Prediction Inference of Time Series with Standard ReLU Deep Neural Networks

We propose a methodology based on the standard ReLU Deep Neural Networks (DNN) to make predictions and quantify their uncertainty. Classically, people rely on linear, non-linear, or non-parametric kernel methods to fit and then predict the time series. As the universal approximation ability was revealed for DNN, its application has become more and more popular for prediction tasks in various scientific areas. However, the corresponding uncertainty quantification has not been studied thoroughly. Particularly, the uncertainty in prediction will consist of two parts: (1) the future variability; (2) the estimation variability within training data. To capture both variabilities, we build the so-called pertinent prediction interval (PPI) with the DNN model estimator. We first explore the consistency property of the DNN estimator with beta-mixing dependent data. Subsequently, we show that the implied forward bootstrap series is still beta-mixing and possesses the same stationary distribution as the original time series in probability, which is a key condition to enable the PPI. Lastly, the desired PPI is built after imposing minimal conditions on the limiting distribution of predictive roots. Simulations and real-data analysis are deployed to challenge our approach with standard non-parametric methods.

Retrieval-Corrected Conformal Prediction for Time Series

Conformal prediction (CP) provides distribution-free prediction intervals for fixed forecasters, but its standard calibration procedure is often inefficient for time series data, where forecast errors are temporally dependent and change across time and operating conditions. Recent time series CP methods improve local calibration using recent, weighted, or localized residuals. Yet local calibration can remain indirect, since broad residual weighting or additional adaptation procedures may dilute the evidence most relevant to the current prediction. This motivates a simple retrieval and correction strategy that selects similar past residuals as local evidence and then corrects the coverage error left by retrieval. In this paper, we propose Retrieval--Corrected Conformal Prediction (RCCP), a retrieval-augmented calibration method for time series prediction intervals. RCCP builds an asymmetric interval from retrieved one-sided residuals and calibrates its normalized retrieval error with a scalar conformal correction. Thus, retrieval provides local residual evidence, while conformal correction determines the final scale needed for coverage. We provide a coverage-gap bound based on the stability of the normalized retrieval error distribution. Across standard benchmarks and backbone forecasters, RCCP attains the target coverage in every setting and achieves the lowest Winkler scores, with fewer severe misses. RCCP also achieves low calibration and inference overhead, showing that retrieval-corrected calibration is an effective and scalable approach to uncertainty quantification in time series forecasting. Code is available at https://github.com/jinsaaang/rccp.

Conformal Calibration for Multi-Modal Regression with Missing Modalities

Prediction intervals for multi-modal regression with tabular variables, text, images, or other input sources are difficult to calibrate when those sources disagree or one is missing. A single global quantile averages these regimes together instead of calibrating to the modality pattern observed at test time. We address this through a modality-aware conformal calibration layer. The layer trains or reuses one predictor per modality, computes a disagreement score from their predictions, and uses that score in split conformal calibration under a strict split protocol. We use the score in two complementary ways. First, a continuous disagreement-scaled method reallocates interval width across examples while preserving the usual marginal split-conformal guarantee. Second, a Mondrian (stratified) method calibrates within groups defined by disagreement or modality availability fixed before calibration, giving group guarantees under joint exchangeability of the calibration and test examples. Across four multi-modal datasets, the disagreement-scaled layer matches or improves the marginal conformal baseline in 59 of 60 paired runs for interval continuous ranked probability score (CRPS) and in 52 of 60 for interval width, while keeping empirical coverage near the 95% target. In stress tests with missing modalities, mask-matched recalibration recovers up to 19.5 percentage points of coverage in the hardest fixed-mask regime. The result is a simple, model-agnostic reliability layer for multi-modal regression systems. A project page is available at https://unco3892.github.io/modality-aware-conformal.

Isotonic Conformal Prediction

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.

TSCoNet: A Two-Stage Copula CNN-LSTM for Uncertainty-Aware Spatio-Temporal Forecasting

Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.

On Optimal Data Splitting for Split 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.

Doubly Robust Adaptive Conformal Inference for Causal Effects Under Temporal Dependence

We propose doubly robust adaptive conformal inference (DR-ACI), which constructs prediction intervals for doubly robust pseudo-outcomes under temporal dependence.

Conformal Prediction Intervals with Tail-Specific Guarantees

This paper extends classical conformal frameworks for constructing prediction intervals with global marginal coverage $1-α$ to intervals that provide explicitly calibrated guarantees for the upper and lower tails separately. Focusing on split conformal prediction, we first construct lower and upper one-sided conformal intervals that achieve marginal validity, and then derive the induced two-sided interval by intersection. Theoretical results prove both tail-specific and global marginal coverage of the induced two-sided interval. Results are presented first for the exchangeable setting, where coverage has finite-sample guarantees, and then for non-exchangeable data, where guarantees are asymptotic. Simulation studies show that the proposed approach achieves improved directional calibration relative to classical two-sided intervals, especially relevant in skewed data. Finally, the benefit of the proposed framework is showcased in a financial application, where one aims for return maximization while seeking strict control on the left tail.

Tail allocation for conformal prediction intervals

We study split-conformal prediction for regression when the reported prediction set must be a single interval, at target marginal coverage $1-α$, where $α$ is the nominal miscoverage level. Under this reporting constraint, the natural conditional target is the shortest interval with conditional mass at least $1-α$, rather than an equal-tailed interval or a possibly disconnected high-probability set. We parameterize this single-interval oracle by a lower-tail allocation, which determines how the nominal miscoverage $α$ is split between the two endpoints, and propose tail-allocation conformalized quantile regression (TA-CQR). TA-CQR estimates this allocation by searching over quantile-defined cores and then applies nonnegative additive split-conformal calibration, retaining exact finite-sample marginal coverage under exchangeability. The main contribution is theoretical. We characterize the oracle geometry, including its highest-density interpretation under unimodality and the positive connectedness cost induced by disconnected highest-density sets. We prove local recovery of the selected allocation and core, establish that calibration radii are asymptotically negligible under endpoint-density conditions, and give a finite-sample calibrated length oracle inequality with explicit grid, endpoint-quantile estimation, and calibration-sampling terms. Simulations and real-data examples report coverage and length jointly.