Kernel-based Optimally Weighted Conformal Time-Series Prediction

TL;DR

The paper tackles uncertainty quantification for time-series under non-exchangeability by proposing KOWCPI, a kernel-based, optimally weighted conformal prediction method that learns data-driven weights via the Reweighted Nadaraya-Watson estimator. By performing nonparametric kernel quantile regression on non-conformity scores within a sliding-window framework, KOWCPI provides sequential prediction intervals with asymptotic conditional coverage under strong mixing. Theoretical contributions include marginal coverage bounds and a formal conditional-coverage guarantee, while empirical results show consistently narrower intervals without sacrificing coverage across real and synthetic time-series. This approach offers a practical, theoretically grounded tool for reliable uncertainty quantification in non-stationary and dependent data settings, with potential for adaptive windowing and multivariate extensions.

Abstract

In this work, we present a novel conformal prediction method for time-series, which we call Kernel-based Optimally Weighted Conformal Prediction Intervals (KOWCPI). Specifically, KOWCPI adapts the classic Reweighted Nadaraya-Watson (RNW) estimator for quantile regression on dependent data and learns optimal data-adaptive weights. Theoretically, we tackle the challenge of establishing a conditional coverage guarantee for non-exchangeable data under strong mixing conditions on the non-conformity scores. We demonstrate the superior performance of KOWCPI on real and synthetic time-series data against state-of-the-art methods, where KOWCPI achieves narrower confidence intervals without losing coverage.

Figures (15)

• Figure 1: Illustration of KOWCPI, a sequential conformal prediction method. In the absence of exchangeability in the data, as indicated by the empirical distribution of residuals and the PACF plot, it is critical to consider the sequentially dependent structure of the data. In KOWCPI, non-conformity score blocks are updated sequentially using a sliding window, which provides prediction intervals for future scores through nonparametric quantile regression.
• Figure 2: Rolling coverage (boxplot)
• Figure 3: Width of prediction intervals (boxplot)
• Figure 4: Rolling coverage over prediction time indices
• Figure 5: $\log(\widehat{W}_i)$ by the time lag

Theorems & Definitions (21)

• Lemma 3.1: hall1999methodscai2001weighted
• Proposition 4.1: Non-asymptotic marginal coverage gap
• Proposition 4.6: Consistency of the RNW estimator
• Corollary 4.7
• Corollary 4.8: Asymptotic conditional coverage guarantee
• Theorem 4.9: Conditional coverage gap
• Definition A.1: Multivariate conditional quantile abdous1992note
• Remark A.2: Compatibility with univariate quantile function
• Lemma B.1: Weights on quantile for non-exchangeable data
• proof : Proof of Proposition \ref{['prop:marginal_coverage']}