AI summaryⓘ
The authors propose a method to predict multiple related time series together, creating uncertainty regions shaped like ellipsoids that adapt to their dependencies. They use a filter to estimate predictions and their uncertainties, then calibrate these predictions using a technique called split-conformal calibration to ensure accurate coverage without assuming normal distribution tails. The paper addresses challenges caused by dependencies in time and hidden states in the filter, providing theoretical guarantees for their method under certain conditions. They test their approach on traffic data graphs and show it produces tighter uncertainty regions than some simpler methods, though other complex models can sometimes perform better on different data.
Filtered conformal predictionMultivariate time seriesState-space filterSplit-conformal calibrationMahalanobis scoreGaussian predictive covarianceContraction analysisGraph convolutional network (GCN)GRU (Gated Recurrent Unit)Ellipsoid prediction sets
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.