Delving into Non-Exchangeability for Conformal Prediction in Graph-Structured Multivariate Time Series

Abstract: Point forecasting for graph-structured multivariate time series is a fundamental problem, but rigorous uncertainty quantification for such predictions is still underexplored. Conformal prediction (CP) offers uncertainty estimation with a solid coverage guarantee under the exchangeability assumption, which requires the joint data distribution to be unchanged under permutation. However, in graph-structured time series, inherent cross-node coupling can violate the exchangeability condition, making direct application of CP unreliable. Inspired by the spectral graph theory, such coupling resides in global trends and can be characterized by the low-frequency components, while high-frequency components are nearly exchangeable. Therefore, we propose a novel concept named Spectral Graph Conditional Exchangeability (SGCE), which conditions exchangeable high-frequency components on low-frequency ones to preserve global trends and enable effective CP in the spectral domain. Based on SGCE, we further propose Spectral Conformal prediction via wAveLEt transform (SCALE). SCALE uses graph wavelets to decompose low/high-frequency components and conformalizes high-frequency residuals via adaptive gating over a low-frequency embedding. Experimental results on real-world traffic datasets show that SCALE not only achieves valid coverage but also consistently improves the coverage-efficiency trade-off over the state-of-the-art CP methods.

• The paper introduces SCALE, leveraging spectral decomposition to separate low-frequency global trends from high-frequency residuals, enabling approximate exchangeability for conformal prediction.
• The method achieves near-nominal coverage and significantly sharper prediction intervals on datasets like METR-LA by calibrating on conditionally exchangeable high-frequency components.
• Robust evaluations and ablation studies affirm that both spectral decomposition and low-frequency conditioning are critical for addressing non-exchangeability in graph-structured time series.

Spectral-Domain Exchangeability in Conformal Prediction for Graph-Structured Multivariate Time Series

Introduction

Graph-structured multivariate time series (MTS) forecasting is central for analyzing spatio-temporal systems such as traffic networks and sensor arrays. While state-of-the-art spatio-temporal graph neural networks (STGNNs) have made significant strides in point forecasting, rigorous uncertainty quantification (UQ) for such models remains challenging. Classical conformal prediction (CP) frameworks offer distribution-free finite-sample coverage guarantees but crucially require exchangeability in the calibration data. For graph-structured MTS, this joint exchangeability assumption is systematically violated due to latent spatial couplings and temporal dependency, severely limiting direct application of standard CP methods.

This paper introduces a new spectral-conditional framework for CP in graph-MTS settings, aiming to recover reliable interval coverage by leveraging structural insights from spectral graph theory.

Problems with Exchangeability in Graph-Structured MTS

Non-exchangeability in graph-structured MTS arises both from temporal dependence and from strong cross-node coupling dictated by the underlying graph topology. While existing CP adaptations for time series (e.g., sequential or weighted calibration) attempt to ameliorate temporal non-exchangeability, spatial coupling remains particularly problematic: correlations between node signals induce system-level dependencies even when individual series appear marginally exchangeable.

Prior approaches include:

• Node-wise independent CP, yielding only marginal (not joint) guarantees.
• Imposing restrictive assumptions such as sparse dependency structures to approximate joint exchangeability.

However, these strategies compromise either theoretical rigor or empirical validity when faced with realistic, densely coupled settings.

Spectral Graph Conditional Exchangeability (SGCE)

The core insight of this work is that spatial couplings are predominantly encoded in the low-frequency components of the graph spectrum, manifesting as smooth, global trends. In contrast, high-frequency components primarily capture localized, weakly correlated fluctuations that approximate exchangeability.

The authors formalize this as Spectral Graph Conditional Exchangeability (SGCE): the high-frequency residuals of the MTS, when conditioned on the low-frequency (global trend) components, satisfy approximate exchangeability across nodes and time. Explicitly, given a sequence of low-frequency components $\{L_t\}$ and corresponding high-frequency components $\{H_t\}$, the exchangeability condition holds for the $H_t$'s conditioned on $L_t$'s:

$$p(H_{t_1}, ..., H_{t_n} | L_{t_1}, ..., L_{t_n}) = p(H_{t_{\sigma(1)}}, ..., H_{t_{\sigma(n)}} | L_{t_{\sigma(1)}}, ..., L_{t_{\sigma(n)}})$$

for any time index permutation $\sigma$.

This construction enables the design of conformal predictors that are valid in the spectral domain even when the original MTS is arbitrarily coupled.

The Proposed Method: SCALE

Spectral Decomposition

The proposed method, Spectral Conformal prediction via wAveLEt transform (SCALE), operationalizes SGCE by first decomposing the residuals of a base spatio-temporal forecaster into low- and high-frequency bands using the Spectral Graph Wavelet Transform (SGWT). The decomposition is:

$X_t = L_t + H_t$

where $L_t$ accumulates the low-frequency (global trend) and $H_t$ represents high-frequency (localized) details.

Calibration and Prediction Intervals

• Low-Frequency Path: A lightweight encoder (STID-derived MLP) produces a conditioning embedding from the low-frequency components.
• High-Frequency Path: Node-wise temporal statistics (standard deviation, root mean square) summarize high-frequency components without re-introducing spatial couplings.
• Adaptive Gated Fusion: The conditioning embedding modulates the high-frequency quantile prediction via an adaptive gating mechanism, enabling the quantile estimator to leverage trend information while calibrating on “quasi-exchangeable” fluctuations.

Training minimizes the pinball (quantile regression) loss.

Theoretical Guarantees

Finite-Sample Validity

Under the SGCE assumption, the authors prove that SCALE achieves correct nominal coverage for the constructed intervals:

$$\mathbb{P}(X_{t+1:t+K} \in C_{t+1:t+K}^\alpha) \geq 1 - \alpha$$

regardless of the non-exchangeable structure of $\{H_t\}$0, as long as $\{H_t\}$1 remains conditionally exchangeable.

Robustness to Imperfect Decomposition

In practice, spectral decomposition is approximate. The theoretical analysis bounds the coverage gap in terms of:

• Wavelet approximation error (spectral leakage)
• Residual spatial dependencies in the high-frequency band

The coverage guarantee thus extends to the case where decompositions are imperfect and high-frequency couplings are small, with the coverage deficit quantifiably controlled.

Experimental Results

Datasets and Protocol

SCALE is evaluated on canonical traffic forecast benchmarks: METR-LA, PEMS04, PEMS07, PEMS08. Backbone forecasters are held fixed (GRU+DiffConv) for fair interval comparison. Metrics include empirical coverage, prediction interval width, and Winkler score.

Main Findings

• Coverage: SCALE consistently achieves empirical coverage within 0.02 of the nominal rate across all datasets and nominal levels ($\{H_t\}$2), supporting its theoretical validity.
• Efficiency: Notably tighter intervals compared to strong baselines. For instance, on METR-LA ($\{H_t\}$3), SCALE reduces interval width by approximately 14.4% versus EnbPI, and is sharper than CoREL and HopCPT.
• Ablations: Removing low-frequency conditioning or the spectral decomposition causes marked coverage and efficiency degradation, confirming the necessity of both components.
• Multi-Horizon Forecasting: SCALE maintains both coverage and efficiency stability across forecast horizons, outperforming multi-horizon methods such as ConForME.
• Robustness: Performance is robust across backbone architectures (Transformer, GWNet, DCRNN).

Interpretation

The spectral analysis reveals strong spatial coupling in the low-frequency graph spectrum and near-independence in high-frequencies, supporting the appropriateness of calibration in the spectral domain. Empirical results confirm that spectral-domain conditioning both sharpens intervals and maintains coverage.

Implications and Future Directions

By shifting the basis of exchangeability from the observation domain to the spectral domain, SCALE unlocks reliable UQ for scenarios characterized by intrinsic spatial structure and dependency. This spectral-conditional perspective may generalize to other structured signals (e.g., images, spatiotemporal medical data).

Future research could target:

• More expressive forms of conditional exchangeability (beyond spectral decomposition)
• Adaptive spectral partitioning for more complex graph topologies
• Extending the framework to non-stationary graph settings or with adaptive graph learning
• Deeper theoretical analysis on the sharpness of conditional coverage bounds

Conclusion

This work introduces a principled approach for conformal UQ in graph-structured multivariate time series by capitalizing on conditional exchangeability in the graph spectral domain. The SCALE framework achieves both theoretical validity and empirical efficiency, significantly advancing interval estimation for coupled MTS forecasting. This spectral-domain framework provides both a new theoretical foundation and practical algorithmic tools for robust prediction in structured dynamical systems (2605.04957).