3.4 Density-based OD estimation
3.4.1 Methodology
The idea is that available speed measurements can help to identify whether a flow measurement represents a congested or uncongested situation. As such they can help to correctly interpret the flow measurements, and to identify which OD flows need to be adjusted, and in which direction. This should help to avoid the occurrence of local optima. The most straightforward approach to include the information provided by speed measurements is to include the deviations from speed measurements in the goal function of the OD estimation problem. The goal function can thus be written as follows:
(
)
(
( )
)
(
( )
)
1 2 3
( ) , , ,
g x =z x x +z y x y +z v x v (3.12)
where v are the measured speeds, and v(x)are the simulated speeds that are a function of the OD flows.
Using the fundamental diagram, another approach to include the same information provided by flow and speed measurements would be to use density data. Indeed, according to the fundamental diagram, there is a one-to-one relationship from density to flow, as well as from density to speed. We explain now that from a theoretical point of view one would expect that using density data, although it provides the same information as the speed-flow case, does a better job in getting to the correct regimes in the OD estimation procedure. Figure 3.11 shows a triangular fundamental diagram, which is used typically in first-order kinematic wave theory (Newel (1993)), and which is also adopted in LTM. Suppose that we start from the point (k’, q’) of the diagram, which is in free flow regime, while the measurement is the point (k’’, q’’), thus in congestion. If one adopts a gradient-based approach for the OD estimation procedure using a combination of flow and speed data in the goal function, the gradient of the flow term will suggest decreasing the OD flow, while the gradient of the speed term will be zero (because in free flow the speed is indifferent to the flow rate for a triangular fundamental diagram). The gradient will thus point in an incorrect direction. For sake of simplicity we used here a triangular relationship between flows, speeds and densities, but the same argument can be made on any concave fundamental diagram. Whether the gradient will point in the correct direction depends on the relative weight given to flow and speed measurements. A density-based approach is not affected by this limitation, because the gradient points into the correct direction. In the previous example, a gradient-based approach therefore has no problem to get to point (k’’, q’’). A second approach is therefore to use the following goal function in the OD estimation problem:
(
)
(
( )
)
1 2
( ) , ,
g x =z x x +z ρ x ρ (3.13)
where ρ are the measured densities, and ρ(x)are the simulated speeds that are a function of the OD flows.
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Figure 3.11: Triangular fundamental diagram used in first-order kinematic wave theory
However, there are some issues that one has to consider when using density data. First of all, in most cases densities are not measured directly, but instead flows and speeds are measured. In that case the densities need to be estimated from the flow and speed measurements. Density can be approximated by dividing flow by speed but this approximation is known to be biased for instance in non-stationary conditions and for low speeds (see e.g. Helbing (1997)). Secondly, the flow rate is highly sensitivity to the density in free flow regime: a very small change in density leads to a large change in terms of flow, as can be understood from the steep slope of the free flow part of the fundamental diagram in Figure 3.11. As a result the estimation becomes highly sensitive to errors in the density. Also, the priority of the solution algorithm to reduce errors in the density in the free flow regime will be very low, since the reduction of error here is limited. To overcome this problem a two-stage approach can be used. First only density data is used, and the goal function (3.13) is optimized. This first stage should help to correctly identify the regime, hereby removing the ambiguity of the traffic flows that was mentioned earlier. Next, this estimate is improved by using flow data in the goal function, possibly complemented with density data. This would result respectively in the following goal functions:
(
)
(
( )
)
1 2 ( ) , , g x =z x x +z y x y (3.14)(
)
(
( )
)
(
( )
)
1 2 3 ( ) , , , g x =z x x +z y x y +z ρ x ρ (3.15)This second stage is used to fine-tune the results, hereby avoiding the aforementioned sensitivity to measurement errors. In the second phase the use of both flow and density data can be preferred over the use of flow data solely, to prevent the estimation to switch back to the incorrect regime, which is likely to happen if for instance only flow data would be used.
It should be noted that the combination of different data types results in issues regarding the weight that should be given to each type of data. When these data types have a different order
of magnitude, less importance is given to the data type of lesser magnitude. This should be corrected for (e.g. by considering normalized values). Also, there might be
between different types of data measured by the same detector. When this is not accounted for, some information in the network might get a higher weight than it should get. To address this problem, it is common practice to weigh with an inverse covariance matrix. However, this covariance matrix assumes the relationship between two data sets to be linear, which does not hold in many cases, for example in the case of the relationship between flows and speeds, or in the case of the relationship between flows and densities.
decide to first assess the proposed methodology without this correction. As will become clear later on, we will decide to continue our research in a different direction. However, should the proposed methodology be further developed , this problem w
In the next section we test the
study, and analyze whether they result in an estimated OD matrix that produces flows and speeds similar to reality.