# Filtered Conformal Ellipsoids for Graph-Native Time Series

arXiv: https://arxiv.org/abs/2606.17014 · HTML: https://mlsanity.com/paper/2606.17014/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 32 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 17 of 17)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Graph WaveNet for Deep Spatial-Temporal Graph Modeling](https://mlsanity.com/paper/1906.00121/) (2019)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [Adaptive Conformal Predictions for Time Series](https://mlsanity.com/paper/2202.07282/) (2022)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [Uncertainty Quantification over Graph with Conformalized Graph Neural Networks](https://mlsanity.com/paper/2305.14535/) (2023)
- [Sequential Predictive Conformal Inference for Time Series](https://mlsanity.com/paper/2212.03463/) (2022)
- [Conformal Prediction for Time Series with Modern Hopfield Networks](https://mlsanity.com/paper/2303.12783/) (2023)
- [Conformal prediction for multi-dimensional time series by ellipsoidal sets](https://mlsanity.com/paper/2403.03850/) (2024)
- [Relational Conformal Prediction for Correlated Time Series](https://mlsanity.com/paper/2502.09443/) (2025)
- [Conformal Inference for Time Series over Graphs](https://mlsanity.com/paper/2510.11049/) (2025)
- [Topology-Aware Conformal Prediction for Stream Networks](https://mlsanity.com/paper/2503.04981/) (2025)

## Abstract (arXiv)

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.