Copula Conformal Prediction for Multi-step Time Series Forecasting

TL;DR

CopulaCPTS addresses the challenge of providing valid and efficient uncertainty quantification for multivariate, multi-step time series forecasts. It combines inductive conformal prediction with a two-stage calibration procedure: stepwise nonconformity scores are calibrated, then a non-parametric empirical copula models their joint dependence to produce compact, horizon-spanning confidence regions with finite-sample validity. The method yields sharper calibration across synthetic and real-world datasets, with larger gains as horizon length increases or dimensionality rises, and can be extended to autoregressive forecasting with copula re-estimation. This approach offers practically useful uncertainty quantification for high-stakes forecasting tasks without relying on strong distributional assumptions.

Abstract

Accurate uncertainty measurement is a key step to building robust and reliable machine learning systems. Conformal prediction is a distribution-free uncertainty quantification algorithm popular for its ease of implementation, statistical coverage guarantees, and versatility for underlying forecasters. However, existing conformal prediction algorithms for time series are limited to single-step prediction without considering the temporal dependency. In this paper, we propose a Copula Conformal Prediction algorithm for multivariate, multi-step Time Series forecasting, CopulaCPTS. We prove that CopulaCPTS has finite sample validity guarantee. On several synthetic and real-world multivariate time series datasets, we show that CopulaCPTS produces more calibrated and sharp confidence intervals for multi-step prediction tasks than existing techniques.

Figures (12)

• Figure 1: Illustration of the multi-step time series forecasting setting. (Left) The timesteps within a time series are temporally dependent, and (Right) the observations in the dataset are independent.
• Figure 2: An example copula, where we express a multivariate Gaussian with correlation $\rho = 0.8$ with two univariate distributions and a copula function $C(u_1, u_2)$.
• Figure 3: Calibration (upper row) and efficiency (lower row) comparison on different $1-\alpha$ levels for synthetic data sets. Shaded regions are $\pm$ 2 standard deviations over 3 runs. For calibration, the goal is to stay above the green dotted (validity) and coincide as closely as possible (calibration). CopulaCPTS is more calibrated across different significance levels. For efficiency, we want the metric to be small. CopulaCPTS outperforms the baselines consistently. (MC-dropout for the right two experiments produces invalid regions, so we don't consider its efficiency.)
• Figure 4: Illustrations of 90% confidence regions given by CF-RNN (blue) and CopulaCPTS (orange) on two real-world datasets, COVID-19 forecast (left 2) and Argoverse (right 2 at time steps 1, 10, 20, and 30). For the Argoverse data, The red dotted lines (ego agent) and blue dotted lines (other agents) are input to the underlying prediction model and the red solid lines are the prediction output. Note that the confidence region produced CF-RNN is uninformatively large, as it covers all the lanes: these examples illustrate the importance of efficiency. Overall, CopulaCPTS is able to produce much more efficient confidence regions while maintaining valid coverage.
• Figure 5: CopulaCPTS remains more calibrated and efficient than baselines over increasing forecast horizons.

Theorems & Definitions (10)

• Definition 3.1: Validity
• Definition 3.2: Exchangeability
• Definition 3.3: Copula
• Theorem 3.4: Sklar's theorem
• Theorem 4.1: Validity of CopulaCPTS
• Theorem A.1: Validity of CopulaCPTS
• proof
• Lemma A.2: Validity of CPD. Theorem 11 of vovk2017nonparametric
• Definition A.3: Vector partial order
• Theorem B.1: The Frechet-Hoeffding Bounds