Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

Zitong Yang; Yu Ming Huang; Emmanuel Candès; Lihua Lei

cs.LGFeb 7, 2024

Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

Authors: Zitong Yang, Yu Ming Huang, Emmanuel Candès, Lihua Lei

Organizations: Department of Statistics, Stanford University · Stanford Graduate School of Bussiness

Abstract

We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides calibrated prediction intervals. Unlike the existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the average interval lengths by solving a one-dimensional stochastic control problem (SCP) at each time step. In particular, we use the dynamic programming algorithm to find the optimal policy for the SCP. We prove that BCI achieves long-term coverage under arbitrary distribution shifts and temporal dependence, even with poor multi-step ahead forecasts. When the raw forecast intervals are approximately calibrated, we prove that BCI is near optimal for its surrogate model predictive control problem. We find empirically that BCI avoids uninformative intervals with infinite lengths under our main experimental configuration and generates substantially shorter prediction intervals when the raw forecast intervals are poorly calibrated.

Figures & tables

Panel of Figure 1 — Figure 1: Online time series forecasting for three different tasks: return forecasting on Amazon stock data, volatility forecasting on Amazon stock data, and Google search popularity data for the keyword “deep learning”. Top panel: moving averages of miscoverage rates over 500 data points. Bottom panel: moving averages of prediction interval lengths. In all figures, the red and green curves correspond to our proposed BCI algorithm and the conformal PID method; the blue curves correspond to the ACI algorithm with stepsize $0.1$ ; the gray curves correspond to setting $\alpha_{t}=\overline{\alpha}$ for all $t$ .

Panel of Figure 2 — Figure 2: Same as Figure 1 , except that the stepsize $\gamma$ for ACI is $0.008$ for looser control.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 3 — Figure 3: Expected calibration curve of nominal prediction intervals for Google trend, volatility, and return forecasting.

Panel of Figure 4 — Figure 4: Additional return forecasting problems.

Panel of Figure 5 — Figure 5: Additional volatility forecasting problems.

	Miscoverage (%) $\pm$ Std.			Avg. length			Frac. $\infty$ (%)
Task/dataset	BCI	ACI	PID	BCI	ACI	PID	BCI	ACI	PID
Return-AMD	$9.95\pm 0.256$	$9.95\pm 0.284$	$10.06\pm 0.441$	0.1051	0.1128	0.1059	0.00	3.37	0.00
Return-Amazon	$9.97\pm 0.270$	$9.99\pm 0.298$	$10.08\pm 0.246$	0.0730	0.0787	0.0891	0.00	2.89	0.00
Return-Nvidia	$10.04\pm 0.255$	$10.04\pm 0.287$	$10.21\pm 0.330$	0.0865	0.0928	0.0886	0.00	4.29	0.00
Volatility-AMD	$9.97\pm 0.235$	$9.99\pm 0.189$	$10.09\pm 0.498$	8.3956	8.2062	9.7662	0.00	10.75	0.00
Volatility-Amazon	$9.97\pm 0.168$	$9.99\pm 0.198$	$10.31\pm 0.288$	6.7463	6.4999	7.0544	0.00	8.85	0.00

Appendix

Table 1: Summary statistics for ACI with

\gamma=0.1

(tight control), BCI with

T=3

, and PID. Miscoverage columns report mean

\pm

std (%); length excludes

\infty

-sized intervals.

	Miscoverage (%) $\pm$ Std.			Avg. length			Frac. $\infty$ (%)
Task/dataset	BCI	ACI	PID	BCI	ACI	PID	BCI	ACI	PID
Return-AMD	$10.20\pm 0.953$	$10.13\pm 1.284$	$10.83\pm 1.463$	0.0985	0.1057	0.1017	0.00	0.00	0.00
Return-Amazon	$9.53\pm 1.067$	$10.02\pm 1.541$	$11.47\pm 2.171$	0.0712	0.0756	0.0720	0.00	0.00	0.00
Return-Nvidia	$10.28\pm 1.390$	$10.07\pm 2.109$	$10.62\pm 2.212$	0.0822	0.0896	0.0854	0.00	0.00	0.00
Volatility-AMD	$9.92\pm 0.762$	$10.03\pm 0.803$	$10.31\pm 0.868$	7.6328	7.6393	7.8388	0.00	0.00	0.00
Volatility-Amazon	$9.88\pm 1.020$	$9.93\pm 0.559$	$10.42\pm 0.505$	6.0761	6.0013	6.3920	0.00	0.00	0.00

Appendix

Table 2: Same as Table 1 , except ACI uses

\gamma=0.008

(loose control).

Explore similar work

Bias-Corrected Adaptive Conformal Inference for Multi-Horizon Time Series Forecasting
Apr 14, 2026Ankit Lade, Sai Krishna J., Indar KumarAdaptive Conformal Inference Time Series Forecasting
Retrieval-Corrected Conformal Prediction for Time Series
Aug 11, 2026Sangjin Jin, Kangmin Kim, Junhyeong Lee +1Time Series Forecasting Conformal Prediction
Ulsan National Institute of Science and Technology Ulsan, Republic of Korea
Dynamic Regime-Aware Conformal Calibration for Reliable Economic Forecast Intervals under Multiple Distribution Shifts
Aug 17, 2026Bogdan OanceaWeighted Conformal Prediction Adaptive Conformal Inference
University of Bucharest, Romania, National Institute of Research and Development for Biological Sciences

Apr 14, 2026J/K move · Enter open · S save

Bias-Corrected Adaptive Conformal Inference for Multi-Horizon Time Series Forecasting

Ankit Lade, Sai Krishna J., Indar Kumar

Adaptive Conformal Inference Time Series Forecasting Conformal Prediction Regression Prediction Intervals Distribution Shift Robustness

Open paper AskPDF arXiv

Adaptive Conformal Inference (ACI) provides distribution-free prediction intervals with asymptotic coverage guarantees for time series under distribution shift. However, ACI only adapts the quantile threshold -- it cannot shift the interval center. When a base forecaster develops persistent bias after a regime change, ACI compensates by widening intervals symmetrically, producing unnecessarily conservative bands. We propose Bias-Corrected ACI (BC-ACI), which augments standard ACI with an online exponentially weighted moving average (EWM) estimate of forecast bias. BC-ACI corrects nonconformity scores before quantile computation and re-centers prediction intervals, addressing the root cause of miscalibration rather than its symptom. An adaptive dead-zone threshold suppresses corrections when estimated bias is indistinguishable from noise, ensuring no degradation on well-calibrated data. In controlled experiments across 688 runs spanning two base models, four synthetic regimes, and three real datasets, BC-ACI reduces Winkler interval scores by 13--17% under mean and compound distribution shifts (Wilcoxon p < 0.001) while maintaining equivalent performance on stationary data (ratio 1.002x). We provide finite-sample analysis showing that coverage guarantees degrade gracefully with bias estimation error.

Abstract

Explore similar work

Bias-Corrected Adaptive Conformal Inference for Multi-Horizon Time Series Forecasting

Retrieval-Corrected Conformal Prediction for Time Series

Dynamic Regime-Aware Conformal Calibration for Reliable Economic Forecast Intervals under Multiple Distribution Shifts