Overview of OD Estimation Techniques

There has been a considerable amount of research conducted on estimating the OD matrix on freeway and urban networks. Traditionally, the OD matrix was obtained from direct measurements, roadside interviews, or license plate surveys (60). However, these approaches are not feasible in most cases because they are often costly and time consuming. In addition, these survey based methods are inappropriate to reflect rapid changes in OD patterns, leading to the outdated OD matrix. Therefore, in recent years, most research have tried to estimate the OD matrix in an indirect way, mainly using traffic counts and sampled OD data.

The OD estimation is to find the most “likely” OD matrix that can reproduce the observed traffic count at any space-time point. Equation 2.12 describes the basic relationship between the estimated OD matrix and observed traffic counts. For a given link a, the sum of all OD pairs passing this link is the observed traffic count on link a.

a v = ij ij a ij tt p

∑

v = traffic counts on link a;

a ij

p = proportion of OD flows between OD pair i and j that use link a;

ij

However, this formulation based on the observed traffic counts leads to an underdetermined OD structure (61, 62, 63). That is to say, there could be a greater number of feasible OD matrices that reproduce the observed traffic counts. This occurs because the number of OD pairs to be estimated is larger than the number of links where traffic counts are measured. Therefore, additional information is required to determine a unique solution. In general, this information is in the form of a priori OD information obtained by a sample OD survey or from an outdated OD matrix.

With this additional information, the problem of estimating the OD matrix can be formulated in the following general form, where the a priori OD matrix is used as a target OD matrix. As shown in Equation 2.13, a unique OD matrix given the target OD matrix is obtained by minimizing the deviation between the estimated and target OD matrices and deviation between the estimated and observed link traffic counts. The values of the weights are determined based on the reliability or accuracy of the target OD matrix and the observed link traffic counts. For example, if traffic counts are observed accurately, then a lager value is assigned to α₂. In turn, this leads to the estimated OD matrix that reproduces link volumes close to the observed traffic counts while allowing a larger deviation between the estimated and target OD matrices.

) , (

min F T v = α1F1(T,Tˆ)+α2F2(v,vˆ) (2.13)

T

T ˆ, = estimated and target OD matrix, respectively ; v

v ˆ, = estimated and observed traffic counts, respectively ;

2 1, F

F = distance measures for OD matrix and traffic counts,

respectively ; and

2 1,α

α = weights for OD matrix and traffic counts, respectively.

Research on the OD estimation can be divided using several categories (64, 65). They includes: 1) time horizon; 2) treatment of congestion; 3) mathematical formulation;

and 4) network configuration. Table 2.2 summarizes important properties that distinguish various approaches. The first category is based on the time horizon. In the steady state OD estimation approach, a dynamic nature of OD patterns is overlooked and the average OD matrix for a relatively long time interval is estimated. In the dynamic model, time dependant traffic counts are taken into account to identify the time varying OD patterns. It is assumed that the travel time for a specific OD pair can span a number of different time intervals by considering various travel times consumed by individual vehicles. Therefore, it allows fractions of OD flows for the current time interval to arrive at their destinations for some future time intervals.

Table 2.2 Classification of OD Estimation Approaches

Category Sub Division

Time Horizon • Steady state • Time dependant Treatment of

Congestion • Proportional assignment • Equilibrium assignment Mathematical

Formulation

• Traffic modeling approach • Statistical inference approach Network

Configuration

• Isolated intersection or freeway • Urban Network or combined network

The second category is the treatment of congestion. The proportional assignment assumes that link costs (or link travel times) are independent of link volumes. Therefore, this approach can be applied to networks with a low level of congestion. The assignment matrix is determined and given exogenously before the estimation of the OD matrix. In the presence of congestion, the equilibrium assignment model can be more representative of a reality. It assumes that link costs depend on link flows. Therefore, the assignment matrix is a function of link volumes, resulting in nonlinear relationship of

where the estimation of the OD matrix is conducted on the upper level whereas a user- equilibrium problem is solved at the lower level.

The third category is based on the mathematical formulation, which is further subdivided into two categories: traffic modeling approach and statistical inference approach. The traffic modeling approach is the minimum information (or entropy maximization) model. It is assumed in this approach that the target OD matrix is typically an old OD matrix and is adjusted to meet the traffic counts. The statistical inference approach includes the maximum likelihood (ML), generalized least squares (GLS), Bayesian approach, and Kalman filtering approach. The statistical approach generally assumes that the target OD matrix is obtained from a sample survey and is considered as an observation of the true OD matrix. In turn, the true OD matrix is assumed to follow a certain probability distribution, thus the estimation of the OD matrix is obtained by estimating the parameters of the statistical distributions (65).

The last category is based on the network configuration. The estimation of the OD matrix at an isolated intersection or a freeway network is a special case. In recent years the research emphasis shifted to the development of the OD matrix estimation approach that is suitable for congested urban networks.