Efficient conformal prediction intervals for time series: Online PID-Expert aggregation

2026-10-02

AI Summary

This study addresses the inefficiency of conformal prediction intervals and unstable coverage caused by single PID calibration in time series forecasting. To overcome these limitations, this work proposes PID-Expert, an online aggregation method that introduces a novel dynamic aggregation mechanism weighted by normalized interval width and miscoverage rate. By integrating multiple expert thresholds to optimize prediction intervals, the approach establishes theoretical guarantees through local regret bounds and path-wise upper bounds. Experiments on both synthetic and real-world datasets demonstrate that the proposed method generates narrower mean prediction intervals while strictly maintaining nominal coverage. The results indicate that PID-Expert significantly outperforms existing baselines and averaging strategies, offering a robust solution for reliable uncertainty quantification in sequential prediction tasks.

Abstract

For a given point forecaster, proportional-integral-derivative (PID) calibration configurations can attain similar overall coverage yet produce different interval widths. We introduce PID-Expert, an online aggregation method for improving the efficiency of conformal prediction intervals for time series. PID-Expert combines thresholds from a fixed library of PID calibrators, each evolving under its own coverage feedback. Aggregation weights depend on normalized interval width and miscoverage, while a shared multiplier adapts the miscoverage penalty using feedback from the reported interval. We establish local regret bounds for weighted expert losses under the realized multiplier sequence and, separately, a pathwise upper bound on time-averaged aggregate miscoverage, with control in expectation and almost surely under a stability condition. Across four simulation settings and two real-data applications, PID-Expert produces narrower mean intervals than a prespecified Conformal PID benchmark while keeping overall empirical coverage close to the nominal level. Compared with expert selection and equal averaging, aggregation generally provides a more balanced coverage--width trade-off across forecasting settings. Rolling analyses further reveal local coverage--efficiency trade-offs, particularly following abrupt distributional shifts. Overall, PID-Expert reduces reliance on a single PID configuration while improving interval efficiency.

Problem

• conformal prediction
• time series
• prediction intervals
• interval efficiency
• coverage

Innovation

• Conformal Prediction
• PID Calibration
• Online Aggregation
• Time Series
• Regret Bounds