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Conformal prediction for multi-dimensional time series by ellipsoidal sets

Chen Xu, Hanyang Jiang, Yao Xie

TL;DR

This work develops MultiDimSPCI, a sequential conformal prediction method that constructs ellipsoidal prediction regions for multivariate time series. By grounding uncertainty sets in residual covariances and adaptive quantile calibration, it addresses non-exchangeability and inter-coordinate dependence while delivering finite-sample conditional coverage guarantees. The approach is model-agnostic with respect to the underlying predictor $\hat{f}$ and can optionally use local ellipsoids to capture non-stationary dynamics. Empirical results on simulated AR/VAR data and real wind, solar, and traffic datasets show valid coverage with markedly smaller prediction sets than CP and non-CP baselines, highlighting its practical appeal for multivariate uncertainty quantification in time series.

Abstract

Conformal prediction (CP) has been a popular method for uncertainty quantification because it is distribution-free, model-agnostic, and theoretically sound. For forecasting problems in supervised learning, most CP methods focus on building prediction intervals for univariate responses. In this work, we develop a sequential CP method called $\texttt{MultiDimSPCI}$ that builds prediction $\textit{regions}$ for a multivariate response, especially in the context of multivariate time series, which are not exchangeable. Theoretically, we estimate $\textit{finite-sample}$ high-probability bounds on the conditional coverage gap. Empirically, we demonstrate that $\texttt{MultiDimSPCI}$ maintains valid coverage on a wide range of multivariate time series while producing smaller prediction regions than CP and non-CP baselines.

Conformal prediction for multi-dimensional time series by ellipsoidal sets

TL;DR

This work develops MultiDimSPCI, a sequential conformal prediction method that constructs ellipsoidal prediction regions for multivariate time series. By grounding uncertainty sets in residual covariances and adaptive quantile calibration, it addresses non-exchangeability and inter-coordinate dependence while delivering finite-sample conditional coverage guarantees. The approach is model-agnostic with respect to the underlying predictor and can optionally use local ellipsoids to capture non-stationary dynamics. Empirical results on simulated AR/VAR data and real wind, solar, and traffic datasets show valid coverage with markedly smaller prediction sets than CP and non-CP baselines, highlighting its practical appeal for multivariate uncertainty quantification in time series.

Abstract

Conformal prediction (CP) has been a popular method for uncertainty quantification because it is distribution-free, model-agnostic, and theoretically sound. For forecasting problems in supervised learning, most CP methods focus on building prediction intervals for univariate responses. In this work, we develop a sequential CP method called that builds prediction for a multivariate response, especially in the context of multivariate time series, which are not exchangeable. Theoretically, we estimate high-probability bounds on the conditional coverage gap. Empirically, we demonstrate that maintains valid coverage on a wide range of multivariate time series while producing smaller prediction regions than CP and non-CP baselines.
Paper Structure (17 sections, 16 theorems, 80 equations, 10 figures, 11 tables, 1 algorithm)

This paper contains 17 sections, 16 theorems, 80 equations, 10 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem setup
  4. Method
  5. Ellipsoidal uncertainty set
  6. Comparison with copula
  7. Benefits of the proposed approach
  8. Improvements using local ellipsoids
  9. Theoretical Analysis
  10. Experiments
  11. Simulation
  12. Real-data
  13. Discussions
  14. Additional experiments
  15. Comparison of non-critical regions.
  16. ...and 2 more sections

Key Result

Lemma 4.9

Under Assumption a1, for any training size T, there is an event $A_T$ which occurs with probability at least $1-\sqrt{\frac{\log(16T)}{T}}$, such that conditioning on $A_T$,

Figures (10)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 5: Wind data
  • Figure 6: Solar data
  • ...and 5 more figures

Theorems & Definitions (34)

  • Remark 4.2
  • Remark 4.4
  • Remark 4.6
  • Remark 4.8
  • Lemma 4.9: Convergence of empirical CDF of $\{\varepsilon_t\}_{t=1}^{T}$ under i.i.d.
  • Remark 4.10
  • Lemma 4.11: Distance between the empirical CDF of $\{\varepsilon_t\}_{t=1}^{T}$ and $\{\hat{\varepsilon}_t\}_{t=1}^{T}$ under i.i.d.
  • Theorem 4.12: Conditional guarantee under i.i.d. assumption
  • Remark 4.13
  • Corollary 4.14: Guarantee with true covariance matrix, and under i.i.d.
  • ...and 24 more