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Copula Conformal prediction for multi-step time series prediction

ICLR 2024 · Time series › Causal and generative time series · approach: Probabilistic and conformal forecasting · 4 citations (OpenAlex)

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Proposes CopulaCPTS, a two-stage conformal prediction method that uses an empirical copula over per-step conformal CDFs to produce valid, sharper joint confidence regions for multivariate multi-step time series forecasts.

Problem

Conformal prediction for multi-step, multivariate time series forecasting yields either only per-step coverage or inefficient joint regions when Bonferroni correction is used over long horizons and high dimensions.

Approach

Split calibration data into two sets: the first builds conformal predictive CDFs per time step; the second fits an empirical copula over the resulting per-step quantile vectors, and a gradient-descent search finds the smallest joint quantile vector satisfying the copula coverage constraint, giving full-horizon finite-sample validity.

Key results

• At 90% confidence on the particle simulation (sigma=0.01), CopulaCPTS (Copula-RNN) achieves 91.3% coverage with area 1.08, versus CF-RNN 97.1%/1.2 and BJRNN 98.9%/2.24 (both more conservative).
• On Argoverse, Copula-LaneGCN reaches 90.4% coverage with area 126.8 versus CF-LaneGCN 98.8% coverage with area 396.9, roughly a 40% sharper region while remaining valid.
• On COVID-19 daily cases, CopulaCPTS reduces the confidence width by about 33% relative to CF-RNN (area ~408.6 vs 610.2) while staying near 90% coverage, though one of three trials was invalid.

Measured effects reported in the paper

Surprising findings

• The Copula-vanilla baseline (multi-target copula without two-step calibration) is often invalid, e.g. 86.9% coverage on the particle sigma=0.01 task and 46.1% for MC-dropout in one setting.
• Autoregressive use of a fixed copula calibrated on early steps gives under-coverage (88.2%) on non-stationary COVID-19 data, so the copula must be re-estimated.

Limitations

• Requires abundant calibration data because the calibration set is split in two; performance deteriorates with scarce calibration (COVID-19 had one invalid trial).
• Validity proof relies on the empirical copula and does not extend to other learnable copula classes.
• Autoregressive extension does not yet guarantee full-horizon coverage for non-stationary series.

Experimental setup

Datasets: Interacting particle simulation (synthetic) Drone trajectory simulation (PythonRobotics) UK COVID-19 daily new cases Argoverse motion forecasting

Benchmarks: Interacting particle simulation Drone simulation COVID-19 daily cases Argoverse trajectory prediction

Metrics: empirical coverage area/volume of confidence region calibration curves across confidence levels

Baselines: MC-dropout BJRNN CF-RNN Copula (multi-target copula CP)

Builds on: inductive conformal prediction Sklar's theorem conformal predictive distributions Bonferroni correction for conformal forecasting (CF-RNN)

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