Conformal prediction gives finite-sample coverage guarantees for regression, but most standard constructions are designed for Euclidean output spaces. When the response lies on a Riemannian manifold, Euclidean residuals and coordinate-based regions can ignore the geometry that defines meaningful error. We propose adaptive geodesic conformal prediction, a simple framework that builds nonconformity scores from geodesic distances and normalizes them with a cross-validated estimate of local prediction difficulty. On the sphere, this produces geodesic caps whose area is independent of position, while their radii still adapt to heteroscedastic noise. In both a synthetic sphere experiment and an IGRF-14 geomagnetic field forecasting task, the adaptive method preserves valid marginal coverage, reduces variation in conditional coverage, and improves worst-case coverage relative to non-adaptive and coordinate-based baselines.
Figures & tables
Figure 1: Visual comparison of the two geodesic conformal methods on the synthetic S2 benchmark. (a) The standard geodesic method assigns a fixed-radius cap to every test point, so easy and hard cases receive identical-size regions. (b) The proposed adaptive method varies cap size by local difficulty σ^(x) , producing tighter regions where prediction is easier and wider regions where prediction is harder.
Figure 2: 300-trial Monte Carlo distributions on the synthetic S2 benchmark. (a) All methods achieve valid marginal coverage near 0.90 . (b) The naive coordinate method requires ≈26% more area. (c) The adaptive method attains the lowest conditional coverage standard deviation. (d) Adaptive worst-case coverage stays closest to the 0.90 target.
Figure 3: Conditional coverage on the synthetic benchmark. (a) Coverage vs. estimated difficulty σ^(x) : standard and naive methods over-cover easy bins ( >0.97 ) and drop below 0.70 on hard bins; the adaptive method stays close to 0.90 across the range. (b) Coverage vs. true vMF concentration κ(x) : at low κ (hard, dispersed targets), both non-adaptive methods fall below 0.90 , while the adaptive method remains comparatively stable.
Figure 4: IGRF-14 data overview. (a) Secular variation rate ( ∘ /yr) at the 352 surface sites. The South Atlantic Anomaly (SAA, dashed ellipse) and magnetic equator exhibit the fastest field evolution, creating intrinsic spatial heteroscedasticity. (b) Distribution of the secular variation rate, spanning a 274× range from 0.0017∘ /yr to 0.467∘ /yr. No synthetic noise is added.
Figure 5: IGRF-14 100-trial Monte Carlo distributions. (a) All methods achieve valid marginal coverage near 0.90 . (b) The naive coordinate method uses substantially more area. (c) The adaptive method reduces conditional coverage standard deviation by ∼3.5× relative to standard geodesic. (d) Worst-case bin coverage improves by 0.166 relative to standard geodesic.
Figure 6: IGRF-14 conditional coverage and diagnostic. (a) Coverage vs. estimated difficulty σ^(x) : the standard geodesic method drops below 0.70 in the hardest bin, while the adaptive method stays near 0.90 . (b) Coverage vs. geographic latitude: the standard method under-covers equatorial latitudes where secular variation is fastest. (c) Difficulty estimator quality: σ^(x) vs. actual prediction error ( r=0.516 , p≈5.6×10−123 ).
We introduce a new conformal prediction method that constructs calibrated prediction sets over collections of spatial events, such as tropical cyclone genesis and earthquake locations. Forecasting natural hazards has become increasingly important, due to their significant economic impact, and quantifying the uncertainty of predictions is critical for accurate risk assessment. Our approach works by representing spatial point clouds as empirical measures so that we can score them using (sliced) Wasserstein distance, then constraining the resulting distribution-valued prediction set to be supported only near the training data manifold. We derive a coverage lower bound for the intersected sets and show that, in practice, this gap can be made small through a simple data-adaptive selection criterion. Because the resulting set is not analytically tractable, we introduce a modified flow-based sampling procedure, which allows us to represent and apply these prediction sets in practice as ensembles. Numerical experiments on synthetic data, tropical cyclone genesis, and earthquake occurrences show that our method achieves near-nominal coverage, with significantly lower energy distance and manifold distance than highest predictive density region (HDR) baselines along with generative model baselines.
Collin Nill, Trevor Harris, Jason Adams
Department of Statistics University of Connecticut Storrs, CT 06269 · Sandia National Laboratories Albuquerque, NM 87123
Split conformal prediction uses prediction errors on held-out (calibration) data to determine how wide the prediction intervals should be. It guarantees distribution-free finite-sample coverage when these errors and the error at the target site are exchangeable. This assumption may fail under spatial dependence and nonrandom sampling geometry. Existing spatial methods use fitting residuals to remove the predictable part of spatial variation from calibration and target errors. However, the spatial variation that only the calibration residuals can predict remains in both the target and calibration errors, reducing the efficiency and stability of the interval. We address this by additionally conditioning on the calibration residuals sequentially, which scales to large networks through nearest-neighbour approximations. Under a correct working covariance and an elliptical residual law, the resulting interval has exact finite-sample coverage under any spatial design, and under further conditions it is asymptotically oracle efficient. We also bound coverage loss under covariance misspecification and develop a diagnostic that identifies regions at risk of undercoverage. In simulated data, our method produces narrower and more stable intervals than global and localized state-of-the-art alternatives. In a national PM2.5 application, it produces narrower intervals within the network and identifies regions at risk of coverage failure.
Ayush Baran Sen, Arkajyoti Saha
Indian Statistical Institute, Kolkata, India · Department of Statistics, University of California, Irvine
This paper studies multivariate conformal prediction (CP), a distribution-free uncertainty quantification framework with finite-sample coverage guarantees. The efficiency of multivariate prediction sets hinges critically on the residual geometry encoded by the nonconformity score, while existing minimum-volume methods rely on quantile thresholds that ignore tail residual severity and implicitly bind geometry learning to coverage level. We propose a tail-aware geometry learning framework for conformal ellipsoids that decouples tail sensitivity in geometry learning from the final coverage guarantee. Using a two-split design, we learn the metric matrix via volume minimization under a CVaR constraint on an estimation split, then apply standard conformal calibration on a held-out calibration split. The resulting problem is convex and admits a bounded-reweighting interpretation that prioritizes high-residual samples. Moreover, we theoretically characterize the trade-off between ellipsoidal volume and tail severity. Experimental results demonstrate the effectiveness of the proposed method.
Xiang Zhang
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore