Skew-adaptive conformal prediction

Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.

A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.

Recent advances in uncertainty quantification for time series forecasting show that conformal prediction can provide reliable prediction intervals, yet standard conformal methods are often inefficient under temporal dependence, drift, and heterogeneous error behavior. Existing methods typically either update miscoverage rates over time or learn unconstrained calibration weights, without explicitly separating two central sources of nonstationarity: smoothly drifting error distributions and co-existing distinct error regimes. We introduce RareCP, a regime-aware retrieval method for adaptive conformal time series prediction. RareCP learns local calibration representations through a mixture of cosine-attention experts that each capture distinct error regimes, while a compact hypernetwork adapts the kernel parameters to track temporal drift. Given a new forecasting context, RareCP retrieves the top-k most relevant calibration examples, assigns similarity weights, and forms a weighted conformal quantile over their signed residuals, yielding asymmetric prediction intervals. The adaptive kernel is trained using a smooth interval score objective, with a parameter-space anchor to a lightweight teacher kernel to preserve stable local representations. On the GIFT-Eval benchmark, RareCP improves interval efficiency over recent conformal baselines and foundation model uncertainty estimates while maintaining empirical coverage. Ablations confirm that regime-specific experts, drift-adaptive kernels, sparse retrieval, and teacher anchoring each contribute to the final performance.