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.     

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.     

Bush-based sensitivity analysis for approximating subnetwork diversion

Subnetwork analysis is often used in traffic assignment problems to reduce the size of the network being analyzed, with a corresponding decrease in computation time. This is particularly important in network design, second-best pricing, or other bilevel problems in which many equilibrium runs must be solved as a subproblem to a master optimization program. A fixed trip table based on an equilibrium path flow solution is often used, but this ignores important attraction and diversion effects as drivers (globally) change routes in response to (local) subnetwork changes. This paper presents an approach for replacing a regional network with a smaller one, containing all of the subnetwork, and zones. Artificial arcs are created to represent “all paths” between each origin and subnetwork boundary node, under the assumption that the set of equilibrium routes does not change. The primary contribution of the paper is a procedure for estimating a cost function on these artificial arcs, using derivatives of the equilibrium travel times between the end nodes to create a Taylor series. A bush-based representation allows rapid calculation of these derivatives. Two methods for calculating these derivatives are presented, one based on network transformations and resembling techniques used in the analysis of resistive circuits, and another based on iterated solution of a nested set of linear equations. These methods are applied to two networks, one small and artificial, and the other a regional network representing the Austin, Texas metropolitan area. These demonstrations show substantial improvement in accuracy as compared to using a fixed table, and demonstrate the efficiency of the proposed approach.     

A model‐based demand‐balancing control for dynamically divided multiple urban subnetworks

Traffic control is an effective and efficient method for the problem of traffic congestion. It is necessary to design a high‐level controller to regulate the network traffic demands, because traffic congestion is not only caused by the improper management of the traffic network but also to a great extent caused by excessive network traffic demands. Therefore, we design a demand‐balance model predictive controller based on the macroscopic fundamental diagram‐based multi‐subnetwork model, which can optimize the network traffic mobility and the network traffic throughput by regulating the input traffic flows of the subnetworks. Because the transferring traffic flows among subnetworks are indirectly controlled and coordinated by the demand‐balance model predictive controller, the subnetwork division can variate dynamically according to real traffic states, and a global optimality can be achieved for the entire traffic network. The simulation results show the effectiveness of the proposed controller in improving the network traffic throughput. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

Estimation of origin-destination matrices from link traffic counts on congested networks

Conventional methods for estimating origin-destination (O-D) trip matrices from link traffic counts assume that route choice proportions are given constants. In a network with realistic congestion levels, this assumption does not hold. This paper shows how existing methods such as the generalized least squares technique can be integrated with an equilibrium traffic assignment in the form of a convex bilevel optimization problem. The presence of measurement errors and time variations in the observed link flows are explicitly considered. The feasibility of the model is always guaranteed without a requirement for estimating consistent link flows from counts. A solution algorithm is provided and numerical simulation experiments are implemented in investigating the model's properties. Some related problems concerning O-D matrix estimation are also discussed.     

Computational experience with an application of a simultaneous transportation equilibrium model to urban travel in Austin,Texas

Safwat and Magnanti (1988) have developed a combined trip generation, trip distribution, modal split, and traffic assignment model that can predict demand and performance levels on large-scale transportation networks simultaneously, i.e. a simultaneous transportation equilibrium model (STEM). The major objective of this paper is to assess the computational efficiency of the STEM approach when applied to an urban large-scale network, namely the urban transportation system of Austin, Texas. The Austin network consisted of 520 zones, 19,214 origin-destination (O-D) pairs, 7,096 links and 2,137 nodes. The Central Processing Unit (CPU) time on an IBM 4381 mainframe computer was 430 seconds for a typical iteration and about 4,734 seconds, less than 79 minutes, to arrive at a reasonably accurate solution in no more than 10 iterations. The computational time at any given iteration is comparable to that of the standard fixed demand traffic assignment procedure. These results encourage further applications of the STEM model to large urban areas.     

Hierarchical travel demand estimation using multiple data sources: A forward and backward propagation algorithmic framework on a layered computational graph

Aiming to develop a theoretically consistent framework to estimate travel demand using multiple data sources, this paper first proposes a multi-layered Hierarchical Flow Network (HFN) representation to structurally model different levels of travel demand variables including trip generation, origin/destination matrices, path/link flows, and individual behavior parameters. Different data channels from household travel surveys, smartphone type devices, global position systems, and sensors can be mapped to different layers of the proposed network structure. We introduce Big data-driven Transportation Computational Graph (BTCG), alternatively Beijing Transportation Computational Graph, as the underlying mathematical modeling tool to perform automatic differentiation on layers of composition functions. A feedforward passing on the HFN sequentially implements 3 steps of the traditional 4-step process: trip generation, spatial distribution estimation, and path flow-based traffic assignment, respectively. BTCG can aggregate different layers of partial first-order gradients and use the back-propagation of “loss errors” to update estimated demand variables. A comparative analysis indicates that the proposed methods can effectively integrate different data sources and offer a consistent representation of demand. The proposed methodology is also evaluated under a demonstration network in a Beijing subnetwork.     

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.     

Traffic assignment by trip type using volume restraint and link restraint for application in small urban areas

This research involved the development of a new traffic assignment model consisting of a set of procedures for an urbanized area with a population of 172,000. Historical, social, and economic data were used as input to conventional trip generation and trip distribution models to produce a trip table for network assignment. This fixed table was divided into three trip types: external-external trips, external-internal trips, and internal-internal trips. The methodology used to develop the new traffic assignment model assigned each of the trip types by varying the diversion of trips from the minimum path. External-external trips were assigned on a minimum path routing and external-internal trips were assigned with a slight diversion from the minimum path. Internal-internal trips were assigned with more diversion than external-internal trips and adjusted by utilizing iterative volume restraint and incremental link restraint. A statistical analysis indicated that assigning trips by trip types using trip diversion and volume and link restraint produces a significant improvement in the accuracy of the assigned traffic volumes.     

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.     

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.     

Multioutput analysis of trucking operations using spatially dissaggregated flows

The operation of a trucking system at a national scale is analyzed by means of an operating multioutput cost function, which recognizes flows with different origin and destination as distinct components of the transportation product. The original, fully disaggregated, information on O-D specific flows is used to generate distance-weighted and pure-flow output vectors, whose dimension allows for the estimation of flexible cost functions. Different models are compared in terms of O-D specific marginal costs and second derivatives in common units, obtained through manipulations of both the estimated parameters and their variance-covariance matrix. Results indicate that marginal costs are not proportional to distance. Constant returns prevail in the operation of the system, simultaneously with economies of spatial scope and cost complementarity between flow pairs, with one exception which suggests some regional specialization. On methodological grounds, the bias on scale economies introduced by the single aggregated output measure (ton-km) becomes evident, and partial aggregation in terms of distance-weighted zonal flows seems more appropriate than pure flow aggregation when distances are heterogeneous. The multioutput formulation with O-D specific flows under a flexible form of the cost function, is recommended for meaningful policy analysis of transportation systems.     

Network equilibrium for congested multi‐mode networks with elastic demand

This paper proposes an elastic demand network equilibrium model for networks with transit and walking modes. In Hong Kong, the multi‐mode transit system services over 90% of the total journeys and the demand on it is continuously increasing. Transit and walking modes are related to each other as transit passengers have to walk to and from transit stops. In this paper, the multi‐mode elastic‐demand network equilibrium problem is formulated as a variational inequality problem where the combined mode and route choices are modeled in a hierarchical logit structures and the total travel demand for each origin‐destination pair is explicitly given by an elastic demand function. In addition, the capacity constraint for transit vehicles and the effects of bi‐directional flows on walkways are considered in the proposed model. All these congestion effects are taken into account for modeling the travel choices. A solution algorithm is developed to solve the multi‐mode elastic‐demand network equilibrium model. It is based on a Block Gauss‐Seidel decomposition approach coupled with the method of successive averages. A numerical example is used to illustrate the application of the proposed model and solution algorithm.     

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     

Modeling the effects of traveler information on freeway origin–destination demand prediction

The primary focus of this research is to develop an approach to capture the effect of travel time information on travelers’ route switching behavior in real-time, based on on-line traffic surveillance data. It also presents a freeway Origin–Destination demand prediction algorithm using an adaptive Kalman Filtering technique, where the effect of travel time information on users’ route diversion behavior has been explicitly modeled using a dynamic, aggregate, route diversion model. The inherent dynamic nature of the traffic flow characteristics is captured using a Kalman Filter modeling framework. Changes in drivers’ perceptions, as well as other randomness in the route diversion behavior, have been modeled using an adaptive, aggregate, dynamic linear model where the model parameters are updated on-line using a Bayesian updating approach. The impact of route diversion on freeway Origin–Destination demands has been integrated in the estimation framework. The proposed methodology is evaluated using data obtained from a microscopic traffic simulator, INTEGRATION. Experimental results on a freeway corridor in northwest Indiana establish that significant improvement in Origin–Destination demand prediction can be achieved by explicitly accounting for route diversion behavior.     

Real-time freeway traffic state estimation based on extended Kalman filter: Adaptive capabilities and real data testing

This paper reports on real data testing of a real-time freeway traffic state estimator, with a particular focus on its adaptive capabilities. The pursued general approach to the real-time adaptive estimation of complete traffic state in freeway stretches or networks is based on stochastic macroscopic traffic flow modeling and extended Kalman filtering. One major innovative feature of the traffic state estimator is the online joint estimation of important model parameters (free speed, critical density, and capacity) and traffic flow variables (flows, mean speeds, and densities), which leads to three significant advantages of the estimator: (1) avoidance of prior model calibration; (2) automatic adaptation to changing external conditions (e.g. weather and lighting conditions, traffic composition, control measures); (3) enabling of incident alarms. These three advantages are demonstrated via suitable real data testing. The achieved testing results are satisfactory and promising for subsequent applications.     

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.     

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.