Considerations on the optimization method

In the proof of concept, a steepest descent method was used as optimization method. A disadvantage of this method is its slower rate of convergence: many iterations are required before convergence. In the context of the sensitivity-based OD estimation, this requires nOD

DNL simulations per iteration, nOD being the total number of OD flows. This constitutes an

enormous computational effort. A faster convergence would help to keep this effort to a minimum. In this case study, the Gauss-Newton method is chosen as optimization method to determine a search direction. It is a part of the family of Newton type optimization methods. It has the advantage that no second derivatives need to be calculated. The search direction (in the case of unconstrained optimization) is determined as follows:

(

)

In this equation F(xk-1) is the vector of deviations between the measurements and the

simulated link flows acquired by assigning xk-1, the OD flows determined in iteration k-1. J is

the Jacobian of F(x) at point xk-1. In the case that the deviations from the a priori matrix are

added as a regularization term, the search direction is determined as follows:

(

) (

)

( ) ( )

T T

k

p = − J J +εΙ − J F x_{k -1} +ε x_{k -1}−x
(4.9) where I is the identity matrix, and ε is the weight for the regularization term. In each iteration, the Gauss-Newton method makes an approximation of the actual goal function, based on the information at point xk. In this approximation it is assumed that the Jacobian J does not alter.

In the case of the SBODE, this can be expected to be a rather good approximation, since in many cases the sensitivity of the link flows to the OD flows does not alter much when there are no regime switches. In fact, when using a triangular fundamental diagram, the sensitivity does not change at all in free flow areas. Therefore, the Gauss-Newton method will have rapid converge properties. Also, because the Jacobian does not alter much in the region of the correct regime, and the Gauss-Newton method assumes a constant Jacobian, it is expected that the Gauss-Newton method has a tendency to stay in the correct regime. Nevertheless, there are also cases where the sensitivity differs a lot, and where furthermore the initial congestion pattern is very sensitive to small changes of the OD flows. In such cases, the

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Gauss-Newton method might take a large jump outside the area of the correct regime, and converge to a local optimum with a different congestion pattern.

In the remainder of this thesis, we limit ourselves to the use of the Gauss-Newton method. We acknowledge that there might be other, more specialized optimization algorithms with better properties (regarding both the convergence rate, and staying in the correct regime). Such a study was outside the scope of this research. In our experience, the Gauss-Newton method proved to be sufficiently efficient, and was not conceived as a major bottleneck in the SBODE methodology.

4.5.3 Results

Unlike for the proof of concept, the actual OD matrix is not known. To compare the accuracy of the different methods, we use the measured flows and speeds for comparison. The performance of the four cases in terms of the RMSE/RMSN of the link flows and speeds is summarized in Table 4.2. Figure 4.11 and Figure 4.12 present plots regarding the speeds and the travel times, and Table 4.3 summarizes these plots quantitatively. Note that the travel times were not measured directly, but are derived from the measured speeds according to the methodology described in Himpe et al. (2011).

Table 4.2 indicates that there is a substantial difference between case a and c, versus case b and d. In the former case, a simplified linear model was used, and the results in terms of link flows are much worse than in the latter case, where a correct linear model was used. This seems to suggest that having a simplified linear model mainly affects the match with the link flows in a negative way. In the case study in the next chapter (subsection 5.3.3.2), this however will not be confirmed, so this finding might be coincidental. Regarding the match with the speed measurements, case c and d perform better than case a and b, although the difference in terms of RMSE and RMSN is not as substantial as for the link flows.

Comparing the RMSE/RMSN values of the speeds in Table 4.2, the difference between the four cases is not that large. The ratio between the performance on the second best case and the performance of the proposed methodology is 1.3. When comparing the travel times and congestion pattern based on Table 4.3, the difference between the four cases is much larger. The ratio between the performance on the second best case and the performance of the proposed methodology ranges from 2 to 3. Nevertheless, it is important to realize that all above-mentioned measures are based on exactly the same data, i.e. the speeds on the links. From these comparisons, it is clear that case d (i.e. our proposed methodology) performs substantially better than all other cases. While there are still some differences with respect to the measured congestion pattern and travel time, the evolution in time is well reproduced in both cases (as can be seen in Figure 4.11 and Figure 4.12), which is an essential requirement that needs to be met.

Case a Case b Case c Case d RMSE link flow (veh/h) 610.9 223 701 232.9

RMSN link flow (%) 18.9 6.9 21.7 7.2

RMSE speed (km/h) 29.4 26.48 23.62 18.24

RMSN speed (%) 42.1 38 33.8 26.1

Table 4.2: RMSE and RMSN of link flows and speeds

(a)

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(c)

(d)

Case a Case b Case c Case d MAE congestion pattern 0.192 0.102 0.102 0.036

RMSE travel time (minutes) 4.5 3.2 4.9 1.6

RMSN travel time (%) 43.3 31.0 46.9 15.3

Table 4.3: MAE of congestion pattern and RMSE and RMSN of travel times

Figure 4.12: Travel time comparison for case (a), (b), (c) and (d)