Fast Calibration of Car Following models to Trajectory data using the Adjoint Method

TL;DR

This work reframes car-following calibration as a gradient-enabled optimization problem where trajectories are fit to data via ODE/DDE-based dynamics. It derives analytic gradients using the adjoint method, enabling computational costs that scale with simulation time rather than the number of parameters, and demonstrates substantial speedups over gradient-free methods such as genetic algorithms. Empirical benchmarks on NGSim data with the optimal velocity model show adjoint-based quasi-Newton methods achieving faster convergence and slightly better accuracy, with large gains as the parameter count grows. The study also explores downstream boundary conditions, reaction-time modeling, and multi-regime/lane-changing discontinuities, and demonstrates that larger platoons can improve calibration results, while highlighting the need for strategies to avoid poor local minima in bigger problems.

Abstract

Before a car-following model can be applied in practice, it must first be validated against real data in a process known as calibration. This paper discusses the formulation of calibration as an optimization problem, and compares different algorithms for its solution. The optimization consists of an arbitrary car following model, posed as either an ordinary or delay differential equation, being calibrated to an arbitrary source of trajectory data which may include lane changes. Typically, the calibration problem is solved using gradient free optimization. In this work, the gradient of the optimization problem is derived analytically using the adjoint method. The computational cost of the adjoint method does not scale with the number of model parameters, which makes it more efficient than evaluating the gradient numerically using finite differences. Numerical results are presented which show that quasi-newton algorithms using the adjoint method are significantly faster than a genetic algorithm, and also achieve slightly better accuracy of the calibrated model.

Figures (11)

• Figure 1: On the left, an excerpt of a single lane from the NGSim data. "Incomplete" trajectories are due to vehicles changing to/out of the lane pictured. One can observe many of the important features of traffic flow such as shockwaves, lane changing events, and individual driver behaviors. Compared to macroscopic data, these features are captured at higher temporal/spatial resolutions and without aggregation. On the right, a car following model (the OVM) has been calibrated to the trajectory data.
• Figure 2: Four different car following models are calibrated to a single empirical vehicle trajectory by minimizing the root mean square error between simulation and measurements. The trajectories are plotted in the speed - headway plane.
• Figure 3: Boxes represent vehicles; vehicle $2$ follows $1$, and $1$ follows $L(1)$. In top panel, the leader $L(1)$ is still in the simulation and so the car following model can be used to update vehicle $1$. In bottom panel, $L(1)$ is no longer in the simulation, and so a downstream boundary condition is used to update vehicle $1$.
• Figure 4: Time for a single evaluation of the gradient for a progressively larger calibration problem. The objective varied from the loss from a single vehicle (5 parameters) to 15 vehicles (75 parameters).
• Figure 5: Compares the accuracy of the gradient for the adjoint method and simultaneous perturbation. The finite differences gradient is treated as the exact gradient.

Theorems & Definitions (6)

• Theorem 1
• Corollary 1
• Definition 1
• Definition 2
• Proof 1: Proof of Theorem 1.
• Proof 2: Proof of Corrolary 1.