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Conformal prediction for multi-dimensional time series by ellipsoidal sets
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Proposes MultiDimSPCI, a sequential conformal method that builds ellipsoidal prediction regions for multivariate time series using a Mahalanobis-type non-conformity score and SPCI-style quantile regression, with finite-sample conditional coverage bounds under mixing assumptions.
Problem
Conformal prediction for multivariate (multi-dimensional) time-series forecasting, where observations are non-exchangeable and existing multi-dimensional CP methods yield hyper-rectangles or assume exchangeability/i.i.d. data.
Approach
Extends the univariate SPCI sequential conformal method to multivariate responses by using a Mahalanobis-distance non-conformity score based on a pseudo-inverse of the residual covariance, so prediction regions are ellipsoids (or differences of ellipsoids) whose radii are sequentially re-estimated via quantile regression.
Key results
- On real data at p=8, MultiDimSPCI gives size 72.10 on wind vs 252.67 for CopulaCPTS, with coverage 0.96, while keeping coverage near the 0.95 target.
- In simulations with p=20 on independent AR(5), MultiDimSPCI keeps 90.0% coverage while the coordinate-wise SPCI size is orders of magnitude larger (e.g., 9.15e+12 vs 8.67e+14 for SPCI).
- Under strong mixing, the coverage-gap bound is O~(T^{-1/3}) plus a term depending on the estimator error delta_T.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ellipsoidal MultiDimSPCI instead of CopulaCPTS | Multivariate time-series prediction · Wind data (p=8) | size: 252.67 -> 72.10 | Helped |
| Use ellipsoidal MultiDimSPCI instead of CopulaCPTS | Multivariate time-series prediction · Solar data (p=8) | size: 3.28e+3 -> 4.97 | Helped |
| Use ellipsoidal MultiDimSPCI instead of CopulaCPTS | Multivariate time-series prediction · Traffic data (p=8) | size: 14.10 -> 2.98 | Helped |
| Use ellipsoidal MultiDimSPCI instead of coordinate-wise SPCI | Independent AR(w) · p=20 | size: 8.67e+14 (2.90e+13) -> 9.15e+12 (2.97e+11) | Helped |
| Use ellipsoidal MultiDimSPCI instead of coordinate-wise SPCI | VAR(w) · p=20 | size: 7.41e+22 (1.68e+21) -> 1.14e+16 (2.11e+14) | Helped |
| Add local covariance ellipsoids to the global-covariance method (ablation) | MultiDimSPCI · Multivariate time-series prediction | average prediction-set size: up to 25% reduction | Helped |
| Replace ellipsoidal prediction regions with convex hulls (ablation) | MultiDimSPCI · Two independent AR(w) sequences | size: 3.92 -> 3.64 | Helped |
| Replace ellipsoidal prediction regions with convex hulls (ablation) | MultiDimSPCI · Two independent AR(w) sequences | coverage: 90.2 -> 90.0 | No clear effect |
| Replace ellipsoidal prediction regions with convex hulls (ablation) | MultiDimSPCI · Uniform errors in [-1,1]^4 (80000 training samples) | coverage: 89.6 -> 85.9 | Hurt |
| Replace ellipsoidal prediction regions with convex hulls (ablation) | MultiDimSPCI · Uniform errors in [-1,1]^4 (800000 training samples) | coverage: 90.2 -> 88.6 | Hurt |
Surprising findings
- On solar data at p=2, the Local ellipsoid baseline yields a smaller average size (1.32) than MultiDimSPCI (1.68).
- Coordinate-wise SPCI and the copula baselines produce hugely inflated regions at higher dimensions (e.g., copula 1.60e+3 at p=8 on wind).
Limitations
- Coverage guarantees rely on assumptions (i.i.d. or stationary strong mixing, Lipschitz CDF, bounded tails, estimator error rate delta_T) that may not hold in practice.
- Conditional-coverage bound for the non-i.i.d. case is weaker (T^{-1/3}) and assumes known covariance for the stated corollary.
- Convex hull extension is only preliminary and becomes computationally costly in higher dimensions.
Experimental setup
Datasets: wind dataset solar dataset traffic dataset simulated AR(5) and VAR(5) series
Benchmarks: simulated AR(w) and VAR(w) series wind solar traffic
Metrics: empirical coverage average prediction region size
Baselines: SPCI (coordinate-wise) CopulaCPTS Local ellipsoid conformal Copula conformal TFT DeepAR Spherical confidence set
Builds on: SPCI copula conformal prediction split conformal prediction
Closest papers by meaning
- Copula Conformal prediction for multi-step time series prediction (ICLR 2024 · similarity 0.89)
- A Kernel Nonconformity Score for Multivariate Conformal Prediction (NeurIPS 2026 · similarity 0.88)
- Kandinsky Conformal Prediction: Beyond Class- and Covariate-Conditional Coverage (ICML 2025 · similarity 0.88)
- Optimal transport-based conformal prediction (ICML 2025 · similarity 0.87)
- Kernel-based Optimally Weighted Conformal Time-Series Prediction (ICLR 2025 · similarity 0.87)
- Probabilistic Conformal Prediction with Approximate Conditional Validity (ICLR 2025 · similarity 0.87)
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