shown to be a proper dynamic extension of Beckmann's mathematical programming problem for a static user equilibrium traffic assignment.     

A comparison of stochastic and deterministic traffic assignment over congested networks

The similarity between link flows obtained from deterministic and stochastic equilibrium traffic assignment models is investigated at different levels of congestion. A probit-based stochastic assignment is used (over a congested network) where the conditions for equilibrium are those given by Daganzo and Sheffi (1977). Stochastic equilibrium flows are generated using an iterative procedure with predetermined step sizes, and the resulting assignment is validated on the basis of the equilibrium criteria. The procedure is intended to assist in the choice of the most appropriate assignment algorithm for a given level of congestion.     

A predictive dynamic traffic assignment model in congested capacity-constrained road networks

In this paper, a predictive dynamic traffic assignment model in congested capacity-constrained road networks is formulated. A traffic simulator is developed to incrementally load the traffic demand onto the network, and updates the traffic conditions dynamically. A time-dependent shortest path algorithm is also given to determine the paths with minimum actual travel time from an origin to all the destinations. The traffic simulator and time-dependent shortest path algorithm are employed in a method of successive averages to solve the dynamic equilibrium solution of the problem. A numerical example is given to illustrate the effectiveness of the proposed method.     

Estimating intersection O-D matrices from observed time-series of traffic counts: The zellner estimator

The problem of estimating intersection O-D matrices from input and output time-series of traffic counts is considered in this paper. Because of possible existence of significant correlation between the error terms across structural equations forming the O-D matrices, the seemingly unrelated estimator (Zellner estimator) was suggested. Estimation results showed evidence of strong correlation between error terms across-equations. Generally, the Zellner estimator produced more efficient estimates than did the ordinary least-squares estimator. Furthermore, the Zellner estimator satisfied all constraints and reproduced turning movements comparable to the actual ones.     

A model of traffic dispersion from a congested road

The study of traffic flow dynamics is developed by defining and clarifying traffic divergence, continuity, congestion and dispersion. Velocity potential is introduced as a gravity function generated by the interaction of two or more motorists occupying neighbouring points in space and describes interference to continuous traffic flow. The relationship between the potential function and carrying capacity is developed and dispersion, when considered as a random walk, satisfies a diffusion equation. A model of traffic dispersion along a maximum congested road in space and time is presented as eigenfunctions of the velocity potential. This suggests that traffic can be dispersed by a series of quantum steps. A probability density function is introduced to define the probability of locating a motorist in a congestion zone.     

Consistent link flow estimation from counts

Partly because of counting errors and partly because counts may be carried out on different days, traffic counts on links of a network are unlikely to satisfy the flow conservation constraint “flow IN = flow out” at every node of the network. Van Zuylen and Willumsen (1980) have described a method of eliminating inconsistencies in traffic counts when a single count is available for each link in the network. In this paper, the method is extended to the case when more than one count is available on some links of the network. In addition, an algorithm is described for application of the method.     

Transport demand model estimation from traffic counts

Many problems in transport planning and management tasks require an origindestination (O-D) matrix to represent the travel pattern. However, O-D matrices obtained through a large scale survey such as home or roadside interviews, tend to be costly, labour intensive and time disruptive to trip makers. Therefore, the use of low cost and easily available data is particularly attractive.The need of low-cost methods to estimate current and future O-D matrices is even more valuable in developing countries because of the rapid changes in population, economic activity and land use. Models of transport demand have been used for many years to synthesize O-D matrices in study areas. A typical example of this is the gravity model; its functional form, plus the appropriate values for the parameters involved, is employed to produce acceptable matrices representing trip making behaviour for many trip purposes and time periods.The work reported in this paper has combined the advantages of acceptable travel demand models with the low cost and availability of traffic counts. Three types of demand models have been used: gravity (GR), opportunity (OP) and gravity-opportunity (GO) models. Three estimation methods have been developed to calibrate these models from traffic counts, namely: non-linear-least-squares (NLLS), weighted-non-linear-least-squares (WNLLS) and maximumlikelihood (ML).The 1978 Ripon (urban vehicle movement) survey was used to test these methods. They were found to perform satisfactorily since each calibrated model reproduced the observed O-D matrix fairly closely. The tests were carried out using two assignment techniques, all-or-nothing and the stochastic method due to Burrell, in determining the routes taken through the network.requests for offprints     

Determining intermediate origin-destination matrices for the analysis of composite mode trips

We consider in this paper the problem of determining intermediate origin-destination matrices for composite mode trips that involve a trip by private car to a parking facility and the continuation of the trip to the destination either by walking or by a transit mode. The intermediate origin-destination matrices relate to each component of the composite mode trip: a matrix from the trip origins to intermediate destinations which are parking lots and a matrix from the parking lots to the final destinations. The approach that we propose to solve this problem is to modify the entropy based trip distribution models to consider inequality constraints related to parking lot capacities. Such models may be easily calibrated by using well known calibration methods or generalization of these methods and may be easily solved by applying a primal feasible direction method of nonlinear programming.     

Estimation of a time‐dependent origin‐destination matrix for congested highway networks

This paper presents a time‐dependent origin‐destination (O‐D) matrix estimation procedure embedded with a dynamic traffic assignment model, in which the predictive dynamic user optimal conditions in congested networks are maintained. Two solution algorithms are proposed, namely: an iterative (ITR) scheme and a method of successive averages (MSA) scheme. It is found that the MSA scheme outperforms the ITR scheme. As a prior O‐D matrix is an important input for the problem, its quality is essential for the reliability of the matrix estimation procedure. Empirical constraints are set in relation to the quality of the prior O‐D matrix for the estimation procedure. Numerical examples are used to demonstrate the effectiveness of the proposed methodology.     

Simplified transport models based on traffic counts

Having accepted the need for the development of simpler and less cumbersome transport demand models, the paper concentrates on one possible line for simplification: estimation of trip matrices from link volume counts. Traffic counts are particularly attractive as a data basis for modelling because of their availability, low cost and nondisruptive character. It is first established that in normal conditions it may be possible to find more than one trip matrix which, when loaded onto a network, reproduces the observed link volumes. The paper then identifies three approaches to reduce this underspecification problem and produce a unique trip matrix consistent with the counts. The first approach consists of assuming that trip-making behaviour can be explained by a gravity model whose parameters can be calibrated from the traffic counts. Several forms of this gravity model have been put forward and they are discussed in Section 3. The second approach uses mathematical programming techniques associated to equilibrium assignment problems to estimate a trip matrix in congested areas. This method can also be supplemented by a special distribution model developed for small areas. The third approach relies on entropy and information theory considerations to estimate the most likely trip matrix consistent with the observed flows. A particular feature of this group is that they can include prior, perhaps outdated, information about the matrix.These three approaches are then compared and their likely areas for application identified. Problems for further research are discussed and finally an assessment is made of the possible role of these models vis-a-vis recent developments in transport planning.     

Estimating total annual traffic flow from short period counts

The total annual traffic flow at a road site is probably the single most important item of data needed in transport studies. In this article we describe a method of estimating this flow from a traffic count of less than 24 hours duration. The method is extended to enable an assessment to be made of the merits of repeating a count. The application of the method is discussed in relation to the United Kingdom, but we believe this method to be generally applicable to most countries.

     

The most likely trip matrix estimated from traffic counts

For a large number of applications conventional methods for estimating an origin destination matrix become too expensive to use. Two models, based on information minimisation and entropy maximisation principles, have been developed by the authors to estimate an O-D matrix from traffic counts. The models assume knowledge of the paths followed by the vehicles over the network. The models then use the traffic counts to estimate the most likely O-D matrix consistent with the link volumes available and any prior information about the trip matrix. Both models can be used to update and improve a previous O-D matrix. An algorithm to find a solution to the model is then described. The models have been tested with artificial data and performed reasonably well. Further research is being carried out to validate the models with real data.     

A shockwave profile model for traffic flow on congested urban arterials

In this paper a new traffic flow model for congested arterial networks, named shockwave profile model (SPM), is presented. Taking advantage of the fact that traffic states within a congested link can be simplified as free-flow, saturated, and jammed conditions, SPM simulates traffic dynamics by analytically deriving the trajectories of four major shockwaves: queuing, discharge, departure, and compression waves. Unlike conventional macroscopic models, in which space is often discretized into small cells for numerical solutions, SPM treats each homogeneous road segment with constant capacity as a section; and the queuing dynamics within each section are described by tracing the shockwave fronts. SPM is particularly suitable for simulating traffic flow on congested signalized arterials especially with queue spillover problems, where the steady-state periodic pattern of queue build-up and dissipation process may break down. Depending on when and where spillover occurs along a signalized arterial, a large number of queuing patterns may be possible. Therefore it becomes difficult to apply the conventional approach directly to track shockwave fronts. To overcome this difficulty, a novel approach is proposed as part of the SPM, in which queue spillover is treated as either extending a red phase or creating new smaller cycles, so that the analytical solutions for tracing the shockwave fronts can be easily applied. Since only the essential features of arterial traffic flow, i.e., queue build-up and dissipation, are considered, SPM significantly reduces the computational load and improves the numerical efficiency. We further validated SPM using real-world traffic signal data collected from a major arterial in the Twin Cities. The results clearly demonstrate the effectiveness and accuracy of the model. We expect that in the future this model can be applied in a number of real-time applications such as arterial performance prediction and signal optimization.     

A note on trip matrix estimation from link traffic count data

In this note, a simple network equilibrium based approach for estimating a trip matrix using link traffic count data is proposed. In essence a combined distribution and assignment model is formulated in which the link flow data serve to furnish an estimate for the sum of the integrals of the link cost functions. A comparison of this model with others proposed previously is made.     

Validation of traffic flow models with respect to the spatiotemporal evolution of congested traffic patterns

We propose a quantitative approach for calibrating and validating key features of traffic instabilities based on speed time series obtained from aggregated data of a series of neighboring stationary detectors. The approach can be used to validate models that are calibrated by other criteria with respect to their collective dynamics. We apply the proposed criteria to historic traffic databases of several freeways in Germany containing about 400 occurrences of congestions thereby providing a reference for model calibration and quality assessment with respect to the spatiotemporal dynamics. First tests with microscopic and macroscopic models indicate that the criteria are both robust and discriminative, i.e., clearly distinguishes between models of higher and lower predictive power.     

Real-time estimation of turning movement proportions from partial counts on urban networks

Modern traffic signal control systems require reliable estimates of turning flows in real time to formulate effective control actions, and accommodate disturbances in traffic demand without deteriorating the system performance. The more accurate the estimation is, the more effective the control plan is. Most of the previous research works assumed that a full set of detector counts is available and employed the least-squares methods to produce unbiased estimates of the turning movement proportions. However, in practice, such a dense detector configuration is expensive to install and maintain. Also, the least-squares estimates are not feasible when the travel time between inflows and outflows is significant, or when intervening traffic conditions change the travel time. This study proposes a nonlinear least-square (NLS) approach and a quasi maximum likelihood (QML) approach to recursively estimate turning movement proportions in a network of intersections where only a partial set of detector counts are available. Using large population approximation technique, a class of nonlinear, discrete-time traffic flow models are transformed into a linear state–space model tractable for on-line applications. The quality of estimates is demonstrated by implementing the proposed algorithms with simulation and real data. As a comparison, the NLS estimator shows less bias but with higher variance than the QML estimator. The QML estimator outperforms the NLS estimator in terms of total mean square error, due to an increase in bias being traded for a decrease in variance.     

Estimation of link performance functions from incomplete flow data

Despite the pivotal importance of link performance functions to models of transport systems, relatively little work has been done on practical aspects of estimating these functions from observed data. Furthermore it is difficult to find any examples in the literature of estimated urban link performance functions faithfully reproducing theoretical travel time-flow relationships. One reason for the paucity of research in this area is the difficulty and expense of obtaining the requisite data. The increase in automatic collection of traffic flow data goes part way to resolving this problem, but matching such flows to manually recorded travel times can present considerable statistical difficulties in the estimation procedure. This paper considers the estimation of link performance functions from a combination of automatically recorded traffic counts and travel collected by hand, using a non-standard statistical methodology. The study is motivated by a set of data of precisely this type, from the UK city of Leicester.     

Equilibrium analysis of the scheduling of tours in congested networks

In this paper, we develop a model of travel in tours that joins several locations by travel through a congested network. We develop a microscopic analysis in continuous time of individual benefits obtained by spending time at each of the locations and costs incurred through travel between them. This is combined with a continuous time macroscopic equilibrium model of travel during congested peak periods to show how individuals' travel choices are influenced by the congestion that result from corresponding choices made by others. We show how different travellers can achieve identical net utilities by making different combinations of choices within the equilibrium. The resulting model can be used to investigate the effect on travel behaviour and individual utility of various transport interventions, and we illustrate this by considering the effect of a peak‐period charge that eliminates congestion.