ResCP: Reservoir Conformal Prediction for Time Series Forecasting

Abstract

Conformal prediction offers a powerful framework for building distribution-free prediction intervals for exchangeable data. Existing methods that extend conformal prediction to sequential data rely on fitting a relatively complex model to capture temporal dependencies. However, these methods can fail if the sample size is small and often require expensive retraining when the underlying data distribution changes. To overcome these limitations, we propose Reservoir Conformal Prediction (ResCP), a novel training-free conformal prediction method for time series. Our approach leverages the efficiency and representation learning capabilities of reservoir computing to dynamically reweight conformity scores. In particular, we compute similarity scores among reservoir states and use them to adaptively reweight the observed residuals at each step. With this approach, ResCP enables us to account for local temporal dynamics when modeling the error distribution without compromising computational scalability. We prove that, under reasonable assumptions, ResCP achieves asymptotic conditional coverage, and we empirically demonstrate its effectiveness across diverse forecasting tasks.

Figures (10)

• Figure 1: Let $\{\bm{x}_t\}_{t=1}^T$ and $\{{r}_t\}_{t=1}^T$ be the time series and residuals of the calibration set, respectively. Each sequence $\bm{x}_{1:t}$ generates the state of the reservoir $\bm{h}_t$. The last state $\bm{h}_T$ is the query state for the prediction of $\hat{y}_{T+H}$. We compute similarity scores between $\bm{h}_T$ and the calibration states $\{\bm{h}_t\}_{t=1}^{T-H}$, which are used to reweight with $\{{w}_t\}_{t=1}^{T-H}$ associated residuals $\{{r}_t\}_{t=H}^T$ by resampling them. Quantiles are computed from the sampled residuals and used to build the for $\hat{y}_{T+H}$.
• Figure 2: Calibration curves of , , and over all datasets with the RNN baseline.
• Figure 3: Calibration curves of and over all datasets and baselines.
• Figure 4: Prediction intervals of (top) and prediction intervals' width of different methods (bottom) for the ACEA dataset.
• Figure 5: Prediction intervals of (top) and prediction intervals' width of different methods (bottom) for the Solar dataset.

Theorems & Definitions (7)

• Definition 3.4: Effective sample size
• Theorem 3.6: Consistency of the weighted empirical
• Corollary 3.7: Asymptotic conditional coverage guarantee
• Lemma A.1: Regularity of the conditional CDF
• proof
• proof : Proof of Theorem \ref{['thm:consistency']}
• proof : Proof of Corollary \ref{['cor:asym_cond_cov']}