Topology-Aware Conformal Prediction for Stream Networks

TL;DR

We address uncertainty quantification for stream networks, where directional flow and inter-site dependencies complicate conformal prediction. STACI integrates topology-aware covariance with data-driven estimates in a Mahalanobis-type nonconformity score and uses Adaptive Conformal Inference to track temporal shifts, delivering valid coverage with reduced prediction set volumes. Theoretical results establish conditional coverage validity and an efficiency-optimal selection of the topology-informed covariance, while empirical results on synthetic and real network data show STACI outperforms baselines in both coverage and efficiency. This framework enables robust, site-cluster level uncertainty quantification for spatio-temporal stream networks and can be extended to broader graph-structured time-series settings.

Abstract

Stream networks, a unique class of spatiotemporal graphs, exhibit complex directional flow constraints and evolving dependencies, making uncertainty quantification a critical yet challenging task. Traditional conformal prediction methods struggle in this setting due to the need for joint predictions across multiple interdependent locations and the intricate spatio-temporal dependencies inherent in stream networks. Existing approaches either neglect dependencies, leading to overly conservative predictions, or rely solely on data-driven estimations, failing to capture the rich topological structure of the network. To address these challenges, we propose Spatio-Temporal Adaptive Conformal Inference (\texttt{STACI}), a novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. \texttt{STACI} introduces a topology-aware nonconformity score that respects directional flow constraints and dynamically adjusts prediction sets to account for temporal distributional shifts. We provide theoretical guarantees on the validity of our approach and demonstrate its superior performance on both synthetic and real-world datasets. Our results show that \texttt{STACI} effectively balances prediction efficiency and coverage, outperforming existing conformal prediction methods for stream networks.

Figures (7)

• Figure 1: An example of stream network ${\mathcal{G}}$. The network segments $\{r_1, \dots, r_5\}$ are denoted by blue lines, and the observation points $\{\ell_1, \dots, \ell_{10}\}$ are marked with green triangles, pointing to the flow directions. The upstream of location $\ell_2$ are segments accompanied by orange area, and the downstream of location $\ell_6$ are blue shaded. The hydrologic distance between $\ell_2$ and $\ell_6$ is calculated through adding lengths of green shaded segments in both $r_1$ and $r_3$.
• Figure 2: Experiment results of synthetic data: (a) Comparison of methods on synthetic datasets with different tail-up parameters $\Theta$ over coverage ($x$-axis) and efficiency ($y$-axis). (b) Trade-off between coverage and efficiency on synthetic data, where the higher the better performance.
• Figure 3: Comparison of Coverage and Efficiency for PeMS data with different belief weight $\lambda$ and calibration set size $n$, with adaptive step size $\gamma = 0.01$ (upper) and $0$ (lower). The pre-determined coverage threshold of $95$% is shown by a horizontal gray dotted line.
• Figure 4: Taxonomy of works in conformal prediction. Among studies that account for both spatial dependency and temporal shift—without assuming spatial and temporal exchangeability—our work is the first to incorporate topology information.
• Figure 5: Real-world road network structures and their abstraction for PEMS-G1 and PEMS-G2. In each sub-figure, the left map displays the road network, where freeways are bold gray lines in blue shade, and ramps off the freeway are represented by blue squares. Based on these ramps and road junctions, the network is divided into different segments. Traffic flow monitoring sensors from $\ell_1$ to $\ell_{12}$ are placed exclusively on those northbound freeways, marked with green transparent triangles. The right map provides an abstract representation of the road network and sensor locations, using the same symbols for consistency.

Theorems & Definitions (22)

• Definition 1: Tail-up model
• Remark 1
• Remark 2
• Theorem 1: Validity
• Remark 3
• Theorem 2: Efficiency
• Remark 4
• Lemma 1
• proof
• Lemma 2