Title: Topology-Aware Conformal Prediction for Stream Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: conformal prediction

stream networks

spatiotemporal graph

uncertainty quantification
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 (STACI), a novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. 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 STACI effectively balances prediction efficiency and coverage, outperforming existing conformal prediction methods for stream networks.

Topology-Aware Conformal Prediction for Stream Networks

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 ( STACI ), a novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. 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 STACI effectively balances prediction efficiency and coverage, outperforming existing conformal prediction methods for stream networks.

1 Introduction

Stream networks represent a distinctive class of spatiotemporal graphs where data observations follow directional pathways and evolve dynamically over both space and time [21]. These networks are prevalent in various domains such as hydrology, transportation, and environmental monitoring, where data exhibit strong flow constraints [9, 18, 33, 57, 36]. For example, in hydrology, river networks dictate the movement of water flow and pollutant dispersion [32], while in transportation, road and rail networks determine congestion and travel times [57, 56]. Understanding and modeling these networks are crucial for infrastructure planning, disaster response, and ecological conservation.

A fundamental challenge in stream network analysis is predicting future observations and quantifying their uncertainty across multiple interconnected locations governed by network topology. Given the dynamic nature of these systems, accurate and reliable uncertainty quantification (UQ) is essential for risk assessment, decision-making, and resource allocation. For example, in transportation, estimating uncertainty in traffic volume forecasts across critical junctions enables optimal routing and congestion management [57]. However, the hierarchical dependencies, directional flow constraints, and evolving conditions inherent in stream networks introduce significant complexities.

Recent advances in machine learning and statistical modeling have enhanced predictive accuracy for spatio-temporal data and enabled effective UQ with statistical guarantees [49, 14, 55]. In particular, conformal prediction (CP) has emerged as a powerful UQ framework, providing finite-sample validity guarantees under mild assumptions [35]. By constructing prediction sets with valid coverage probabilities, CP ensures that future observations fall within specified confidence intervals, enhancing reliability in decision-support systems [27, 23, 5, 54].

Despite its success in various domains, traditional CP methods face significant limitations when applied to stream networks due to two key challenges: ( i ) Multivariate prediction : Unlike standard time-series predictions that focus on a single target variable, stream networks require joint predictions at multiple locations, where observations are highly interdependent. Applying CP independently at each location neglects network-wide dependencies, leading to inefficiencies in prediction set construction and potential loss of coverage guarantees. ( ii ) Intricate spatio-temporal flow constraints : Traditional CP assumes exchangeable data, an assumption that fails in stream networks due to directional flow constraints. While graph-based and spatial models account for topological relationships, stream networks exhibit unique dependency structures that neither conventional graph-based approaches nor purely data-driven models fully capture. Existing CP approaches either completely ignore dependencies without considering the spatio-temporal dynamics [37, 10] or attempt to learn dependencies solely from data without incorporating topological constraints [51, 40]. The former results in overly conservative or miscalibrated prediction sets, while the latter risks overfitting to specific network conditions, reducing generalizability.

To address these challenges, we propose a novel framework, Spatio-Temporal Adaptive Conformal Inference ( STACI ), for constructing uncertainty sets in stream networks. Our method integrates network topology and temporal dynamics into the conformal prediction framework, yielding more efficient and reliable UQ. Specifically, we develop a nonconformity score that explicitly incorporates spatial dependencies across multiple locations on the stream network as determined by their underlying topology, balancing observational correlations with topology-induced dependencies. To achieve this balance, we introduce a weighting parameter that regulates the contribution of topology-based covariance and data-driven estimates. A greater reliance on the topology-induced covariance structure improves coverage guarantees, assuming it accurately reflects underlying dependencies. Conversely, prioritizing sample-based estimates mitigates potential misspecifications in the topology-induced covariance, often leading to better predictive efficiency. Additionally, we consider a dynamic adjustment mechanism that accounts for temporal distributional shifts, allowing prediction intervals to adapt over time and maintain valid coverage in non-stationary environments.

We provide a theoretical analysis of STACI , demonstrating that it maximizes prediction efficiency by reducing uncertainty set volume while maintaining valid coverage guarantees. To validate its effectiveness, we evaluate STACI on synthetic data with a stationary covariance matrix and real-world data with time-varying covariance, comparing its performance against state-of-the-art baseline. 3 Both our theoretical and empirical results underscore the importance of the weighting parameter that balances data-driven insights with topology-induced knowledge, optimizing performance and enhancing predictive reliability in stream network applications.

Our contribution can be summarized as follows:

We propose a novel conformal prediction framework specifically designed for stream networks, integrating both spatial topology and temporal dynamics to enhance uncertainty quantification.

We highlight the limitations of purely data-driven dependency estimation in stream networks and introduce a principled approach that leverages both observational data and inherent network structure.

We provide a theoretical analysis establishing STACI 's validity and efficiency, and empirically demonstrate its superior performance in achieving an optimal balance between coverage and prediction efficiency on both synthetic and real-world datasets.

2 Related Works

Stream networks, such as hydrology [21, 18, 36], transportation networks [14, 28], and environmental science networks [25], have been extensively studied due to their critical role in natural and engineered systems. Forecasting for stream network can be approached from two perspectives: as a graph prediction problem or as a multivariate time series prediction problem. In this work, we focus on the latter one, with the aim of predicting future data based on historical network data.

Many approaches to stream network analysis rely on domain-specific statistical and physical models. [18] introduced spatial stream network models for hydrology, emphasizing the importance of flowconnected relationships and spatial autocorrelation. The tail-up model [43] generalized spatial covariance structures to stream networks by weighting observations based on flow connectivity. Recent advances in machine learning have spurred innovative approaches for modeling stream networks for point forecasts, particularly through graph-based frameworks that leverage their inherent spatio-temporal (ST) graph dynamics [20, 11].

While effective and widely adopted, models without uncertainty quantification often lack considerations for reliability, posing limitations particularly in safety-critical scenarios. To address this, some studies [49, 58, 38, 48, 33] turn to explore interval prediction, which ensures that prediction intervals cover the ground-truth values with a pre-defined high probability, offering a more reliable alternative. Among these approaches, the majority of studies employ Bayesian methods to construct prediction intervals for ST forecasting problems [47]. These methods commonly utilize Monte Carlo Dropout [49, 33] or Probabilistic Graph Neural Networks [58, 48]. However, the performance of Bayesian methods has been found to be sensitive to the choice of prediction models and priors, particularly the type of probabilistic distributions [48]. To address these limitations, classic Frequentist-based methods have been employed, such as conformal prediction, which generally offer more robust coverage across data and model variations.

Conformal prediction (CP) [46] has recently gained significant attention across various domains, including graph-structured data [8, 19, 29] and multi-dimensional time series [40, 30, 50]. Existing CP methods for graphs [8, 19, 29] have primarily addressed node/edge classification and ranking tasks, where UQ concerns discrete labels or scores on static graph structures. In contrast, our work focuses on a fundamentally different problem: UQ for spatio-temporal forecasting on dynamic networks. Given that spatio-temporal graphs can be naturally formulated as a special case of multivariate time series, we discuss related CP methodologies developed for the latter setting [40, 30, 50]. [40] assumes that data samples for each entire time series are drawn independently from the same distribution, while [30] assumes exchangeability in the data. Both approaches fail to capture the complex temporal and spatial dependencies inherent in ST graphs, limiting their applicability. [50] construct ellipsoidal prediction regions for non-exchangeable multi-dimensional time series, but their model neglects the inherent graph structure embedded within the multi-dimensional time series and overlook scenarios where the error process (see eq. (1)) is non-stationary, a prevalent feature in real-world data. Section B in the Appendix provides a taxonomy of existing CP methods, highlighting our unique positioning within the CP literature. To the best of our knowledge, no previous work has specifically tailored CP for stream networks or other spatio-temporal graphs, reinforcing the novelty to this domain.

3 Problem

Consider a stream network G with fixed flow direction at time t ∈ { 1 , . . . , T } , with I observational sites indexed by I = { 1 , . . . , I } . Let L ⊂ R 2 denote the set of all geolocations on the network, and let the geographical location of site i ∈ I be represented as ℓ i ∈ L . The stream network consists of segments { r j ⊂ L , j ∈ J} , where J is the index set of all stream segments. Each site i ∈ I is located within a specific segment r j for some j ∈ J , and a segment may contain multiple or no observational sites. For any location u ∈ L , we define ∧ u as the set of all upstream segments of location u , and ∨ u as the set of all downstream segments of location u . The hydrologic distance between two locations v, u ∈ L , denoted as d ( v, u ) , is the distance measured along the stream. If v and u belong to the same segment r i , d ( v, u ) is simply the Euclidean distance between v and u . See Figure 1 for an illustration.

Now, consider a multivariate time series observed at the I sites. We denote the dataset as D := { ( X t , Y t ) } t ∈ [ T ] . Here Y t := [ Y t ( ℓ 1 ) , Y t ( ℓ 2 ) , . . . , Y t ( ℓ I )] ⊤ ∈ R I , and Y t ( ℓ i ) (or simply

Y i t ) represents the observation at location ℓ i at time t . The historical observations are given by X t ∈ R I × h , defined as X t := [ Y t -1 , Y t -2 , . . . , Y t -h ] ⊤ ∈ R I × h . We assume that Y t follows an unknown true model f ( X t ) with additive noise ϵ t , such that:

where ϵ t ∈ R I has zero mean and a positive definite covariance matrix Σ ≻ 0 .

The goal is to construct a prediction set for Y T +1 given the new history X T +1 , denoted by C ( X T +1 ) , such that, for a predefined confidence level α , the following coverage guarantee holds:

This objective can be achieved using split conformal prediction (CP) [46], a widely used framework for uncertainty quantification. Split CP operates by first partitioning the data into a training set and a calibration set. The prediction model ˆ f is trained exclusively on the training set. To assess the reliability of predictions, a nonconformity score is computed, which quantifies the deviation of each calibration sample from the ground truth. Given a target confidence level α , the method determines the (1 -α ) -th quantile of the nonconformity scores from the calibration data. This quantile is then used to adjust ˆ f 's predictions for test samples, ensuring the constructed prediction sets maintain valid

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coverage. Under the assumption that the calibration and test data are exchangeable, the prediction sets are guaranteed to achieve a coverage rate of at least 1 -α on the test data.

The challenges of performing multivariate time-series prediction over a stream network are twofold: ( i ) Multi-dimensionality : The response variable Y t is multivariate and potentially high-dimensional, significantly increasing the complexity of constructing accurate prediction sets. Standard CP methods, when applied to multi-dimensional variables without a carefully designed nonconformity score, often produce overly conservative prediction sets. This leads to inefficiencies, as the prediction set size |C ( X t ) | becomes too large to provide meaningful uncertainty quantification. ( ii ) Non-exchangeability : Observational sites exhibit complex spatial and temporal dependencies due to strong correlations imposed by the network topology. As a result, traditional CP methods, which rely on exchangeability assumptions, cannot be readily applied.

4 Proposed Framework

This paper proposes a novel framework, referred to as spatio-temporal adaptive conformal inference ( STACI ), for constructing uncertainty sets in spatio-temporal stream networks. Our approach consists of two key components: ( i ) We develop a nonconformity score that explicitly captures spatial dependencies induced by the stream network's topology, leading to more efficient prediction sets. ( ii ) We account for temporal distributional shifts to refine prediction sets dynamically, ensuring reliable coverage over time. We demonstrate that STACI significantly improves prediction efficiency while maintaining valid coverage guarantees, making it a robust and effective approach for uncertainty quantification in spatio-temporal settings.

Topology-aware Nonconformity Score We use the most recent n < T data points to construct the calibration dataset. Specifically, we denote the calibration dataset as D cal := { ( X t , Y t ) , t = T -n +1 , · · · T -1 , T } , and define ˆ Y t := ˆ f ( X t ) , where ˆ f is the fitted model trained on the rest of the data D \ D cal. For each calibration data point ( X t , Y t ) ∈ D cal, we compute its nonconformity score, denoted by s ( X t , Y t ) .

To account for the intricate spatio-temporal dependencies, we consider a general class of nonconformity score functions based on the Mahalanobis distance [22]:

where A is an I × I symmetric positive definite matrix and ˆ ϵ t := Y t -ˆ Y t -¯ ϵ t is the centered prediction error, with ¯ ϵ t denoting the sample average of errors on D cal .

The core idea of our method is a linearly weighted representation for A , which integrates both topology-induced and sample-based covariance estimates. Formally,

Here ˆ Σ n is the sample covariance matrix computed from the residuals { ˆ ϵ t , t ∈ D cal } , and ˆ Σ G represents the covariance structure induced by the stream network topology. The weighting parameter λ ∈ [0 , 1] balances these two estimates. A higher value of λ places greater reliance on the topologydriven covariance structure, assuming it accurately captures the underlying dependencies. Conversely, a lower λ shifts reliance toward the sample-based estimate, mitigating potential misspecifications in the topology-induced covariance.

Unlike the method proposed by [51], which relies solely on the sample covariance estimate, this formulation incorporates the underlying topology of the stream network. By balancing data-driven and structural information, it provides a more robust covariance estimation, leading to better prediction efficiency without sacrificing coverage validity.

Topology-induced Covariance Estimation We develop a novel method to estimate the topologyinduced covariance ˆ Σ G used in eq. (3) by assuming the observations on the stream network can be captured by a tail-up model [18, 43, 17]. The tail-up model is formally defined as follows:

Definition 1 (Tail-up model) . Given a stream network G , the observation at any location u on the network can be modeled as a white-noise random process, which is constructed by integrating a moving average function over the upstream process, i.e.,

where ∧ u denotes all the segments that are the upstream of u . Here, µ ( u ) is the deterministic mean at u , and m ( v -u ) is a moving average function capturing the influence from upstream location v to u . Both w ( v ) and w ( u ) are weights that satisfy the additivity constraint such that the variance remains constant across sites.

We note that the tail-up model only requires the assumptions of ergodicity and spatial stationarity [42], which is highly flexible and can be broadly applied to a wide range of stream network data. Also, the choice of the moving average function m (∆) remains adaptable, as long as it has a finite volume, allowing the model to accommodate different spatial structures effectively.

To estimate ˆ Σ G , we model B ( v ) using Brownian motion and adopt an exponential moving average function for m (∆) = β exp( -∆ /ϕ ) . Therefore, the topology-induced covariance between any two locations u, v can be expressed as follows (See Lemma 5 in Section A of the Appendix):

where ϕ and σ 2 are estimated scaling parameters of the tail-up model. In practice, weights w can be obtained by estimating the intensity of the flow through the observational, for instance, using normalized traffic counts as the weights for traffic stream network data.

Intuitively, the covariance structure reflects how information propagates along the stream network. The exponential decay in eq. (5) models diminishing influence with increasing hydrologic distance d ( u, v ) , while the weight term √ w ( u ) /w ( v ) modulates this effect based on flow intensity. This formulation naturally aligns with real-world stream dynamics, where proximal upstream sites exert stronger influence than distant or disconnected ones.

Adaptive Uncertainty Set Construction We construct a spatio-temporally adaptive prediction set for a new observed history X T +1 using our proposed nonconformity score, defined in eq. (2), as follows:

where ˆ Q 1 -α is the (1 -α ) -th quantile of the empirical cumulative distribution function of { s ( X t , Y t ) , t ∈ D cal } .

To account for potential temporal distribution shifts in the predictive error of eq. (1), we adopt the Adaptive Conformal Inference (ACI) framework proposed in [15]. This approach dynamically updates the confidence level α t over time, ensuring that the prediction set remains responsive to evolving data distributions. Specifically, we iteratively update α t , and reconstruct the prediction set C ( X t , α t ) accordingly. At the initial test time T +1 , the confidence level is set as α T +1 = α . For subsequent time steps t > T +1 , α t , we update α t with a step size γ ≥ 0 as follows:

The rationale behind ACI is that if the prediction set fails to cover the true value at time t , the effective error level is reduced, leading to a wider prediction interval at time t +1 , thereby increasing the likelihood of coverage. A larger step size γ makes the method more responsive to observed distribution shifts but also introduces greater fluctuations in α t . When γ = 0 , the method reduces to standard conformal prediction with fixed α . The detailed analysis of coverage guarantee of Adaptive Uncertainty Set construction is discussed in Section C in Appendix.

5 Theoretical Analysis

Our theoretical analysis focuses on establishing two key properties for the proposed STACI :

Optimal Efficiency: We establish that STACI maximizes predictive efficiency by reducing the uncertainty set volume, justifying the need for accurate covariance estimation in spatio-temporal stream networks (Theorem 2).

Validity Guarantees under Stationarity and Adaptation to Distribution Shifts: We prove that STACI ensures valid conditional coverage under stationary assumptions (Theorem 1) and extend the framework to handle non-stationary settings via an ACI adjustment, ensuring approximate average coverage (Proposition 1 in Section C of the Appendix).

Our analysis is based on the Mahalanobis distance framework in eq. (2), which enables the construction of arbitrary ellipsoidal uncertainty sets, with greater flexibility for nonconformity scores. For example, standard CP with spherical uncertainty sets arises as a special case when A is an identity matrix. Another instance is the approach in [51], where A is set as the sample covariance matrix.

We adopt standard asymptotic notation and norm definitions. The bigO notation O ( · ) characterizes an upper bound on a function's growth rate: if f ( n ) = O ( g ( n )) , then there exists a positive constant C such that f ( n ) ≤ Cg ( n ) , for all n ≥ n 0 . The littleo notation o ( · ) denotes strictly smaller asymptotic growth, with f ( n ) = o ( g ( n )) implying lim n →∞ f ( n ) /g ( n ) = 0 . Additionally, we use standard ℓ 2 norms for quantifying vector and matrix magnitudes.

5.1 Coverage Validity

We analyze the conditional coverage validity of the proposed method. Consider the additive error model described in eq. (1) where the errors, ϵ t , are i.i.d. . We introduce the following assumption and, for simplicity, denote the nonconformity score ϵ ⊤ t Aϵ t as s t .

Assumption 1 (Estimation quality) . There exists a sequence { ν n } , n ≥ 1 such that 1 n ∑ T t = T -n +1 || ϵ t -ˆ ϵ t || 2 ≤ ν 2 n , || ϵ T +1 -ˆ ϵ T +1 || ≤ ν n .

Remark 1. The assumption ensures that the prediction error is bounded by ν 2 n . For many estimators, the ν n vanishes as n →∞ , indicating improved estimation accuracy with larger sample sizes [7].

Assumption 2 (Convergence of A n ) . The sequence { A n } associated with the nonconformity score converges to a fixed matrix A as n increases, with an upper-bounded convergence rate o ( g ( n )) , i.e. ∥ A n -A ∥ = o ( g ( n ) ) . Additionally, there exists a constant r > 0 such that ∥ A ∥ ≤ r .

Remark 2. When designing nonconformity scores, the matrix A can be chosen to either remain constant or converge to a fixed matrix. The flexibility in selecting A allows for adaptability across different scenarios. For example, if the true covariance matrix of the error ϵ is known, A can be set as its inverse. Alternatively, if only sample estimates are available, A can be chosen as the inverse of the estimated sample covariance matrix of ϵ , provided it converges under proper tail behavior conditions [45]. The major difference between choices of A n lies in the respective convergence rates.

Assumption 3 (Regularity conditions for s t and ϵ t ) . Assume that the cumulative distribution function (CDF) of the true nonconformity score, F s ( x ) , is Lipschitz continuous with a constant L > 0 . Suppose there exist constants κ 1 , κ 2 > 0 such that ∥ ϵ t ∥ ≤ κ 1 I almost surely, and Var[ ∥ ϵ t ∥ 2 ] ≤ κ 2 I. .

Theorem 1 (Validity) . Under the assumptions stated above, the proposed method satisfies the following conditional coverage guarantee:

where

Remark 3. The finite-sample bound on the coverage gap is directly influenced by the estimation quality and the convergence rate of A n , which is given by max( O ( log n n ) , O ( ν n ) , O ( √ g ( n ))) . In general, reducing the coverage gap requires high-accuracy estimations ( i.e. , a rapidly vanishing ν n ) and a well-chosen nonconformity score matrix A n that converges quickly.

Theorem 1 highlights the importance of incorporating topology-based estimators in STACI . Relying solely on the sample covariance matrix often leads to coverage gaps in finite samples, undermining validity. In contrast, the topology-based matrix acts as a covariance estimator with topology-informed regularization, generally achieving faster convergence than the sample covariance estimator. A hybrid approach that combines both estimators provides an optimal trade-off between validity and efficiency.

5.2 Prediction Efficiency

We evaluate the efficiency of STACI via the volume of the prediction set in I -dimensional space, defined as V ( A,r ) = π I/ 2 Γ ( I 2 +1 ) · r I/ 2 · det( A ) -1 / 2 . The radius of the prediction set is determined by the (1 -α ) -th quantile of the empirical CDF, computed from n data points in the calibration dataset. This radius is denoted as ˆ Q 1 -α ( { ˆ ϵ ⊤ t A ˆ ϵ t , t ∈ D cal } ) . In the ideal case where ˆ f ( X t ) = f ( X t ) and ˆ ϵ t = ϵ t , minimizing inefficiency reduces to: min A ≻ 0 V ( A, ˆ Q 1 -α ( { ϵ ⊤ t Aϵ t , t ∈ [ n ] } )) . To simplify computation, we approximate the empirical quantile with the true quantile, justified by the Glivenko-Cantelli Theorem [41], which ensures the convergence lim n →∞ ˆ Q 1 -α ( { ϵ ⊤ t Aϵ t , t ∈ [ n ] } ) = Q 1 -α ( ϵ ⊤ Aϵ ) , where Q 1 -α ( · ) , the 1 -α quantile of ϵ ⊤ Aϵ is assumed to be continuous.

In the limiting case, we formulate the following minimization problem, presented in Theorem 2, and use its solution as the guiding criterion for selecting the matrix A :

Theorem 2 (Efficiency) . There exists 0 < α 0 < 1 , such that when α < α 0 , the optimal solution to the minimization problem is given by:

where ϵ ∼ N (0 , A -1 ∗ ) .

Remark 4. The optimization problem is invariant to scalar rescaling ( i.e. , A and any positive scalar multiple cA , where c > 0 , yield the same mathematical solution). Empirically, we find α 0 > 0 . 2 for Gaussian noises when I ≤ 30 , ensuring that the result is applicable to conformal prediction settings with typical coverage levels of 90% or 95% in the focused setting. The analysis can go beyond the Gaussian noise assumption for ϵ and be extended to broader distributions that satisfy appropriate tail-bound conditions.

Theorem 2 underscores the importance of selecting A optimally in eq. (2) and highlights that accurately estimating the inverse of the error covariance matrix is key to minimizing inefficiency in CP. In practice, designing an optimal A ∗ is often challenging due to empirical limitations. For example, the estimated residuals ˆ ϵ t may deviate significantly from the true errors ϵ t . Additionally, when the sample size n is small, the empirical CDF may differ considerably from the true CDF. Despite these challenges, constructing A based on an estimate of the inverse covariance matrix offers substantial improvements in high-dimensional settings compared to CP methods that ignore variable dependencies, such as those that set A as the identity matrix.

6 Experiments

To demonstrate the suitability of STACI , we evaluate its performance on both synthetic data with a stationary covariance matrix and real-world data with time-varying covariance. By default, the first 60 %of observations are used for training, the calibration set consists of the most recent n = 500 observations, and the test contains the sequentially revealed observations n ′ = 5000 in simulation and n ′ = 5000 in real study. The desired confidence rate α is fixed at 0 . 95 . Our method is compared

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against the following conformal prediction and learning-based uncertainty quantification baselines: ( i ) Sphere : Spherical confidence set, where the covariance matrix is an identity matrix. In other words, the prediction error at different locations are not considered to have correlations. ( ii ) Sphere-ACI ( γ = 0 . 01 ): Spherical confidence set with adaptive conformal inference (ACI). ( iii ) Square : Square confidence set. This equals to computing different nonconformity scores for each dimension, and then calibrate accordingly. ( iv ) GT : Ellipsoidal confidence set using the ground-truth covariance matrix. ( v ) MultiDimSPCI : Ellipsoidal confidence set using the sample covariance matrix [50], alongside its localized variant using the most recent observations, MultiDimSPCI (local) . ( vi ) CopulaCPTS : Prediction region based on modeling the joint distribution of forecast errors with a copula function [39]. ( vii ) DeepSTUQ : A Bayesian deep learning model that quantifies uncertainty in spatio-temporal graphs by using graph convolutions and Monte Carlo dropout [33].

We consider both validity and efficiency to evaluate the uncertainty quantification performance: ( i ) Coverage quantifies the likelihood that the prediction set includes the true target, i.e. , Coverage := ∑ t ∈D test 1 { Y t / ∈ C ( X t ; α t ) } / n ′ ; ( ii ) Efficiency is evaluated based on the size (or volume) of the prediction set, with smaller sets indicating higher efficiency. The volume of the prediction set, Vol ( C ( X t ; α t )) , is measured by the size of the ellipsoid determined by A . Formally, Efficiency :=

∑ t ∈D test ( Vol ( C ( X t ; α t )) ) 1 /I / n ′ . An optimal method should achieve the predefined coverage with high efficiency/low inefficiency.

6.1 Simulation

In this section, we conduct simulation experiments on synthetic data generated by a tail-up model. Specifically, we follow [31] and construct the stream network as shown in Figure 1. The details of synthetic network is provided in Section D.1 of the Appendix. We generate the observation of site u at time point t by simulating stochastic integration from all upstream points r ∈ ∧ u to downstream point u according to eq. (4), where we set µ t ( u ) = ∑ w i =1 θ i Y t -i ( u ) following the AR ( w ) structure and m (∆) = exp( -∆) as the exponential moving average function. The process is repeated until 5000 time steps. This experiment simulates the stream network data without any misspecification.

Experiment Configuration In synthetic data, the prediction model f is simply a linear regression model. We first estimate parameters of in AR ( w ) structure, i.e., Θ = ( θ i ) i ∈ [ w ] , through linear regression and then parameters in eq. (29), ϕ and σ 2 through ℓ 1 -loss. Parameters of Θ = (0 , 0) and Θ = (0 . 7 , 0 . 3) are selected for data generation. When Θ = (0 , 0) , the observations consist of pure noise, thus stationary; when Θ = (0 . 7 , 0 . 3) , the process resembles a second-order autoregressive model. The weighting factor λ is set to 0 . 6 .

Results Our numerical results demonstrate that our method enhance the predictive efficiency significantly without sacrificing the coverage guarantee, by considering both sample-based and topology-based covariance. From Figure 2(a), we observe that CP methods employing ellipsoidal uncertainty sets tend to cluster towards the upper region of the plot, indicating higher efficiency compared to CP methods based on spherical or square uncertainty sets. Although MultiDimSPCI achieves the lowest efficiency, its coverage drops significantly below the required threshold, highlighting its instability when relying solely on the sample covariance matrix. This issue persists even

in simulated data designed to align with its error process assumptions. Its local variant, designed to capture temporal correlations, provides only a marginal improvement and similarly fails to achieve the required coverage. In contrast, with λ fixed at 0 . 6 , our method STACI is positioned near the upper-right corner alongside GT, which leverages the ground-truth covariance matrix. This suggests that our method achieves performance comparable to GT, balancing low efficiency (smaller volume) while maintaining the necessary coverage guarantees. Among all methods that surpass the coverage threshold, our method, STACI ( γ = 0 . 01 ), demonstrates the best efficiency with the smallest variance, further reinforcing its robustness and effectiveness.

Figure 2(b) reveals a clear trade-off trend between coverage and efficiency: the higher λ , the confidence level rises, but efficiency is worse. This suggests that λ should be carefully chosen: if too large, our method over-relies on topology and fails to adapt to covariance shift; if too small, it depends more on sample covariance matrices, which are purely data-driven and thus unstable, leading to a coverage drop. Nonetheless, no matter whether adapting confidence level, setting a larger λ in STACI can efficiently increase coverage and maintain it near the pre-determined level, while only slightly increasing volume, which remains comparable to GT.

6.2 Real Data Study

We further conduct experiments on a real-world traffic dataset, Performance Measurement System (PeMS) [6], which contains the data collected from the California highway network, providing 5 -minute interval traffic flow counts by multiple sensors, along with flow directions and distances between sensors. To model it into stream network, we also rely on [4] to check accurate road connection information. We select two highway forks, each equipped with 12 sensors, named PeMS-G1 and PeMS-G2, and plot their locations and corresponding road segments in Figure 5 in the Appendix.

Experiment Configuration We adopt Adaptive Graph Convolutional Recurrent Network (AGCRN) [2] as the default backbone model f . To demonstrate STACI's generality under post-hoc conformal prediction framework, we evaluate over alternative GNN backbones, including attentionbased ASTGCN [16] and continuous-time STGODE [13]. We set our default λ = 0 . 6 . For simplicity, we only use fixed weights with all equal values, without requiring any additional information. Multiple hyperparameter and ablation study are also provided over the key parameters in our framework: ( i ) λ from 0 to 1 with step of 0 . 02 ; ( ii ) n = 100 , 200 , 300 , 400 , 500 ; ( iii ) γ = 0 or 0 . 01 .

Result Table 1 clearly demonstrates that our method consistently outperforms all baseline CP techniques across diverse backbone models and road topologies, achieving significantly higher coverage and superior efficiency. Further comparisons against CopulaTSCP and DeepSTUQ are detailed in Appendix D.5, as those methods are less fitted for our problem setting.

Our method exhibits robustness to the choice of the hyperparameter λ . In the first two subplots of Figure 3, setting λ = 0 reduces our algorithm to sole usage of sample covariance matrix, equivalent to our strongest baseline, MultiDimSPCI. Our coverage is greater than MultiDimSPCI with arbitrary hyperparameter λ . Further, across all calibration sample sizes n , selecting any λ ∈ [0 . 3 , 0 . 9] consistently yields both higher coverage and greater(lower) efficiency. This underscores the critical contribution of topological information and demonstrates STACI 's insensitivity to λ .

97

Coverage Ratewithy=0.01

60

Efficiencywithy=0.01

97

CoverageRatewithy=0

60

Efficiencywithy=0

96

96

95

70

95

70

94

94

(%)

80

80

Coverage

92

Efficiency

Coverage

92

iency

90

Efficie

90

06

n=100

100

90

n=100

100

n=200

n=200

88

n=300

n=400

110

88

n=300

n=400

110

n=500

n=500

86

0.0

0.2

0.4

0.6

0.8

1.0

120

0.0

0.2

0.4

0.6

0.8

1.0

86

0.0

0.2

0.4

0.6

0.8

1.0

120

0.0

0.2

0.4

0.6

0.8

1.0

入

入

入

入

The incorporation of topology-induced covariance matrix is pivotal even without ACI ( γ = 0 ), as shown in the last two subplots in Figure 3. STACI can obviously lift coverage from under 87 %to surpass the desired 95 %level with only a marginal efficiency cost. This indicates that under inherent temporal covariance shifts, utilizing topological information offers a robust remedy to under-coverage problem while maintaining highly informative predictions. Moreover, as demonstrated in Section D.4 in the Appendix, STACI can also outperform in an offline setting, where the covariance matrix is estimated once and then held fixed at the beginning of test time. We discuss the scalability of STACI in term of time length and graph size in Section 7 and in Appendix D.7, respectively.

In conclusion, the estimate of the covariance matrix can benefit from both topology and samples, compared with relying on any single resource. On one hand, with limited finite calibration sample n , the topology-based estimator offers a more stable structure as it possess fewer parameters. It can alleviate the temporal distribution shift and the resulting under-cover problem, consistent with our theoretical analysis. The sample covariance, on the other hand, captures the actual spatial patterns from samples in the calibration data and gives higher efficiency, but it can lose coverage guarantee if the distribution changes. By blending these two estimates and adaptively adjusting the significance level, STACI can effectively maintain desired coverage and smaller volumes.

7 Limitation

STACI targets joint uncertainty quantification on moderate -size subgraphs ( 2 ∼ 30 D). This choice is deliberate: performing UQ in very high-dimensional output spaces (hundreds of coordinates or more) is statistically ill-posed. Directly constructing geometric prediction sets in the full output space is vulnerable to the curse of dimensionality [44]. In particular, hyper-rectangles formed by marginal intervals can inflate to near-vacuous volumes even at low dimensions as in Table 1.

In high-dimensional regimes, common practice is to avoid explicit geometric sets and instead use generative or sampling -based UQ [26, 3]. For instance, image UQ, a prototypical high-dimensional setting, typically samples plausible outcomes from a learned posterior [12] rather than delineating a set in pixel space. In the same spirit, STACI focuses on subgraphs of manageable joint dimensionality so that (i) the resulting joint sets remain informative, (ii) the coverage-efficiency trade-off is nondegenerate. This design also matches how practitioners assess risk: traffic engineers often analyze corridors with 10-30 sensors rather than entire city-wide networks, and hydrologists frequently study clusters of ∼ 20 gauges when evaluating localized flood risk.

8 Conclusion

We proposed STACI , an adaptive conformal prediction framework for stream networks. Theoretically, we established coverage guarantees and demonstrated the model's ability to minimize inefficiency under mild conditions. Empirically, STACI produced smaller prediction sets while maintaining valid coverage across both (stationary) simulated data and (non-stationary) real-world traffic data.

Future work includes three potential directions. (i) STACI can be extended to general spatio-temporal graphs by replacing Σ G with alternative network parameterizations, enabling the development of novel methods that effectively exploit topological structures in broader spatio-temporal settings; (ii) stronger theoretical guarantees such as finite-sample coverage bounds when adaptively calibrating the significance level, could be developed by imposing assumptions on error distribution shifts (e.g., first-order differencing stationarity); and (iii) the present formulation does not aim to provide full-graph joint sets over hundreds of nodes. One might build corridor-level joint sets and composing them with rigorous controls on cross-correlation to scale coverage beyond the subgraph level.

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Contents

A Theoretical proofs

A.1 Proof of Theorem 1

Theorem 1 (validity). Under the assumptions stated above, the proposed method satisfies the following conditional coverage guarantee:

where

For easy notation, denote ˆ A = A n , ∆ t = ˆ ϵ t -ϵ t and sometimes we drop subscript t .

Lemma 1. For any test conformity score ˆ s t = ˆ ϵ t T ˆ A ˆ ϵ t and the true conformity score s t = ϵ T t Aϵ t , with probability at least 1 -δ ,

where

Proof. We have:

The inequality (i) exists because of the Cauchy-schwartz inequality and Assumption 2. Hence, by Assumption 1, we have

From Assumption 3, we have

Using Chebyshev's inequality, we have

which means that with probability higher than 1 -δ ,

The last inequality is because we can set δ such that δnI < 1 . Plug into eq. (10), we have with probability higher than 1 -δ , we obtain eq. (8) and the lemma follows.

Denote the empirical CDF: ̂ F n +1 ( x ) = 1 n ∑ T i = T -n +1 1 ˆ s i ≤ x , ˜ F n +1 ( x ) = 1 n ∑ T i = T -n +1 1 s i ≤ x and true CDF of score function F s ( x ) = P ( s ≤ x ) .

Lemma 2. Under Assumption 3, for any n , there exists an event A n which occurs with probability at least 1 -√ log(16 n ) n , such that, conditioning on A n ,

Proof. The proof follows Lemma 1 in [51] that utilizes Dvoretzky-Kiefer-Wolfowitz inequality in [24].

Lemma 3. Under Assumption 1, Assumption 3,with high probability,

Proof. The proof is similar to Lemma B. 6 in [50], and is written here for completeness.

Using Lemma 1 we have that with probability 1 -δ ,

Let S = { t : | s t -ˆ s t | ≥ √ ω } . Then,

So | S | ≤ n √ ω . Then,

where ( i ) is because | 1 { a ≤ x } -1 { b ≤ x }| ≤ 1 {| b -x | ≤ | a -b |} for a, b ∈ R , and ( ii ) is due to the Lipschitz continuity of F s ( x ) .

Proof of Theorem 1

Proof. Look at the conditional coverage of Y T +1 given X T +1 :

Based on Lemma 2, we can define the event A n , P ( A n ) ≥ 1 -log(16 n ) n , conditional on A n , we have:

Hence, we can write eq. (18) as

Conditional on A n :

The last equation exists because of Lemma 3 1 and eq. (19).

Note that F s ( s T +1 ) ∼ Unif (0 , 1) , we have

Plug into eq. (18), we have

A.2 Proof of Theorem 2

Theorem 2 (Efficiency). There exists 0 < α 0 < 1 , such that when α < α 0 , the optimal solution to the minimization problem is given by:

where ϵ ∼ N (0 , A -1 ∗ ) .

Lemma 4 (Uniform tail threshold via Dirichlet-Jensen) . Let S λ = ∑ I i =1 λ i Z 2 i with Z i i.i.d ∼ N (0 , 1) and λ ∈ ∆ I -1 . Then S λ d = T U λ with T ∼ χ 2 I independent of U λ = ∑ i λ i V i , where V ∼ Dirichlet( 1 2 , . . . , 1 2 ) . For all λ we have 0 < U λ ≤ 1 and E [ U λ ] = 1 /I . Moreover, for s ≥ I +2 , the map u ↦→ ¯ F I ( s/u ) is convex on (0 , 1] (where ¯ F I is the χ 2 I survival function). Consequently,

Equivalently, with p ∗ ( I ) := F χ 2 I ( I ( I +2) ) ∈ (0 , 1) , we have

Proof. The Dirichletχ 2 factorization and E [ U λ ] = 1 /I are standard. Write g s ( u ) := ¯ F I ( s/u ) . Using χ 2 I density f I ( x ) ∝ x I/ 2 -1 e -x/ 2 , one computes g ′′ s ( u ) ≥ 0 iff f ′ I ( x ) ≤ -2 x f I ( x ) at x = s/u . Since f ′ I /f I = ( I 2 -1) 1 x -1 2 , the inequality holds whenever x ≥ I + 2 . If s ≥ I + 2 then x = s/u ≥ s ≥ I + 2 for all u ∈ (0 , 1] , hence g s is convex on (0 , 1] . Apply Jensen and note S λ ⋆ d = 1 I χ 2 I to conclude.

Proof of Theorem 2

Proof. By the definition of ellipsoid volume, we have

Since the optimization problem is invariant to the rescaling of the scalar (that is, A and any positive scalar multiple cA , where c > 0 , produce the same mathematical solution), additional constraints, such as the bounding of the matrix norm A ≤ 1 , can be imposed without loss of generality. Note that ϵ ∼ N (0 , A -1 ∗ ) and A ≻ 0 by Cholesky decomposition, we can write A -1 ∗ = LL ⊤ where L is a lower triangular matrix. Define the matrix B = L ⊤ AL , we can rewrite the eq. (25) as

where x ∼ N (0 , Id ) and det( L ) is a constant independent of B .

To further solve the optimization problem, we look at the eigenvalue of B , suppose B = O diag ( λ 1 , λ 2 , · · · , λ I ) O ⊤ and O is an orthogonal matrix. Since the optimization value is invariant to different scaling of B , we are imposing additional constrains on λ i .

and { x i } 1 ≤ i ≤ I are i.i.d. random variables x i ∼ N (0 , 1) .

Now we would like to prove that the above optimization problem is solved when λ ∗ = ( 1 I , · · · , 1 I ) . Note that by Cauchy-Schwartz Inequality ∏ I i =1 λ -1 2 i is minimized when λ 1 = λ 2 · · · = λ d = 1 I , by Lemma 4, Q 1 -α ( ∑ λ i x 2 i ) is minimized in λ ∗ , for α < α 0 and α 0 is decided according to I . Theorem 2 follows.

A.3 Tail-up model

Lemma 5. The spatial covariance Σ of any node pair ( u, v ) is:

Proof. Note that

Due to the independence of increments for Brownian motion, only when r = s ∈ ∧ u ∩ ∧ v , the covariance is non-zero. Note that for Brownian motion, we have Cov ( dB ( r ) , dB ( s )) = 1 r = s drds . Hence Lemma 5 follows.

Lemma 6. If we set the moving function m ( r -u ) = β exp ( -d ( r,u ) ϕ ) , with parameters β > 0 (a scale factor) and ϕ > 0 (a range or decay parameter), then the covariance matrix between two locations u, v can be expressed as:

Proof. By Lemma 5 and substitute m ( r -u ) = β e -d ( r,u ) /ϕ and m ( r -v ) = β e -d ( r,v ) /ϕ , we have

The set ∧ u ∩ ∧ v are the segments of networks that flow into both u and v . Consider the following cases:

If u and v are not flow-connected , then ∧ u ∩ ∧ v = ∅ and hence Σ( u, v ) = 0 .

If u and v are flow-connected , without loss of generality assume v is downstream of u . Then ∧ u ∩ ∧ v = ∧ u , and for each r in ∧ u , d ( r, v ) = d ( r, u ) + d ( u, v ) . Hence we have

leads to a remaining integral over r ∈ ∧ u . We can write

1

Topology

7

3

5

2

Multi-

Spatial

6

Temporal

Time

variate

Dependency

Shift

Series

[46] 3. [10] 4. [30]

[15, 53, 51, 1, 52]

[50] 7. Our method, STACI

Note that ∫ ∧ u exp ( -2 d ( r,u ) ϕ ) w ( r ) w ( u ) dr = Σ( u, u ) is a constant, since the additivity constraint on w ( u ) assures the constant variance of site u . Thus, the tail-up exponential model yields a covariance of the form

B Taxonomy for Related Works

Figure 4 provides an overview of the conformal prediction (CP) literature, complementing the discussion in the related work section. The Venn diagram categorizes existing CP methods based on the type of data they are designed for-time series, multivariate data, or both-and the nature of the assumptions they make, particularly regarding exchangeability, temporal stationarity, and spatial independence.

Traditional CP methods, such as split conformal prediction for i.i.d. regression[46], assume full exchangeability, and thus cannot be directly applied to time series or spatially structured data without modification. In the time series domain, recent works such as [1, 53, 15, 52] relax the exchangeability assumption via online calibration, sliding window methods, scorecasters or dynamic programming. These methods handle distribution shift over time but typically operate in univariate or low-dimensional settings. In contrast, CP methods for multivariate or high-dimensional data [30, 10] often focus on constructing prediction sets that exploit geometry or sparsity, but generally assume no temporal structure.

Our proposed method, STACI , is positioned at the intersection of these axes in the Venn diagram. It is designed for high-dimensional time-indexed multivariate data arising from spatio-temporal stream networks. Our method explicitly accounts for both spatial dependencies and temporal shifts, leveraging the underlying topological structure of the network to enhance predictive performance.

C Adaptive Uncertainty Set Construction

We present the analysis of the average coverage guarantee of STACI without any assumption about ϵ t . The proof follows from Proposition 4.1 in [15].

Algorithm 1 STACI

Input: Data D ; Network topology G ; Model f ( · ) ; Hyper-parameters λ ; Confidence level α . Output: Prediction set C ( X T +1 ; α ) .

1: // Training

2: ˆ f ← Fit f using D\D cal ;

6:

ˆ Σ

G

Compute (29) for

(

ℓ

i

, ℓ

9: ˆ Q 1 -α ← Compute ⌈ (1 -α )( n +1) ⌉ n -th largest element of S ;

11: C ( X T +1 ; α ) ←{ y : s ( X T +1 , y ) ≤ ˆ Q 1 -α } ;

Proposition 1. Consider n ′ test data points as the n ′ realizations of ( X T +1 , Y T +1 ) , denoted by D test . We have the asymptotic coverage guarantee:

While Proposition 1 provides a weaker coverage guarantee compared to Theorem 1, it offers broader applicability, remaining valid even in adversarial online settings. Empirical results suggest that when the error process exhibits minimal distribution shift and the assumptions of Theorems 2 and 1 are only slightly violated, STACI maintains the predefined coverage level ( γ = 0 . 01 ) while achieving efficient prediction sets. However, when γ > 0 , Proposition 1 does not ensure a finite-sample coverage gap. Understanding this limitation and developing methods to control the finite-sample coverage gap presents an interesting direction for future research.

D Additional Experiment Details and Results

D.1 Computational Resource

Experiments were conducted on a single NVIDIA GeForce RTX 4080 Super GPU, an AMD Ryzen 9 7950X 16-Core Processor CPU, 64GB Memory and 2TB SSD.

D.2 Datasets and Codes

The PeMS03 dataset used in this paper is collectd by California Transportation Agencies (CalTrans) Performance Measurement System (PeMS). It contains three months of statistics on traffic flow every 5 minutes ranging from Sept. 1st 2018 to Nov. 30th 2018, including 358 sensors. The datasets are from https://github.com/guoshnBJTU/ASTGNN/tree/main without available license.

The first backbone ST-Graph model is AGCRN [2], which we adapted the official implementation from https://github.com/LeiBAI/AGCRN under MIT license. ASTGCN [16] was implemented with PyTorch Geometric Temporal package [34] from https://github.com/ benedekrozemberczki/pytorch_geometric_temporal under MIT license. STGODE [13] was adapted from the official implementation https://github.com/square-coder/STGODE under Apache-2.0 license.

D.3 Simulation Data Generation Details

Segment r 1 and r 2 starts with (0, 1) and (0.5, 0.8), respectively, and both end with (0.3, 0.5). The next segment r 3 also starts with (0.3, 0.5), and end with (0.2, 0.1). Segment r 4 start from (0.6, 0.6), and ends at the same location as r 3 . Starting from this location, r 5 ends at (0.4, 0). The weights for

i

′

)

,

∀

′

i, i

∈ I

given

G

;

segment 1 -5 are set as 0 . 35 , 0 . 5 , 0 . 85 , 0 . 15 and 1 , respectively. Each segment has two observation locations - one at the start point, another at the middle point.

To approximate the integral, each segment is uniformly divided into 300 smaller sub-intervals. For segments without parent nodes ( r 1 , r 2 and r 4 in our example), the source nodes are treated as infinitely distant. In implementation, the source node of each segment is extended 10 times in the same direction to simulate infinity.

D.4 Real-world Data Details

As shown in Figure 5, we construct two subgraphs from PeMS03 dataset. We construct PEMS03-G1 and PEMS03-G2 with temporal distribution shift shown in Table 2. Note that road network structure are required in stream networks. Among all PeMS datasets, PeMS03 is the only one that contains a mapping to real sensor identification number in PeMS. Therefore, other datasets are not available for our setting.

D.5 Main Experiment Continued

We evaluated two additional baselines, CopulaTSCP and DeepSTUQ, on the PeMS-G1 dataset, with the results summarized in the Table 3. Using the same backbone model AGCRN, CopulaTSCP significantly underperforms our method and even general CP methods. This is because that learning the joint cumulative distribution function, a requirement for CopulaTSCP, is highly unstable and unreliable in high-dimensional settings like our 12-dimensional joint prediction task. DeepSTUQ, a Bayesian GNN method, provided better efficiency but still failed to outperform STACI. Since DeepSTUQ produces marginal intervals, we aggregated them into a hyper-rectangle for a joint comparison. However, this aggregation lacks a formal guarantee of joint coverage, and applying a formal correction like Bonferroni would result in overly conservative and inefficient prediction sets. These results underscore the limitations of existing methods in high-dimensional joint prediction and highlight the need for specialized approaches.

D.6 Robustness under imperfect topological information

To evaluate our model's robustness against graph structure noise, we conducted an experiment by perturbing the graphs of the PeMS dataset. A "noisy" graph was constructed by adding 12 spurious edges: for each of the 12 nodes in the original graph, we randomly selected another node and added an edge between them. As shown in the Table 4, the performance of all GNN-based methods degraded when using the noisy graph. Notably, our method, STACI, demonstrated the strongest robustness to these perturbations. It experienced the least performance degradation among all models, showcasing its resilience to imperfect graph structures.

D.7 Running Time

On a single Nvidia Geforce 4080S, the training time of AGCRN with PEMS-G1 is 1419 s . In table 5, we show the computation time of our and baseline CP methods on AGCRN model and PEMS-G1 dataset. Our method STACI has comparable running time with other methods. Here we provide a detailed complexity analysis. Compare to MultiDimSPCI, for each test time point we

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involved. However, MultiDimSPCI also needs matrix inverse for Mahalanobis distance calculation. Additionally, estimation of only two parameters can converge fast. Therefore, our method does not need significantly more time than the baseline method, consistent to the Table 5. STACI provides fast, reliable, and interpretable joint UQ over localized subgraphs across long time horizons.

D.8 Method Details

We provide pseudo-codes for our proposed method in 1. This applies all experiments in this paper, excluding offline setting detailed in Section D.10.

D.9 Hyperparameter Study

The studies for hyperparameter are presented in Figure 6.

D.10 Additional Ablation Study: Offline Experiment

In both synthetic and real-world data, MultiDimSPCI achieves the closest to our proposed method, STACI , in efficiency. Therefore, we focus our comparison on four specific variants: vanilla MultiDimSPCI( γ = 0 ), MultiDimSPCI( γ = 0 . 01 ), STACI ( γ = 0 ), and STACI ( γ = 0 . 01 ). In the offline setting, STACI does not update the covariance matrix estimation. To ensure a fair comparison, we similarly fix the covariance matrix for MultiDimSPCI methods at the beginning of the test phase.

The results are illustrated in Figure 7. As seen in the left figure, fixing the covariance matrix significantly improves the coverage rates of all methods, bringing them close to the desired 95% level. However, despite having the same γ , STACI consistently outperforms MultiDimSPCI in efficiency. Notably, when ACI is not applied ( γ = 0 ), both methods tend to be overly conservative, resulting in coverage rates well above the desired 95%. Therefore, since STACI ( γ = 0 ) achieves a higher coverage rate, MultiDimSPCI ( γ = 0 . 01 ) and STACI ( γ = 0 ) exhibit similar efficiency.

In conclusion, regardless of whether the covariance matrix is fixed or not, STACI consistently surpasses MultiDimSPCI in both coverage and efficiency. Furthermore, to achieve an exact coverage rate, incorporating ACI ( γ = 0 . 01 ) is recommended.

(1a)-(1d)

: AGCRN on PeMS-G1

(2a)-(2d)

: ASTGCN on PeMS-G1

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: STGODE on PeMS-G1

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: AGCRN on PeMS-G2

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: STGODE on PeMS-G2

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EfficiencyofOfflineMethods

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a Conformal Prediction model for spatiotemporal data generated from sensors monitoring locations in a stream network. The idea is to use the structure of the network to regularize the learning of the (spatial) covariance structure of the error distribution. The model design is well-motivated, and I believe the paper will be of interest to researchers working in the area. However, the discussion of the method's limitations and related work should be improved in several dimensions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The idea to use a tail-up model to regularize the estimation of elipsoidal conformal sets is, to the best of my knowledge, novel, and the proposed design makes sense.\n- The theoretical analysis is interesting, although the authors should be more open in discussing the strong assumptions being made.\n- Decent empirical results compared to a quite limited selection of baselines.\n\nMajor issues:\n\n- **Discussion of limitations must be improved.** There are many aspects of the work that should have been discussed more, including some limitations.\n    - The theoretical analysis in section 5 assumes both a specific system model (eq. 1) and iid noise. These are very strong assumptions (much stronger than stationarity alone) and these should be acknowledged and discussed at length. I'd suggest adding a Limitations section to the paper.\n    - The volume of the interval is not locally adaptive (does not depend on X_t). This is different than MultidimSPCI [1], where the quantile is predicted using a model conditioned on the last observations. Indeed, the authors seem to suggest that MultidimSPCI simply consists in using the covariance matrix estimated from data, but that is not the case (see comment on related work).\n    - The framework includes basically no components that account for serial correlation. If the authors believe that such dependencies can be modeled by integrating elements from other approaches (e.g., [1]), this should be explicitly discussed, and a comparison with existing methods that do address serial correlations should be provided. The theoretical analysis itself assumes no serial correlation in the noise.\n    - There's little discussion of computational scalability issues. Data coming from sensor networks can have hundreds or thousands of nodes. This issue should be, again, discussed in the paper.\n- **Discussion on related work could be improved.** Discussion of related work could be improved in different dimensions.\n    - The paper is constantly misrepresenting [1] as simply relying on estimating the covariance matrix from data, but on top of that [1] adds a local term to the covariance estimate (inspired by [2]) and estimates the quantile level at each time step to exploit serial correlation by using a regressor (potentially accounting for nonstationarity, differently from what the paper claims in line 113). Furthermore, similarly to the proposed method, MultidimSPCI uses the approach introduced in [2] to regularize the covariance matrix by using local errors in the last $k$ time steps.\n    - The paper should clarify what portion of the theoretical results is simply derived from previous work (e.g., [1]).\n    - The author should compare to the actual MultidimSPCI (or at least elipsoidal sets accounting for local serial correlations) or explain why such a comparison is not appropriate.\n    - The idea of regularizing the covariance matrix for ellipsoidal confidence sets in CP has been introduced in [2]. This should be acknowledged and discussed. I'm a bit puzzled by the fact that the paper does not discuss this related work.\n- **Somewhat limited selection of baselines.** The selection of baselines is quite limited (see comments on related work).\n\nAdditional comments:\n\n- It looks like [41] is referenced in place of [40] in many cases (e.g., lines 193 and 241).\n- The paper claims to be the first approach to use a graph structure for CP for spatiotemporal observations, but prior work [3] does this in the general setting of correlated time series and spatiotemporal data. [3] came out in February, so it's quite close to the threshold for being considered concurrent work. I don't think this paper should be penalized for missing this, plus the two approaches are very different. However, I do believe that [3] should be discussed and some of the novelty claims adjusted accordingly. \n- Looking at the plot, I wouldn't say that STACI is  \"insensitive to \\lambda\". Maybe I'd say that is robust to different configurations.\n\n[1] Xu et al. , \"Conformal prediction for multi-dimensional time series by ellipsoidal sets\" ICML 2024\\\n[2] Messoudi et al., \"Ellipsoidal conformal inference for multi-target regression\" Conformal and Probabilistic Prediction with Applications 2022\\\n[3] Cini et al., \"Relational Conformal Prediction for Correlated Time Series\" ICML 2025\n\n---\n\nThe rebuttal addressed most of my concerns. However, although I believe the paper is interesting, I feel that resolving the issues identified in the rebuttal will require substantial revisions, which may require another round of peer review.\n\nQUESTIONS:\nPlease comment on the issues mentioned above.\n\nLIMITATIONS:\nSee above.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces STACI, a conformal prediction framework designed for stream networks, which are spatiotemporal graphs with flow directionality and complex dependency structures (e.g., river or traffic systems). STACI integrates both topological constraints and temporal dynamics using a weighted covariance-based Mahalanobis distance for nonconformity scoring, and adopts dynamic adjustment via Adaptive Conformal Inference (ACI). It is supported by theoretical guarantees and experiments on synthetic and real-world data.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength: \n-\tThe problem is well-motivated, as conformal prediction for stream networks is understudied.\n-\tIt seems a novel idea that the work introduces a hybrid nonconformity score incorporating both topology-induced covariance and data-driven estimates.\n-\tValidity and efficiency of the proposed method are rigorously analyzed, along with tail-up modeling and dynamic distribution shift.\n-\tThe empirical performance shows improvements in prediction set tightness and valid coverage vs. baselines.\n\n## Weakness:\n-\tThe need to estimate both sample and topology-induced covariance may limit the applicability to high-dimensional or large-scale networks. Model complexity analysis is missing.\n-\tThe empirical experiments are limited. For example: ACI is introduced but its empirical benefits are under-discussed.\n-\tThe set of CP baselines (those handling structured outputs or graph-structured data) could be expanded. e.g. [1]\n-\tReal-world results are useful, but broader evaluations across different stream domains would improve generalizability claims.\n-\tAblation study on $\\lambda$ (weight between topology and data), especially analysis on how performance varies with poor topology estimation,would be helpful.\n-\tSensitivity analyses for tail-up model hyperparameters (e.g., $\\phi$) are missing.\n\n\n## References:\n\n[1] Schlembach, F., Smirnov, E., Koprinska, I., & Winands, M. H. M. (2025). Conformal multistep‑ahead multivariate time‑series forecasting. Machine Learning, 114(7), Article 165.\n\nQUESTIONS:\n-\tFor equation 2, why is the definition of $\\hat{\\epsilon}_t$ different from the one in reference [2] mentioned in the paper?\n-\tWhat are the sizes of the two datasets(PeMS-G1 and PeMS-G2)? How big does the temporal distribution shift exist in the dataset?\n-\tHow scalable is STACI to very large graphs with hundreds or thousands of nodes and variable flow topology? [3]\n-\tHow robust is performance when the assumed topology does not fully reflect true dependencies (e.g., due to missing links or misaligned directionality)?\n-\tHow does STACI perform in non-flow-based graphs (e.g., social diffusion), and is the framework still appropriate?\n-\tExpand the set of baseline methods, such as GNN-based uncertainty models or graph-based deep ensembles, would be more informative.\n-\tClarify the real-world dataset types (e.g., traffic vs. hydrology) and the topological structure used in each.\n\nI'm open to increase the score if the questions are addressed reasonably.\n\n\n## References:\n\n[2] Katsios K, Papadopoulos H. Multi-label conformal prediction with a mahalanobis distance nonconformity measure[J]. Proceedings of Machine Learning Research, 2024, 230: 1-14.\n\n[3] Liu X, Xia Y, Liang Y, et al. Largest: A benchmark dataset for large-scale traffic forecasting[J]. Advances in Neural Information Processing Systems, 2023, 36: 75354-75371.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses uncertainty quantification on stream networks by extending the conformal prediction (CP) framework with a topology-aware Mahalanobis non-conformity score and the adaptive conformal inference (ACI) mechanism to handle temporal distribution shifts. Treating the problem as a multivariate time series one, and encoding spatial relationship directly in the covariance matrix estimation, the proposed method constructs prediction sets according to the graph structure while adapting to temporal distribution shifts. Authors further prove that these extensions preserve CP’s finite-sample coverage guarantees and empirically show the improvement in efficiency of the prediction sets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The manuscript is coherently structured, leading the reader smoothly from the problem statement through method, theory, and experiments.\n- Embedding graph topology within the Mahalanobis-based non-conformity score is a well-motivated advance that extends recent graph-aware conformal prediction research as well. Applying this idea to multivariate time-series data on stream networks appears novel. Especially since the few prior works in this area overlooks the graph topology.\n\nWeaknesses:\n\n- The lack of clarity is my main concern at the moment: several key definitions, symbols, and implementation choices are missing or only briefly sketched (see Questions), which makes the core ideas hard to follow. Mathematical notation is also inconsistent, and some symbols, e.g. the $\\gamma$ mentioned in the real-data study, are never formally introduced in the main manuscript. A thorough pass to standardise notation and supply the missing details is needed to improve readability and reproducibility.\n    - Adaptive Conformal Inference (ACI) is central to the method, yet it is introduced only in Appendix C. Adding even just a brief description of its mechanics in the main text (around line 222) would help the reader follow the subsequent Coverage Validity subsection. Appendix C itself would also benefit from a notational cleanup and removal of redundancies (line 169 and line 176 both mention the “adversarial online setting”; line 172 and line 179 both mention that “Proposition 1 does not guarantee a finite coverage gap”).\n- The experimental scope is a bit narrow.\n    - Synthetic benchmarks are usually designed to embed a specific properties , e.g. type of uncertainty or distribution shift so that a new method’s advantages can be demonstrated transparently. Here, only the generation process is sketched in the appendix and it is unclear what aspect of uncertainty is being tested. A more explicit description of the synthetic design along with an explanation of how it isolates the advantages of the topology-aware score would make the results on the synthetic dataset far more convincing.\n    - For the real-world data, both PeMS-G1 and PeMS-G2 are subsets of the same highway-traffic dataset, so the empirical scope is effectively a single domain.  Moreover, the related work section cites three CP methods for stream networks [1,2,3], yet the comparison in Table 1 includes only one of them [3]. Expanding to additional datasets or adding the omitted baselines would strengthen the evidence.\n    - Lines 375–377 claim “significantly higher coverage and superior efficiency” but Table 1 shows STACI’s coverage exceeds the best baseline in only one cell (ASTGCN, PeMS-G1) and by just $0.12\\%$. I think more evidence is needed to better support the claim.\n    - Including a baseline that applies ACI with the vanilla sample covariance matrix would isolate the incremental benefit of the topology-based estimator. Without it, one cannot tell whether gains arise from ACI, from the topology-aware covariance, or from their combination.\n    - Details such as the number of nodes, edges, and time steps are absent, making it difficult to judge whether the method scales to large real-world stream networks.\n\n[1] Sun et al., Copula conformal prediction for multi-step time series forecasting\n\n[2] Messoudi et al., Copula-based conformal prediction for multi-target regression\n\n[3] Xu and Jiang et al., Conformal prediction for multi-dimensional time series by ellipsoidal sets\n\nQUESTIONS:\n1. I found the problem definition (section 3) a bit difficult to digest. It would definitively benefit from using a more consistent notation and clarifying the relationships among locations, observation sites, and stream segments (maybe with a real-world analogy?).\n\n    a. Lines 125-126 introduce both $\\ell_i$ and $r_j$ as subsets of $\\mathcal{L}$, yet Figure 1 suggests that each segment $r_j$ contains several geographical location sites $\\ell_i$. Could author clarify distinction between these two objects?\n\n    b. Segments across the section, are referenced using different notations, e.g. $\\ell_i$, $r_j$, $u$, and $s_i$. Is there any actual difference between those symbols or are they referencing the same thing? In case, it would be better to state the difference in the description otherwise using a consistent notation would improve the readability.\n2. Equation (5) introduces the tail-up scaling parameters $\\phi$ and $\\sigma^2$ described only as “estimated by the tail-up model”. Are these terms learned jointly with the model’s weights? Could you also expand more on the purpose of these two terms?\n3. In Appendix C, $\\gamma$ is introduced in equation (27) as the step-size parameter of the ACI update. However, the paragraph following Proposition 1 (lines 166 – 180) states that “[…] STACI maintains the predefined coverage level ($\\gamma$ = 0.01)” (line 171). Should this reference be to $\\alpha$ instead of $\\gamma$? If not, please clarify the distinction and ensure the notation is used consistently throughout the section.\n4. In appendix C, the paragraph following Proposition 1 (lines 168–169) claims that the asymptotic guarantee “remains valid even in adversarial online settings”, yet no theoretical argument or empirical evidence is provided. Could the authors substantiate this claim? It would be helpful to better describe the mentioned adversarial online setting.\n5. Similarly, in line 277, authors mention that the topology-based matrix “generally achieves faster convergence than the sample covariance estimator”. Could the authors provide supporting evidence for this claim?\n6. Which calibration set size $n$ was used to generate the results shown in Table 1?\n7. Which backbone model was used to generate the curves in Figure 3, and do the same coverage-efficiency trends hold for different backbones (e.g., ASTGCN or STGODE)?\n8. The conclusion proposes extending STACI to “general spatio-temporal graphs” and exploring “alternative network parameterizations,” but these notions are still somewhat vague. Could the authors elaborate on what they regard as a *general spatio-temporal graph* and give a concrete example of a *network parameterization* (similar to how the second future work suggestion is presented)?\n\nTypos:\n\n- Line 217, “Intuitively, The” -> the\n- Line 330, details of synthetic network are in D.3\n- Line 367, no Figure 5 in the appendix, I think authors refer to Figure 2\n- Line 389 Appendix D.8\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper introduces a novel graph-specific conformal prediction framework that accounts for the topological structure of graphs when calibrating prediction sets for GNN. Unlike standard CP which assumes exchangeability of data samples, TopoCP adapts the calibration strategy by modeling inter-graph correlations via graph kernels. Empirical studies show it maintains valid coverage and achieves tighter prediction sets across multiple graph classification benchmarks compared to existing conformal and Bayesian uncertainty methods

STRENGTHS AND WEAKNESSES:
Strengths: 

1. The CP method for graphs explicitly incorporates topological similarity using graph kernels in the calibration process. The approach is well-motivated.
2. The framework extends its validity guarantees to graphs under topological dependence, with detailed theoretical proofs and assumptions.
3. Empirical evaluation demonstrate the effectiveness of the approach. 

Weakness: 
1. The usage of graph kernel matrices introduces O(n^2) memory and compute complexity, limiting scalability to large graph. 
2. The framework depends on the choice of graph kernel to compute influence scores. It is unclear how to select or learn the best kernel for a given task.
3. The experiments lack ablation on the major components of the framework.

QUESTIONS:
Please refer to weakness part.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-06