Spectral Adaptive Conformal Prediction for Structured Non-Exchangeable Data

Panel data, in which multiple units are repeatedly observed over time, arise throughout science and engineering. Quantifying predictive uncertainty in such settings is challenging because conformal prediction, while distribution-free and model-agnostic, classically relies on exchangeability assumptions that fail under temporal dependence and unit heterogeneity. We propose a simple online conformal framework for non-exchangeable panel data. The method exploits a key feature of online panel prediction: when a forecast is required for one unit, contemporaneous outcomes from related units may already be observed and can serve as a calibration panel. At each round, prediction sets are formed using currently observed calibration units together with two adaptive quantities: history-based similarity weights that emphasize calibration units resembling the target, and an adaptive miscoverage level that is updated whenever target feedback is revealed. This two-state design yields a stepwise coverage bound and a long-run coverage guarantee. Empirically, across synthetic and real panel data sets, the method improves coverage on the worst-covered target units through adaptive interval-width allocation rather than uniform inflation. The two states are complementary: similarity weights protect coverage when target feedback is sparse, while the adaptive level further improves coverage as feedback accumulates.

Online conformal prediction must balance fast adaptation to distribution shift against stable coverage: feedback-driven methods react quickly but become volatile, while strongly discounted Bayesian methods lag and inflate intervals at tight coverage. We introduce \textbf{State-Adaptive Bayesian Conformal Prediction (SA-BCP)}, which forms the predictive quantile as a gated convex combination of long-term temporal inertia and local spatial evidence from a kernel density estimate, controlled by a single interpretable evidence threshold $K$. We establish three results: (i) asymptotic marginal validity of the resulting intervals up to a gate-controlled bias that vanishes as spatial evidence accumulates (exact under recurrent states); (ii) a closed-form expression for the MSE-optimal threshold, $K^*_{\mathrm{MSE}}=α(1-α)/M^{\mathcal{T}}$, trading the coverage-indicator (Bernoulli) variance against the temporal structural bias $M^{\mathcal{T}}$; and (iii) a rolling-origin procedure for selecting $K$ online -- consistent under stationarity, with $O(\sqrt{T\log N})$ regret against the best fixed $K$ and, for a segmented variant, a sublinear dynamic-regret bound under sublinearly many ($B_T=o(T)$) threshold shifts. Across four financial-volatility and weather datasets, three target coverage levels, and eight baselines, SA-BCP attains at-or-above-nominal coverage in most settings while producing substantially sharper intervals -- up to roughly $3\times$ lower Winkler score than discounted Bayesian CP at the tightest coverage -- and a coverage-matched audit confirms these efficiency gains are not an artifact of under-coverage. We disclose our principal limitation: a volatility-specialized CF-GARCH competitor remains more efficient on its home volatility-base series, though it does not transfer across domains.

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.