Abstract
Drivers in different countries have different driving styles, drive different types of vehicles, and are subject to different traffic regulations. This means, models need to be adapted to the situations they are to describe by varying their parameters (calibration). Furthermore, it must be verified that this procedure is successful (validation). After introducing the mathematical principles behind calibration, we discuss nonlinear optimization and give hints of how to run a calibration task. We explain the various calibration methods by means of example and also discuss the necessary data preparation. Finally, we introduce validation techniques and point to interpretation pitfalls and the limits of the predictive power of models.
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
Attributed to von Neumann
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Notes
- 1.
The superscript T denotes transposition, i.e., a row rather than a standard column vector.
- 2.
By this step, the ML method looses its first-principles nature and assumes the same ad-hoc nature as the LSE calibration.
- 3.
iid is an abbreviation for independently and identically distributed.
- 4.
Obviously, this assumption is not fulfilled as explicitly described for the Human Driver Model by serially-correlated estimation errors obeying Eq. (12.9). However, it can be shown that violation of this assumption does not change the estimates \(\hat{\varvec{\beta }}\) but only their statistical properties.
- 5.
Of course, there are no measurements for the parameter vector. However, the Kalman filter does not need empirical data for all state variables.
- 6.
Since maximizing a function means minimizing the negative function, we will only speak of minimization, henceforth.
- 7.
In contrast, the objective function (negative log-likelihood) obtained by a local maximum-likelihood calibration remains smooth.
- 8.
It iterates to the exact minimum in one step if the function is purely quadratic, i.e., the Hessian \(\mathbf{\mathsf{{H}} }\) does not depend on \(\varvec{\beta }\).
- 9.
This is also a desirable property. However, orthogonality in the errors of the parameter estimates depends more on the fitting data and on the objective function than on the model.
- 10.
For notational simplicity, we drop the vehicle index \(\alpha \).
- 11.
The length drops out: Following the leading vehicle is equivalent to following its rear bumper.
- 12.
In a popular test data set, the sign of the acceleration changes in 80 % of the 0.1-s time intervals.
- 13.
Sometimes, the data are already organized in this form organized in this form, as is the case of the data of the well-known NGSIM initiative.
- 14.
We could trace back such a jump in a common data set to the boundary of a shadow cast from a tall building onto the road which obviously “bamboozled” the tracking software.
- 15.
If the model allows for a simulation with the data time step, no re-sampling should be undertaken. Otherwise, the simulation time step should be as near as possible to the time step of the data.
- 16.
Smoothing and differentiation are linear operators, so their order can be exchanged.
- 17.
In fact, the central ACC control logic must provide consistent acceleration responses to cut-ins, cut-outs, and active lane changes.
- 18.
The vehicle length \(l\) drops out in calibrations to trajectory or extended floating-car data.
- 19.
Notice that we gave the opposite recommendation when calibrating to extended floating-car data or single trajectory data: There, the integrated speed is externally fixed by the leader while no constraints apply to the gap, i.e., the microscopic density.
- 20.
This includes fluctuations of the inflow from one minute to the next, whether a driver entering the freeway via the ramp is able to merge at once, or whether there are trucks overtaking each other, or not.
- 21.
Even for state prediction rather than calibration, we do not recommend using speed detector data to estimate densities and use them directly by calculating \(c_{12}=(Q_2-Q_1)/(\rho _2-\rho _1)\). The bias in estimating the densities makes this approach impractical.
- 22.
There may be a small drift as a consequence of systematic counting errors at the two detectors. This can be taken care of by applying Eq. (16.45) only if the downstream detector indicates congested traffic. However, this measure was not necessary in our example.
- 23.
Generalizing the method to include ramps is straightforward but requires ramp detector data which are rarely available.
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Further Reading
Further Reading
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Brockfeld, E., Kühne, R.D., Wagner, P.: Calibration and validation of microscopic traffic flow models. Transportation Research Record: Journal of the Transportation Research Board 1876 (2004) 62–70
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Hoogendoorn, S., Hoogendoorn, R.: Calibration of microscopic traffic-flow models using multiple data sources. Philosophical Transactions of the Royal Society A: Mathematical, Physical, Engineering Sciences 368 (2010) 4497–4517
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Kesting, A., Treiber, M.: Calibrating car-following models by using trajectory data: Methodological study. Transportation Research Record: Journal of the Transportation Research Board 2088 (2008) 148–156
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Ngoduy, D., Maher, M.J.: Calibration of second order traffic models using continuous cross entropy method. Transportation Research Part C: Emerging Technologies 24 (2012) 102–121
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Ossen, S., Hoogendoorn, S.P.: Heterogeneity in car-following behavior: Theory and empirics. Transportation Research Part C: Emerging Technologies 19 (2011) 182–195
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Punzo, V., Borzacchiello, M.T., Ciuffo, B.: On the assessment of vehicle trajectory data accuracy and application to the Next Generation SIMulation (NGSIM) program data. Transportation Research Part C: Emerging Technologies 19 (2011) 1243–1262
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Thiemann, C., Treiber, M., Kesting, A.: Estimating acceleration and lane-changing dynamics from next generation simulation trajectory data. Transportation Research Record: Journal of the Transportation Research Board 2088 (2008) 90–101
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Treiber, M., Kesting, A.: Validation of traffic flow models with respect to the spatiotemporal evolution of congested traffic patterns. Transportation Research Part C: Emerging Technologies 21 (2012) 31–41
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Treiber, M., Kesting, A. (2013). Calibration and Validation. In: Traffic Flow Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32460-4_16
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DOI: https://doi.org/10.1007/978-3-642-32460-4_16
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