A Stationarity-and-Coupling Criterion for Training-Free Time-Lagged Spectral Embeddings of Multivariate Time Series

2026-06-11

AI Summary

This study investigates the conditions under which training-free, fixed-length time series embedding methods are effective, rather than merely evaluating their performance. The authors propose an embedding \( D(\tau) \) derived from the time-lagged correlation matrix, truncated at the Marchenko–Pastur boundary, and paired with cosine similarity for zero-parameter classification. Their key contribution is the first actionable applicability criterion: the method succeeds only when the signal is approximately stationary and class-discriminative information arises from inter-channel temporal couplings. This criterion is validated through pre-screening using an augmented Dickey–Fuller test and power baseline saturation. The approach achieves strong performance on four qualifying datasets (e.g., 88.5 ± 4.5% accuracy on Sleep-EDF) and predictably fails on three datasets violating the stated conditions, thereby confirming the reliability of the proposed criterion.

Abstract

We study training-free fixed-length descriptors for multivariate time series and ask not merely whether such a descriptor performs well, but when it can be expected to work at all. Our object of study is $D(τ)$, built from a time-lagged correlation matrix truncated at the Marchenko-Pastur edge so that only signal-bearing eigenvalues survive and classified by cosine similarity to class centroids with zero learned parameters. The central contribution is not the descriptor but a falsifiable applicability criterion for it. Working from a stationary Gaussian VAR(1) model, we argue that $D(τ)$ separates two classes when the signals are approximately stationary and the class information lives in their cross-channel temporal coupling rather than in marginal per-channel power. We derive, semi-formally, three consequences: a distinguishability condition, why the static ($τ=0$) covariance collapses to chance, and why a stationary but power-discriminated paradigm defeats the descriptor. The criterion is operational: a two-part pre-flight test -- an augmented Dickey-Fuller stationarity check and a power-baseline saturation check -- predicts applicability before any training. We validate both halves on a mixed assortment. On four paradigms that satisfy the criterion (Sleep-EDF, BCI-IV-2a, MIT-BIH, ESC-50) the descriptor is competitive with strong baselines at a fraction of their cost, reaching $88.5\pm4.5\%$ under 20-subject leave-one-subject-out on Sleep-EDF on a single CPU thread. On three that violate it -- non-stationary ERPs, and financial-volatility and wearable-stress regimes that are power-discriminated -- it fails exactly as the pre-flight predicts, and these negatives are the more informative half. We are explicit that $D(τ)$ is not the most accurate representation; its value is a compact, training-free embedding whose domain of validity is known in advance.

Problem

• stationarity
• temporal coupling
• training-free embedding
• multivariate time series
• applicability criterion

Innovation

• training-free embedding
• time-lagged spectral embedding
• stationarity criterion
• cross-channel coupling
• applicability prediction