The Spectrum Is Not Enough: When Context Helps Time-Series Forecasting

2026-07-14

AI Summary

This work addresses the limitation of existing power spectrum–based predictability metrics in evaluating the genuine benefit conferred by contextual information—such as long look-back windows, retrieval mechanisms, or pretraining—on time series forecasting. The authors propose generating surrogate sequence pairs via phase randomization that preserve both spectral and marginal distributions, thereby enabling, for the first time, a clear distinction between in-spectrum and out-of-spectrum prediction mechanisms. They introduce an unsupervised, configuration-level diagnostic metric termed “coverage gap” to quantify predictive gains beyond spectral structure. Experiments across seven benchmarks reveal that retrieval-based methods exhibit a dramatic collapse in performance gains on these surrogate pairs (e.g., median gain on ECL drops from +33% to −35%, p < 10⁻⁴⁰), while conventional spectral metrics remain unchanged. The coverage gap effectively predicts the sign of out-of-spectrum gains and significantly outperforms existing indicators.

Abstract

A growing family of indices scores how predictable a series is from its spectrum. Practitioners increasingly read these scores as answering a different question: whether \emph{adding context}, a longer lookback, a retrieval plug-in, or a pretrained model, will help. These are not the same question. The value of context is a property of the operating point, not of the series. Any index built from the power spectrum is invariant under phase randomization, whereas the beyond-second-order value that retrieval and foundation models supply is not, because a phase-randomized series is asymptotically Gaussian. We state this as an impossibility result and isolate it with surrogate pairs that fix the spectrum and the marginal by construction. We then give a label-free, configuration-level diagnostic, the coverage deficit, whose principal term measures beyond-spectrum structure as the gain of analog over linear prediction. On seven benchmarks the prediction holds: window-keyed retrieval's value collapses across surrogate pairs (ECL median $+33\%\!\to\!-35\%$, $p{<}10^{-40}$) while every spectral index stays frozen; a foundation model's value splits into a surviving second-order part and a small beyond-linear margin that collapses; a longer linear window's value survives. Leave-one-dataset-out, the structure term predicts the sign of beyond-spectrum value where the spectral indices trail it, and the reverse holds for the second-order mechanism. We introduce no new forecaster; the contribution is the distinction, a controlled comparison, and a diagnostic for the deployment decision. Code: https://anonymous.4open.science/r/SINE.

Problem

• time-series forecasting
• spectrum
• context
• predictability
• beyond-second-order structure

Innovation

• spectrum
• context
• surrogate pairs
• coverage deficit
• beyond-second-order structure