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.     

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.     

Confidence interval estimation for path flow estimator

The uncertainty of an origin-destination (O-D) trip table estimate is affected by two factors: (i) the multiplicity of solutions due to the underspecified nature of the problem, and (ii) the errors of traffic counts. In this paper, a confidence interval estimation procedure for path flow estimator (PFE) is developed for assessing the quality of O-D trip tables estimated from traffic counts. The confidence interval estimation consists of two parts: (i) a generalized demand scale (GDS) measure for quantifying the intrinsic underspecified nature of the O-D estimation problem at various spatial levels, and (ii) an error bound to quantify the contribution of input errors (traffic counts) to the estimation results. Numerical results using PFE as the O-D estimator show that the proposed confidence interval estimation procedure is able to separate the two sources of uncertainty in constructing the confidence intervals at various spatial levels. Simulation results also confirm that the proposed quality measure indeed contain the true estimates within the defined confidence intervals.     

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     

On the variance of recurrent traffic flow for statistical traffic assignment

This paper generalizes and extends classical traffic assignment models to characterize the statistical features of Origin-Destination (O-D) demands, link/path flow and link/path costs, all of which vary from day to day. The generalized statistical traffic assignment (GESTA) model has a clear multi-level variance structure. Flow variance is analytically decomposed into three sources, O-D demands, route choices and measurement errors. Consequently, optimal decisions on roadway design, maintenance, operations and planning can be made using estimated probability distributions of link/path flow and system performance. The statistical equilibrium in GESTA is mathematically defined. Its multi-level statistical structure well fits large-scale data mining techniques. The embedded route choice model is consistent with the settings of O-D demands considering link costs that vary from day to day. We propose a Method of Successive Averages (MSA) based solution algorithm to solve for GESTA. Its convergence and computational complexity are analyzed. Three example networks including a large-scale network are solved to provide insights for decision making and to demonstrate computational efficiency.     

Link choice proportions from trip distribution and assignment models: An overview and comparison

A number of estimation procedures have been suggested for the situation where a prior estimate of an origin-destination matrix is to be updated on the basis of recently-acquired traffic counts. These procedures assume that both the link flows and the proportionate usage of each link made by each origin-destination flow (referred to collectively as the link choice proportions) are known. This paper examines the possibility and methods for estimating the link choice proportions. Three methods are presented: (1) using ad hoc iteration between trip distribution and traffic assignment; (2) combining trip distribution and assignment in one step; (3) solving a new optimization problem in which the path flows are directly considered as variables and its optimal solution is governed by a logit type formula. The algorithms, covergencies and computational efficiencies of these methods are investigated. Results of testing the three methods on example networks are discussed.     

Origin-destination trip table estimation based on subarea network OD flow and vehicle trajectory data

Identifying accurate origin-destination (O-D) travel demand is one of the most important and challenging tasks in the transportation planning field. Recently, a wide range of traffic data has been made available. This paper proposes an O-D estimation model using multiple field data. This study takes advantage of emerging technologies – car navigation systems, highway toll collecting systems and link traffic counts – to determine O-D demand. The proposed method is unique since these multiple data are combined to improve the accuracy of O-D estimation for an entire network. We tested our model on a sample network and found great potential for using multiple data as a means of O-D estimation. The errors of a single input data source do not critically affect the model’s overall accuracy, meaning that combining multiple data provides resilience to these errors. It is suggested that the model is a feasible means for more reliable O-D estimation.     

An intersection turning movement estimation procedure based on path flow estimator

Estimation of intersection turning movements is one of the key inputs required for a variety of transportation analysis, including intersection geometric design, signal timing design, traffic impact assessment, and transportation planning. Conventional approaches that use manual techniques for estimation of turning movements are insensitive to congestion. The drawbacks of the manual techniques can be amended by integrating a network traffic model with a computation procedure capable of estimating turning movements from a set of link traffic counts and intersection turning movement counts. This study proposes using the path flow estimator, originally used to estimate path flows (hence origin–destination flows), to derive not only complete link flows, but also turning movements for the whole road network given some counts at selected roads and intersections. Two case studies using actual traffic counts are used to demonstrate the proposed intersection turning movement estimation procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Inferences on trip matrices from observations on link volumes: A Bayesian statistical approach

Previous methods for estimating a trip matrix from traffic volume counts have used the principles of maximum entropy and minimum information. These techniques implicitly give as little weight to prior information on the trip matrix as possible. The new method proposed here is based on Bayesian statistical inference and has several advantages over these earlier approaches. It allows complete flexibility in the degree of belief placed on the prior estimate of the trip matrix and also allows for different degrees of belief in diffeent parts of the prior estimate. Furthermore under certain assumptions the method reduces to a simple updating scheme in which observations on the link flows successively modify the trip matrix. At the end of the scheme confidence intervals are available for the estimates of the trip matrix elements.     

A maximum entropy-least squares estimator for elastic origin-destination trip matrix estimation

In transportation subnetwork-supernetwork analysis, it is well known that the origin-destination (O-D) flow table of a subnetwork is not only determined by trip generation and distribution, but also a result from traffic routing and diversion, due to the existence of internal-external, external-internal and external-external flows. This result indicates the variable nature of subnetwork O-D flows. This paper discusses an elastic O-D flow table estimation problem for subnetwork analysis. The underlying assumption is that each cell of the subnetwork O-D flow table contains an elastic demand function rather than a fixed demand rate and the demand function can capture all traffic diversion effect under various network changes. We propose a combined maximum entropy-least squares estimator, by which O-D flows are distributed over the subnetwork in terms of the maximum entropy principle, while demand function parameters are estimated for achieving the least sum of squared estimation errors. While the estimator is powered by the classic convex combination algorithm, computational difficulties emerge within the algorithm implementation until we incorporate partial optimality conditions and a column generation procedure into the algorithmic framework. Numerical results from applying the combined estimator to a couple of subnetwork examples show that an elastic O-D flow table, when used as input for subnetwork flow evaluations, reflects network flow changes significantly better than its fixed counterpart.     

A dynamic stochastic assignment model for the analysis of general networks

The paper adopts the framework employed by the existing dynamic assignment models, which analyse specific network forms, and develops a methodology for analysing general networks. Traffic conditions within a link are assumed to be homogeneous, and the time varying O-D travel times and traffic flow patterns are calculated using elementary relationships from traffic flow theory and link volume conservation equations. Each individual is assumed to select a departure time and a route by trading off the travel time and schedule delay associated with each alternative. A route is considered as reasonable if it includes only links which do not take the traveller back to the origin. The set of reasonable routes is not consistant but depends on the time that an individual decides to depart from his origin. Equilibrium distributions are derived from a Markovian model which describes the evolution of travel patterns from day to day. Numerical simulation experiments are conducted to analyse the impact of different work start time flexibilities on the time dependent travel patterns. The similarity between link flows and travel times obtained from static and dynamic stochastic assignment is investigated. It is shown that in congested networks the application of static assignment results in travel times which are lower than the ones predicted by dynamic assignment.     

Norm approximation method for handling traffic count inconsistencies in path flow estimator

Path flow estimator (PFE) is a one-stage network observer proposed to estimate path flows and hence origin–destination (O–D) flows from traffic counts in a transportation network. Although PFE does not require traffic counts to be collected on all network links when inferring unmeasured traffic conditions, it does require all available counts to be reasonably consistent. This requirement is difficult to fulfill in practice due to errors inherited in data collection and processing. The original PFE model handles this issue by relaxing the requirement of perfect replication of traffic counts through the specification of error bounds. This method enhances the flexibility of PFE by allowing the incorporation of local knowledge, regarding the traffic conditions and the nature of traffic data, into the estimation process. However, specifying appropriate error bounds for all observed links in real networks turns out to be a difficult and time-consuming task. In addition, improper specification of the error bounds could lead to a biased estimation of total travel demand in the network. This paper therefore proposes the norm approximation method capable of internally handling inconsistent traffic counts in PFE. Specifically, three norm approximation criteria are adopted to formulate three L_p-PFE models for estimating consistent path flows and O–D flows that simultaneously minimize the deviation between the estimated and observed link volumes. A partial linearization algorithm embedded with an iterative balancing scheme and a column generation procedure is developed to solve the three L_p-PFE models. In addition, the proposed L_p-PFE models are illustrated with numerical examples and the characteristics of solutions obtained by these models are discussed.     

Traffic assignment in a two-dimensional continuous representation of a traffic network with flow-dependent speeds

The work deals with the assignment of traffic to a two-dimensional continuous representation of a traffic network. An important aspect of the treatment is that the reciprocal of the speed on each road in the network is at all times a linear function of the flow on that road. This speed-flow relationship is generalized to two-dimensional space using travel intensities and taking account of road densities, so that there is direct dependence of speeds upon flows at all points regardless of their location. There is also dependence of flows upon speeds at all points because Wardrop's first assignment principle is adopted. That is, for a given O-D pair, journey times on all routes actually used are identical, and less than journey times on all other possible routes. This results in the identification for each O-D pair of an “assignment zone”, an area within which all trips between that O-D pair are made, and beyond which no such trips are made. For a single O-D pair the assignment zone is identified by ?_m, the maximum angular divergence of a path from the straight line between O and D. Paths are then assumed to be bilinear so that for a single O-D pair the assignment zone is a parallelogram. Journey times, speeds, lateral displacement and other related quantities are obtained as functions of the flow Q between O and D. The work is extended to three O-D pairs located at the extremities of an equilateral triangle and four O-D pairs located at the corners of a square. At low flows these two configurations are trivial extensions of the single O-D pair problem because assignment zones do not overlap. At higher flows account is taken of this tendency to overlapping, so that although they do not overlap they do touch, becoming kite-shaped. Origins and destinations are assumed to be at the periphery of small circles of arbitrary radius. The work is inelegant to the extent that it involves a numerical integration but it is possible that this might eventually be circumvented.     

Selection of a trip table which reproduces observed link flows

In urban traffic management and planning, an important problem is estimating the number of drivers traveling between each origin-destination zone. We review a model due to Nguyen for estimating these numbers of drivers, based on counts of the traffic flows on each street, and develop an effective algorithm for solving it. The multiplicity of solutions of this model poses the additional question of which solution to use; we introduce a secondary optimization problem to overcome this difficulty. Efficient solution techniques are described for these problems and computational results are reported. It is noted that the most efficient solution methods involve user interaction to specify values of parameters which improve the convergence rates.     

Simultaneous estimation of an origin–destination matrix and link choice proportions using traffic counts

A procedure for the simultaneous estimation of an origin–destination (OD) matrix and link choice proportions from OD survey data and traffic counts for congested network is proposed in this paper. Recognizing that link choice proportions in a network change with traffic conditions, and that the dispersion parameter of the route choice model should be updated for a current data set, this procedure performs statistical estimation and traffic assignment alternately until convergence in order to obtain the best estimators for both the OD matrix and link choice proportions, which are consistent with the survey data and traffic counts.Results from a numerical study using a hypothetical network have shown that a model allowing θ to be estimated simultaneously with an OD matrix from the observed data performs better than the model with a fixed predetermined θ. The application of the proposed model to the Tuen Mun Corridor network in Hong Kong is also presented in this paper. A reasonable estimate of the dispersion parameter θ for this network is obtained.     

Real-time estimation of turning movement counts at signalized intersections using signal phase information

A variety of sensor technologies, such as loop detectors, traffic cameras, and radar have been developed for real-time traffic monitoring at intersections most of which are limited to providing link traffic information with few being capable of detecting turning movements. Accurate real-time information on turning movement counts at signalized intersections is a critical requirement for applications such as adaptive traffic signal control. Several attempts have been made in the past to develop algorithms for inferring turning movements at intersections from entry and exit counts; however, the estimation quality of these algorithms varies considerably. This paper introduces a method to improve accuracy and robustness of turning movement estimation at signalized intersections. The new algorithm makes use of signal phase status to minimize the underlying estimation ambiguity. A case study was conducted based on turning movement data obtained from a four-leg signalized intersection to evaluate the performance of the proposed method and compare it with two other existing well-known estimation methods. The results show that the algorithm is accurate, robust and fairly straightforward for real world implementation.     

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 new fuzzy approach to estimate the O–D matrix from link volumes

Abstract

Estimation of the origin–destination (O–D) trip demand matrix plays a key role in travel analysis and transportation planning and operations. Many researchers have developed different O–D matrix estimation methods using traffic counts, which allow simple data collection as opposed to the costly traditional direct estimation methods based on home and roadside interviews.

In this paper, we present a new fuzzy model to estimate the O–D matrix from traffic counts. Since link data only represent a snapshot situation, resulting in inconsistency of data and poor quality of the estimated O–Ds, the proposed method considers the link data as a fuzzy number that varies within a certain bandwidth. Shafahi and Ramezani's fuzzy assignment method is improved upon and used to assign the estimated O–D matrix, which causes the assigned volumes to be fuzzy numbers similar to what is proposed for observed link counts. The shortest path algorithm of the proposed method is similar to the Floyd–Warshall algorithm, and we call it the Fuzzy Floyd–Warshall Algorithm. A new fuzzy comparing index is proposed by improving the fuzzy comparison method developed by Dubois and Prade to estimate and compare the distance between the assigned and observed link volumes. The O–D estimation model is formulated as a convex minimization problem based on the proposed fuzzy index to minimize the fuzzy distance between the observed and assigned link volumes. A gradient-based method is used to solve the problem. To ensure the original O–D matrix does not change more than necessary during the iterations, a fuzzy rule-based approach is proposed to control the matrix changes.     

A unified framework for estimating or updating origin/destination matrices from traffic counts

This paper presents an in-depth study of the methodology for estimating or updating origin-to-destination trip matrices from traffic counts. Following an analysis of the statistical foundation of the estimation and updating problems, various basic approaches are reviewed using a generic traffic assignment map. Computational issues related to specific assignment maps and estimation models for both road and transit networks are then discussed. Finally, additional insight into the relative performance of several estimators is provided by a set of test problems with varying input data.