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Conformalized Adaptive Forecasting of Heterogeneous Trajectories

Yanfei Zhou, Lars Lindemann, Matteo Sesia

TL;DR

This work addresses the need for dependable uncertainty estimates in forecasting heterogeneous trajectories, as in motion planning, by introducing CAFHT, a conformalized adaptive forecaster that delivers simultaneous finite-sample guarantees for the entire predicted trajectory. The approach merges a black-box forecaster with Adaptive Conformal Inference (ACI) and a calibration-based conformity scheme to produce trajectory-specific prediction bands at level $1-\\alpha$, automatically widening or narrowing in response to apparent difficulty. It extends to multi-dimensional trajectories, offers additive and multiplicative conformity-score variants, and can incorporate Conformal PID, while providing data-driven parameter selection. Empirical results on synthetic and pedestrian data show CAFHT yields narrower, more informative bands with higher conditional coverage than baselines like CFRNN and NCTP, while maintaining the prescribed marginal coverage, demonstrating practical value for reliable uncertainty quantification in autonomous systems.

Abstract

This paper presents a new conformal method for generating simultaneous forecasting bands guaranteed to cover the entire path of a new random trajectory with sufficiently high probability. Prompted by the need for dependable uncertainty estimates in motion planning applications where the behavior of diverse objects may be more or less unpredictable, we blend different techniques from online conformal prediction of single and multiple time series, as well as ideas for addressing heteroscedasticity in regression. This solution is both principled, providing precise finite-sample guarantees, and effective, often leading to more informative predictions than prior methods.

Conformalized Adaptive Forecasting of Heterogeneous Trajectories

TL;DR

This work addresses the need for dependable uncertainty estimates in forecasting heterogeneous trajectories, as in motion planning, by introducing CAFHT, a conformalized adaptive forecaster that delivers simultaneous finite-sample guarantees for the entire predicted trajectory. The approach merges a black-box forecaster with Adaptive Conformal Inference (ACI) and a calibration-based conformity scheme to produce trajectory-specific prediction bands at level , automatically widening or narrowing in response to apparent difficulty. It extends to multi-dimensional trajectories, offers additive and multiplicative conformity-score variants, and can incorporate Conformal PID, while providing data-driven parameter selection. Empirical results on synthetic and pedestrian data show CAFHT yields narrower, more informative bands with higher conditional coverage than baselines like CFRNN and NCTP, while maintaining the prescribed marginal coverage, demonstrating practical value for reliable uncertainty quantification in autonomous systems.

Abstract

This paper presents a new conformal method for generating simultaneous forecasting bands guaranteed to cover the entire path of a new random trajectory with sufficiently high probability. Prompted by the need for dependable uncertainty estimates in motion planning applications where the behavior of diverse objects may be more or less unpredictable, we blend different techniques from online conformal prediction of single and multiple time series, as well as ideas for addressing heteroscedasticity in regression. This solution is both principled, providing precise finite-sample guarantees, and effective, often leading to more informative predictions than prior methods.
Paper Structure (43 sections, 2 theorems, 28 equations, 39 figures, 35 tables, 11 algorithms)

This paper contains 43 sections, 2 theorems, 28 equations, 39 figures, 35 tables, 11 algorithms.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Problem Statement and Notation
  4. Benefits and Limitations of Marginal Coverage
  5. Towards Approximate Conditional Coverage
  6. Preview of Main Contributions
  7. Methodology
  8. Training a Black-Box Forecasting Model
  9. Initializing the Adaptive Prediction Bands
  10. Calibrating the Adaptive Prediction Bands
  11. Data-Driven Parameter Selection
  12. CAFHT with Multiplicative Scores
  13. Extension to Multi-Dimensional Trajectories
  14. Leveraging Conformal PID Prediction Bands
  15. Direct Comparison to ACI
  16. ...and 28 more sections

Key Result

Theorem 1

Assume that the calibration trajectories in $\mathcal{D}_{\text{cal}}$ are exchangeable with $\bm{Y}^{(n+1)}$. Then, for any $\alpha \in (0,1)$, the prediction band output by CAFHT, applied with fixed parameters $\alpha$, $\alpha_{\mathrm{ACI}}$, and $\gamma$, satisfies eq:simu_coverage.

Figures (39)

  • Figure 1: One-dimensional representations of 10 pedestrian trajectories, one of which is intrinsically less predictable.
  • Figure 2: Conformal forecasting bands constructed using different methods, for the heterogeneous pedestrian trajectories from Figure \ref{['fig:individual_path']}. All methods guarantee simultaneous marginal coverage at the 90% level. Our method (CAFHT) can automatically adapt to the unpredictability of each trajectory. Here, the CFRNN bands so wide as to be uninformative, spanning from -1 to +1.
  • Figure 3: Performance on simulated heterogeneous trajectories of prediction bands constructed by different methods, as a function of the total number of training and calibration trajectories (of which 25% are utilized for calibration). All methods achieve 90% simultaneous marginal coverage. Our method (CAFHT) leads to more informative bands with lower average width and higher conditional coverage. The error bars indicate 2 standard errors. Note that the CFRNN bands here are so wide as to be uninformative.
  • Figure 4: Performance on simulated heterogeneous trajectories of prediction bands constructed by different methods, as a function of the prediction horizon. Other details are as in Figure \ref{['fig:main_exp_sim_ndata']}. For large prediction horizon, the CFRNN bands so wide as to be uninformative.
  • Figure 5: Performance on heterogeneous pedestrian trajectories of conformal prediction bands constructed by different methods, as a function of the noise level controlling the intrinsic unpredictability of the more difficult trajectories. Note that the CFRNN bands so wide as to be uninformative.
  • ...and 34 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof : Proof of Theorem \ref{['theorem:coverage']}
  • Theorem A1
  • proof