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Conformal Prediction Regions for Time Series Using Linear Complementarity Programming
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Proposes a parameterized multi-step nonconformity score whose weights are optimized via an LCP/MILCP on a first calibration split, yielding tighter valid conformal prediction regions for time-series predictors than union-bound approaches.
Problem
Split conformal prediction for time series with simultaneous coverage over T steps requires union bounding (confidence 1-δ/T per step), producing conservative, impractical prediction regions for long-horizon planning and verification.
Approach
Replace per-step union bounding with a single parameterized score R = max_t(alpha_t R_t), choose alpha_t by minimizing the empirical quantile on a first calibration split, then apply standard split conformal on a second split; the min-max problem is cast as a MILCP and relaxed to an LCP via KKT conditions, with equal optimal cost.
Key results
- Pedestrian case study: average trajectory coverage 0.9558 for the proposed method vs 0.9945 for the union bound approach over 100 trials, with significantly smaller regions.
- F16 case study: average coverage 0.953 for the proposed method vs 0.993 for union bound; region widths smaller for the proposed method across calibration sizes.
- LCP relaxation matched MILCP: average objective difference 1.847e-9; LCP runtime 0.207 s vs 0.525 s for the original problem on ORCA (F16 alphas exactly matched).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the optimized parameterized score instead of union-bound calibration | Time-series conformal prediction with a Social LSTM · ORCA pedestrian validation | prediction-region size | Helped |
| use the optimized parameterized score instead of union-bound calibration | Time-series conformal prediction with a Social LSTM · ORCA pedestrian validation | trajectory-level coverage: 0.9945 -> 0.9558 | Mixed |
| use the optimized parameterized score instead of union-bound calibration | Time-series conformal prediction with an LSTM · F16 altitude validation | prediction-region width | Helped |
| use the optimized parameterized score instead of union-bound calibration | Time-series conformal prediction with an LSTM · F16 altitude validation | trajectory-level coverage: 0.993 -> 0.953 | Mixed |
| increase |D_cal,1| from 100 to 400 (ablation) | Proposed method · F16 altitude validation | prediction-region width | Helped |
| increase |D_cal,1| from 100 to 400 (ablation) | Proposed method · F16 altitude validation | coverage variance: 5.89*10^{-5} -> 9.04*10^{-5} | Hurt |
| replace the original MILCP with the relaxed LCP (ablation) | Proposed method · ORCA optimization | runtime: 0.525 seconds -> 0.207 seconds | Helped |
| replace the original MILCP with the relaxed LCP (ablation) | Proposed method · ORCA optimization | objective value: 1.847*10^{-9} | No clear effect |
| replace the original MILCP with the relaxed LCP (ablation) | Proposed method · F16 optimization | optimized alpha values | No clear effect |
Surprising findings
- The proposed method's coverage is below the nominal 0.95 target only slightly in some cases, while union bound coverage is much higher, showing conservatism of union bounding.
- Larger first-split calibration data slightly shrinks regions but increases coverage variance.
Limitations
- Exchangeability via trajectory-level splitting requires separate full-rollout calibration data and splitting reduces data for the conformal step.
- Coverage variance increases when the first calibration split is larger.
- Prediction regions are balls under Euclidean error; other shapes require different choices of R_p.
Experimental setup
Datasets: ORCA simulator pedestrian trajectories F16 ground collision avoidance trajectories (Aerobench)
Benchmarks: ORCA pedestrian case study F16 altitude case study
Metrics: trajectory-level coverage conformal region size/area objective and alpha differences between MILCP and LCP runtime
Baselines: union bound approach (per-step conformal with delta/T) MILCP formulation
Builds on: split conformal prediction union bound approach for time-series conformal prediction Social LSTM
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