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Flow-based Conformal Prediction for Multi-dimensional Time Series

Junghwan Lee, Chen Xu, Yao Xie

TL;DR

This work addresses uncertainty quantification for multi-dimensional time series with non-exchangeable temporal structure by introducing Flow-based Conformal Prediction (FCP) using classifier-free guidance. A guided flow maps residuals conditioned on historical context to a target distribution, enabling flexible, high-dimensional prediction sets with exact marginal coverage and finite-sample conditional guarantees. Training uses flow matching with a Transformer encoder to capture dependencies in past features and residuals, while prediction-set construction relies on transforming an isotropic Gaussian residual ball via the learned flow. Empirical results on real-world datasets demonstrate that FCP produces significantly smaller, yet well-calibrated prediction sets across varying outcome dimensions and base predictors, outperforming existing conformal prediction baselines.

Abstract

Time series prediction underpins a broad range of downstream tasks across many scientific domains. Recent advances and increasing adoption of black-box machine learning models for time series prediction highlight the critical need for reliable uncertainty quantification. While conformal prediction has gained attention as a reliable uncertainty quantification method, conformal prediction for time series faces two key challenges: (1) adaptively leveraging correlations in features and non-conformity scores to overcome the exchangeability assumption, and (2) constructing prediction sets for multi-dimensional outcomes. To address these challenges jointly, we propose a novel conformal prediction method for time series using flow with classifier-free guidance. We provide coverage guarantees by establishing exact non-asymptotic marginal coverage and a finite-sample bound on conditional coverage for the proposed method. Evaluations on real-world time series datasets demonstrate that our method constructs significantly smaller prediction sets than existing conformal prediction methods while maintaining target coverage.

Flow-based Conformal Prediction for Multi-dimensional Time Series

TL;DR

This work addresses uncertainty quantification for multi-dimensional time series with non-exchangeable temporal structure by introducing Flow-based Conformal Prediction (FCP) using classifier-free guidance. A guided flow maps residuals conditioned on historical context to a target distribution, enabling flexible, high-dimensional prediction sets with exact marginal coverage and finite-sample conditional guarantees. Training uses flow matching with a Transformer encoder to capture dependencies in past features and residuals, while prediction-set construction relies on transforming an isotropic Gaussian residual ball via the learned flow. Empirical results on real-world datasets demonstrate that FCP produces significantly smaller, yet well-calibrated prediction sets across varying outcome dimensions and base predictors, outperforming existing conformal prediction baselines.

Abstract

Time series prediction underpins a broad range of downstream tasks across many scientific domains. Recent advances and increasing adoption of black-box machine learning models for time series prediction highlight the critical need for reliable uncertainty quantification. While conformal prediction has gained attention as a reliable uncertainty quantification method, conformal prediction for time series faces two key challenges: (1) adaptively leveraging correlations in features and non-conformity scores to overcome the exchangeability assumption, and (2) constructing prediction sets for multi-dimensional outcomes. To address these challenges jointly, we propose a novel conformal prediction method for time series using flow with classifier-free guidance. We provide coverage guarantees by establishing exact non-asymptotic marginal coverage and a finite-sample bound on conditional coverage for the proposed method. Evaluations on real-world time series datasets demonstrate that our method constructs significantly smaller prediction sets than existing conformal prediction methods while maintaining target coverage.

Paper Structure

This paper contains 41 sections, 17 theorems, 122 equations, 3 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Method
  4. Preliminary: Guided Flow
  5. Conformal Prediction for Time Series using Guided Flow
  6. Guided flow design.
  7. Prediction set.
  8. Theory
  9. Marginal Coverage
  10. Conditional Coverage
  11. Experiments
  12. Baselines.
  13. Datasets and base predictor.
  14. Evaluation metrics.
  15. Results.
  16. ...and 26 more sections

Key Result

Lemma 4.3

Let $X \sim p_X$ be a continuous random variable on $\mathbb{R}^d$, and let $\psi: \mathbb{R}^d \to \mathbb{R}^d$ be a $C^1$ diffeomorphism. Define $Y := \psi(X)$ with density $p_Y$ given by the push-forward of $p_X$ under $\psi$. Then, for any measurable set $\mathcal{A} \subset \mathbb{R}^d$, the

Figures (3)

  • Figure 1: Our method adaptively constructs the prediction set at time $i$ using a flow transformation $\psi$ conditioned on guidance $h_i$, which encodes contextual information extracted from past features and residuals.
  • Figure 2: Comparison of the prediction sets with a target coverage of 0.95, constructed by FCP (ours), MultiDimSPCI xu2024conformal, and conformal prediction using copula messoudi2021copula on (a) wind, (b) traffic, and (c) solar datasets. Prediction sets are manually selected from the test set for visual clarity. Two prediction sets are shown for the wind dataset.
  • Figure 3: Rolling coverage results on the wind dataset with rolling window size 20.

Theorems & Definitions (37)

  • Remark 4.2
  • Lemma 4.3: Probability mass preserving property of flows
  • Proposition 4.4: Marginal coverage
  • Remark 4.7
  • Lemma 4.10: Convergence of empirical CDF of i.i.d. $\{ e_i \}_{i=1}^{T}$
  • Lemma 4.11: Norm concentration of isotropic Gaussian random vectors
  • Lemma 4.12: Distance between the empirical CDF of $\{ e_i \}_{i=1}^{T}$ and $\{ \hat{e}_i \}_{i=1}^{T}$
  • Theorem 4.13: Conditional coverage bound under i.i.d. non-conformity scores
  • Definition 4.14
  • Definition 4.15
  • ...and 27 more