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Error-quantified Conformal Inference for Time Series

ICLR 2025 · Time series › Causal and generative time series · approach: Probabilistic and conformal forecasting · 0 citations (OpenAlex)

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Proposes Error-quantified Conformal Inference (ECI), which augments the online conformal quantile update with a smooth error-quantification term so the threshold reacts to the magnitude of miscoverage, yielding tighter time-series prediction sets with long-term coverage guarantees.

Problem

Uncertainty quantification for time series prediction under temporal dependence and distribution shift, where existing online conformal methods (ACI, OGD) use only binary miscoverage feedback and so cannot distinguish the size of the error.

Approach

Smooth the quantile loss with a sigmoid approximation to the indicator and keep the true miscoverage indicator, adding an error-quantification term (s_t - q_t) f'(s_t - q_t) that gives continuous, magnitude-dependent feedback to the threshold update; variants add a cutoff on small errors and an integral over past steps.

Key results

• On Amazon stock with Prophet at alpha=10%, ECI-integral reaches 89.8% coverage with average width 42.01 versus 55.15 for OGD and 97.22 for decay-OGD, and ECI-cutoff gives 43.46.
• On Delhi temperature with AR, ECI-cutoff gives average width 6.28 versus 6.82 for OGD and 8.92 for PID, all near 90% coverage.
• Theory: for fixed learning rate, every miscoverage step is followed by N-1 coverage steps (N=floor(1/alpha)), and an averaged miscoverage bound is given for adaptive learning rates.

Measured effects reported in the paper

Surprising findings

• The fully smoothed update (smoothing both the indicator and the loss) is overly conservative, giving overly high early coverage and wider sets, which motivates keeping the hard miscoverage indicator.
• PID's scorecasting helps the weaker Prophet base predictor but widens sets with better base predictors.

Limitations

• Theoretical guarantee for fixed learning rate holds only within short intervals and requires a learning-rate and sigmoid-scale condition (c small relative to eta).
• Performance depends on hyperparameters such as scale c, window length w and learning rate range, which are selected per dataset.
• Assumes bounded non-conformity scores (s_t in [0,B]).

Experimental setup

Datasets: Amazon stock Google stock NSW electricity demand Delhi temperature synthetic changepoint data

Benchmarks: Amazon stock Google stock NSW electricity demand Delhi temperature synthetic changepoint data

Metrics: coverage (%) average interval width median interval width

Baselines: ACI OGD SF-OGD decay-OGD PID

Builds on: Adaptive Conformal Inference Conformal PID online gradient descent

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