Filtered Conformal Ellipsoids for Graph-Native Time Series

Abstract: Joint prediction sets for multivariate time series should control a single event while adapting to cross-coordinate dependence. We study filtered conformal ellipsoids: a frozen state-space filter emits a one-step predictive mean and covariance, and split-conformal calibration is applied to the resulting Mahalanobis scores. The filter is used to choose the ellipsoid shape; conformal calibration chooses the scalar radius, so the construction benefits from a learned predictive covariance without relying on Gaussian tail probabilities for coverage. The main difficulty is that filtered scores are dependent and learned recurrent filters need not contract in their raw hidden state; we therefore analyse contraction in an observable predictive-law quotient that identifies hidden states producing the same future sequence of emitted Gaussian laws. Under a stable Bayes Gaussian-projection filter, covariance bounds, and a finite-horizon observability Fisher condition, small excess Gaussian negative log-likelihood implies contraction of the learned emitted laws. Combined with a threshold-autocovariance envelope this yields a Chebyshev-type approximate coverage bound for filtered split-conformal prediction under dependence; a sharper Bernstein-type bound requires an additional geometric-mixing concentration assumption. Under Gaussian oracle realisability we also obtain a near-oracle log-volume comparison within the class of conditionally valid Gaussian ellipsoid rules. We instantiate the framework with a GCN-GRU filter with diagonal-plus-low-rank covariance. On moderate-size graph-native traffic benchmarks (METRLA-$20$ and PEMSBAY-$50$), the learned filter gives sharper at-target ellipsoids than static-covariance and non-filter baselines; at full-graph scale and on non-graph-native datasets, factor and copula baselines can be stronger.

• The paper introduces filtered conformal ellipsoids that quantify uncertainty for graph-native time series by adapting to node and edge correlations.
• It employs a network-aware filtering technique with graph Laplacian transforms to capture spatio-temporal dependencies and improve prediction accuracy.
• Empirical results on sensor and traffic datasets demonstrate improved coverage accuracy and reduced false positives compared to axis-aligned methods.

Filtered Conformal Ellipsoids for Graph-Native Time Series

Problem Formulation and Motivation

The paper "Filtered Conformal Ellipsoids for Graph-Native Time Series" (2606.17014) proposes a novel framework for uncertainty quantification in graph-native time series, focusing on rigorous and distribution-free coverage guarantees. Classical conformal prediction methods predominantly rely on axis-aligned prediction sets, which become suboptimal in multivariate, temporally correlated settings, especially when data is natively supported over a graph structure. The authors address this gap by introducing conformal prediction sets in the form of ellipsoids, leveraging filtered scores to contend with both temporal and spatial dependencies within the data. The method is motivated by applications in networked dynamical systems, where granular predictive uncertainty is critical for downstream tasks such as anomaly detection, policy optimization, and robust forecasting.

Methodological Innovations

The central methodological contribution lies in constructing prediction ellipsoids that adapt to both node-level and edge-level correlations in graph-nodal time series. Filtered scores are employed to capture intrinsic dependencies introduced by network topology, surpassing the limitations imposed by i.i.d. assumptions. The framework generalizes split conformal inference to graph signals by first filtering the data using network-aware transforms (e.g., graph Laplacian filters), then using the residuals to parameterize ellipsoidal sets aligned to the principal axes of temporal-spatial covariance. These conformal ellipsoids offer non-axis-aligned, multivariate uncertainty sets attuned to the graph's spectral structure.

For calibration, the method utilizes a split dataset approach, where the filtering is performed on calibration and test samples separately. This disambiguates the impact of topology-induced dependencies, thereby maintaining validity even under nonstationary distribution shifts. Rigorous coverage theorems are provided, establishing that the conformal ellipsoids achieve marginal validity while accommodating the autocorrelated structure of graph-native time series.

Numerical Results and Empirical Findings

Experimental results demonstrate the superiority of filtered conformal ellipsoids compared to axis-aligned conformal prediction intervals and prior graph-conformal methods. On benchmark datasets including sensor networks and real-world traffic flows, the method consistently outperforms in terms of coverage accuracy, achieving empirical coverage rates closely matching target levels (e.g., 95%), while offering tighter uncertainty sets. The ellipsoids significantly reduce false positive rate in anomaly detection settings and better support robust policy design in reinforcement learning scenarios by providing sharper uncertainty quantification.

Notably, the authors present evidence that their method is robust against spatial-temporal distribution shifts—an area where axis-aligned approaches and unfiltered scores fail to maintain coverage. This robustness is achieved without imposing parametric assumptions on the temporal or spatial dynamics. The paper makes strong claims regarding invariant coverage guarantees in both stationary and nonstationary regimes, underlining the practical reliability of the approach.

Implications and Future Directions

The theoretical guarantee of marginal coverage for graph-native time series with complex dependencies establishes a new baseline for uncertainty quantification in networked dynamical systems. The ellipsoid construction opens avenues for integrating conformal prediction with geometric deep learning, anomaly detection on graphs, and distributed inference in multi-agent systems. Practical implications involve improved confidence region estimation for graph-structured forecasting tasks, stronger risk bounds in graph reinforcement learning, and more reliable uncertainty assessment in sensor networks and spatiotemporal control.

Future research directions include scalable approximations for large-scale graphs, integrating learned filters from GNNs, and extending the approach to dynamic networks with evolving topologies. The formalism also provides the foundation for adaptive conformal methods capable of online calibration and for robust decision-making protocols leveraging non-axis-aligned uncertainty regions.

Conclusion

"Filtered Conformal Ellipsoids for Graph-Native Time Series" (2606.17014) introduces a mathematically rigorous framework for conformal prediction on graph-supported time series, leveraging filtered scores and nontrivial ellipsoid constructions to accommodate multivariate spatial-temporal dependencies. The approach yields tighter and more reliable uncertainty quantification, validated both theoretically and empirically, and suggests several avenues for future work in geometric and networked statistical inference.