# Error-quantified Conformal Inference for Time Series

arXiv: https://arxiv.org/abs/2502.00818 · HTML: https://mlsanity.com/paper/2502.00818/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 42 references that are in the map.
- First posted on arXiv 2 Feb 2025.

## Builds on (most cited first, 30 of 30)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Enhancing the Locality and Breaking the Memory Bottleneck of Transformer on Time Series Forecasting](https://mlsanity.com/paper/1907.00235/) (2019)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [Prediction-Powered Inference](https://mlsanity.com/paper/2301.09633/) (2023)
- [Predictive inference with the jackknife+](https://mlsanity.com/paper/1905.02928/) (2019)
- [Adaptive Conformal Predictions for Time Series](https://mlsanity.com/paper/2202.07282/) (2022)
- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [Conformal PID Control for Time Series Prediction](https://mlsanity.com/paper/2307.16895/) (2023)
- [Improved Online Conformal Prediction via Strongly Adaptive Online Learning](https://mlsanity.com/paper/2302.07869/) (2023)
- [Inducing High Energy-Latency of Large Vision-Language Models with Verbose Images](https://mlsanity.com/paper/2401.11170/) (2024)
- [Exact and Robust Conformal Inference Methods for Predictive Machine Learning With Dependent Data](https://mlsanity.com/paper/1802.06300/) (2018)
- [Online conformal prediction with decaying step sizes](https://mlsanity.com/paper/2402.01139/) (2024)
- [Conformal prediction for time series](https://mlsanity.com/paper/2010.09107/) (2020)
- [Improved Strongly Adaptive Online Learning using Coin Betting](https://mlsanity.com/paper/1610.04578/) (2016)
- [Doubly Robust Calibration of Prediction Sets under Covariate Shift](https://mlsanity.com/paper/2203.01761/) (2022)
- [Bayesian Recurrent Neural Networks](https://mlsanity.com/paper/1704.02798/) (2017)
- [Adaptive Regret for Control of Time-Varying Dynamics](https://mlsanity.com/paper/2007.04393/) (2020)
- [Smoothing quantile regressions](https://mlsanity.com/paper/1905.08535/) (2019)
- [High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization](https://mlsanity.com/paper/2109.05640/) (2021)
- [Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series](https://mlsanity.com/paper/2402.05203/) (2024)
- [Adaptive Conformal Inference by Betting](https://mlsanity.com/paper/2412.19318/) (2024)
- [Online Control of the False Coverage Rate and False Sign Rate](https://mlsanity.com/paper/1905.01059/) (2019)
- [CAP: A General Algorithm for Online Selective Conformal Prediction with FCR Control](https://mlsanity.com/paper/2403.07728/) (2024)
- [Smoothed estimating equations for instrumental variables quantile regression](https://mlsanity.com/paper/1609.09033/) (2016)

## Built on by (most cited first, 12 of 12)

- [Conformal Prediction for Time-series Forecasting with Change Points](https://mlsanity.com/paper/2509.02844/) (2025)
- [Boost, Disentangle, and Customize: A Robust System2-to-System1 Pipeline for Code Generation](https://mlsanity.com/paper/2502.12492/) (2025)
- [MoGU: Mixture-of-Gaussians with Uncertainty-based Gating for Time Series Forecasting](https://mlsanity.com/paper/2510.07459/) (2025)
- [Distribution-informed Online Conformal Prediction](https://mlsanity.com/paper/2512.07770/) (2025)
- [Feature Fitted Online Conformal Prediction for Deep Time Series Forecasting Model](https://mlsanity.com/paper/2505.08158/) (2025)
- [Relevance-Aware Thresholding in Online Conformal Prediction for Time Series](https://mlsanity.com/paper/2510.02809/) (2025)
- [Decomposition-Based Modular Conformal Prediction for Two-Stage Modeling](https://mlsanity.com/paper/2510.04406/) (2025)
- [Delving into Non-Exchangeability for Conformal Prediction in Graph-Structured Multivariate Time Series](https://mlsanity.com/paper/2605.04957/) (2026)
- [Online Conformal Prediction with Corrupted Feedback](https://mlsanity.com/paper/2605.20515/) (2026)
- [DistMatch: Adaptive Binning via Distribution Matching for Robust Sequential Conformal Prediction](https://mlsanity.com/paper/2606.00690/) (2026)
- [Counterfactual Online Conformal Prediction Under Adaptive Logging](https://mlsanity.com/paper/2609.30811/) (2026)
- [State-Matched Sequence Conformal Prediction](https://mlsanity.com/paper/2610.19484/) (2026)

## Abstract (arXiv)

Uncertainty quantification in time series prediction is challenging due to the temporal dependence and distribution shift on sequential data. Conformal inference provides a pivotal and flexible instrument for assessing the uncertainty of machine learning models through prediction sets. Recently, a series of online conformal inference methods updated thresholds of prediction sets by performing online gradient descent on a sequence of quantile loss functions. A drawback of such methods is that they only use the information of revealed non-conformity scores via miscoverage indicators but ignore error quantification, namely the distance between the non-conformity score and the current threshold. To accurately leverage the dynamic of miscoverage error, we propose textitError-quantified Conformal Inference (ECI) by smoothing the quantile loss function. ECI introduces a continuous and adaptive feedback scale with the miscoverage error, rather than simple binary feedback in existing methods. We establish a long-term coverage guarantee for ECI under arbitrary dependence and distribution shift. The extensive experimental results show that ECI can achieve valid miscoverage control and output tighter prediction sets than other baselines.