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Conformal Predictions under Markovian Data

Frédéric Zheng, Alexandre Proutiere

TL;DR

This work extends Conformal Prediction to Markovian data by deriving explicit coverage-gap bounds that scale with the mixing time $t_{\mathrm{mix}}$, showing that split CP incurs a gap of $O\left(\sqrt{t_{\mathrm{mix}}\frac{\ln(n)}{n}}\right)$ under ergodicity, and proposing $K$-split CP to reduce the gap to $O\left(\frac{t_{\mathrm{mix}}}{n\ln(n)}\right)$ with minimal impact on set size. It provides concentration results for empirical quantiles and proves asymptotic optimality when the model satisfies stability and density regularity, including thinning-based improvements via the blocking technique. The authors also develop adaptive $K$-split CP by online mixing-rate estimation, with theoretical guarantees, and validate the approach on synthetic scenarios such as a lazy random walk and Gaussian AR(1), as well as real-world data like EUR/SEK and electricity prices. Overall, the paper offers a principled, scalable framework for reliable conformal prediction in dependent data, with practical guidance for choosing thinning levels and adapting to unknown mixing properties.

Abstract

We study the split Conformal Prediction method when applied to Markovian data. We quantify the gap in terms of coverage induced by the correlations in the data (compared to exchangeable data). This gap strongly depends on the mixing properties of the underlying Markov chain, and we prove that it typically scales as $\sqrt{t_\mathrm{mix}\ln(n)/n}$ (where $t_\mathrm{mix}$ is the mixing time of the chain). We also derive upper bounds on the impact of the correlations on the size of the prediction set. Finally we present $K$-split CP, a method that consists in thinning the calibration dataset and that adapts to the mixing properties of the chain. Its coverage gap is reduced to $t_\mathrm{mix}/(n\ln(n))$ without really affecting the size of the prediction set. We finally test our algorithms on synthetic and real-world datasets.

Conformal Predictions under Markovian Data

TL;DR

This work extends Conformal Prediction to Markovian data by deriving explicit coverage-gap bounds that scale with the mixing time , showing that split CP incurs a gap of under ergodicity, and proposing -split CP to reduce the gap to with minimal impact on set size. It provides concentration results for empirical quantiles and proves asymptotic optimality when the model satisfies stability and density regularity, including thinning-based improvements via the blocking technique. The authors also develop adaptive -split CP by online mixing-rate estimation, with theoretical guarantees, and validate the approach on synthetic scenarios such as a lazy random walk and Gaussian AR(1), as well as real-world data like EUR/SEK and electricity prices. Overall, the paper offers a principled, scalable framework for reliable conformal prediction in dependent data, with practical guidance for choosing thinning levels and adapting to unknown mixing properties.

Abstract

We study the split Conformal Prediction method when applied to Markovian data. We quantify the gap in terms of coverage induced by the correlations in the data (compared to exchangeable data). This gap strongly depends on the mixing properties of the underlying Markov chain, and we prove that it typically scales as (where is the mixing time of the chain). We also derive upper bounds on the impact of the correlations on the size of the prediction set. Finally we present -split CP, a method that consists in thinning the calibration dataset and that adapts to the mixing properties of the chain. Its coverage gap is reduced to without really affecting the size of the prediction set. We finally test our algorithms on synthetic and real-world datasets.
Paper Structure (39 sections, 10 theorems, 91 equations, 7 figures)

This paper contains 39 sections, 10 theorems, 91 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Markovian data
  5. $\beta$-mixing processes
  6. Split Conformal Prediction
  7. Notation
  8. Split CP for Markov Chains
  9. Marginal coverage
  10. Size of the conformal prediction set
  11. Concentration of empirical quantiles
  12. Asymptotic optimality
  13. $K$-Split Conformal Prediction
  14. Marginal coverage guarantees via the blocking technique
  15. $\beta$-mixing processes
  16. ...and 24 more sections

Key Result

Proposition 4.1

(1. With restart) Applying Split CP with restart yields a coverage gap $\gamma$ satisfying, for any $u>{1\over n}\sum_{a=1}^n \delta_1(a)$, $\gamma\le \gamma(u)$ where (2. Without restart) Applying Split CP yields a coverage gap $\gamma$ satisfying, for any $u> \delta(N)$ and for any $r\in [n]$, $\gamma\le \gamma(u,r)$ where

Figures (7)

  • Figure 1: Coverage for the lazy walk ($w=20$) as a function of $n$. The second values on the $x$-axis represent the (optimal) values of $K^\star(n)$.
  • Figure 2: $\frac{|q_{n,\alpha}-q_\alpha|}{q_\alpha}$ for the lazy walk as a function of $n$. The second values on the $x$-axis represent the (optimal) values of $K^\star(n)$.
  • Figure 3: Coverage for Gaussian AR $(\theta=0.9, \omega=1)$ as a function of $n$. The second values on the $x$-axis represent the (optimal) values of $K^\star(n)$.
  • Figure 4: Daily coverage for EUR/SEK exchange rate $(n=43200, N=43200)$ during one month
  • Figure 5: Monthly coverage for French electricity price forecast $(n=12 960, N=12 960)$
  • ...and 2 more figures

Theorems & Definitions (14)

  • Proposition 4.1
  • Theorem 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 5.1
  • Theorem 5.1
  • Proposition 5.2
  • Proposition 5.3
  • proof
  • proof
  • ...and 4 more