An Empirical Bayes Analysis of Object Trajectory Representation Models

TL;DR

The paper investigates whether simple linear trajectory representations, specifically polynomial basis functions, can faithfully model real-world object motion across vehicles, cyclists, and pedestrians. By applying Empirical Bayes to estimate observation-noise covariances and priors over trajectory parameters from large public datasets, the authors quantify the trade-off between model complexity and fit error, using AIC/BIC to select optimal complexity. They report that moderate polynomial degrees yield very low fit error (often in the centimeter range) and that the resulting representation error is small compared to the total displacement error of current state-of-the-art predictors, implying linear models are both effective and computationally advantageous. The findings support using linear trajectory representations in motion-prediction systems, providing principled regularization and leveraging the mathematical benefits of linear models for tracking and filtering.

Abstract

Linear trajectory models provide mathematical advantages to autonomous driving applications such as motion prediction. However, linear models' expressive power and bias for real-world trajectories have not been thoroughly analyzed. We present an in-depth empirical analysis of the trade-off between model complexity and fit error in modelling object trajectories. We analyze vehicle, cyclist, and pedestrian trajectories. Our methodology estimates observation noise and prior distributions over model parameters from several large-scale datasets. Incorporating these priors can then regularize prediction models. Our results show that linear models do represent real-world trajectories with high fidelity at very moderate model complexity. This suggests the feasibility of using linear trajectory models in future motion prediction systems with inherent mathematical advantages.

Figures (7)

• Figure 1: The decomposition of the total displacement error from prediction system.
• Figure 2: An example of expression flexibility of sequence-based representations and model-based representations.
• Figure 3: Left: A typical scene from a trajectory prediction dataset, here Argoverse Motion Forecasting v1.1 chang_argoverse_2019. Data is gathered by a moving sensor platform (ego vehicle) and subsequently transformed into a fixed world coordinate frame. As distance and angle between sensor and agent change during recording, the observation covariance of agent locations stretches and rotates over time. We show all sample points and a few $95\%$ confidence ellipses for agent position, enlarged by a factor of $4$ for better visibility. Right: The same agent trajectory as in the left figure but fitted with a $5$-degree polynomial trajectory representation estimated via eq. (\ref{['eqn:posterior_mean']}). The resulting posterior covariances for agent positions are also shown, enlarged by a factor of $8$ for better visibility.
• Figure 4: The longitudinal (left) and lateral (right) fit error of models for vehicle trajectories in A1, A2 and WO with $T \in [3\mathrm{s}, 5\mathrm{s}, 8\mathrm{s}]$. "A, B" denote the model complexity $n=\hat{n}$ that maximizes AIC and BIC, respectively. The upper whisker denotes the 99.9% percentile.
• Figure 5: The fit error of models for cyclist (left) and pedestrian (right) trajectories in A2 and WO with $T \in [3\mathrm{s}, 5\mathrm{s}, 8\mathrm{s}]$. "A, B" denote the model complexity $n=\hat{n}$ that maximizes AIC and BIC, respectively. The upper whisker denotes the 99.9% percentile.