Topology-Aware Conformal Prediction for Stream Networks
Authors: Jifan Zhang
Fangxin Wang
Zihe Song
Philip S. Yu
Kaize Ding
Shixiang Zhu
Venue: NeurIPS 2025 poster ()
Keywords: conformal prediction
stream networks
spatiotemporal graph
uncertainty quantification
Primary area: general_machine_learning
TL;DR: We propose a conformalized framework for constructing prediction sets in stream networks, a specialized class of spatiotemporal graphs.

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
Jifan Zhang
Fangxin Wang University of Illinois Chicago Chicago, IL 60607 fwang51 @uic.edu
Northwestern University
Evanston, IL 60208
ji fttaahhang2266@uunorrrvveeer
Zihe Song
Philip Yu
Kaize Ding
University of Illinois Chicago Chicago, IL 60607 zsong29 @ uic.edu
University of Illinois Chicago Chicago, IL 60607 psyu@uic.cod
Northwestern University Evanston, IL 60208 kaize.dinge @ notthwestttwweddd
Shixiang Zhu
Carnegie Mellon University Pittsburgh, PA 15213 Thixiangzhu@cmu..
Abstract
Stream networks, a unique class of spatiotemporal graphs, exhibit complex direc- tional flow constraints and evolving dependencies, making uncertainty quantifi- cation a critical yet challenging task. Traditional conformal prediction methods struggle in this setting due the need for joint predictions across multiple interde- pendent locations and the intricate mentioltmmporl dependencies inherent in stream networks. Existing approaches either neglect dependencies, leading overly con- servative predictions, or rely solely on data-driven estimations, failing to capture the rich topological structure of the network. To address these challenges, we propose Opatio-Tempone Adaptive Conformal Inference (STACI), novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. STACI introduces appology-wware 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 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 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 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.
"quual Contribution.
CCo-corespooding Authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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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, srciisionmmkinn 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 data and enabled effective UQ with statistical guarantees [49 [4 [5]. In particular,
conformal prediction (CP) has emerged a powerful UQ framework, providing snite-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-supporr systems 27
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 single target variable, stream networks require joint predictions at multiple locations, where observations are highly interdependent. Applying CP independently each location neglects network wide dependencies, leading to inefficiencies in prediction set construction and potential loss of coverage guarantees. (ti) Intricate 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 relation- ships, 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 [0] or attempt to learn dependencies solely from data without incorporating topological constraints [5] 40). The former results in overly conservative or miscalibrated prediction sets, while the latter risks overfitting to specific network conditions, reducing generalizability.
flow constraints:
To address these challenges, we propose novel framework. Opatio-Tempore Adaptive Conformal In- ference (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 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- covariance and data-driven estimates. A greater reliance on the topology-induced covariance structure improves coverage guarantees, assuming accurately reflects underlying dependencies. Conversely, prioritizing sample-bases estimates mitigates potential misspecifications in the sopology-inducee covariance, often leading to better predictive efficiency. Additionally, we consider a dynamic ad- justment 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 synthetic data with a stationary covariance matrix and real-world data with time-varying covariance, comparing its performance against state-of-the-ar baseline. Both our theoretical and empirical results underscore the importance of the weighting parameter that balances Sata-driven insights with topology-inde -ndduce 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 principled approach that leverages both observational data and inherent network structure.
Our code is publicly available at huthtps:: puttps://gte http://ggghuuuu..mmmegge.eee hotpa///.o.boeeeaiiieaaaan/////s

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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 [4] 28 and environmental
science networks [25], have been extensively studied due their critical role in natural and engineered systems. Forecasting for stream network can be approached from two perspectives: as a graph prediction problem or 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. [8] introduced spatial stream network models for hydrology, emphasizing the importance of flow- connected 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-tempore (ST) graph dynamics 220
While effective and widely adopted, models without uncertainty quantification often lack consid- erations for reliability, posing limitations particularly in safety-critical scenarios. To address this, some studies 449 558 38 48 (33] turn to explore interval prediction, which ensures that prediction intervals cover the ground-truth values with pre-defined high probability, offering 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 the choice of prediction models and pri- ors, particularly the type of probabilistic distributions [48]. To address these limitations, classic drequentist-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 graph-structured data 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. contrast, our work focuses on fundamentally different problem: UQ for
forecasting 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-exchange nulti-dimmnsional time series, but their model neglects the inherent graph structure embedded within the multi-dimennional time series and overlook scenarios where the error process (see eq. (I) is non-stationary. 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 9 with fixed flow direction at time E (1,..., .......7h T}, with observational sites indexed by I= (1..... .. /). Let L. C IR2 denote the set of all geolocations on the network, and let the geographical location of site E I be represented as É, E L. The stream network consists of segments {r> C. C.j E JJ. where J is the index set of all stream segments. Each site e I is located within a specific segment r, for some j E J, and a segment may contain multiple or no observational sites. For any location E L. we define Au as the set of all upstream segments of location u, and Vu as the set of all downstream segments of location u. The hydrologic distance between two locations v, E L, denoted as d(v, u), is the distance measured along the stream. If and u belong to the same segment r;, d(u, u) is simply the Euclidean distance between v and u. See Figure I for an illustration.
Now, consider a multivariate time series observed at the sites. We denote the dataset as D = ((X,, Here Y. = Y((B), ((())) R², and (E,) (or simply
3

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Y7) represents the observation at location & at time t. The historical observations are given by X E / defined as X == [Y-1, 2 b E Rixh. We assume that Y, follows an unknown true model f(x,) with additive noise Et, such that:
Y f(x)) + Ct:
where E, E R' has zero mean and a positive definite covariance matrix Σ > O.
The goal is to construct prediction set for +1+1 given the new history XT+1, denoted by C(XT+1), such that, for a predefined confidence level a, the following coverage guarantee holds:
r1
2
00
P(Y++1 E c((T++))
a.
00
This objective can be achieved using split confor- mal prediction (CP) 46), a widely used frame- work for uncertainty quantification. Split CP operates by first partitioning the data into train- ing set and calibration set. The prediction model f is trained exclusively the training set. To assess the reliability of predictions, a nonconformity score is computed, which quan- tifies the deviation of each calibration sample from the ground truth. Given target confidence level a, the method determines the (1 - a-- quantile of the nonconformity scores from the calibration data. This quantile is then used to adjust f's predictions for test samples, ensuring the constructed prediction sets maintain valid coverage. Under the assumption that the calibration and test data are exchangeable, the prediction sets are guaranteed to achieve coverage rate of least 1 a on the test data.
A42
T3
fs
T4
8
v&a
Eg
10
T5
Figure I: An example of stream network G. The network segments {ri,...,,Te rt} are denoted by blue lines, and the observation points ( (10) are marked with green triangles, pointing to the flow directions. The upstream of location fg are segments accompanied by orange area, and the downstream of location la are blue shaded. The hydrologic distance between &g and lg is calculated through adding lengths of green shaded segments in both 1 and r3--
The challenges of performing multivariate time-series prediction over a stream network are twofold: (i) Mulii-diieensonnaly:: The response variable Y, is multivariate and potentially high-dimensional, significantly increasing the complexity of constructing accurate prediction sets. Standard CP methods, when applied multi-dimensional variables without carefully designed nonconformity score, often produce overly conservative prediction sets. This leads to inefficiencies, as the prediction set size (c(x,) becomes too large to provide meaningful uncertainty quantification. (ti) Non-exchangeabiitty Observational sites exhibit complex spatial and temporal dependencies due to strong correlations imposed by the network topology. As result, traditional CP methods, which rely on exchangeability assumptions, cannot be readily applied.
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 Deai := {X,, Yt). t T 1, ... T 1, 1,,, and define Y; == f(x)), where f is the fitted model trained on the rest of the data D \ Deat- For each calibration data point (X,, Y,) E Deats we compute its nonconformity score, denoted by Yi).
To account for the intricate spatio- aatio-temppral dependencies. we consider general class of nonconfor- mity score functions based on the Mahalanobis distance [22]
s (X,, Yi) E, Ae Vt E Deal:
(2)

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where A is an I x I symmetric positive definite matrix and é : Y, - Y, Eg is the centered prediction error, with & denoting the sample average of enrors on Deat.
The core idea of our method is a linearly weighted representation for A, which integrates both ropoloyy-induces and sample-based covariance estimates. Formally,
(3) Here E,, is the sample covariance matrix computed from the residuals {&,, Deal, and Eg represents the covariance structure induced by the stream network topology. The weighting parameter A E [0, 1] balances these two estimates. A higher value of places greater reliance the topology- driven covariance structure, assuming accurately captures the underlying dependencies. Conversely, lower À shifts reliance toward the sample-based estimate, mitigating potential misspecifications in the topology-induced covariance.
A= (1- (1-)),) + +5
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 more robust covariance estimation, leading to better prediction efficiency without sacrificing coverage validity.
hopology-induced Covariance Estimation
We develop a novel method to estimate the topology- by assuming the observations on the stream network can be The tail-up model is formally defined as follows:
induced covariance En used in eq.
3
captured by a tail-up model 18 43
Definition 1 (Tail-up model). Given a stream network G, the observation at any location u on the network can be modeled as white-noise random process, which is constructed by integrating moving average function over the upstream process, i.e.,
m(v- - u) Y(u) p(l) + / m(v
w(U) dB(r), w(u)
where Au denotes all the segments that are the upstream of u. Here, p(u) is the deterministic mean at u, and m(u u) is moving average function capturing the influence from upstream location to u. Both w(r) 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 [4]. which is highly flexible and can be broadly applied to wide range of stream network data. Also, the choice of the moving average function m(A) remains adaptable, as long as has a finite volume, allowing the model to accommodate different spatial structures effectively.
To estimate Σg. we model (v)) using Brownian motion and adopt an exponential moving average function for m (A) 8 (xp(-AA/)) Therefore. the topology-induced covariance between any two locations u, v can be expressed as follows (See Lemma in Section A of the Appendix):
w(u) w(r)
Ea(u, v) = 0²
v)/6). if U
(5)
where é and 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 / //(u)/w(( 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 spatio-temporally adaptive prediction
set for a new observed history Xr+1 using our proposed nonconformity score, defined in eq. (2) as follows:
C(X++1:a) {v s(X7+1 v) QQ)=))
where Qi-a is the (1 - a)-th quantile of the empirical cumulative distribution function of {s(X,, Yi), E Death.
5

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To account for potential temporal distribution shifts in the predictive error of eq. ① we adopt the Adaptive Conformal Inference (ACI) framework proposed in [[5]. This approach dynamically updates the confidence level over time, ensuring that the prediction set remains responsive to evolving data distributions. Specifically, we iteratively update (e and reconstruct the prediction set C(X,, (x) accordingly. At the initial test time T + 1, the confidence level is set as 0T+1 = a. For subsequent time steps t T + 1, at, we update ar with a step size
2 as follows:
a++1 = at + 7(a 1(Y, & c()); Vt 2 T
(6)
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 wider prediction interval at time t + 1, thereby increasing the likelihood of coverage. A larger step size y makes the method more responsive to observed distribution shifts but also introduces greater fluctuations in ag- When y = O, the method reduces to standard conformal prediction with fixed a. 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:
1. 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: ensures valid conditional coverage under stationary assumptions (Theorem [ and extend the framework to handle non-stationary settings via an ACI adjustment, ensuring approximate average coverage (Proposition in Section C of the Appendix).
We prove that STACI
Our analysis is based on the Mahalanobis distance framework in eq. (2). which enables the construc- tion of arbitrary ellipsoidal uncertainty sets, with greater flexibility for nonconformity scores. For example, standard CP with spherical uncertainty sets arises as 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 big-O notation (-)) characterizes an upper bound on a function's 's growth rate: if f(n) (g(n)), then there exists a positive constant C' such that f(n < Cg(n), for all n 2 no- The little-o notation o(-) denotes strictly smaller asymptotic growth, with f(n) (n)(n) implying lim,->oe (g)) 0. Additionally, we use standard & 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. ( where the errors, c,, are i.i.d. We introduce the following assumption and, for simplicity, denote the nonconformity score e, Ac; as st.
Assumption 1 (Estimation quality). There exists sequence {vr}. n "E=r--41 ||e, Ee1 S v lle+1 éx+ill S Vn.
>
1 such that
Remark 1. The assumption ensures that the prediction error is bounded by u2. For many estimators, the Un vanishes as n - oo, indicating improved estimation accuracy with larger sample sizes (Z).
Assumption 2 (Convergence of An). The sequence (An) associated with the nonconformity score converges to a fixed matrix as n increases, with an upper-bounded convergence rate o(g(n)), i.e.
A| All o(g(n)). Additionally, there exists a conslant r > O such that ||A||| Sr.
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 e is known, can be set as its inverse. Alternatively, if only sample estimates available, A can be chosen as the inverse of the estimated sample covariance matrix of e, provided converges under proper tail behavior conditions (45). The major difference between choices of An lies in the respective convergence rates.
Remark 2.
function Assumption 3 (Regularity conditions for 9t and ex). Assume that the cumulative distribution fustribution,function
(CDF) of the true nonconformity score, F. (x), is Lipschitz continuous with a constant L > 0. Suppose there exist constants K1, K2 such that || S 11 almost surely, and Var[]]ee]??] Kg!...

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Theorem 1 (Validity). Under the assumptions stated above, the proposed method satisfies the following conditional coverage guarantee:
log (16m)
log (16n)
[P(Y++1 E Cr+1(a)|Xr+1 = =x) -1 (1-aa) ( + 22 -V
where
- u 2r v (K) VF2) +(g(n))))) + VR2)I.
The samite-sample bound on the coverage gap is directly influenced by the estimation quality and the convergence rate of Ag, which is given by max(O logn), (vv)), O(vg(n))) In general, reducing the coverage gap requires high-accuracy estimations (i.e., rapidly vanishing v.) and a well-chosen nonconformity score matrix An that converges quickly.
Remark 3.
Theorem Thighlights the importance of incorporating topology-basee 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-base -ased matrix acts a covariance estimator with -opology-infomeed regularization, generally achieving faster convergence than the sample covariance estimator. A hybrid approach that combines both estimators provides 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 ((Arr) n($+1)
1/2 r/// det())) 1/2, The radius of the prediction set determined
by the (1 a- quantile of the empirical CDF, computed from data points in the calibration dataset. This radius is denoted as ((2 A&, ). In the ideal case where f(X.) fX,,) and Eg. minimizing inefficiency reduces to: minA>- V(A, Qi= Ae,, t [m]])). To
simplify computation, we approximate the empirical quantile with the true quantile, justified by
the Glivenko-Cante Theorem [41], which ensures the convergence limm- Q Qi-a (fe, Ae,, E
[n(}) = Q1-a(E Ae), where Q1-a(-), the 1 - a quantile of "e is assumed to be continuous.
In the limiting case, formulate the following minimization problem, presented in Theorem and use its solution as the guiding criterion for selecting the matrix A:
Theorem (Efficiency). There exists 0 < ag < 1, such that when < ae, the optimal solution to the minimization problem is given by:
A. == arg min Q(A,Qi-a(" Ac)), A>
where (O,,A,1)
Remark 4. The optimization problem is invariant to scalar rescaling (i.e., A and any positive scalar multiple cA, where >> yield the same mathematical solution). Empirically, we find oq > 0.2 for Gaussian noises when I S 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 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 é, may deviate significantly from the true errors ez. Additionally, when the sample size n. is small, the empirical CDF may differ considerably from the true CDF. Despite these challenges, constructing A based an estimate of the inverse covariance matrix offers substantial improvements in high- dighh-dimnniinall settings compared to CP methods that ignore variable dependencies, such as those that set as the identity matrix.
6 Experiments
To demonstrate the suitability of STACI, we evaluate its performance both synthetic data with 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 simulation and n'' 5000 in real study. The desired confidence rate is fixed at 0.95. Our method is compared

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10.7, 0.31
0-10.01
0- 103.0.3)
96.0
2.5
STACRy=D.C2)
STD 011
1.0
Coveraee ltt
Coveragel
24
Efficiency
64.5
3,5
64.0
sphors ACT lyed OLI
93.5
2.8
Square
30623
$760
93.0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
(86)
(3)
(a) Comparison STACI with baselines.
(b) Impact of on STACI.
Figure 2: Experiment results of synthetic data: (a) Comparison of methods on synthetic datasets with different tail-up parameters & over coverage (I-axis) and efficiency (y-axis). (b) Tradc-off between coverage and efficiency on synthetic data, where the higher the better performance.
against the following conformal prediction and learning-bais -baii uncertainty quantification baselines: (i) Sphere: Spherical confidence set, where the covariance matrix is an identity matrix. In other words, the prediction error different locations
considered have correlations. (ii) Sphere-ACI (7 = 0.01): Spherical confidence set with adaptive conformal inference (ACT). (iii) Square: Square confidence set. This equals computing different nonconformity scores for each dimension, and then calibrate accordingly.
(iv) GT: Ellipsoidal confidence set using the ground-truth covariance matrix. (v) MultiDimSPCII: Ellipsoidal confidence set using the sample covariance matrix (50), alongside its localized variant using the most recent observations, MultiDimSPCI (local). (wi)
CopulaCPTS: Prediction region based on modeling the joint distribution of forecast errors with copula function [39]. (vit) 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: (1) Coverage quantifies the likelihood that the prediction set includes the true target, ie., Coverage EneD 1{Y, & C(X; (LXX;aa))//i //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, (ol(C(Xe; d.)), is measured by the size of the ellipsoid determined by A. Formally, Efficiency ==
ELEDO (Vol(((X+; 1/1
/n'. An optimal method should achieve the predefined coverage with high efficieney/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 The details of synthetic network is provided in Section D. of the Appendix. We generate the observation of site at time point t by simulating stochastic integration from all upstream points Au to downstream point u according to eq. (4). where we set Me (u) Ei=1 (u) following the AR( (w) structure and m A) exp(- - A) 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 linear regression model. We first estimate parameters of in AR(w) structure, i.c., = (0;)se(w); through lincar regression and then parameters in eq. 22 & and o² through Ellloss. Parameters of θ = (0, 0) and + = (0.7, 0.3) are selected for data generation. When θ (0, 0), the observations consist of pure noise, thus stationary; when = (0.7, 0.3), the process resembles a second-order autoregressive 0.66
model. The weighting factor À is
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 [(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 spherical or square uncertainty sets. Although MultiDimSPCI achieves the lowest efficiency, coverage drops significantly below the required threshold, high- lighting its instability when relying solely the sample covariance matrix. This issue persists even
8

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Table 1: Comparison of Different CP Methods over Different GNN Models for two PeMS Datasets. Coverage below 94.5% italicized. Inefficiency is reported as mean + standard deviation, with the lowest value in bold.
Backbone Models
AGCRN
ASTGCN
STGODE
Dataset
CP Methods
Coverage
Effeciency L
Covorage
Efficioncy
Coverage
Efficiency 1
Sphere
97.76
133.60
96.64
128.23
14.2
96.08
145.40
Sphere-ACT E 0.01)
95.26
108.93
15.96
95.24
114.64 + 19.54
95.03
136.36
19:47
PeMS-GI -I
MuCr
Square
95.98
92.92
155.84
T3.82
23.92
8.16
96.41
93.19
155.62
74.68 +: 7.67
23.74
96.38
92.70
172.30
18
23.51
4.26
MuliDimSPCI (locul)
93.01
74.23 = 8.27
93.57
75.18 7.09
92.98
88 99
4.26
STACI
97.12
88
88.1 19.07
77.34
STACI ( 0.01)
95.75
67.54 + 10.94
95.54
67.92
10.22
95.14
73.62 :: 9.83
Sphere
95.64
130.69
95.83
147.13
Sphere-ACI Cy 0.01)
95.03
133.63 + 14.18
96.07
122.72 + 16.67
95.14
122.34 + 19.72
PeMS-G2
MPCI
Square
95.26
91.14
172.18
101.42
19.43
7.71
95.37
90.93
180.60
W2.12 1: 6.67
19.95
95.12
90.74
138.97
107.23
17.78
7.34
MulriDimSPC7 flocal)
91:141
102,
7.90
91.12
92.81 + 6.95
91.10
107. .96 7 7.58
STACI()
0)
95.39
73.36
10:35
98.18
95.33
77.14 8.65
STACK = 0.01)
95.01
69.62
+ 12.85
95.05
58.07 + 9.83
93.21
74,86 9.63
in simulated data designed to align with its error process assumptions. Its local variant, designed to capture temporal correlations, provides only marginal improvement and similarly fails to achieve the required coverage. In contrast, with 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(? = 0.01), demonstrates the best efficiency with the smallest variance, further reinforeing its robustness and effectiveness.
Figure 2(b) reveals a clear trade-off trend between coverage and efficiency: the higher x, the confidence level rises, but efficiency is worse. This suggests that À should be carefully chosen: if too large, our method over-relies on topology and fails to adapt to covariance shift; if too small, depends more sample covariance matrices, which are purely data-driven and thus unstable, leading to coverage drop. Nonetheless, no matter whether adapting confidence level, setting a larger À 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-GI 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 attention- based ASTGCN [66 and continuous-time STGODE [3]. We set our default 0.6. For simplicity, only use fixed weights with all equal values, without requiring any additional information. Multi- ple hyperparameter and ablation study are also provided over the key parameters in our framework: (i) À from 0 to with step of 0.02; (ii) 100, 200, 300, 400, 500; (iid) y 0 or 0.01.
Result Table [ clearly demonstrates that our method consistently outperforms all bascline 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 Figure setting À = 0 reduces our algorithm to sole usage of sample covariance matrix, equivalent to our strongest baseline, MultiDimSPPI. Our coverage is greater than MultiDimSPCI with arbitrary hyperparameter X. Further, across all calibration sample sizes n,, selecting any À E [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 's insensitivity to A.
In the first two subplots of

---

97
Coverage 0.00
Effieenc
Coverage Rate V
Effilency y
95-
94-
Cowneey
02
02
a-300
100
80
100
0-220
0-226
88
0-400
110
86
110
86
0.2
120
0.6
86
0.2
0.6
130
0.2
0.6
Figure 3: Comparison of Coverage and Efficiency for PeMS data with different belief weight À and calibration set size n, with adaptive step size ? 0.01 (upper) and 0 (lower). The pre-determined coverage threshold of 95% is shown by horizontal gray dotted line.
The incorporation of topology-induce
covariance matrix is pivotal even without ACI (? = 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 robust remedy to under-coverage problem while maintaining highly informative predictions. Moreover, as demonstrated Section D.4 in the Appendix, STACI can also outperform in 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 any single resource. On one hand, with limited finite calibration sample n, the topology-base -baeed 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 ~ 30D). This choice is deliberate: performing UQ in very sigh-dimensional output spaces (hundreds of coordinates or more) is statistically ill ill-posed. Directly constructing geometric prediction sets in the full output space is vulnerable the curse dimensionality 44) particular, hyper-rectangles formed by marginal intervals can inflate to near-vacuous volumes even at low dimensions as in Table
In nigh-dimensional regimes, common practice is to avoid explicit geometric sets and instead use generative or sampling-basee UQ 26 3. For instance, image UQ a prototypical high-dimensions
setting, typically samples plausible outcomes from a learned posterior [12] rather than delineating set in pixel space. In the same spirit, STACI focuses on subgraphs of manageable joint dimensionality so that (i) the resulting joint remain informative, (ii) the coverage-efficiency trade-off is non- degenerate. 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 ~0 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 general spatio-temporal graphs by replacing Σg with alternative network parameterizations, enabling the development of novel methods that effectively exploit topological structures 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 guap- joint sets over hundreds of nodes. One might build corridor-level joint sets and composing them with rigorous controls cors-correlation to scale coverage beyond the subgraph level.
10

---

Acknowledgements
We thank Aravindan Vijayaraghavan, Shuwen Chai, Yifan Wu for helpful discussions. We also thank anonymous reviewers for constructive feedback.
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14

---

Contents
Introduction
2
Related Works
Problem
4
Proposed Framework
4
Theoretical Analysis
5.1
Coverage Validity
5.2
Prediction Efficiency
Experiments
6.1
Simulation
6.2
Real Data Study
Limitation
10
8
Conclusion
10
Theoretical proofs
16
Proof of Theorem
16
A.2 Proof of Theorem
19
A.3 Tail-up model
20
Taxonomy for Related Works
21
Adaptive Uncertainty Set Construction
D Additional Experiment Details and Results
22
D.I Computational Resource
22
D22 Datasets and Codes
22
D.3
3
Simulation Data Generation Details
................................
22
D.4
Real-world Data Details
................................
23
D.5
Main Experiment Continued
23
D.6
Robustness under imperfect topological information
23
D.7
Running Time
23
D.8
Method Details
25
D.9
Hyperparameter Study
25
D.10
Additional Ablation Study: Offline Experiment
25
15

---

A Theoretical proofs
A.1 Proof of Theorem
Theorem l(validity). Under the assumptions stated above, the proposed method satisfies the following conditional coverage guarantee:
log(16n)
log (16n)
[P(Y++1 Cr+I(a)|XT+I x) -1 -1-aa) a)) (4L 2LV +2 + -6
71
n
where
w v 2r V (K1 Vx2)1 ((g(n))((( + VR2)
For easy notation, denote À An, D = Et and sometimes we drop subscript
Lemma 1. For any test conformity score 8: E:¹ Åéi and the true conformity score &t of Ae with probability at least
Σ
15 sel wnn
(8)
1=T-n+1
where
w = var+ 27V 20 /
(11 1 + VR2)1 + o(g(n))((M) + VM2)l.
Proof. We have:
[8 s1 = le Ac; - E A S 1e Ac; - E Aeil + 1e, Ae, - El -Aé
< A T ÁΔ + A T Åe, + le, (A - A)Ezl
|| + 2|| Al|le| + |||||||| All
<r|| + |||| + =(g(n))]lel...
(9)
The inequality ! exists because of the Cauchy- inequality and Assumption 2. Hence, by Assumption 1, we have
s: S T + 2r
|| + ((g(n))
||a.ll
(=T-++
(=T-n
T
Srnu, + 2r
1AIl2)
Σ
[|<al?2) + olg(n))
|leel1
(
TT-
T
T
Srnu? + 2r
nug(
[|el|22) + (((((n)
Σ
1
(10)
T+-
From Assumption 3, we have
|
2
1
Σ
Bille 1 II
(11)
n
n
Using Chebyshev's inequality, we have
T'
Var[l|c||1]]
Varllee
||e 2
[[]
2
8,
(12)
nd
Var [cc ?
t=T- n+n
16

---

which means that with probability higher than 1 - 6,
Var ||
Σ
leell S [[[ler11]] +
nd
1=T-n+1
K21
S skIl+ +
nō
S (N1 + V
K2
) S (K1 + v
(13)
nål
The last inequality is because we can set such that dn/ < 1. Plug into eq. 10 we have with probability higher than we obtain eq. (8) and the lemma follows.
Denote the empirical CDF: (xr) = ET true CDF of score function (2 = x))
-i lasa, (x+i(x)= - laga and
Lemma 2.
Under Assumption 3. for any n,, there exists event An which occurs with probability
least / log 16n) succh that, conditioning on An.
sup F (x) - F(x) I (16(16n)
Proof.
The proof follows Lemma I in 51 that utilizes WvoretakyKKiefer-Wollee inequality in
24
Lemma 3. Under Assumption 1, Assumption 3,with high probability,
sup Pn+1(1) Fx))(z) s S2L 1) + ssup Fatz(x) F((x)
Proof. The proof similar to Lemma B.6 in [50] and is written here for completeness.
Using Lemma T we have that with probability - 8,
T
|st - So S nw.
(14)
++1
Let S = {t: 18: &el 2 Vw). Then,
15|/ <
Σ
13t 8,1 S mas
(15)
17

---

So |S| S nyw. Then,
(2-1(2)- Fn+1(1))
S
7 Σ
[1[8 [) - 50 x]
S
+
Sz) 115, s
71
1g8
5 +
Σ
11 x/ S Vw)
n
1=T-n+1
I Vw= + P([ST+1-2
- x š VW)
T
1
+ sup
- S [W} - P(|ST+1- - x x/ Vv)
m+l
v + (2 vw) F2 [w]]
F vw) Fnt1 Vw) - (Fs(x Vw) Fs vw))]
+ sup
(ii)
< (22+1)V++2)) Fatz (z) - F.((///
(16)
where (i) is because [1fa S x² 1(b =3| S 11 x| la b/} for a, b E R, and (it) is due to the Lipschitz continuity of F2(r).
Proof of Theorem 1
Proof. Look at the conditional coverage Yr+1 given X741:
P (Yr+1 E CC++1 Xr+1 .r+1) (1 a1]
(17)
(sT-1 S 1 - a quantile of Fo+1 |T+1 = =x) -(1 -a) ail
(Fnti (3r+1) s 1 - a) - (F(((8))))) $1- aa
B[ Fat (8T+1) - a
a) - ((F((5++1) $1 - a3l
<P (+T=1) (1 - -0) $ | (8T-1) - F,((T+1)l)
(18)
Based on Lemma we can define the event A,, (AA)) 2 log(16n)
conditional on A,, we have:
log (16)
sup (x+1 Fs(x)) S
(19)
n
Hence, we can write eq. (8 as
P((F((9441) - (1 - -1-a)1 (a41() F((s4+)))
<P((F((94+))- -1- a) ( F((ST+))llAn)++++ +
<P((F((9T41) (1 (11--00)) a)l < ( F.(3T+1) + F(( + log(16n)
(20)
Conditional on A.:
(BT-1) - (| - F,(844))
< sup Fati(x) - F.(x)) + L|5T+1 8T41
x
log(16n) Lw.
<(2L + +ww + 3//
(21)
18

---

The last equation exists because of Lemma 3 and eq. (99)
Note that (8(++1)) Uni 1), have
P((F((5++1) - (1 a)] [o)s - Fn+1((r+1) Fa(3T+1)l - +
(F,(3r+1)
(s(
66/ (16()6n)
+ 2Lw.
(22)
n
Plug into eq.
18
we have
P Yrrt1 C741 X1+1= = = (1 aa)
+2 + +6V log(16m)
+ 2Lw + log(16n)
(23)
A.2 Proof of Theorem 2
Theorem (Efficiency). There exists 0 < og < 1, such that when a < an, the optimal solution to the minimization problem is given by:
A. == arg min Qi-,Qi-a(e" Ac)), A>U
(24)
where (O0,A,1)
Lemma (Uniform tail threshold via Dirichlet-lensen). Let Sx = -Et 7:77 with Za 3.dd (O0 1) and E A-1 Then Sx & TUx with ~ x² independent of Ux = A.V. where V ~ Dirichlet 1 2 2..... 2) For all we have 0 < Ux S 1 and E[Ux] = [[]UA]=1/1
Moreover, for s 2 I 2, the map u + (s(u/u) is convex on (0, 1] (where Fr is the x² survival function). Consequently,
P(SA 2 8) = E(F((s/U))] 2 F(s/E(Ux)) FI(Is) P(Sx s).
Equivalently, with p²)))=====)))))))))) = Pxt (1(1 + 2)) E (0, we have
(x()x) & (SA), AAEA' <p> VAEA'.NVp'''U) (I).
Proof. The Dirichlet-x² factorization and R[UA)] 1/T standard. Write g,(n) Ff(s/u). Using x² density (z) x² 2/
#/2 one computes (") 2 iff ff(x) < 2f((z) at s/u. Since J/S ( 1) 1. the inequality holds whenever x 2 + 2. If I+ then
8/u 2 x 7 + for all E (0. 1). hence ys is convex on (0, 1]. Apply Jensen and note
Sy. tx? to conclude.
Proof of Theorem 2
Proof. By the definition of ellipsoid volume, we have
V(A, Qi-a(i--eTTA
Constant x Qi-a(e (de()((et(()))-1/
(25)
Since the optimization problem is invariant to the rescaling of the scalar (that is, and any positive scalar multiple cA, where с >> 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 e ~ N (0, 1 and > by Cholesky decomposition, we can write A, LL where L is a
lower triangular matrix. Define the matrix B = L AL, we can rewrite the eq. 25 as
min Q/22 (x Bx) det (et()) B+0
(26)
where (0, Id)and det(L) is constant independent of B.
19

---

To further solve the optimization problem, we look at the eigenvalue of B, suppose B O AA) and an orthogonal matrix. Since the optimization value is invariant to different scaling of B, we are imposing additional constrains on X;-
min 2. 1 A==
Σ A
A
(27)
i=1
and (aihsisr are i..d. random variables ().
Now we would like to prove that the above optimization problem is solved when X = (7 .. 7).
Note that by Cauchy-Schwartz Inequality IT X, is minimized when Y A2... Xa 7,
by Lemma 4 QI } is minimized in X", for ag and ag is decided according to I. Theorem tollows.
A.3 Tail-up model
Lemma 5. The spatial covariance E of any node pair (u, v) is:
E(u,v) S/ mannn m(r u)m(( v) w(r) Vw(u)w(o))
dr.
(28)
Proof. Note that
E(u, v) = = / m(s w w(s) dB(s), / m(r mi(--o) dB(r)) w(u)
w(r) w(v)
Due to the independence of increments for Brownian motion, only when E Au O Av, the covariance is non-zero. Note that for Brownian motion, we have Cov(dB(r),dB(3) = drds.
Hence Lemma Followw..
D
Lemma we set the moving function m(r - u) B exp d(r.u) with parameters 3 (a scale factor) and & >> 0 (a range or decay parameter), then the covariance matrix between two locations u, v can be expressed as:
w(u) exp(-d(u, 1 (u-ro).
E(u, v) = 02
(29)
w(v)
Proof. By Lemma and substitute m(r - u) = Be d(r,a// and m)(-v)) = B e---errrrp d(r,u)/o, we have
E(u,v) = / JannA 3² exp
d(r,a))
exp
d(nww)
w(r)
dr.
(w(u) w(v)
The set Au Ni Ae are the segments of networks that flow into both and Consider the following cases:
Ifu u and u are not flow-connected, then Au O Ae and hence E(u, v) = 0.
If and are are fow-cconeccted, without loss of generality assume is downstream of u. Then Au Au, and for each Au, d(r, v) d(r, u) + d(u, v). Hence we have
expl d(r, u) + d(r:
exp(
d(u, v) exp
2 2d(r, u)
Ieads to remaining integral over r E Au. We can write
w(u) w(v)
exp d(u, v) /
w(r) dr. w(u)
E(u v) = 82
exp 2d((..))
20

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1
Topology
3
4 Spatial Dependency
2
Multi- variate
6
Temporal Shift
Time Series
1.
46
10
4.
30
5.
15]
533
51
52
6.
50
7. Our method, STACI
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- work the first to incorporate topology information.
w(rr dr E(u, u) is constant, since the additivity constraint on w((u) assures the constant variance of site Thus, the tail-up exponential model yields a covariance of the form
Note that JAu exp( 2d(r.u)
/ 0.
[onntant factors) expl d(u.u) if u and v are flow-connected, otherwise.
E(u, v)
B Taxonomy for Related Works
Figure 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-att the nature of 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
relax the exchangeability assumption via online calibration, sliding window methods, scorecasters or dynamic programming. These methods handle distribution shift over time but typically operate univariate or diw-dimensional settings. In contrast, CP methods for multivariate or sigh--imenssinal data (30) [0]] often focus on constructing prediction sets that exploit geometry or sparsity, but generally assume no temporal structure.
15 52
Our proposed method, STACI, is positioned at the intersection of these in the Venn diagram. 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 e The proof follows from Proposition 4.1 in [[].
21

---

Algorithm STACI
Input: Data D; Network topology G; Model f(:); Hyper-paramellrs X; Confidence level a.
Output: Prediction set C(X++15a).
1: // Training
2: f f using D Deal
3: // Calibration
4: & + =ē= Yi f(Xi) Dieo, (X-- JED:
Daa
5: En + ELEDe. Geg (n - 1) given E:
6: Σg Compute 29 for (l,, (v). Vi, E given
7: +- Aāg1 +(1 A)
8: S + (4 A given E;
9: Q1-a + Compute [I-a)(++I]] th largest element of S;
10: // Testing
11: C(XT+1;0) t (y::8(Xx+1.))) s sQ:=)::
Proposition 1. Consider n' test data points as the realizations of (Xr+1. Yr41), denoted by Diest- We have the asymptotic coverage guarantee:
lim E 1(Y, C(Xnao))//'' = =1-a. a. LED
While Proposition[| 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 and are only slightly violated, STACI maintains the predefined coverage level (7 0.01) while achieving efficient prediction sets. However, when > 0. Proposition [ does not ensure 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 single NVIDIA GeForce RTX 4080 Super GPU, AMD Ryzen 7950X 16-Core Processor CPU, 64GB Memory and 2TB SSD.
D.2 Datasets and Codes
The PcMS03 dataset used in this paper is collectd by California Transportation Agencies (CalTrans) Performance Measurement System (PeMS), It contains three months statistics on traffic flow every 5 minutes ranging from Sept. 1st 2018 to Nov. 30th 2018, including 358 sensors. The datasets from hom/httpss//githhi cow htw.a//a.oo.oogggooo/////////// wwt.://TUUU/SSTN/TTEeeeeee tttp:///ttteeeeeeein/naaaaaaaaaa without available license.
The first backbone ST-Graph model is AGCRN 2 which we adapted the official implemen- tation from httpp:///ithul/ Btt.://..comeeeeB////eAaa..
under MIT license. ASTGCN (66 was
implemented with Py'Torch Geometric Temporal package 34 from https://gi thubb com/ httpre/reeeeeeeecee////rrrcereee
under MIT license. STGODE [13] was
adapted from the official implementation Apache-2.0 2.0 license.
square coder under
D.3 Simulation Data Generation Details
Segment i1 and starts with (0, 1) and (0.5, 0.8), respectively, and both end with (0.3, 0.5). The next segment 13 also starts with (0.3, 0.5), and end with (0.2, 0. Segment ra start from (0.6, 0.6), and ends at the same location 73. Starting from this location, ends at (0.4, 0). The weights for
22

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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 (r1, r2 and in our example), the source nodes treated as infinitely distant. In implementation, the source node of each segment is extended 10 times in the same direction simulate infinity.
D.4 Real-world Data Details
As shown in Figure S we construct two subgraphs from PeMSO3 dataset. We construct PEMS03-GE and PEMS03-G2 with temporal distribution shift shown in Table 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.
Table 2: Temporal distribution shift of constructed datasets
Dataset
Intra-Period Variance
Inter-Period Flow Shift
PEMS03-G1
117.4614
21.5453
PEMS03-G2
106.2113
17.4750
D.5 Main Experiment Continued
We evaluated two additional baselines, CopulaTSCP and DeepSTUQ. on the PeMS-G1 dataset, with the results summarized 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, 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 hyper-rectangle for 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 high- nigh-dimensional joint prediction and highlight the need for specialized approaches.
highly unstable and
Table 3: Comparison over additional baselines on the PeMS-GI dataset.
Method
Coverage
Efficiency
CopulCCPSS/AGRRN)
70.60
668.00
DeepSTUQ
93.82
66.51
STACKAGCRN)
95.75
67.54
D.6 Robustness under imperfect topological information
To evaluate our model's robustness against graph structure noise, 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, randomly selected another node and added an edge between them. As shown in the Table 4 the performance of all GNN-based based methods degraded method, STACI, demonstrated the strongest robustness to
when using the noisy graph. Notably, these perturbations. experienced the least performance degradation among all models, showcasing its resilience imperfect graph structures.
D.7 Running Time
a single Nvidia Geforce 4080S, the training time of AGCRN with PEMS-GI 1419s. In table 5 we show the computation time of our and baseline CP methods on AGCRN model and PEMS-GI dataset. Our method STACI has comparable running time with other methods. Here we provide detailed complexity analysis. Compare MuliDimSPCI, for each test time point
23

---

41
L2
fz
l
L0
2
L
6
Bg
12
L1
(a) PEMS-G1
12
L 10
L1
8
2
L
bg
83
l5
(b) PEMS-GE G2
Figure 5: Real-world road network structures and their abstraction for PEMS-GI 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 & to l12 are placed exclusively on those northbound freeways, marked with green transparent triangles. The right map provides abstract representation of the road network and sensor locations, using the same symbols for consistency.
Table 4: Robustness STACI to noisy sensor locations on the PeMS-G1 and PeMS-G2 datasets across various GNN backbones. Noise Scale represents the standard deviation of the Gaussian noise added to location coordinates. The bolded rows indicate performance on the original, clean data.
AGCRN
ASTGCN
STGODE
Dataset Noise Scale
Coverage
Efficiency
Coverage
Efficiency
Coverage
Efficiency
0.0
95.75
67.54
10.94
95.54
67.92
10.22
95.14
73,62
9.83
0.5
95.83
71.94
11.39
95.05
85.24
14,68
95.09
91.73 + 25.11
GI
1.0
95.81
64.12 1 10,25
95.05
85.24
= 14.68
95.12
99.33 4: 22.42
95.58
12.73
17.29
3.0
95.
119.40 + 15.65
95.12
97.44
14.67
95.12
105.80 + 21.06
0.0
95.01
69.62 + 12.85
95.05
58.07 +: 9.83
95.20
74.86
9.63
11.06
95.12
34
13.67
95.16
79.08
G2
1.0
94.97
69.45 + 11.37
95.12
69.34
13.67
95.14
80.04 : 11.29
2.0
95.01
92.31
+: 13.12
95.03
13.18 69.56
68.97
95.09
79 68
+ 11.27
3.0
95.00 09
117.99
25.89
95.12
11.77
95.12
80.06
11.31
need: additional estimation of tail-up parameters d, o² and using historical covariance matrix weighted addition of spatial covariance and empirical covariance. Consider the node size is and the optimization methods (e.g., least square our implementation) iteration round N The former estimation takes O(Nr²), and the latter addition takes O(n²), pseudo-inverse of matrix is

---

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 STACI provides fast, reliable, and interpretable joint UQ over localized subgraphs across long time horizons.
Table 5: Computation Time (seconds) for Different CP Methods with Different Calibration Set Size
Calibration Set Size
Sphere-ACI
STACI 0.01)
Sphere
Square
MudiDinSPCI
0.01)
0)
100
142
144
24
24
25
25
200
152
153
24
21
26
25
400
132
135
25
24
24
25
500
135
136
23
21
23
24
D.8
Method Details
We provide pscudo-codes for our proposed method in excluding offline setting detailed in Section D.10
This applies all experiments in this paper,
D.9
Hyperparameter Study
The studies for hyperparameter are presented Figure
D.10 Additional Ablation Study: Offline Experiment
In both synthetic and real-world data, MultiDimSPCI achieves the closest to proposed method, STACI, efficiency. Therefore, we focus our comparison on four specific variants: vanilla MultiDinSPCI(> 0), MultiDinSPCI(y the offline setting, STACI does not update the covariance matrix estimation. To ensure a fair compari- son, similarly fix the covariance matrix for MultiDimSPCI methods at the beginning of the test phase.
0.01), STACI(> 0), and STACI(? = 0.01). In
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 y, STACI consistently outperforms MultiDimSPCI in efficiency. Notably, when ACI is not applied (? 0), both methods tend overly conservative, resulting in coverage rates well above the desired 95%. Therefore, since STACI (7 0) achieves higher coverage rate, MultiDimSPPI (y 0.01) and STACI(? 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, achieve an exact coverage rate, incorporating ACI (7 0.01) is recommended.
25

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Coveraje Rote vizh y
60
Eficensyy w6th y-
Coverage Rats with y=001
60
Effilensy y y-00
80
Commige 150
122
P-200
130
86
0.0
04
06
0.5
100
140
0.0
02
04
08
1.0
0.2
0.6
08
10
140
0.2
0.4
06
10
(Ia)-(1d): AGCRN on PeMS-G1
Coverage Rote with y
Emicenny with V
Coverage Rate with 0-001
Efficlncy w y-0001
150
110
120
120
86
130
88
130
00
0.0
02
0.4
0.
0.5
1.0
240
0.0
0.7
0.4
0.6
0.8
1.0
0.8
67
04
0.6
0.5
1.0
140
00
02
0.4
06
0.8
3.0
(2a)-22d): ASTGCN on PeMS-G1
Crvenage Bale y-o
Ifficency
Ooyarage with
Elickency with
Crowegge 10
100
110
90
110
120
120
20
130
88
2m0000
130
86
0.0
0.4
0.
08
1.0
140
0.0
02
0.4
0.8
3.0
0.0
0.2
0.6
1.0
02
0.4
0.
3.0
(3a)-(3d): STGODE on PeMS-GI
Coverage
Coverage
Efficency y
80
100
100
Conmrepe
120
12
92
1/0
00
160
160
0-300
180
1-361
200
Ubb
(4a)-(4d): AGCRN PeMS-G2
y-00
30
180
300
22
3.20
60
140
100
160
10000
(5a)-(5d): ASTGCN on PeMS-GB
Crverace Rate with y
Tflicincy 0
Pllcclnny p-00
96
300
Churegge 8
120
120
12
32
10
200
180
1-4E0
180
0.6
(6a)-(6d): STGODE on PeMS-G2
Figure 6: Comparison of Coverage and Efficiency for all PeMS experiments with different backbone GNN models and datasets. Each figure show results for different belief weight À and calibration set size n, with adaptive step size y = 0 (a&b) and 0.01 (c&d). Pre-determined coverage threshold of 95% is shown by horizontal gray dotted lines.
26

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Coverage of Offiine Nethods
Efficiency Offine Methods
wuttmtRRay-100 1.0
05
(12111-11)
STACAy
95.3
85.0
04.5
230
500
Figure 7: Coverage and efficiency of different methods with different calibration set size Multi- DimSPCI methods are in blue, and our methods are in red. Methods with = are in darker colors; while those with adaptive coverage, y 0.01, shown in shallow colors.
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Justification: As stated in the abstract and introduction, the main contribution of this paper is that we propose Spatio-Temporal Adaptive Conformal Inference (STACT), a novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. We provide theoretical analysis and empirically show that STACI balances prediction efficiency and coverage well.
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Reviews:
[{"review_id": "nx4bAOILIm", "reviewer": "Reviewer_mDnW", "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", "questions": "Please refer to weakness part.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1. The CP method for graphs explicitly incorporates topological similarity using graph kernels in the calibration process. The approach is well-motivated.\n2. The framework extends its validity guarantees to graphs under topological dependence, with detailed theoretical proofs and assumptions.\n3. Empirical evaluation demonstrate the effectiveness of the approach. \n\nWeakness: \n1. The usage of graph kernel matrices introduces O(n^2) memory and compute complexity, limiting scalability to large graph. \n2. 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.\n3. The experiments lack ablation on the major components of the framework.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the feedback. Here we provide response to weaknesses:\n\n**W1: The usage of graph kernel matrices introduces O(n^2) memory and compute complexity, limiting scalability to large graph.**\n\nA: Here we provide a detailed complexity analysis. The psuedo-codes of our algorithm can be found in Algorithm 1 of Appendix. Compare to MultiDimSPCI, for each test time point we need: 1) additional estimation of tailup parameters ($\\phi$  and $\\sigma^2$) using historical covariance matrix (line 6), 2) weighted addition of spatial covaraince and emperical covaraince (line 7). Consider the node size is $n$ and the optimization methods (e.g., least square in our implementation) iteration round $N_i$. The former estimation takes $O(N_i⋅n^2)$, and the later addition takes $O(n^3)$, as pseudo-inverse of matrix is involved. However, MultiDimSPCI also needs matrix inverse for mahalabis distance calculation (line 8), which also takes $O(n^3)$. Additionally, estimation of only two parameters $\\phi$  and $\\sigma^2$ can converge fast. Therefore, our method do not need significantly more time than the baseline method, consistent to the Table 1 running time results in Appendix D.5. \n\n**W2: 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.**\n\nA: Our method currently employs the tail-up model, which is well-suited for directed flow networks with upstream–downstream relationships. While this choice is task-specific, we emphasize that the STACI framework is modular: the topology-induced covariance matrix $\\hat{\\Sigma}_{\\mathcal{G}}$  can in principle be replaced by any other graph-based kernel or learned spatial structure. It is an interesting research direction to explore data-driven alternatives, such as graph diffusion kernels or attention-based learned structures.\n\n**W3: The experiments lack ablation on the major components of the framework.**\n\nA: We agree that ablation is important and have provided relevant experiments in both the main paper and appendix (e.g., Figure 3), where we vary the reliance on the graph-induced covariance via the mixing parameter $\\lambda \\in [0, 1]$ . These results demonstrate that the topology-aware component improves coverage and stability, particularly under limited calibration samples. Additionally, we study the effect of adaptive conformal inference (ACI) by varying  $\\gamma$  and show how its interaction with the graph structure affects both coverage and efficiency. We will revise the manuscript to make this connection more explicit and clarify which components are responsible for observed gains.\n\nOnce again, we thank the reviewer for the comments and positive assessment of the technical contribution, theoretical guarantees, and practical motivation of our framework."}, {"review_id": "w6ipMtZusA", "reviewer": "Reviewer_q8kS", "summary": "This 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.", "questions": "1. 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", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\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", "was_revised": true, "final_justification": "I initially scored 3 due to limited clarity, a narrow empirical scope, missing baselines, unclear simulation goals, and absent scalability details. The rebuttal substantially improved things: the authors clarified notation and problem setup, added results for the CopulaTS baseline, and committed to include dataset statistics and a clearer discussion of limitations. These address most clarity and methodological concerns, so I raise the score to 4. That said, the evaluation remains single-domain and small-scale, the simulation study’s representativeness is still debatable, and all promised clarifications and results must appear in the main paper. My recommendation reflects practical promise with the expectation that these revisions are incorporated.", "rebuttal": "**Q1a: Problem definition clarification**  \nA: We appreciate the reviewer’s suggestion to improve clarity. The stream network is represented as a directed acyclic graph, where each node corresponds to a spatial observation site located on a stream segment. For example, in the traffic dataset, stream segments represent roads, and observation sites are traffic sensors placed along those roads. A visual illustration is provided in Figure 1 of the main paper. Note that each location belongs to exactly one segment, while a segment can contain multiple observation sites.\n\n**Q1b: Notations for segments**  \nA:We use i to index sites, and j to index segments. a network consists of multiple stream segments, where all the possible locations on a segment $j$ is represented by a geolocation set $r_j \\subset L$. A segment may contain one or multiple observational sites. The geolocation of a site $i$ is denoted by $l_i \\in L$. $u, v \\in L$ are dummy variables for arbitrary locations on the network. The use of $s_i$ was a typo and should be corrected to $r_i$ and thanks for pointing out.\n\n**Q2: Tail-up scaling parameters  $\\phi$  and $\\sigma^2$: design and learning**  \nA: The parameters $\\phi$ and $\\sigma^2$ in Equation (5) are scaling factors estimated during the construction of the topology-induced covariance matrix $\\hat{\\Sigma_G}$ used in the Mahalanobis non-conformity score. They are not learned jointly with the predictive model’s weights. The weights of backbone GNNs are learned on training data and fixed, but $\\phi$ and $\\sigma^2$ are optimized on the calibration data. Specifically, $\\phi$ and $\\sigma^2$ are selected by minimizing the discrepancy between the sample covariance matrix and $\\hat{\\Sigma}_{{G}}$, allowing the tail-up model to better reflect the empirical correlation structure. $\\sigma^2$ controls the overall magnitude of spatial dependence, while $\\phi$ determines the rate of decay of spatial correlation with hydrologic distance.\n\n**Q3: In Appendix C, Equation (27) introduces ($\\eta_t$) as the ACI step size, but line 171 refers to “($\\alpha = 0.01$)” as the predefined coverage level. Should this reference be to ($\\eta_t$) instead? Please clarify the notation.**  \nA: Thanks for pointing it out. It should refer to $\\alpha=0.01$, and we will fix this in the final version.\n\n**Q4: Line 169 mentions the guarantee holds even in adversarial online settings. However, no theoretical or empirical support is given. Could you substantiate this claim and describe the adversarial setting more clearly?**  \nA: Yes, we clarify that the asymptotic average coverage guarantee in Proposition 1 is theoretically grounded in Proposition 4.1 of [4], which analyzes Adaptive Conformal Inference (ACI) under arbitary time series data, even chosen in an adversarial manner, without assuming i.i.d. or stationarity. The key mechanism is the dynamic calibration of the quantile threshold $\\alpha_t$ based on past coverage. If recent prediction sets are too conservative, the threshold is decreased to improve efficiency; if they under-cover, the threshold is increased to restore validity. This mechanism ensures that the long-run average coverage converges to the target level, regardless of the underlying sequence.\nThough the proof in [4] is presented for univariate outputs, the argument extends naturally to our setting with high-dimensional multivariate responses, as the calibration procedure operates on scalar-valued conformity scores, regardless of the dimensionality of the input space.\n\n**Q5: Line 277 states that the topology-based matrix converges faster than the sample covariance estimator. Could you provide evidence for this claim?** \n\nA: The topology-based covariance estimate benefits from strong structural priors. Specifically, the tail-up estimator introduces strong structural regularization by encoding known topological relationships, which parameterizes the covariance structure using two scalars, $\\sigma^2$ and $\\phi$, in contrast to the $O(I^2)$ parameters needed for a full sample covariance. This regularization improves sample efficiency and stabilizes estimation in high-dimensional, low-sample settings. Empirically, we observe that the topology-regularized estimator achieves better coverage under small calibration sizes. As shown in Figure 3 of the appendix, when using only the sample covariance matrix (i.e., $\\lambda = 0$), the coverage deteriorates sharply as the sample size decreases. In contrast, the topology-based estimator maintains reliable coverage even when few residuals are available, supporting the claim of faster empirical convergence. While theoretical convergence rates for topology-aware estimators like the tail-up model remain an open area of research, our observations are consistent with the general statistical principle that incorporating structural priors enables more robust estimation with fewer samples. \n\n**Q6: What calibration set size was used to generate the results in Table 1?** \n\nA: Calibration set size $n$ is set to be 300, as specified in the default experimental setup (line 310 of the main paper).\n\n**Q7: Which backbone model was used to generate the results in Figure 3? Do the trends hold for other backbones like ASTGCN or STGODE?** \n\nA: The results in Figure 3 (main paper) were generated using AGCRN, as stated in lines 368–369. We confirm that similar trends hold across other backbones, including ASTGCN and STGODE. Plots for all datasets and backbones are provided in the Appendix (Figure 3).\n\n**Q8: The conclusion suggests extending STACI to general spatio-temporal graphs and alternative network parameterizations. Could you elaborate on what you mean by “general spatio-temporal graphs” and give a concrete example of such parameterizations?**  \n\nA:  By general spatio-temporal graphs, we refer to settings that go beyond directed stream networks and include arbitrary graph topologies where node features evolve over time such as social networks, etc. In these cases, spatial dependencies may arise from learned or domain-specific connectivity patterns rather than physical flow, and may not follow upstream-downstream constraints.\n\nThe main idea is that incorporating topological structure rather than treating the data as purely multivariate time series can yield more structured and stable estimators of covariance or other quantities used in high-dimensional conformal prediction. For example, one could use diffusion kernels or graph Laplacians to construct covariance structures that reflect node influence or proximity in a soft, learnable manner. This opens the door to extending STACI-like methods to broader domains with complex spatio-temporal dependencies.\n\n**Response to weakness**\n\nIn addition to the specific questions raised, we would like to address several points mentioned in the weaknesses.\n\n**w1.2: lack of ACI introduction in the main paper**\n\nA: We agree that the role of ACI deserves more attention. The main contribution of our work lies in the topology-based regularization of the covariance structure. Due to space constraints, we included its description in the Appendix. However, we will include a brief summary in the main text to provide readers with a clearer understanding of how ACI interacts with our topology-based regularization. \n\n\n**W2.2: Expanding to additional datasets or baselines**\n\nA: Regarding the baselines, as shown in Xu et al. [3], those methods significantly underperform compared to MultiDimSPCI, especially in high-dimensional joint prediction tasks. For this reason, we focused our comparisons primarily on MultiDimSPCI, which represents the strongest known baseline in this setting. Notably, our method outperforms MultiDimSPCI.\n\n**W2.3: 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\nA: We respectfully clarify that in conformal prediction, the primary objective is to achieve the target coverage level (e.g., 95%) while minimizing the size of the prediction sets. Once valid coverage is attained, efficiency becomes the key metric: a more efficient method yields tighter prediction sets without sacrificing validity. Therefore, coverage above the target is not inherently better; in fact, methods that consistently exceed 0.95 may be overly conservative and less informative. In this context, STACI achieves coverage close to the nominal level while producing smaller prediction sets in most scenarios, indicating a more efficient uncertainty quantification.\n\n**W2.4: 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\nA: We included such ablation experiments in Figure 3 in the main paper and in the Appendix to disentangle the effects of ACI (via $\\gamma = 0$ vs. $\\gamma = 0.01$) and the use of the topology-induced covariance matrix (via varying $\\lambda$). When $\\gamma$ is 0, ACI is not applied, but STACI can still achieve better performance than most other baselines.\n\n\n**w2.5: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\nA: Detailed dataset statistics, including the number of nodes, edges, and time steps, are provided in Appendix D.2. We will ensure this is made clearer in the main text as well.\n\n[4] Gibbs, Isaac, and Emmanuel Candes. \"Adaptive conformal inference under distribution shift.\""}, {"review_id": "28vyv9vTfk", "reviewer": "Reviewer_2Eta", "summary": "This 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.", "questions": "-\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.", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "Some of my concerns have been addressed.", "rebuttal": "**Q1: Difference of definition with [2] in Equation 2**\n\n A: The quantity $\\hat{\\epsilon}_t$ in Equation (2) represents the prediction error at time $t$, defined as the difference between the observed value $y_t$  and the prediction $f(x_t)$. While the notation differs from the error vector $r_t$ used in [2], the underlying semantics are equivalent. Both quantify the residuals between predicted and true values at each time step.\n \n**Q2: Sizes of datasets and magnitute of temporal distribution shift**\n\nA: PeMS-G1 and PeMS-G2 are spatial subsets of the larger California PeMS traffic dataset, each containing 12 nodes and spanning 26208 time steps, as stated in Appendix D.2. We select a localized subgraph within the full traffic network to define a coherent spatial region for conformal prediction.\n\nTo assess temporal distribution shifts, we employ two metrics leveraging daily periodicity (288 5-minute observations/day): Average Intra-Period Flow Variance (average daily flow variability) and Average Inter-Period Mean Flow Shift (average day-to-day change in mean daily flow). Due to space limit, we only show the average value over all observation points as follows.\n\n| Dataset|Intra-Period Variance|\tInter-Period Flow Shift|\n|--|--|--|\n|PEMS03-G1|\t117.4614\t|21.5453|\n|PEMS03-G2|\t106.2113\t|17.4750 |\n\nThe analysis reveals substantial temporal shifts, with strong intra-day flow pattern shifts (high intra-period variance) and non-negligible inter-day mean flow shifts.\n\nFurther, empirical evidence strongly suggests its presence. In particular, as shown in Figure 3, the coverage achieved by ACI with adaptive thresholding $\\gamma_t = 0.01$) is substantially higher than that obtained with a fixed threshold ($\\gamma = 0$ ). This performance gap reflects ACI’s ability to correct for temporal variation in the error distribution, thereby indirectly confirming that distribution shift exists in the data.\n\n**Q4: Scalability of STACI to very large graphs and variable flow topology? [3]**\n\nA: STACI scales well for graphs with a moderate number of nodes (typically fewer than 30). In general applying joint conformal prediction or consider joint uncertainty quantification to hundreds or thousands of nodes simultaneously is not statistically meaningful, due to two reasons: 1. They are not interpretable. One cannot obtain any useful insights from a uncertainty set in a super high-dimensional space. In such setting, it is usually done by sampling, i.e., presenting samples from a high-dimensional space. 2. They are usually overly conservative. \n\nBy focusing on high-dimensional uncertainty quantification within spatially coherent subregions such as traffic corridors or watershed areas, STACI aligns well with practical needs in domains where stakeholders are primarily concerned with the joint behavior of a specific region rather than the entire network. For example, in freeway traffic management, operators may monitor 10–30 sensor locations along a congested segment to assess regional flow stability; in hydrology, flood risk is often evaluated jointly across 15–25 stream gauges within a watershed. These are typical dimensionalities where joint prediction is both statistically meaningful and computationally tractable, and where STACI is most effectively applied.\n\n**Q5: Robustness when the assumed topology does not fully reflect true dependencies**\n\nA: When the topology does not fully reflect true dependencies, STACI is designed to be robust to imperfect topological information through the weighting parameter $\\lambda \\in [0, 1]$, which interpolates between the sample-based covariance matrix and the topology-induced one. The topology-induced covariance may lead to suboptimal efficiency, but the overall coverage guarantee remains valid due to the conformal framework. \n\nWe conducted additional ablation studies and observe that even when the topology is only partially accurate, it still contributes useful inductive bias especially in capturing local dependencies and reducing estimation variance, compared to using the sample covariance alone. In Tables 1, we introduced random noise (mean=0, std = noise scale) to observation locations; clean G1 and G2 have longest connected distances of 8.75 and 3.41 miles. Bold rows highlight clean data performance. Results show STACI's robustness to moderate noise, with higher noise levels leading to prediction degradation, mainly through increased inefficiency.\n\nTable 1. Performance of STACI with different backbone GNNs on noisy PEMS03-G1& G2\n\nDataset| Noise Scale | AGCRN Coverage | AGCRN Efficiency   | ASTGCN Coverage | ASTGCN Efficiency   | STGODE Coverage | STGODE Efficiency   |\n|--|--|--|--|--|--|--|--|\n|G1| **0.0**   |  **95.75**          | **67.54 ± 10.94**      | **95.54**           | **67.92 ± 10.22**        | **95.14**           | **73.62 ± 9.83**   |\n|G1| 0.5   | 95.83          | 71.94 ± 11.39      | 95.05           | 85.24 ± 14.68        | 95.09           | 91.73 ± 25.11       |\n|G1| 1.0   |  95.81          | 64.12 ± 10.25      | 95.05           | 85.24 ± 14.68        | 95.12           | 99.33 ± 22.42       |\n|G1| 2.0   | 95.58          | 81.82 ± 12.73      | 94.92           | 90.38 ± 17.29        | 95.12           | 107.26 ± 24.30      |\n|G1| 3.0   | 95.26          | 119.40 ± 15.65     | 95.12           | 97.44 ± 14.67        | 95.12           | 105.80 ± 21.06      |\n|G2| **0.0**   |**95.01**          | **69.62 ± 12.85**      | **95.05**           | **58.07 ± 9.83**         | **95.20**           | **74.86 ± 9.63**        |\n|G2| 0.5   |  95.01          | 68.04 ± 11.06      | 95.12           | 69.34 ± 13.67        | 95.16           | 79.08 ± 10.81       |\n|G2| 1.0   | 94.97          | 69.45 ± 11.37      | 95.12           | 69.34 ± 13.67        | 95.14           | 80.04 ± 11.29       |\n|G2| 2.0   |  95.01          | 92.31 ± 13.12      | 95.03           | 68.97 ± 13.18        | 95.09           | 79.68 ± 11.27       |\n|G2| 3.0  | 95.09          | 117.99 ± 25.89     | 95.12           | 69.56 ± 11.77        | 95.12           | 80.06 ± 11.31       |\n\n**Q6: Extension to non-flow-based graphs**\n\nA: The current formulation of STACI is tailored to flow-based structures such as stream networks, where directionality and upstream-downstream relationships are well-defined. In non-flow-based graphs, such as social networks, the corresponding topology-induced covariance structure equires modeling efforts and current tail up model may not be applicable. To apply STACI in such settings, one would need to adopt alternative network parameterizations such as diffusion kernels or graph Laplacians that better capture the relevant spatial dependencies. We view this as a promising future direction and have outlined it in our conclusion, where we suggest extending STACI to general spatio-temporal graphs using more flexible graph-based covariance models.\n\n**Q7: Expansion of baseline methods**\n\nA: Unlike most GNN-based methods that provide independent marginal prediction intervals, STACI performs joint uncertainty quantification for all nodes in a local region, yielding multivariate prediction sets in ${R}^I$ that capture spatial correlations. Table 2 compares STACI (with various GNN backbones) against DeepSTUQ [1], a state-of-the-art Bayesian GNN for uncertainty quantification. DeepSTUQ failed to achieve 95% coverage on dataset G1. With desired coverage met, our methods demonstrate superior efficiency.\n\nTo enable a meaningful comparison of set efficiency, we aggregate the marginal intervals from DeepSTUQ into a hyper-rectangle and compute its volume as a proxy. Theoretically, to ensure joint coverage $\\geq95\\%$, each marginal interval would need to be Bonferroni-adjusted to $1 - \\alpha/I$ ( 99.58% when $I = 12$), which leads to overly inflated volumes and high coverage. Instead, we report volumes using unadjusted 95% marginal intervals, which better reflects how DeepSTUQ might be used in practice. We acknowledge that this does not guarantee joint coverage, but it provides a practical and meaningful basis for comparison.\n\nTable 2. Performance of DeepSTUQ and STACI on PEMS03-G1 and G2\n| Dataset     | DeepSTUQ Coverage | DeepSTUQ Efficiency   | STACI(AGCRN) Coverage | STACI(AGCRN) Efficiency|STACI(ASTGCN) Coverage | STACI(ASTGCN) Efficiency|STACI(STGODE) Coverage | STACI(STGODE) Efficiency|\n|--|--|--|--|--|--|--|--|--|\n| G1   | 93.82       | 66.51±22.40       | 95.75| 67.54±10.94 | 95.54 | 67.92±10.22 | 95.14 | 73.62±9.83|\n| G2   | 95.59       | 77.27±20.93       | 95.01 | 69.62±12.85| 95.05| 58.07±9.83| 95.20 |  74.86±9.63|\n\n**Q8: Clarification of dataset types and topological structure**\n  \nA: In our experiments, we use the traffic dataset, where sensor observations are recorded at fixed locations along freeway segments. The underlying topology is constructed from the physical road network, with directed edges representing traffic flow direction between road segments.\n\nWhile our empirical evaluation focuses on traffic data, the STACI framework is applicable more broadly to domains that share the following structural characteristics: (1) observations are collected at discrete spatial sites located on segments (e.g., roads or stream reaches), and (2) the observed data reflects flow with a meaningful directionality (e.g., vehicle movement or water discharge). This includes applications in hydrology, transportation, and other spatio-temporal systems where directional flow and location-based measurements define the graph structure. \n\n**Response to weakness**:\n\n- **Complexity analysis** & **ACI and Topology Estimator Evaluation**. Due to space limit, please refer to our response to W1 & W3 of reviewer mDnW. \n\n- **CP Baseline Coverage**. Please see response to W2.2 of reviewer q8kS. \n\nWe thank the reviewer again for your helpful suggestions.\n\n[1] Qian, Weizhu, et al. \"Uncertainty quantification for traffic forecasting: A unified approach.\"\n\n[2] Xu et al. , \"Conformal prediction for multi-dimensional time series by ellipsoidal sets\""}, {"review_id": "y8i1uw9CFa", "reviewer": "Reviewer_b5rT", "summary": "The 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.", "questions": "Please comment on the issues mentioned above.", "limitations": "See above.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "As stated in my review, the submitted paper has serious issues in presentation (e.g., discussion of limitations and assumptions) and in the discussion of related work.\n\nThe authors’ rebuttal addressed these concerns and acknowledged the issues in the initial submission. Although they state that these issues will be addressed in a revision, I believe that the implementation of the necessary changes should be verified through another round of review. For this reason (along with limitations in the selection of datasets), I will keep my score (“borderline reject”). Nonetheless, I would be fine with the paper being accepted if the other reviewers and area chairs believe it can be accepted on the basis of the revisions proposed in the rebuttal.", "rebuttal": "We sincerely thank for the reviewer’s comment. \n\n**Assumptions and Limitations**\n- The i.i.d. error assumption underlying Theorem 5.6 is a standard setup when establishing conditional coverage gap guarantees, which is also used in [1] (see Assumption 4.1). The assumption ensures tractability in the theoretical analysis of time-varying nonconformity scores, as it allows the score at a new time point to be meaningfully compared to those from past calibration time points. Importantly, the i.i.d. assumption in our analysis applies specifically to the true error process $\\varepsilon_t$, not to the observed time series $y_t$ or the predicted residuals $\\hat{\\varepsilon}_t$. This setting covers a broad class of time series models, including non-stationary processes where the innovations are i.i.d., such as autoregressive models. Additionally, when the i.i.d. assumption may not hold, our method resorts to the Adaptive Conformal Inference (ACI) mechanism, which adaptively adjusts the threshold $\\alpha_t$. While this approach no longer guarantees conditional coverage, it retains asymptotic average coverage without requiring distributional assumptions on $\\varepsilon_t$. We will clarify this distinction in the revised manuscript.\n\n- The volume of the prediction set is locally adaptive in STACI and STACI accounts for serial correlation. Please see more in the related work discussion.\n\n-  STACI scales well for graphs with a moderate number of nodes (typically fewer than 30). Applying joint conformal prediction, or general uncertainty quantification, to hundreds of nodes is statistically uninformative due to the curse of dimensionality, as the resulting prediction set volumes become overwhelmingly large. Our framework focuses on localized subgraphs where uncertainty quantification remains both tractable and practically meaningful. This also aligns with practical needs in domains like traffic or hydrology where stakeholders are often concerned with the overall status or joint safety of a spatial region.\n\n**Related Work Discussion** \n\nHere we would like to put some clarifications on our work as well as the discussion for the related work, particularly [1]&[2] as mentioned. In short, while both methods incorporate temporal adaptivity, STACI inherently distinguishes local MultidimSPCI [1] and show superior performance in our setting.\n\nSpecifically, in the local version of [1], the algorithm constructs the sample covariance matrix and nonconformity scores at each time step using the most recent $T$ observations, and suggests the possible introduction of an additional local term using the most recent $k = 0.1T$ observations to regularize short-term variability (Section 3.4, Equation 11). This design indeed provides temporal regularization and adaptivity to local serial correlation. Similarly, STACI also incorporates local adaptivity and serial correlation. We compute conformity scores using covariance matrix based on a rolling window of the most recent $n$ residuals, which reflects temporal local variation. \n\nThe key distinction is in the type of regularization: [1] imposes local regularization in time based on residual windows; in contrast, STACI introduces regularization in space, via a topology-induced covariance structure that encodes spatial dependencies through a tail-up model. This spatial structure imposes stronger constraints with fewer parameters, often leading to improved empirical stability and faster convergence. As illustrated in Figure 3 of our appendix, when using only the empirical sample covariance matrix $\\lambda = 0$, which corresponds to MultiDimSPCI, the prediction sets suffer from under-coverage unless the calibration window size $n$  (or $T$ in [1]'s notation) is sufficiently large. This issue persists regardless of whether adaptive thresholding (ACI) is applied $\\gamma = 0.01$ or not $\\gamma = 0$, highlighting the sample inefficiency of unstructured covariance estimation. In contrast, by incorporating topological priors, STACI achieves valid coverage even with a relatively small number of calibration samples, demonstrating improved robustness and data efficiency.\n\nIn the original paper, we compare with the classic version of MultidimSPCI [1] instead of the local variant, following the authors experiment setup (Section5, [1]). In response to your suggestion, we provide the comparisons in Tables 1 and 2, following the setting in Sec 6.2 and hyperparameters suggested by [1]. In this problem, the local version of MultidimSPCI do not bring sigificant benefit over its original version, let alone supurpassing our methods. STACI is advantageous in both coverage and efficiency. \n\nTable 1: Coverage Comparison over different datasets and backbone models\n\n| Setting | MultiDimSPCI | MultiDimSPCI(local) | STACI|\n| -------- | -------- | -------- | -------- | \n| AGCRN + G1  | 92.92   |  93.04   | **95.75** | \n| ASTGCN + G1  |  93.19   | 93.57   | **95.54** | \n| STGODE + G1  |   92.70  |  92.98  | **95.14** | \n| AGCRN + G2  |  91.14   |   91.44 | **95.01** | \n| ASTGCN + G2  |   90.93  |   91.12 | **95.05** | \n| STGODE + G2  |   90.74  |  91.10  |  **95.20** | \n\n\nTable 2: Efficiency Comparison over different datasets and backbone models\n\n| Setting | MultiDimSPCI | MultiDimSPCI(local) | STACI|\n| -------- | -------- | -------- | -------- | \n| AGCRN + G1  |  73.82±8.16   |  74.23±8.27   | **67.54±10.94** | \n| ASTGCN + G1  |   74.68±7.67  |   75.18±7.69 | **67.92±10.22** | \n| STGODE + G1  |   88.48±4.26  |  88.99±4.26  | **73.62±9.83** | \n| AGCRN + G2  |   101.42±7.71  | 102.25±7.90   | **69.62±12.85** | \n| ASTGCN + G2  |  92.12±6.67   |  92.84±6.95  | **58.07±9.83** | \n| STGODE + G2  |  107.23±7.34   |   107.96±7.58 | **94.86±9.63**| \n\n\nWe clarify that while Theorem 1 in our paper shares some similarities in proof structure with Theorem 4.12 in [1], the statement, setting, and generality of the result are different. Specifically, [1] focuses on convergence guarantees when the Mahalanobis nonconformity score is constructed using the empirical sample covariance matrix. Our result establishes convergence guarantees for a general positive semi-definite matrix $A$ or a convergent matrix sequence $A_T$, which includes but is not limited to the sample covariance matrix. This generalization allows us to theoretically justify the use of topology - induced covariance structures or other regularized estimators within the conformal prediction framework that are not discussed in [1]. Furthermore, our analysis includes theoretical conditions under which the choice of $A$ leads to efficient conformal prediction, while [1] does not address how to select $A$ beyond using empirical estimates. \n\nWe'll revise the related work to include the suggested concurrent research and refine the discussion on MultidimSPCI to accurately reflect its contributions and our clarifications. Thank you again for your valuable opinions!"}]

Author rebuttal:

Meta-review: This paper presents STACI, a topology-aware conformal prediction framework for stream networks that integrates spatial topology and temporal dynamics to achieve efficient and valid uncertainty quantification. Reviewers initially raised concerns about clarity, limited baselines, scalability, and the scope of evaluation, with some giving borderline or negative ratings. Through rebuttal and additional experiments, the authors clarified notation, added missing baselines such as CopulaTS and quantile regression variants, and discussed limitations and scalability, which convinced most reviewers to raise their scores, though one reviewer remained cautious about presentation and related work.
Decision: Accept (poster)