Flow-based Conformal Prediction for Multi-dimensional Time Series
Authors: Junghwan Lee
Chen Xu
Yao Xie
Venue: ICLR 2026 Poster (2026)
Keywords: Conformal Prediction
Time Series Prediction
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We propose a novel conformal prediction method for time series using flow with classifier-free guidance.

Abstract: Time series prediction underpins a broad range of downstream tasks across many scientific domains. Recent advances and increasing adoption of black-box machine learning models for time series prediction highlight the critical need for uncertainty quantification. While conformal prediction has gained attention as a reliable uncertainty quantification method, conformal prediction for time series faces two key challenges: (1) \textbf{leveraging correlations in observations and non-conformity scores to overcome the exchangeability assumption}, and (2) \textbf{constructing prediction sets for multi-dimensional outcomes}. To address these challenges, we propose a novel conformal prediction method for time series using flow with classifier-free guidance. We provide coverage guarantees by establishing exact non-asymptotic marginal coverage and a finite-sample bound on conditional coverage for the proposed method. Evaluations on real-world time series datasets demonstrate that our method constructs significantly smaller prediction sets than existing conformal prediction methods, maintaining target coverage.

Published as a conference paper at ICLR 2026
FLOW-BASED CONFORMAL PREDICTION FOR MULTI-DIMENSIONAL TIME SERIES
Junghwan Lee, Chen Xu & Yao Xie
H. Milton Stewart School of Industrial and Systems Engineering
Georgia Institute of Technology {j {jlee3541 3 gatech..
yao. yao.xielisye. wat.syy...aac.e.e.....
ABSTRACT
Time series prediction underpins a broad range of downstream tasks across many scientific domains. Recent advances and increasing adoption of black-box ma- chine learning models for time series prediction highlight the critical need for uncertainty quantification. While conformal prediction has gained attention a reliable uncertainty quantification method, conformal prediction for time se- ries faces two key challenges: (1) leveraging correlations in observations and non-conformity scores to overcome the exchangeability assumption, and (2) constructing prediction sets for multi-dimensional outcomes. To address these challenges, we propose novel conformal prediction method for time series using flow with classifier-free guidance. We provide coverage guarantees by establish- ing exact non -on-ysympi marginal coverage and finite-sample bound on condi- tional coverage for the proposed method. Evaluations on real-world time series datasets demonstrate that our method constructs significantly smaller prediction sets than existing conformal prediction methods, maintaining target coverage.
INTRODUCTION
Uncertainty quantification has become essential in scientific fields where black-box machine learn- ing models are widely deployed (Angelopoulos & Bates, 2021). Conformal prediction (CP) has emerged as a distribution-free framework for uncertainty quantification with coverage guarantees, ensuring prediction sets contain the true outcome at specified confidence level (Shafer & Vovk, 2008; Vovk et al., 2005). By constructing uncertainty sets using non-conformity scores that quantify how atypical predictions are, CP generates reliable prediction sets that satisfy a specified confidence level.
Time series prediction aims to forecast future out- comes from past sequential observations of features and outcomes (Box et al., 2015), underpinning a broad range of downstream tasks. Recent advances in machine learning have led to the development of various foundation models designed for time series prediction (Kim et al., 2025; Miller et al., 2024; Wen et al., 2023). The growing adoption of such mod- els for time series prediction highlights the press- ing need for reliable uncertainty quantification. Al- though CP has been actively studied for reliable un- certainty quantification, most cxisting CP methods rely on the assumption of data exchangeability (Bar- ber et al., 2023). The exchangeability assumption is frequently violated in time series data, where obser- vations exhibit complex temporal dependencies that induce correlations in the non-conformity scores, thereby making the direct application of CP to time series prediction particularly challenging. An addi- tional challenge is that modern time series data of-
Enc
(+112.1-3
0.00
-0.25
-0.50
Po
(xalhh
0.0
i-3
-0.5
-1.0
1-1
-1.5
Figure 1: Our method adaptively constructs the prediction set using flow transforma- tion conditioned on guidance h4, which en- codes the past context.

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24
Ergakir MettinnFCI
RCP
Ro
04
-15
-04
-15
(a) Wind dataset
(b) Traffic datasct
(c) Solar dataset
Figure 2: Comparison of the prediction sets at a target coverage of 0.95, constructed by FCP (ours), MultiDimSPPI (Xu et al., 2024), and conformal prediction using empirical copula (Messoudi al., 2021) on (a) wind, (b) traffic, and (c) solar datasets. Prediction sets are manually selected from the test set for visual clarity. Two prediction sets are shown for the wind dataset.
ten contain multi-dimensional outcomes. While CP methods for univariate outcomes are well- established, extending these methods to generate prediction sets for multi-dimensional outcomes not straightforward and requires careful consideration in constructing prediction sets.
There has been substantial effort to extend CP beyond the exchangeability assumption. One line of research focuses addressing distribution shifts in the data (Barber et al., 2023; Tibshirani et al., 2019). More recently, several works have developed CP methods for time series. For example, Xu & Xie (2021a) proposed method to construct sequential prediction intervals for time series based on a bootstrap ensemble estimator, which were later extended to incorporate conditional quantile estimation in order to exploit correlations in non-confornity scores (Xu & Xie, 2023b). Auer al. (2024) used modern Hopfield networks to capture temporal dependencies for sample reweighting and constructed prediction intervals based on these weights. Another line of work has proposed multi-step conformal prediction methods for time series, but these approaches assume access to multiple i.i.d. time series sequences (Stankeviciute al., 2021; Sun Yu, 2022), which may limit their applicability in practical settings. Despite these efforts, most of the existing methods remain limited to univariate outcomes.
Constructing prediction sets for multi-dimensional outcomes has been active area of research. Early approaches used copulas (Messoudi al., 2021) and ellipsoidal uncertainty sets (Hender- son et al., 2024; Johnstone & Ndiaye, 2022; Messoudi et al., 2022), yielding hyper-rectangular and ellipsoidal prediction sets, respectively. Subsequent research has aimed to move beyond specific ge- ometric shapes of prediction sets: Braun et al. (2025) formulated structured non-convex optimization to obtain minimum-voliu Recent works have focused on transporting multi-dimeessonal non-conformity scores to a reference distribution from which prediction sets can be constructed. For example, Klein et al. (2025) and ranks (Chernozhukov et al., 2017; Hallin al., 2021)
sets; and Tumu et al. (2024) used convex templates for prediction sets.
Thurin et al. (2025) used Monge-Kantone to map multi-dimensional non- non-conformity scores onto reference distribution to construct predic- tion sets, by solving optimal transport problems. Fang et (2025) applied conditional normalizing flows to map multi-dimensional non-conformity scores to the source distribution and construct pre- diction sets using calibration set with the source distribution.
Consequently, an effective CP method for time series prediction should address the two aforemen- tioned challenges: overcoming the exchangeability assumption and constructing prediction sets for multi-dimensional outcomes. To the best of our knowledge, Xu et al. (2024) is the only work that seeks to jointly addresses both challenges, constructing ellipsoidal prediction sets by defining non- conformity scores as the radii of the sets and predicting these non-conformity scores conditionally.
In this work, we propose a novel conformal prediction method for time series prediction with multi- dimensional outcomes. Our method is designed to effectively address the two challenges by using flow with classifier-fres guidance. Specifically, we use flow to model the distribution of predic- tion residuals and their transformations conditioned on past context. which is encoded by using Transformer (Vaswani et al., 2017). We define the non confoormnmtt score as the Euclidean distance between the transformed prediction residual and the mean of Gaussian source distribution of the
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flow, which allows to construct prediction sets desired confidence level. We provide theo- retical coverage guarantees by establishing an exact non -symmtootiiic marginal coverage and a finite- sample bound conditional coverage for the proposed method. Empirical evaluations three real-world multi-dimensional time series datasets demonstrate that the proposed method constructs significantly smaller prediction sets while maintaining target coverage, outperforming existing base- lines.
2 PROBLEM SETUP
We consider a sequence of observations {( V:) : 1,2,...J. where E IRd represents dæ- dimensional feature, and y/ IRdv represents dy dj-diimnsioonl continuous outcome. We assume that we have a base predictor that provides point prediction O for ye. given by ú: = (()()))=))) where & specifies the size of the past observation window. The base predictor f can be any black-box model and is not restricted to any specific constraints.
Suppose that the first T examples, {(r,, yr) 3:)] are used for training. Our goal is sequentially construct a prediction set C,(2,, a) for the next step, beginning time i T + 1. Here, 2, denotes the features used to construct C, and a E [0. 1] denotes pre-specified significance level. In the simplest setting, 2 consists only of 24. but it may also include past features, outcomes, conformity scores. We aim to construct prediction sets that satisfy marginal coverage:
IP U E C((2),)) z -, Vi,
(1)
and ideally conditional coverage:
P P( P E ((22, 2 aa Vi.
Although trivially large prediction sets can always satisfy the target coverage, they do not provide useful information for uncertainty quantification. Therefore, the meaningful objective is to construct efficient prediction sets the prediction sets that are as small as possible while satisfying the target coverage (Vovk al., 2005).
Throughout this paper, we distinguish between the indices i and to avoid confusion: the subscript i refers to the discrete time index of the sequence of observations, while the subscript / is reserved to refer to continuous time in ODEs. We use uppercase letters (e.g., X) to denote random variables and lowercase letters (c.g., x)) denote their realizations.
METHOD
3.1 PRELIMINARY: GUIDED FLOW
We use x as a generic variable in this section, distinct from the time series feature x; introduced in the problem setup. A flow is a time-dependent mapping á [0, x Ie Idd that push-forward a random variable Xo E Rd from source distribution to X₁ E Rdd from time-dependent probability density (i.e., probability path) Pt at time E [o, 1] as follows:
(Iv)).p))<ze) (1 det
(ar)
(3)
Os.
where * denotes the push-forward operator, det(-) denotes the determinant, and (x) := (tt, Flow &b is defined by vector field [0, 1] Id Rdd through the following ordinary differential equation (ODE):
d (10) (z(b)
(flow ODE)
vo()o) 20.
(initial condition)
A guided flow exppa : [0, x Rd x ah + Rd enables conditional generation by learning a mapping from source distribution target conditional distribution, and is defined by guided vector field
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Mrja : [0, 1] x IR Rh - Rd with the following ODE:
di (an fi) weth (Nrea (zo (t) kt,,
(guided flow ODE)
(5)
11=0jh (vo (.) :0:
(initial condition)
where h E Rd denotes the guidance. By appropriately designing a conditional probability path per sample 3i interpolating Po|z, (ar x1)) = po and Pi|e, (ar 21) òzz, where 6z, denoting the Dirac
delta distribution centered at 21, we can obtain the marginal guided probability path:
Paja (2 h) f Palay (or (1) dri;
(6)
which interpolates the source distribution Po and the target conditional distribution q( h). Given the conditional vector field Ut|z, that generates each conditional path Pt|z, the marginal guided vector field is obtained as:
f Uujon (x x1) Pelar (xr (1))((11 h) dr1.
Ueja (z h)) -
(aja h)
One can verify the marginal guided vector field generates the marginal guided probability path using the continuity equation (see Proposition A.I). Therefore, in order to learn the target conditional distribution, parameterize the guided vector field with neural networks and train it to approximate the marginal guided vector field as closely as possible. A simple and effective way to train the guided vector field is through flow matching, which minimizes the mean-squared error between the conditional guided vector field and the parameterized guided vector field (Lipman et al., 2022):
LCFM (x JuI (xc h) "el (x 221
(8)
where t Unif[0, 1], h) Idata and u/ is the guided vector field with parameters 0.
We consider Gaussian conditional probability path defined as Pela (x x1)= = N( a,31, La), where N denotes the Gaussian kernel and Ia E Rdxdd denotes the identity matrix. on or [0. 1] [0, interpolating scheduler, which are smooth functions satisfying Qg, 01 0, or du 1, and d dt 0r _0, 0 for E (0, 1). The guided vector field ugja(z h) can be reformulated as:
uaja h) (x) + V. log (h Ix),
where ut (z) is unguided vector field, b, is a scalar constant regarding a and og (see Proposi- tion A.2). Based on this reformulation, early approaches trained separate classifier (Song al., 2020) with a classifier scale w is beneficial in conditional generation in practice (Dhariwal & Nichol, 2021):
C h)) = (x) ) + wb, v log Phte(h x). By using the identity V. log Pt A (x h) = V log (x) + Va log Pa (h x)), equation (10) can be equivalently rewritten as:
(10)
Ten (x h) = (1 - (x) + wrge (x h).
(1I)
Instead of modeling (xr) and (x h) separately, Ho & Salimans (2022) proposed using single vector field to model both cases by assigning a null condition ha to represent the unguided vector field, which known as classifier-free guidance (CFG):
depa (x h) = (1 w w)4n(( hg) + wurja (x: h),
(12)
where his denotes the guidance representing the unguided state of the vector field. The guided vector field can be trained using flow matching with the loss:
[| (1 n) + nha) - uaja, (x mm)
CPG CFM
= Be.miter.hh)
(13)
where 7 ~ Bernoulli(pz) and pa denotes the probability of assigning hg. The resulting guided vector field (njn(x h) in equation (12) enables conditional generation by solving the guided flow ODE and has been widely used in various tasks such as image generation (Esser et al., 2024) and video generation (Polyak et al., 2024).
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Algorithm 1: Training Guided Flow using Flow Matching
Input: Pe, initialized udj.. and Enc"
while not converged do
& < Vi
> obtain prediction residuals
h, <- Enc" (z.)
P obtain guidance
h. + ho with probability pe
> assign unguided state with probability pa
2n po()))
t Unif(0,
2t f oft + 0,300
Male + d aus,, + 00
dt
Update with Vollugja (Ix, (s), uesell?
flow matching loss
Output: trained w/y, and Enc"
3.2 CONFORMAL PREDICTION FOR TIME SERIES USING GUIDED FLOW
We use guided flow to learn a mapping from the source distribution to the distribution of predic- tion residuals é y D. conditioned on past context. The prediction set is then defined through this transformation using guided flow to achieve the target coverage. This construction effectively addresses the two aforementioned challenges in conformal prediction for time series. First, guided flow leverages guidance from the encoder derived from past context, enabling accurate estimation of the conditional distribution of é and mitigating the reliance on the exchangeability assumption. Sec- ond, since guided flow defines transformations between random variables in arbitrary dimensions, it enables the generation of prediction sets for any multi-dimensiona outcomes. Figure provides a visual illustration of the method. We describe the method in detail in this section.
Guided flow design. We use Gaussian probability path with interpolating scheduler a; = / and a (1 t). The source distribution is set to isotropic Gaussian with zero mean and covariance scale y > 0, i.e., 7l, For each time index i, we construct by concatenating the past w features and prediction residuals, and use an encoder to obtain contextual representation h; Enc(2)) The guided vector field as defined in equation (12) uses h2 as the guidance to model the conditional distribution of é, In our method, we use Transformer as the encoder (Vaswani et al., 2017). though alternative sequence models such as recurrent neural networks (RNNs) are also applicable. The guided vector field trained via flow matching as defined in equation (13), and the encoder is jointly trained with it. The overall training procedure is summarized in Algorithm 1.
Prediction set. The trained guided flow models the conditional distribution of the prediction resid- ual by mapping from the source Gaussian distribution to residuals conditioned on the guidance. Since this transformation is bijective. we can define prediction sets for the residuals directly through the transformation. Let (y)) ;= || i (ē h4)ll be the Euclidean distance between the trans- formed residual and the origin, then the prediction set at significance level a can be defined as:
Ci(2,, a)) = (v 4vv) S ri-a).
(14)
where 71-a is the radius of the ball B1-a that contains 1 - probability mass. Since we use (0, as the source distribution, the radius r r is given by r1-a = v1xz,(1-a), where xā, denotes the inverse cumulative distribution function (CDF) of the chi distribution with dy degrees of freedom. Intuitively, the prediction set is obtained by taking the ball that contains the same amount of probability mass as the target coverage and transforming it to the prediction set for the residual using the guided flow. Although this construction directly uses ê(y) to construct the prediction set, é(y) is computed from the transformed residual and therefore serves proxy non-conformity score, consistent with treating residuals as non-conformity scores.
Since the prediction set is obtained through the transformation using guided flow, it can take on flexible shapes without being constrained to follow any fixed geometric form, such as convex or ellipsoidal sets. We believe this enables the guided flow to generate smaller prediction sets that are better aligned with the data. Although the prediction sets do not have any fixed geometric shape, some useful topological properties can still be inferred. In particular, Theorem A.4 and A.5 ensure
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that the prediction sets are closed and connected. Figure 2 shows prediction sets in R² generated by our proposed method alongside two other methods that produce hyper-rectangular prediction sets (Messoudi et al., 2021) and ellipsoidal prediction sets (Xu et al., 2024). The figure visually demonstrates the flexible shapes of the prediction sets constructed by our proposed method.
The size of the prediction set is computed as:
J |eet (Juc-nn (z) h))l dr = Size
1
det [Jt(S (ay h)]]
(15)
where 1i represents the flow transformation from = to 1, and Joh (xr h)) denotes the Jacobian of 1 at æ E B& conditioned on h. The right-hand side provides a Monte Carlo approx- imation, where 2j ii.d. samples drawn from B, and N is the number of samples. However, directly computing det (Jg, (x h)) is computationally expensive, as requires solving the guided flow ODE and evaluating the full Jacobian. Instead, we compute the reg-determinant of the Jacobian by solving the following ODE:
log det Juins (xr hi] = div (1e (h) 1t)
(Jacobian ODE)
(16)
log |et (Ju--a, (x h)) = 0,
(initial condition)
where div(.) denotes the divergence operator. A detailed derivation is provided in Proposition A.3. The accuracy of the prediction set size estimate depends on the Monte Carlo approximation. Purely random sampling from Ba may introduce bias due to uneven coverage of the sampling space, and a small sample size can result in high variance. To reduce sampling bias, we use quasi-Mons Carlo sampling based Sobol sequences (Sobol, 1967; Owen, 2023), which provides more uniform sampling from B.. To control variance from finite sampling, we monitor the relative error in terms of sample size N. Additional implementation details are provided in the experiments section.
THEORY
In this section, we present exact non-asymptotic marginal coverage and Enite-sample bound on conditional coverage. We assume that y/s E R4 is generated from unknown true function f with additive noise e, Rd according to yı = (x=))));)) + ej. Proofs are presented in Appendix A.
4.1 MARGINAL COVERAGE
We first establish that prediction sets generated by our method achieve exact non-asymptotic marginal coverage. This result follows from fundamental property of flow: probability mass preservation under push-forward operations. Lemma 4.3 formalizes this property and suffices to prove the exact non-asymptotic marginal coverage stated in Proposition 4.4.
Assumption 4.1 (Flow existence and uniqueness). The guided vector field ux( h)) is continuously differentiable and Lipschitz continuous in for all and h. That there exists a constant Lg such that
[[ h)) (a(" A)ll T-llix- - x'I], Vt,
(17)
Remark 4.2. Assumption 4.1 ensures the existence and uniqueness of solutions of the guided flow ODE. In practice, the guided vector field can be modeled using neural network architectures that satisfy this assumption, such as multi-layer perceptrons (MLP) with smooth activation functions.
Lemma 4.3 (Probability mass preserving property of flows). Let ~ px be a continuous random variable on IRd, and let vb Bdd > TR. be C¹l diffeomorphism. Define Y == (((X) with density py given by the fosh-forward of Px under vb. Then, for any measurable set C TRd, the transformed
set A' : (AA) satisfies:
P(X E A) P(Y E A')
(18)
Proposition 4.4 (Marginal coverage). Let a E (0, 1) be a pre-specified significance level. Under Assumption 4.1, suppose the guided flow provides sufficiently accurate approximation of the target distribution from from the source distribution. the ball B, defining the prediction set in equation (14)
has probability mass a, then the prediction set achieves exact marginal coverage of 1 a.

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4.2 CONDITIONAL COVERAGE
We next establish a finite-sample bound on conditional coverage. We define the non-conformity score based on the prediction residual é; (| h.)ll. and the non-conformity score based on the true noise as e, = [e 1-]l-- The guided flow tb is trained on the training set until convergence and then fixed for computing e and é. The empirical CDF of and are defined as:
(T+ - - T
1 E S..
1(: Se Fr+i(u)= -
(19)
We denote Fe(u) = PPe S u) as the CDF of the true non-conformity scores. Since the source dis- tribution of the guided flow in our method is set to be identical across time, the marginal distribution for e; can be considered to be identical for all i. However, while the marginal distribution of e, is identical for all i, they may exhibit dependence through h, Therefore, we consider two settings: (1) when { + are ii.d., and (2) when {er J T++ are stationary and strongly mixing. We first establish a finite-sample bound conditional coverage under the assumption of i..d. non-conformity scores. Assumption 4.5 (i.i.d. non-conformity scores). The true non-conformity scores {c:]7=1 are i.i.d.
Assumption 4.6 (Bi-Lipschitz flow). We assume that the guided flow ( h)) is bi-Lipschitz continuous in for all and h. That is, there exist constants Ligt O and L >> O, such that
[1 h)) V/(x' | hill S Læ11 2'11, "t,
(20)
and
1% (ar h) - 'a Alll S ---111> 10'1,
(21)
Remark 4.7. Lemma A.7 shows that bi-Lipschitz guided vector field results in bi-Lipschitz guided flow. Consequently, we can model the guided vector field to satisfy this assumption. For example, one may use invertible Residual Networks (iResNet) (Behrmann et al., 2019; Chen et al., 2019) with smooth activation functions.
Assumption 4.8 (Lipschitz continuous of the CDF of true non-conformity scores). Fa(u) is Lips- chitz continuous with Lipschitz constant LT+1 > 0, and that F. is strictly increasing in u. Assumption 4.9 (Estimation quality). Define A; = é EG. There exists sequence (ôr)ra) such that
1 T
Σ
|a..
S 87,
| S 8r.
(22)
As a result of Lemma A.9 and 1.1,, Theorem 4.10 establishes the finite-sample hound for condi- tional coverage under i.i.d. non-conformity scores.
Theorem 4.10 (Conditional coverage bound under i..d non-conformity scores). Under Assump- tion 4.5, 4.6, 4.8, and 4.9, suppose the guided flow provides a sufficiently accurate approximation of the target distribution. With probability 1 1- 6, we have:
|((Y+1
ZT+1 2T+1) (1-a)
(23)
log(167))
+ 4(LT+1++ + ! + &8).
s 12
T
Next, we present a finite-sample bound on conditional coverage under the assumption of stationary and strongly mixing non-conformity scores.
Assumption 4.11 (Strictly stationary and strongly mixing non-conformity scores). {e:)7- are
strictly stationary and strongly mixing, with mixing coefficients satisfying 0 < Eiso (k) M <
Corollary 4.12 (Conditional coverage bound under stationary and strongly mixing non-conformity scores). Under Assumption 4.6, 4.8, 4.9, and 4.11, suppose the guided flow provides a sufficiently accurate approximation of the target distribution. With probability
we have:
PP(Y+11 E C
Zr+1 =x-1) - -(1 (a)
(24)
1/133 (log ()
+ 4(LT+1 + 1 1 + +T))
< 12
T1/3
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The bounds in Theorem 4.10 and Corollary 4.12 depend the sample size T and the estimation error ōy. Both bounds converge to 1 a as T + oo, provided that 87 = ( ) for some a > 0. Intuitively, this implies the conditional coverage is guaranteed with sufficiently large training data and an accurate base predictor f. The condition őt can be satisfied by broad class of estimators. For example, sieve estimators based on general neural networks achieve by op (T(T)//4 when is sufficiently smooth (Chen & White, 1999). The Lasso estimator and Dantzig selector achieve ör = op(TT 1/2) when f is a sparse high-dimensional linear model (Bickel et al., 2009).
5 EXPERIMENTS
For notational convenience, we refer to our method as FCP, which stands for Flow-based Conformal Prediction. The guided vector field is modeled by an MLP with Softplus activations. We concatenate the guidance and time with the input for the MLP. dopri5 (Dormand & Prince, 1980) at absolute and relative tolerances of 10-5 is used to solve ODEs. A grid search con- ducted to select the optimal hyperparameters for FCP. To determine an appropriate sample size N. we compute the relative standard error (SE) of the Jacobian determinants of ét, defined as SE ((det J (zz 1n)}) /Avg ((det Jplet (æ( h)),"-)) then choose the smallest N such that the average relative SE across all falls below 0.01. Additional details on hyperparameter selection and the choice of N are provided in Appendix B.
Baselines. Wc evaluate FCP against several conformal prediction methods: MultiDimSPCI (Xu et al., 2024), OT-CP (Thurin et al., 2025), CONTRA (Fang et al., 2025), conformal prediction using local ellipsoids (Messoudi al., 2022), CopulaCPTS (Sun & Yu, 2022), and conformal predic- tion using empirical and Gaussian copulas (Messoudi et al., 2021). We also include two widely used probabilistic time series forecasting methods as baselines: Temporal Fusion Transformer (TFT) (Lim et al., 2021) and DeepAR (Salinas et al., 2020). Although TFT and DeepAR are orig- inally developed for time series with univariate outcomes, adapt them our multi-dimensional setting by constructing independent copulas using the predicted intervals for each output dimension. Further details on the baseline methods and their experimental setup are provided in Appendix B.
Datasets and base predictor. We evaluate FCP and baselines on three real-world time series datasets: wind, traffic, and solar datasets. Additional details on the datasets are provided in Ap- pendix C. For the wind and traffic datasets, we randomly select dy E {2, 4, 8) locations construct five sequences of d, didiemnniooa time series. For the solar dataset, we use dy E {2,4] and sim- ilarly construct five sequences. Base predictor f is required to provide point prediction ÿ. We use two types of base predictors: (1) leave-one-out (LOO) bootstrap ensemble of 15 multivariate linear regressors, and (2) recurrent neural network (RNN) with long short-term memory (LSTM) units (Hochreiter & Schmidhuber, 1997). Since the RNN base predictor requires part of the se- quence for training, whereas the LOO bootstrap predictor can leverage the full sequence, the effec- tive sequence length available for evaluation varies by base predictor. The base predictor is trained independently for each of the five constructed sequences. For the RNN base predictor, the first 50% of each sequence was allocated for training, and predictions were made for the remaining 50%, which served as the evaluation sequence. Within this evaluation sequence, the first 80% was used as training set, and the final 20% was evenly divided into validation and test sets. Since FCP does not require a calibration set to construct prediction sets, the validation set was used for model selection during training. To ensure fair evaluation in terms of data utilization, we combined the training and validation sets into a single calibration set for methods that require a calibration set.
Evaluation metrics. Efficient prediction sets are those that are as small as possible while satis- fying the desired coverage. Therefore, average prediction set size. The empirical coverage at a target confidence level a is defined as:
two evaluation metrics: empirical coverage and the
Σ 1
( E C(a,))))
(25)
Drest
where Dies denotes the test set. The average prediction set size is computed by averaging the sizes of C2 over the test set, with the specific definition of the set size depending on each method.
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Table I: Average empirical coverage and prediction sets size obtained by FCP and all baselines on three real-world datasets, evaluated under different base predictors and varying outcome dimensions dy. Reported values represent the average and standard deviation over five independent experiments conducted on five constructed sequences. The target confidence level is set to 0.95. Results with average empirical coverage below the target confidence level are grayed out, and the smallest pre- diction set sizes, excluding the grayed-out results, are highlighted in bold.
Dataset
Base Predictor
Method
dy 2
dy = 4
dy
Coverage
Size
Coverage
Size
Coverage
Size
RCP
0.951 g.ois
0.88 nos
3.43 11:a7
0.956
19.4 1102
MuleDimSPCI
0.953g.01
1.31
0.968g.01
6.39 oo
0.951 024
205.5g o0.5
CopulaCPTS
1.0g.000
22.5 22.3-215.0
1.0
611.3:246477
1.0g.
3.50 x 10
OT-CP
2.961
2.71
0.958
y.u.4
0.927 2.007
Bootstrap
CONTRA
0.970
32.9 3229.
1.0002..00m
7.80 x 10% 1i-
0.904
5.88 10 11-00t10
Local Ellipsoid
0.964_.00
419
2.18
8.63
0.974
394. 42522
Empirical Copula
0.951..01
1.22 316
4.94 57
0.948 012
27.4
Goussian Copula
0.945_.017
1.17
0.968
5.1 Tl
0.948=.012
77 77-4225.1
TFT
0.728..1v2
0.515
4. 1.266
111877
6.70g9..
Wind
DespAR
0.9001_.000
1.322.445
0.6722..20
4.84.a3.0
0.320
FCP
0.952 161
1188 R15
0.967
10.8
0.953
2.48 10
MuleDinsPPCI
0.974g.mmm
a.79
0.926
0.896g.o01
5.58
10 1107
CopulaCPTS
1.0g.000
25.7445.4
100
4.82 x 10 max
1.0g.000
2.83 x 10 2a00o
OT-CP
0.970_.m0
9.18
g.989g.ooa
212.3 2.2.9.
0.913g..0
8.39
LSTM
CONTRA
0.317 3177
0.
2.204
0. 0.1512e.124
0.761
25, 25.0.am.
Local Ellipsoid
0.978
10.5
354.4
1.0g.com
2.63 10 vo-cio
Empirical Copula
0.983 005
14.2
1.0
(94.5_2190
1.0g
4.46 x 10 20822100
Gaussian Copala
0.083_.06
14.1 28.18
1.0g000
409.1 2199s
1.0g.co
5.24 10 sa0tto
TFT
0.550
1.90 o96
0.355 166
3.93 3.98ga.o
11136gg
28.7gats
DeepAR
0.780
1 1.09
0.305
9.88
0.00
RCP
0:957 =.014
0.918
0.963..co
1.06
0.965
1.53
MultDinnPPCI
0.963 20s
1. 1.58 2.40
0.968 oo0
2. 2.62 c0s
0.971
10.7g4.0
CopulaCPTS
1.000
21.6
1.000..oe
645.8 645.8..65.55
1.000
3.18 x 10 200000000000
OT-CP
0.966 000
2.00 uss
32 12.0g200
0.954g..o om
3.90
10 10.1022
LOO Bootstrap
CONTRA
0.950
1.32
0.963_
L,58 g
0.931
6.21 11gss
Local Ellipsoid
0.970
2.04
0.975
2.90
0.980
3.82
Empirical Copula
0.973 003
2,35 c46
0.972 201
5.61 48
0.970 005
40.420.04
Goussian Copula
0.9732_.00
2.37
0.9722.004
5.61 as
0.970=..00
40-420.04
TFT
D 0.407=.00
0.2922:000
0.189
0. 0.07=
0 0.09
O 0.0092
Traffle
DespAR
0:443..0o1
0.3088 oos
0.197
0.07 sucon
0.00
0.00
FCP
0.968 2.02
0.859
0 8,560 022
1.05
0.9500g.00 0000
1.82 257
MaIdDimSPCI
0.957
0.570
0.960
1.59
0.952
14.2
CopulaCPTS
1.000_.00
21.9
1,000
330.0 230.0
0.9022.000
4.47
OT-CP
0.953 006
0.920 370
0.539 000
11.8 L
0.50 0.921
730. 20022aaee
LSTM
CONTRA
0.340
0.222
0.9422.00
0. 0.106
0.910
0,050
Ellipsoid
0:957
0,987 415
0.9482.00
1.48g.500
0.922g..01
3.3 905
Empirical Copula
0.955 000
3.81 629
0.948 o00
25.8 1.5
0.920 0017
1.22
10
Gaussian Copula
0.953
3,7
0.952z.011
26 16.424.00
0.9200..00
1.22
101
0.3742.110
0.1922..om 012
0.06k..co2
0.06222.013
0.0008- 002
DeepAR
0 0663
0,266 oo0
0.211 n06
0.00 L.10
0.09 0.09__cmm
0.003, 001
FCP
0.957 1.007
148
2013
0.969
4.18
MalnDimSPCI
0.968_.00 005
1.97 076
0.971g.00m
11.4gg:00
CopulaCPTS
1.000__0000
Pe126
1.000
7:25 x 107, 10111900000
OT-CP
0.984_.00
3.69
0.971
248. 9
1G0 Bootstrap
CONTRA
0.950=..01
3.08
80.8
Local Ellipsoid
0.917=.004
1.41
0.918
1.87=.240
Empirical Copula
0.986
174
0.988
36.524410
Gaussian Copula
0.986
4.47 171
0.989
38. 28.2
TFT
0.779 o56
0.722_.com cco
1.18
Solar
DeepAR
0.802
1.03g 114
0.7 d60
6.75
FCP
D. 0.968_.co
1.16
0.961
2.09
MulniDimsPCC
0.9081
1.31 20000
0.976
6.465 81
CopulaCPTS
1.000 200
44.8
1,000 200
3.34 x 10
OT-CP
0.979
2.25
0.963
142.0g
LSTM
CONTRA
0.009_.002
0.1004
0.
0.0222-.014
Local Ellipsoid
0.972=.0m. 005
1.27 148
0.978 cod
2.432.000
Empiriaal Copula
0.987
6.472.105
0.5000=.0m
67.7 67.7=109
Gaussian Copula
0.992=.00
7.11+:216
0.997=.001
89.9 89.9-4.
0.740-.m.:
0.6844..0s 0.651.2.m
1.63
DeepAR
0.839 008
1.0
0.715g.043
3.57 m0

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Published as a conference paper at ICLR 2026
Results. Table presents the results of experiments three real-world datasets. FCP consis- tently obtains smaller prediction sets than all baselines while maintaining the target coverage. The performance gains of FCP are especially notable for higher outcome dimensions, showing signifi- cantly smaller prediction set sizes with lower variability. Moreover, FCP maintains stable coverage across varying dy. whereas baseline methods often suffer from undercoverage overcoverage cou- pled with either overly contracted or inflated prediction sets. In particular, methods relying on the exchangeability assumption often exhibit severe miscoverages and unstable prediction set sizes.
MultiDimSPCI and CP using local ellipsoids generally show good performance. In particular, on the solar dataset, CP with local ellipsoids achieves performance comparable FCP. This is possibly due to their ability to capture temporal or local correlations, respectively. OT-CP and CONTRA also perform well in certain experiments, indicating some potential to adapt beyond the exchangeability assumption. We observe that increasing the guidance scale w often reduces the prediction set size, though at the cost of slightly lower coverage. In practice, an effective range for w is typically between I and 1.5 across our experiments.
Ablation study. We conduct ablation study to assess the impact of the encoder. Specifically, we evaluate FCP with and without the encoder, where in the latter case the guidance is replaced by the concatenation of the feature time and residual at time 1. Table 2 reports the average empirical coverage and prediction set sizes of FCP with and without the encoder on the wind dataset. We observe that removing the encoder led to less stable coverage and noticeably larger prediction set sizes.
Since the conditional coverage bound of FCP relies on the bi-Lipschitz flow assumption (Assump- tion 4.6), we conduct an additional ablation study using iResNet (Behrmann et al., 2019) to model the guided vector field, ensuring this assumption satisfied. Table 7 reports the average empirical coverage and prediction set sizes of FCP with MLP and iResNet across the three datasets with vary- ing dy. We observe that imposing bi-Lipschitzness in the vector field does not degrade coverage or prediction set size.
Table 2: Average empirical coverage and prediction sets size obtained by FCP with and without the encoder on the wind dataset, evaluated under different base predictors and varying outcome dimensions dy. Reported values represent the average and standard deviation over five independent experiments conducted on five constructed sequences. The target confidence level is set to 0.95.
Base Predictor
Method
d, =
d, =
dy
Coverage
Size
Coverage
Size
Coverage
Size
FCP with Encoder
0.951
0.88
0,953 oo6
3.43
0.956 a1o
19. 19.4
LOO Bootstrap
FCP w/a Eieende
0.9484.02a
1.13
0.9644 000
3.99
0.964_.010 .010
35-3414.0
FCP with Encoder
0.9522.064
1.18
0.957
10.8
0.953 ose
2.48 10
LSTM
FCP w/o Encoder
0.965
1.92:..97
0.957 014
12.2
0.935 007
5.55
10 47.470221
CONCLUSION
In this study, we proposed novel conformal prediction method for multi-dimensional time series using flow with classifier-free guidance. We provided coverage guarantees of our method by estab- lishing exact non-asymptotic marginal coverage and a finite-sample bound on conditional coverage. Experiments on real-world datasets with broad set of baselines demonstrated that our method con- structs smaller prediction sets while satisfying the target coverage, consistently outperforming the baselines. Future work will focus investigating flow with optimal transport coupling construct more efficient prediction sets, and on conducting extensive evaluations across diverse datasets with higher-dimensional outcomes.
10

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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
The authors would like to thank Hanyang Jiang and Jonghyeok Lee for the insightful discussions. This work is partially supported by NSF CAREER CCF-1650913, NSF DMS-2134037, CMMI- 2015787, CMMI-2112533, DMS- 106, DMS- DMS-1830210, and the Coca-Cola Foundation.
ETHICS STATEMENT
We confirm that study complies with the ICLR Code of Ethics and does present additional ethical issues.
REPRODUCIBILITY STATEMENT
The source code for the method and experiments is available at https://github. com/ Jayaos/ / /flow_pp. Details of the experimental setup, including hyperparameters, datasets, are provided in the Experiments section and the Appendix B.
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A PROOFS
Proposition A.1. Let us m, (x x1) be the vector field generating the probability path Pile, (x vi).
Then, the vector field (q (x] h) a valid vector field generating Pejk (x: h).
Proof. Since Vaja (x xi) generates the probability path Pt (x 21), the continuity equation holds as follows:
a
Palar (x xı) + div (urjan (ar xı Pi) (x x1)) 0.
(26)
Ot
By the definition of the guided marginal probability path, we have:
Pena (2 h)) = Dee (r x1)g((1 | h)dø1.
(27)
The time derivative Paja (x h) expressed as:
/ Piles (x xi)a(() h) da
0i Pajh (x ())
Ø1.
/
ai Peles (o mn)g(an h) dai
(28)
/ div ( (o an Pat (w xi)) g( h) dai
-div / Jumen (a 1 (r x1)g((1) | h) dzl.
The marginal guided vector field is defined as follows:
(x h) = unlon (xr xx) Pelas (xr xn (1)4(11) h)
(an h) dx1.
(29)
We can rewrite the marginal vector field as:
Uy k (x (> h) / uelor (xr a Petee (x xr Da(a h) dz
(30)
Substituting equation (30) into equation (28), we have:
ot Pala (r h) -div (ueja(1) fa) (2) [h)),
(31)
which is the continuity equation for Pija (z h) under the guided vector field unja h). Therefore, uqjaa: (x h) is valid vector field generating (ap( (r] h).
14

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Published as a conference paper at ICLR 2026
Proposition A.2. With a given Gaussian probability path Pe|zz (z (1) Na of the guided vector field (an( h)) can be reformulated as:
uapa( h) ux(x) + b. log 1x).
(32)
Proof. By the definition of the guided marginal probability path, we have:
Pala h) Pe (xr x1)g((1 h)d21,
(33)
where Pela, (z 21) (a o
We can express the score function of the guided marginal probability path as:
V Peja h) = (aPin( h))
(34)
Pepa (x (h)
V Pela (xr x1)4(21 h)dr1
(35)
(1 h)
vx logptta (x x1) Ptla (x) xn 21)9(21 h))
da1,
(36)
Pijo h)
where (i) follows from the definition of the guided marginal probability path.
Since Pale, (x x11) = Nx (4 of I), we have:
à; (x a1t1) + +
uelzr (x x1) =
(37)
Tt
à
å
r a1 + 2
(38)
σt
at
à: 04 )
+ (ùt
(39)
at
0t
åi
1
x + (àngt - (zăāt)
(x - axx1)
(40)
att
di x + (áror - anăā) ot v log Pilz, 2 (xr x1),
(41)
where di denotes de 0r, and de denotes d (z axT1).
de o. The last equality holds since Vz log (x x1)
The guided vector field is defined as:
ueja(x (x h)) = f Mejer
(x Jujj
x1) Ptlar (I x1 (1))((11 h) (e h)
dr
(42)
Therefore, by pugging uaaa (x x1) into equation (42), we have:
ueja (x h) = ax + b, V. log h),
(43)
where a, dعe and b, (ájg, and a. ar
By using the identity V. (ej h) = V Vologhijeh 1)) + V log (x)), we obtain:
ux(x h) a, + b, (V (Vlogphehh x) + Vlogp( u, (x) + V, log logenj((h (2). x)).
(44)
Proposition A.3. The log deternnnnnn Jacobian ODE defined in equation (16) is equivalent to the divergence the guided vector field.
15

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Published as a conference paper at ICLR 2026
Proof. The Jacobian ODE is defined as:
di Junn (a ht) (Vein hi) 1 h))
d =
(venaa (1))) (1pe c (t)
Jeina (x) h),
(45)
Oeep (2r
dx
with the initial condition:
Jun (x 12) I. By using Jacobi's formula, we have:
(46)
d det Junn (x fi) det Juhn (or h) (Jrin (x h)) d 11) dt
Jeen dea (x hi)
(47)
dt
Substituting equation (45) into equation (47), obtain:
Oueea (vee (2: h)) Oupen( h)
dt det Juin (x lt) det Jonn (x h) tr
(48)
Therefore,
(
(Dun [1))
log [og|det.e Juija (x (x h)] = tr
(49)
dt
Oxpej(( h)
Since the trace of the Jacobian of a vector field corresponds to its divergence, we have:
tr Queej (1i (x: (b)) = div (ueja (aja h)))
(50)
Ovan(1 h)
Therefore, the log-determinant of the Jacobian ODE is defined as:
log det Juen (x [n] div (Ven((aa h)))
(51)
dt
with the initial condition:
log det Josroja (xr [b) 0.
(52)
L
Theorem A.4 (Continuous images of connected sets, Rudin (1953)). Let X and be topological spuces and v/ X + Y be continuous. If E C X is connected, then v/ (E) is connected.
Theorem A.5 (Closed sets under homeomorphism, Munkres (2000)). Let X and Y be topological be homeomorphism. If X closed, then (EE) C Y is closed.
spaces and X
Lemma A.6 (Lipschitz continuous of the guided flow). Let the denote the guided flow defined by guided vector field wt. If the guided vector field ux(x h)) is Lipschitz continuous in x for all l and h, iee, there exists constant In 0 such that
[| h) ux(a' hill S Lullx -are-x1] x'll
Vx, x' h.
(53)
then the guided flow (/ h) is Lipschitz continuous in x for all I and h. That is, there exists constant Lop such that
[xx hb) - -//x' |b)ll < Lolk x'll Vx, h,
(54)
Proof. Let d(t) = [x h)) (x bill and x(t) = (e( 11) - Ve(x' | [)). Since the guided
vector field is Lipschitz continuous, there exists L. such that
|| 1.) un (a" Aill < Tulli a'll:
1l, h x'
(55)
This is equivalent to
[ (1 1) h) (v h)) Mill Ten (22 M) shela' [AIlls v,, (56)
16

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Published as a conference paper at ICLR 2026
Since d(t) + 0, we have:
d
d(() ||=((]] (=((), d =()) - - (22() d
1
2(t)
'=(OIl' dt =())-
(57)
dt
di
Since =1) = (1 h)) h)) (1 h)) h)), by Cauchy- Sannch-Schwwaz.e inequality, we have: x(t) | at =()) - (1r(( h) h) (1/((( h)) |b]l
(58)
which implies:
dt d(t)) - (1/ 1) h)) (1i [h) Ri]l S L,d(t),
(59)
where the last inequality follows from the Lipschitz continuity of the guided vector field.
Based on Gronwall's inequality (Gronwall, 1919; Hirsch et al., 2013), it follows that
d(t) d(0) el
(60)
This can be proved by defining (1)) == e L. d(1). Then,
d
dt (t) c liat d(t) Lud(t)) <
(61)
dt
which implies o is non-increasing on t. Since (tt) S &(0) = d(0) for all t E [0, 1). multiplying both sides by chat yields:
d(t) S d(0)etut
(62)
Therefore,
log d(() - log d(0) S Luth.
(63)
Exponentiating both sides gives
d(t)
d(0) S Lut
(64)
and hence
d(t) S d(0)e
(65)
Since d(0)- |] [h) vola' [b]ll || r'11, we conclude that
[<v((> h) ve(2'' |b]ll e || x'1l
(66)
Proof of Lemma 4.3. Since the probability density function of Y (X) the push-forward of Px, we have:
(y) Px (v- (1y)) (dct Jm-:())
(67)
where det A denotes the determinant A and J 1 8y '(v) probability mass of the transformed set A' (AA is:
is the Jacobian of -ー The
oy
P(Y E A') / (v( dy.
(68)
Using the change-of- y = (x) with dy = det Jo (x)(dz, we have:
[ pv(() dy / py (v(2)) /det Jo())) dr.
(69)
Substituting from equation (67), we have:
f py (1)))) /det J (z) dx / px px(x)) /det J (1b2)) |det Jx) dz.
(70)
Since J (1(x)) = -1 we know that det J det (x)) = 1. Hence,
pr()) dy = f px (x)
(71)
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Lemma A.7 (bi-Lipschitz guided flow). Assume that the guided vector field is bi-Lipschitz in z over for all t and h, i.e., there exists Lu and Lu such that
La112 x'1l - |lue( h)) vela' Aill S Lull x'll
Vt, 2,,,,
(72)
Then the guided flow & is bi- bipsipschit There exists Ly and Lo such that
1011 2'll 11b( hb) velel Rt]l S Lø 2'll Vt, vt,
(73)
Proof. Proof follows similarly to Lemma A.6. Let z(1) = (3r ht) (be(a') h)) and d(() = 11vx(z h)) - v/(a' [b)ll |
dt |=(()]12 2(2(t), d
di (t))
(74)
By Cauchy-Schward inequality,
dt 1=00] d(t)² 2 2||=(()|||| मt (t)|
(75)
dt
Since d =t) ue( h)) h) Ve' 1b)l, we have:
ua h) and || h) ue(a' hill 2 lallx x'1l
Lull Ve
de d(t))²
-(tlll((t)1 = - = d(t)
(76)
Using Gronwall's inequality,
| 11) v'e(2' [n]l 2 tuutf| x'll
(77)
Therefore, we know that
1<< h)) ve(a'' |b]ll 2 - =''l Vx, 2e, h
(78)
Combining with the upper Lipschitz bound obtained from Lemma A.6, we conclude that
tu || x'1l S (< h) - ( | < e e || - x'll
Ya, 22,2,,,,,, 1,
(79)
Lemma A.8. Under Assumption 4.8, Fe(eT+1) ~ Unif 1].
Proof. Since F. is strictly increasing and continuous under Assumption 4.8, the Lemma holds for eT41 Fa.
Lemma A.9 (Convergence of empirical CDF of i.i.d. (e:) Under Assumption 4.5 and 4.6. for any T, there exists an event Ar with probability at least v Ar,
[og(162)
such that conditioned on
log(16T)
sup Fr+1(2) ( Fo(x) s
(80)
T
Proof. This proof follows the proof of Lemma in Xu & Xie (2023a). Under the assumption that {er [+1 are i.i.d., the Kinetzky-Kiifer--olllel (DKW) inequality (Dvoretzky al., 1956; Kosorok, 2008) implies:
IP sup FT+1(x) - Fo(2)) > 8T
< 2c 2T
(81)
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Choose sT (W(16T)((2V)) where (T) denotes the Lambert W function satisfying ((T) T. Since (16T)) I 1og(16T)) it follows that ST Vlog(16T)/T. Define
the event AT which sup, (x))(x)= Fx(x) /log(16T)/T, so that we have:
1og(166)
sup
Fr+1(x) Fe(x)) Ar V
(82)
T
x
and
l0g(167)
P(Ar) 1
(83)
T
Lemma A.10 (Gaussian concentration inequality, Theorem 5.6 in Boucheron et al. (2003)). Let
X (O, La) be standard Gaussian random vector in Rd and let f IR + R be an Lipsschhi continuous function. Then, for all t O,
P((() 2 BIJ(X)) exp
(84)
2T.
Proposition A.11 (Gaussian concentration inequality with isotropic covariance). Let ~ (O Ta) be an isotropic Gaussian random vector in Rd with covariance matrix ala E Rd for some > 0. Let J Rd - IR be an L--Lisschitz continuous function. Then, for all / > 0,
-t2
P(f(X) [[f(X) +t) exp
(85)
2y
2
Proof. Let X' ~ NO,, Ta), and define = VTX', so that X ~ (T) Define the function fz(x) f(/īx). Then fy is -ipsccitt Using Lemma A.10 to fX'), we obtain:
-12
P(L(X)) 2 E()( exp
(86)
2yL,²
Since f(X)= = fx('),
-12 + exp
2 2((x)) =
(87)
2 2
Lemma A.12 (Norm concentration of isotropic Gaussian random vectors). Let X; N/0, 7Ia) be
an isotropic Gaussian random vector in Rd and be 2-norm. Then for any d E (0, 1), with probability least 8, we have:
max 1SSS
X < Mr.
(88)
where My= V7 (Vd+ /llgg(T/)))
Proof. Using Proposition A.11 and since f is 1-Lipschitz continuous, we have for
t2
[[| 2 X + <
exp
(89)
2>
Using Jensen's 's inequality and since X ~ = =
E ||||| < |E[|
Vyd.
(90)
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Therefore, for any > 0,
(-
IP (|XII vod+t) s exp
(91)
27)
By the union bound.
1.2
P
max ISiST
|x
2 Vnd +1 2 Σ
(1xill
Vtd +1)++1 s
-xxp
exp
(92)
27
By setting 7 exp (-12/27) 1 6, we obtain:
T
t 2 V
27 log
(93)
Therefore, with probability at least 8,
T 27 log
max S5STT
|Xi|| S Vdd +
(94)
Defining Mr VT (vd+ / (2log(T/5) we conclude:
|X,|| S Mr. SSST
max
(95)
Lemma A.13 (Bound on the sum of differences between true and estimated non-conformity scores).
Under Assumption 4.6 and 4.9, with probability at least 1 8,
Σ - - T 1 07)
(96)
Proof. Since the encoder is fixed after convergence, it generates the same h for é and e. Let 8; 2b (&s h) and s; - (c. h).
Using the identity for the difference of squared norms, we have:
|| || = ||s + (8: - sz)ll
(97)
= ||s;|| + 2(5s 6: .s)) + ||s, - -
Therefore,
|| ||s; 2 = 2(8 84 81) + ||5, - s111?
(98)
which implies
|ē - e:l |||&,112 - ||sa|12
(99)
= (2 $4 - -st) + ||8: - 111
By the Cauchy-Schward inequality,
(8;, 83 8:)1 S ||s:|| ||6; sill-
(100)
Since -1 is Lipschitz continuous with Lipschitz constant Lg-1, we have:
|| - s < Lg- ||e e = L |||
(101)
Substituting inequality (101) into the inner product bound in equation (100),
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1s 8, - 81) $ ||s:|| || - sell - L ||s:||] A
(102)
Then, by the triangle inequality.
jē eil S 1 ||++|||AAaI LB .11.112
(103)
By Lemma A.12. have with probability least 1 1 that || S Mr for all and by Assump- tion 4.9, |||,|| & 8y. Substituting these into the inequality (103), we have:
|ē eal 2MT + L 667.
(104)
Summing over all i = ,....7, T, we conclude that
Σ [ē: eal s T (2M₂ 1 10-1097 07))
(105)
Lemma A.14 (Distance between the empirical CDF of {ei)==1 and (&:)7=)) Under Assump- tion 4.6, 4.8, and 4. 9, with probability 5. Fr41 (x) and FT+1(x) satisfy
sup (x) - (T+1(t) S (2LT41 + 2 ssu PT+1(1) - Fa(x))
(106)
where C = MrL 184 + 12 87.
Proof. By Lemma A.13, with probability at least 1 Ó Ó, we have:
Σ l eal T (2Mr-gg- Iz.
(107)
1/2
Let C L2 184 12 1
Then,
T Σ 1=1
le eil TC2.
(108)
Define S = : :ê, eel 2 C}. Then,
[S| C <
Σ 1=1
é eal ST22
(109)
which implies |S| TC.
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We can bound the difference between the empirical CDFs of é, and e, follows:
(F+1(0)) -F)+1()) -
[[ S - 1e - x)|
|S|
Σ
[ S < 1fee s x))
IgS
||| +
Σ
life -
7
LES
$ T
[S] Σ
1 - sc
< C P((eT+1 SC)
1
+ sup
L - x| S 0) Pale-PP((+111/111
- -1
T
(
C + [Fa(x C) - Pa(x Ci]
+ sup
[T+ C) - FT+1(x -1+1 ci] -) F c)]
(m) (2IT+1 1)C + 2sup (T+1(x) - Pa(x))
(110)
Here, (i) follows from the inequality [1 < 2| (ii) follows from the identity P(eer+1 x $ C) = F. C) Fe(x - C), and (tii uses the
x| la la-hi b|} for a, b E R.
Lipschitz continuity of Fe))..
Proof of Theorem 4.10. For any 8 E [0,
YT+ C7 ZTT1 = =2++1 ) - (1 - aa)
p (e(e--1 E
(Tr1 Fr (1 - a 8)] ZT+1 = = - -a)
(1II)
P B Pre1((r41) $ a + 0) - P P Fe(e++1) $ 3)
Equality (i) follows from Lemma A.8, which states that (e Unif(0, 1]. This can be further
bounded by:
s P
P
$ 1 { 3 < Fr+ $1 1B3 Faleri)) 11 -
(112)
< E 1 B < FT 1 Fe(eT+1))
113
| Fr+ $ -1-a+ 3} - (F((err))
31
Here, inequality (1) follows from the fact that for any a, R and real values x, y/ E IR
[a < x < 6) - laa s 2 [1(a < x) laa s w + 11 < 1 s 6)}. (113)
By triangle inequality,
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1 B Fr+1((++1)) - 1 (3 Fe(er+1)]
( - (
5 - a +3)
E [18 Fr+l(er+)) -13 =
(114)
(0)
+ [ (Feler+1) a a+ +3)
For term (a), we have:
E (110 s Frt1(er+)) 118 (((er+))))
(115)
SP (P(((eTTT1)) 31 (Pr-- -(eeer-)))(((TT
This inequality follows from the fact that for a, e R. [1{a 1 s x s ((b x| Ja - bl}, and [[]{ P(A).
s
Similarly, for term (b), we have:
E ( (1 -1
(116)
P (Fa(et+1) - -1 (
Therefore,
PP YT+ E C |-+++= - = - (1 as|
P (IFo(er+a) - 31 (Fr+1(er+1) Feler+1)l)
(117)
+ (P((et)))) - (1 8)1 S (Tr4)) Fc(ert1))
In Lemma A.9, we defined Ar as the event on which
log (167)
sup
T
x
where ((AA)) > 1
logí
Let AC denote the complement of the event Ar. For any E [0, 1].
we have:
P (( -1 (0r41) Fe Fe
P (Ieo(eT+1) - 71 1 (6r41) Fe(eT+1)l Ar) PP((T)
(118)
P (((F((eT+1)) - < ( F.(6++1) + + (e(+1)) F (e(eTz))] Ar)
log(167)
T
To bound the conditional probability above, we note that with probability 1 6, conditioning on the event Ar,
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(er1() - Fe(eT+1)) (f((+r) - Fe(
(
S sup Pr+1 Fe(x)) Ar + L1+1ler41 er+il
suup (T+1(1) PT+ (x)) Ar + sup Fr+1 (ar) Fa(x))l Ar + LTtllTT - er+il
(119)
(8)
S (2LT41 + 11C + 3sup (x) - Fe(x)) Ar + LTt1
(66)
10g()6T) (LT+1+ 5) (2C + ôT).
3V
T
Here, inequality (t) holds due to the supremum of [T++ over and Lipschitz continuity of F. from Assumption 4.8. Inequality (ii) follows from Lemma A. 14. Inequality (rii) follows from Lemma A.9.
Since Fe(eT+1) ~ Unif 1].
P [F - >1 Fi+z(er+1) Fe(er41) + [e((r)))) - Fe(er+1)l Ar)
(120)
log(161"))
1 2 + |
s 6V
(2C + 8r).
Therefore, by substituting inequality (120) inequality (117), obtain:
VT+ E C++1|
7T+1 = -2++1 - -(1 aa)
(121)
1og(16T) T
+ +4[l++++ 1 ) + or).
W v
Definition A.15. A sequence of random variables (X,, said be strictly stationary if for every k 2 1, any integers n1..... Pg, and any integer h, the joint distribution of the random variables X,,) (X,,....,,,, is the same as the joint distribution (Xn,-h:... Xna+h)-
Definition A.16. sequence of random variables {X,,} is said be strongly mixing (or a-mixing) if the mixing coefficients (k) defined by
(kk) sup sup nEN AEFT BEF wtt
(122)
satisfy a(k) 0 as k 00, where F&th denotes the o-algebra generated by (Xa..... Xo).
Lemma A.17 (Convergence of empirical CDF of stationary and strongly mixing (es (ei) Under
Assumption 1, for any T, there exists an event AT with probubility at least (llo TY2 such that conditioned Ar.
1/3
21
sup -F()) (M) (log 2/3
(123)
T1/3
Proof. The proof follows similarly in the proof of Lemma B. in Xu et al. (2024). Define cr(x) = (T(Fr+1(z) Fz(z)). By using Proposition 7.1 in Rio et al. (2017), we have:
(3
log 2
E sup Jur (2))
<
1
Σ (Kk)
3
(124)
2 log 2
O==0
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where akk denots the k-th mixing coefficient. Under Assumption 4.11, have Σxpo a(k) 1 M < co. Applying Markov's inequality yields:
E (sup, (T(x))//TT)
1 + 4M
(3 log I
P
sup (x) Fe(xt)
2 8T
(125)
24
Ts4
2 log 2
By setting
4M
(3. +
log T
2
)
1/3
M (log T7)
1/3
8T =
23
(126)
T
log
27
we then have:
P sup Pr+1(1) -c(o) < (Molg 2)2 1/3
M M(lgg T)
1/3
2
(127)
2T
2T
Define the event Ar on which sup, Fx) - Fc(c)) 1
Millog T)² 27
1/3 that we have:
MM T)2
1/3
sup (r) - -Fc(o))) Ar
(128)
2T
and
[Moo T)2
1/3
P(AT) > 1
(129)
2T
Proof of Corollary 4.12. Under Assumption 4.11, the result follows by combining Lemma A.14
and A.1 using an argument analogous to the proof of Theorem 4.10.
B EXPERIMENT DETAILS
B.1 EXPERIMENT SETUP
OT-CP. We implement OT-CP using the source code released by the authors (Thurin et al., 2025). The training and validation sets combined to form calibration set. Following the setup in the original publication, 75% of the calibration set is used to solve OT, and the remaining 25% used to calibrate the prediction sets.
As the source code from the original publication (Fang et al., 2025) was not released, implement CONTRA following the methodology and details provided in the original publication. Consistent with the original setup, we use six coupling layers with a hidden dimension of 128 and train the model for 100 epochs with the same batch size as FCP and learning rate of 0.001. The training and validation sets are combined into calibration set, of which 50% is used to train the model and the remaining 50% is used to calibrate the prediction sets.
CONTRA.
MultiDimSPCI. We implement MultiDimSPCI (Xu et al., 2024) using the source code released by the authors. The context window size set to 50 for all real-world datasets, consistent with the setup used for FCP. Following the original publication, the number of trees for the quantile random forest set to 15. The training and validation sets are combined into a single training set.
Conformal prediction using copulas. We implement the method using the source code released by the authors (Messoudi et al., 2021). Following the setup described in the original publication, the training and validation sets combined form calibration set.
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Conformal prediction using local ellipsoids We implement the method using the source code provided by the authors (Messoudi et al., 2022). Following the setup in the original publication, the training set is used the proper training set and the validation set the calibration set. The number of neighbors for kNN is set to 5% of the proper training set size, as suggested in the original publication. We also conduct experiments with different numbers of neighbors, but does not lead to meaningful differences in performance.
CopUlaCPTS We implemented CopulaCPT'S using the source code provided by the authors (Sun & Yu. 2022). Following the setup described in the original publication, the training and validation sets are combined to form a calibration set.
Temporal Fusion Transformer We implement Temporal Fusion Transformer (TFT) (Lim et al., 2021) using Pytorch Forecasting. A hyperparameter grid search is conducted on the training set of each dataset with dy = 2 to determine the optimal configuration. We believe this hyperparame- ter search generalizes well to higher d within each dataset, since TFT makes predictions for each outcome dimension independently in our setup. Performance is observed to saturate at model dimension of 32, with two attention heads and two layers, therefore this setting used for ex- periments. For consistency with FCP, the context window size is fixed at 50 across all experiments. We train the models using the Adam (Kingma, 2014) with a learning rate of 0.001, a maximum of 50 epochs, and dropout of 0.1. Quantile loss with q {0.025, 0.975) is used for 0.95 target coverage.
DeepAR We implement DeepAR (Salinas et al., 2020) using Pytorch Forecasting. A hyperparam- eter grid search is conducted on the training set of each dataset with d,, determine the optimal configuration similarly to TFT. Performance is observed to saturate at a model dimension of 32 with two layers, therefore this setting is used for all experiments. For consistency with FCP, the context window size is fixed at 50 across all experiments. We train the models using the Adam (Kingma, 2014) with a learning rate of 0.001, a maximum of 50 epochs, and dropout rate of 0.1. Multivariate normal distribution loss with q E f0.025, 0.975) is used for 0.95 target coverage.
Table The hyperparameter search space for FCP.
Hyperparameter
Search space
the number of layers hidden dimension
[2 4, 6 32, 64)
Vector field
the number of layers the number of heads
2, 4, 6
2, 4, 8
Encoder
model dimension dropout
16. 32, 64 }
0.1
covariance scale learning rate batch size
{ 1, 2, 4, 8 0.0005, 0.0001 88 16
General
FCP We use multilayer perceptions (MLP) model the guided vector field. The time variable E [0, 1] and guidance h E Ih are concatenated with the input and fed into the guided vector field. hyperparameter grid search conducted on the training set of each dataset with different d, to determine the optimal configuration. We set the hidden dimension of the vector field identical to the model dimension of the encoder, so that additional layer not required between the vector field and the encoder. Table 3 shows the hyperparameter search space and Table 4 shows the optimized hyperparameter configuration. The context window size for the encoder is set to 50. We train the model with Adam (Kingma, 2014) with maximum of 50 epochs for all experiments and use the validation set to select the best model.
To determine appropriate sample size N for the set size estimation using quasi-Monte Carlo sampling. we compute the relative standard enror of the Jacobian determinants of sh defined as SE(det J/.n)// Avg(det J, where det. Ja.h {det Jo (ar h)) are the sampled Jacobian
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determinants conditioned on h. We select the smallest N such that the average relative standard error across all h falls below 0.01. As a result, we use N 4096 for experiments with dy 2, N 8192 for experiments with dy = 4, and N 16384 for experiments with dy 8.
Table The optimized hyperparameter configuration for FCP based on the hyperparameter search.
Dataset
Hyperparameter
dy 2
dy
dy
the number of layers of the vector field the number of heads of the encoder the number of layers of the encoder the hidden dimension of the vector field and encoder
2
32
32
32
covariance scale y encoder dropout batch size
Wind
0.1
0.1
0.1
learning rate null condition probability
0,0005
0.0005
0.0005
0.05
0.05
0.05
guidance scale w (LO0/LSTM base predictor)
1.1/1.1
1.1/1.1
1.1/1.1
the number of layers of the vector field
4
4
the number of heads of the encoder
2
the number of layers of the encoder the hidden dimension of the vector field and encoder
4
4
4
32
covariance scale encoder dropout batch size
1
Traffic
0.1
0.1
0.1
8
learning rate null condition probability
0.0001
0.0001
0.0001
0.05
0.05
0.05
guidance scale w (LOO/LSTM base predictor)
1.1/1.2
1.1/1.2
1.05/1.5
the number of layers the vector field
4
the number of heads of the encoder
2
the number of layers the encoder the hidden dimension of the vector field and encoder
4
32
32
covariance scale y encoder dropout batch size
1
Solar
0.1
0.1
8
learning rate null condition probability
0.0005
0.0005
0.05
0.05
guidance scale (LO0/LSTM base predictor)
1.5/1.2
1.2/1.1
B.2 COMPUTATIONAL COST
Training time. Table 5 reports the wall-clock training time for all methods, computed as the sum All models were trained machine equipped with dual Intel Xeon Gold 6226 CPUs and a single NVIDIA A100 GPU. For the methods that do not use neural networks, only CPU used.
five independent
C DATASET DETAILS
Wind dataset The wind dataset contains wind speed records measured 30 different wind farms (Zhu et al., 2021). Each wind farm location provides 768 records with features at each times- tamp. We randomly select dy {2,4, 2,,488 8} locations construct five sequences of dy di-mmennioonl time series.
Traffic dataset The traffic dataset contains traffic flow collected at 15 different traffic sensor cations (Xu & Xie, 2021b). Each sensor location provides 8778 observations with 5 features each timestamp. We randomly select d, E {2,4, 4] locations to construct five sequences of dy- dimensional time series.
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Table The wall-clock training time (hrs) for all methods, computed as the sum over five indepen- dent runs.
Dataset
Method
d
d=
d
FCP
S 0.2
0.2
CONTRA
S 0.2
X 0.2
S 0.2
MultiDimSPCI
X 0.05
S 0.05
0.05
Local Ellipsoid
3 0.01
3.0.1
0.01
Wind
Empirical Copula
0.01
0.01
Gaussian Copula
< 0.01
S 0.01
0.01
CopulaCPTS
S 0.01
& 0.01
0.01
TFT
S
DeepAR
2
S 4
FCP
0.5
S 0.5
< 0.5
CONTRA
0.5
$.5
0.5
MultiDimSPCI
21
21
<1
Local Ellipsoid
< 0.01
< 0.01
0.00
Traffic
Empirical Copula
W
0.01
< 0.01
0.01
Gaussian Copula
0.01
0.01
0.00
CopulaCPTS
0.01
0.01
TFT
$4
8
< 16
DeepAR
34
88
16
FCP
S 0.5
$ 0.5
CONTRA
< 0.5
0.5
MultiDimSPCI
31
S1
Local Ellipsoid
< 0.01
0.00
Solar
Empirical Copula
S 0.01
S 0.01
Gaussian Copula
S 0.01
S 0.01
CopulaCPTS
& 0.01
& 0.01
TFT
88
DeepAR
S 5S
Solar dataset The solar dataset considers solar radiation in Diffused Horizontal Irradiance (DHI) units at different solar sensor locations (Zhang et al., 2021). Fach location provides 8755 records with features each timestamp. For the solar dataset, randomly selected dy E (2, 4) locations to construct five sequences of dyyddimesioona time series. We did not construct sequences with d, 8 due the limited number of unique locations, which could lead to overlapping sequences across different trials of experiments.
D ADDITIONAL EXPERIMENTS
D. EXPERIMENTS AT DIFFERENT CONFIDENCE LEVELS
Table reports the results on the three real-world datasets the 0.9 confidence level. We exclude TFT and DeepAR, they did not demonstrate competitive performance in the experiment at the 0.95 confidence level. The overall results remain consistent with those the 0.95 confidence level. Notably, the gap in average prediction set sizes between FCP and other strong baselines-such as MultiDimSPCI, CP using local ellipsoids, and OT-CP for dy E 2, on the traffic solar datasets decreases at the 0.9 confidence level.
D.2 ROLLING COVERAGE
Since conditional coverage is difficult to assess on real-world data, we use rolling coverage as proxy for conditional coverage at specific time index. The rolling coverage at time index is defined as:
m-1 RC,
1 m
(130)
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Table 6: Average empirical coverage and prediction sets size obtained by FCP and all baselines three real-world datasets, evaluated under different base predictors and varying outcome dimensions dy. Reported values represent the average and standard deviation over five independent experiments conducted on five constructed sequences. The target confidence level is set to 0.9.
Dataset
Base Prodictor
Method
d,
dy 4
4,
8
Coverage
Size
Coverage
Size
Coverage
Size
PCP
p.90
200
0.9251 our
0.734
10:3
5.24.1.45
MultiDimPPII
0.9172.015
0.790
0.919
2.26
0.9
47.7 47.7a58.
CopulaCPTS
1.000..000
22.3+19.0
1.000s
611 Te.4444
1.00f:
3.50
107 raxtes
Boctstrup
OT-CP
0.919g.ms
0.90
0.951, ozo
23. 28.9.2n
0.853
1.00
10
CONTRA
0.9192.045
6.53es.17
0.9744
4.12 10 10
0.9742.0.6
4.12
10" Lanssto
Ellipsoid
0.949g.ms m2
0,952
0.958 ous
3.58 18
0.961 008
53.2498.1
Empiriaal Copula
0.914s.025
0.507
0.917 0.607
1.23
0.8062.002
7.38-42.04
Wind
Goussian Copula
0.913s.mx
0.622e.100
0.917
1.54..20
0.919
17.02444
1.0
g.88
0.924 oz4
5.72
1
MultiDimSPCI
0.948 02
2.
0.9044
41.92.0
0.839 024
2.37 x x10 225xxst
CrpulaCPTS
1.000g.000
46.7
1.000
4.82 102
1.000
2.83 10 31.0t0""
LSTM
OT-CP
0.909 2.909
5 5.98
9000
188 1
0.978..0:9
7.21 x 10
CONTRA
0.7
0.22
0.686 247
0.052.00s
2.7
7.71
Local Ellipsoid
0.978g.013
2.40
1.000 oco
167 131mn
1.000g.000 000
1.28 x 10
Empiriaal Copula
0.97
10.61500
1.000
325.9
2.38
102 10220000004
Goussian Copula
0.978
10.725.00
1.000
331, 4gia1
0.901
3.01 x 11
PCP
0.913g.or
0.613
0.9362
0.453
ane
1.082.100
MulhiDiSPPII
0.9220g.005
1.01
0.929 011
1.48 408
0.934 006
2.922
CopulaCPTS
1.000
21.61163
1.000
645.8
1.000..000
3.18 x 102 soxiot
Bootstrap
OT-CP
1.00 pm
0.927. on
2.30 915
0.914 00e
0.414 10s.
CONTRA
0.8022.007
0.806
0.902
0.565
0.848 o4e
Local Ellipsoid
0.927=..02
1.22+.101
0.9422.00
1.17=.301
0.945
0.9644.370
Empirical Copula
m.
1.24 200
0,930, 004
1.17
0.951
9.63
Ganssian Copala
0.915g.ma
1.26
0.904 ocT
2.38
0.936 oos
10.9
PCP
0.953
0.633
0.945
0.623
0.9231
0.6733
MulliDiSPPC1
0:914
CopulaCPTS
g.000g.m
11.
1.000 oco
330 1200
0.999 o02
4.47 x 102
LSTM
OT-CP
0.804
0.575
0.870
1.99
0.850
356-5
CONTRA
1.m
0,880 o13
0,031 1,00
1000
0,007 1.406
Local Ellipsoid
0.915 oz
0.625
0.899
0.706 0.706
0.871 ore
1.12;
Empirical Copula
015
2.5
0.912
13.9
0.890
516-22105.7
Gaussian Copula
0.910
2.62+.366
0.908 ou1
13.342.0 6
0.8742.0.5
479.0g.1i..1
FCP
0.905 1.5mmmm
0.589 2m
0.900 oo00
1.66
MultiDimSPCI
0.930
1.10
0.942
5.13
CopuloCPSS
1.000g.000
67.9+12
1.000
7.25 10 1.s0xt
LOO Bootstrap
OT-CP
0.936
1.44g
0.928
8.54g1.4
CONTRA
0.88912.004
1.38
0.878
7.16.4.00
Ellipsoid
0.897 1m
0.719
one
0,320_.cm .oon
Empirical Copula
0.949g.mm
1.98
0.955 ocs
7.87
Solar
Goussian Copula
0.953
2.12
0.962
9.66
KCP
0.911
0.673
0.907
0.503%
MuliiDimPPCI
0.938 0.938
0.733
0.937 064
2. 7.60z.
CopulaCPTS
1.000
44.8
1000
3.34 x 10
LSTM
OT-PP
0.914
0.585=.254
0.9244 0.9
10.6 50
CONTRA
0.835g.m
0.112 267
0.858 017
0.031 083
Local Hllippoid
0.9212.012
0.5822.203
0.9334
0.514
Empirical Copula
0.925_.mm
2,96
0.939
17.3
Gonssian Copala
0.909 002
3.50 2003
0.9644 ocs
28.0g2.0
29

---

Published as a conference paper at ICLR 2026
1.000
1.03
1.000
0.355
0.93
0.925
0.93
0.925
0.020
0.000
Biling
0.875
0.83
0.82
0:825
0.830
0.75
0.80
FCP
OTCPP
MSPCI
FCP
OTCP
MiRDinsKK
F3P
OTEP
MunDmmSKI
(a) 1.00 base predictor, dy
(b) I.O0 hase predictor, dy
(c) LOO base predictor, dy
1.00
0.95
0.95
0.35
To covclags
0.05
0.90
0.30
0.11
0.85
0.9
0.80
0.30
0.03
0.73
0.75
OT-CP
M.RDD
FCP
OT:CP
MuRDin5PI
FCP
OTCPP
MuRDinsKI
(d) LSTM base predictor, dy
(c) LSTM base predictor, d, =
LSTM base predictor, dy
Figure 3: Boxplots of the rolling coverages on the wind dataset with rolling window size 20 for FCP,
OT-CP, and MultiDimSPCI.
Table 7:
Average empirical coverage and prediction sets sizes obtained by FCP using MLP vector field and iResNet vector field on three real-world datasets, evaluated under different base predictors and varying outcome dimensions dy. Reported values represent the average and standard deviation over five independent experiments. The target confidence level was set 0.95. Results with average empirical coverage below the target confidence level are grayed out, and the smallest prediction set sizes, excluding the grayed-out results, are highlighted in bold.
Datasel
Base Predictor
Method
dy 2
dy
dy
Coverage
Coverage
Coverage
FCP (MLP)
0.951
0.88
0.953
1.43
0.956
19.4-10.2
100 Bootstrap
Wind
(ReeNNet)
0.951 2.11
1.14g. 000
0. 0.9544.om
1.79 s 700
0.953 01s
14.8-
LSTM
FCP (iResNet)
(MIP)
0.9
0.9 0.55220054
997 2.957.004
1.8 1.84
211
0.957 022
0.9 0.957
6.37
1..8
0.953 o6e
0.978
2.55
10 10000
10? on
FCP (MLP)
014
0.915
0.953
1.06
D.5655
1.53
Traffic
LO0 Boutstrap
FCP ((ResNet)
1.01
0.939
1.33 111
0.970 ou7
2.72_.215
FCP (MLP)
0. 1.9688.om
0.859
0.966
1.05
0.950
1.82_
LSTM
(iResNet)
0.9672.024
g.51
0.970 and
1.31
0.0564
2.50- 428
FCP (MLP)
1.957
1. 18
0.969
4.18
Solar
Bootstrap
FCP (iResNet)
0.9
1.42 g.
0.956 oza
2.69
FCP (MLP)
1.9888.00
1.162.002
0.961
2.09
FCP (iResNet)
955
1. 1.21
0. 0.935
2. 12,
where m rolling window size. Figure 3 presents boxplots of RC, computed with m 20 on the wind dataset.
30


Reviews:
[{"review_id": "2gDRXUbXlK", "reviewer": "Reviewer_s5Xm", "summary": "The proposed work presents a novel conformal production method for multidimensional time series by combining a transformer and a flow model. The obtained prediction region incorporates the historical context of the time series and is based on the transformation of residuals, creating an Euclidean ball in the latent space of flow residuals.  The efficacy of the proposed method is evaluated across several settings and various datasets.", "questions": "1. Line 016, \"leveraging correlations in features and non-conformity scores\" Did the authors want to allude to the exchangeability assumption in the traditional CP setup?\n\n2. The proposal is to use a flow with Classifier-free guidance; however, the idea only requires a kind of flow-based model that can transform a given distribution to another and is bijective. This makes the idea somewhat restrictive, given that a practitioner might want to use the Diffusion Model, classifier-based guidance, or a discrete flow. \n\n3. The idea to model the residual distribution is already present in the literature. For example, Res-CONTRA [1] transforms the residuals as a means of learning the calibration, while also utilising an Euclidean ball to define conformity scores. \n\n4. Line 045: If I am not wrong, the cited Barber et al 2023 does not require the exchangeability assumption; I believe this is a mis-citation.\n\n5. Line 077: \"Despite these efforts, existing methods remain limited to univariate outcomes or assume access to multiple i.i.d. time series.\" This statement seems misleading, given that many methods nowadays discuss multivariate outcomes or single time series setups.\n\n6. Line 095: There are more works apart from Xu et. al 2024 that also discuss single time series settings with multiple step-ahead forecasting. CAFHT, JANET [2, 3] is one such example. Note that their work is also applicable to multiple trajectories. More importantly, any idea, such as ACI, can incorporate multivariate responses by simply using a non-conformity score that caters to multivariate responses. \n\n7. Line 113: with y_i = f(x_i), it looks like there is no dependence on history y_{i-1}. Is that intentional?\n\n8. Line 119: Similarly,  z_i seems only to include x. I mean to say there is a lack of clarity around the notations.\n\n9. The idea of using a ball is already present in CONTRA, such as in Eq. 7. Furthermore, they have a computationally efficient version, as one only needs to work around the boundary of the ball. \n\n10. One issue with using Flow-matching is solving ODE or computing the divergence for the area computation. From this perspective, it seems more natural to use discrete flows. \n\n11. Theorem B.4, Lemma 4.5, and Proposition 4.6 present a similar result to one in CONTRA. \n\n12. Conditional coverage results, while important, depend on strong assumptions such as strongly mixing.\n\n13. Line 373-377: I am confused about the data splits here. Why is it that there is no calibration set needed for FCP? Furthermore, using the training and the validation sets for calibration skews the coverage guarantees, as far as I can understand. \n\n14. Line 385: The target confidence is set as high as 0.95, which is okay. But it is necessary to show the performance of the proposed method with different significance levels. A calibration curve might be helpful to see if the proposed method works well in all cases.\n\n15. The theoretical statements are there for conditional coverage, but there is no empirical evidence for the same.\n\n\n[1] CONTRA: Conformal Prediction Region via Normalizing Flow Transformation Zhenhan FANG · Aixin Tan · Jian Huang\n[2] Conformalized Adaptive Forecasting of Heterogeneous Trajectories Yanfei Zhou, Lars Lindemann, Matteo Sesia\n[3] JANET: Joint Adaptive predictioN-region Estimation for Time-series Eshant English, Eliot Wong-Toi, Matteo Fontana, Stephan Mandt, Padhraic Smyth, Christoph Lippert", "limitations": "", "rating": null, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "The experiments are conducted with multiple base predictors, and various baselines have been compared against, with FCP coming out as the best in the experiments.\n\nThe experiments are repeated across several runs, providing uncertainty statements that give a better picture of the method's efficacy.", "weaknesses": "To best judge a conformal method, it is imperative to compare its performance across different significance levels, and plotting a calibration curve is particularly helpful. \n\nMost of the theoretical results are well-established or fundamental. \n\nSee the \"Questions\" below for more.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tKeajW5gPA", "reviewer": "Reviewer_PND7", "summary": "This paper studies the construction of calibrated prediction intervals for multivariate time-series \ndata.  Building on Guided Flow, the proposed method transforms residuals \ninto a Gaussian variable and then derives a confidence region by inverting the  \nGaussian distribution. Some theoretical guarantees for the method are provided in the paper, \nand the proposed method is evaluated on numerical experiments.", "questions": "Please refer to the \"Weaknesses\" section.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper considers an important yet challenging problem: constructing \ncalibrated prediction intervals for multivariate time-series data.\n2. The proposed method has provided some fresh perspectives on the problem.", "weaknesses": "1. Although the goal is to construct prediction intervals, the proposed method does not seem to \nbe a \"conformal prediction\" method in the usual sense: it does not leverage any (approximate) exchangeability \nof the conformity scores to determine the prediction region.\nInstead, it seems closer to a nonparametric method that directly models the residual distribution and then constructs prediction sets.\n\n2. The theoretical guarantees are unclear. For marginal coverage, Proposition 4.6 states that the prediction set produced by the algorithm satisfies coverage guarantee \"if the ball $B_{\\alpha}$ defining the prediction set in equation (5) has probability mass $1-\\alpha$\". This is a strong \"if\", since $B_\\alpha$ is derived under Gaussian distribution, while $\\hat e(y_i)$ need not be exactly or even close to Gaussian in the presence of approximation error or model misspecification. As written, it is unclear what unconditional coverage guarantee the method provides and under what assumptions (or calibration procedures) the condition is satisfied. \n\n3. The theoretical results rely on numerous assumptions that are not fully justified. A more thorough discussion—motivating each assumption, assessing plausibility in typical applications, and exploring sensitivity to violations—would greatly strengthen the work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "o8nAwImGqW", "reviewer": "Reviewer_bXWm", "summary": "This paper introduces a method for uncertainty quantification in multidimensional time series forecasting. The approach begins by using any base forecaster to generate point predictions and residuals. It then encodes recent historical data as contextual information to condition a continuous normalizing flow, which maps an isotropic Gaussian source distribution to the situational residual distribution.", "questions": "(1) Is the precise marginal coverage established solely on theoretical grounds? In practice, do flow estimation errors and base predictor errors influence coverage bias? If so, can an upper bound or consistency conclusion be provided?  \n\n(2) At a given nominal level, does the constructed forecast set size exhibit near-optimality (minimal volume)? As the sample size approaches infinity, does it converge consistently to the optimal solution? What assumptions or conditions are required?  \n\n(3) Are the assumptions for achieving conditional coverage, such as the strong mixing condition, empirically supported in real data? If not satisfied, can a robust adaptation scheme be proposed that accommodates both the data and the proposed method?  \n\n(4) If a distribution shift occurs during the testing phase, will the context encoder and conditional stream cause coverage degradation? Does the method adaptively handle distribution shifts while still meeting the specified coverage rate?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "(1) The paper features a clear structure and rigorous logic, facilitating readers' comprehensive understanding of the methodology.  \n\n(2) The paper introduces a multidimensional uncertainty quantification approach centered on conditional continuous flow. By mapping source distributions to residual distributions, it establishes a unified pathway for constructing prediction sets, demonstrating novelty and general applicability.  \n\n(3) The paper presents a comprehensive argumentative framework, rigorously defining the assumptions and applicability of marginal coverage and conditional coverage respectively, accompanied by rigorous proofs.   \n\n(4)The experimental design encompasses multiple datasets, diverse output dimensions, and various base predictors. Evaluation metrics are clearly defined, and results effectively validate the proposed methodology", "weaknesses": "(1) The paper lacks a corresponding overview of methods and a concise algorithmic workflow.  \n\n(2) The rationale for selecting the source distribution and scoring metric is not sufficiently discussed. The paper adopts an isotropic Gaussian distribution as its core design without presenting alternative source distributions or concluding on differences in coverage and set size.  \n\n(3) The claim of conditional coverage lacks empirical support. While presented as a theoretical contribution, the experiments only report overall coverage and set size without showing coverage performance grouped by scenario or context.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "v6iZfZI35B", "reviewer": "Reviewer_W2dB", "summary": "This paper introduces a conformal prediction method for multi-dimensional time series that addresses two main challenges: temporal dependencies and multi-output prediction. The approach combines a Flow model with Transformer-based context encoding to model residual distributions, using Euclidean distance for non-conformity scoring. Trained via Flow Matching, the method provides theoretical coverage guarantees and achieves tighter prediction sets than baselines on real-world datasets.", "questions": "1. Please compare computational cost against baselines and report sensitivity to hyperparameters.\n2. Please empirically validate Assumptions 4.1, 4.3, and 4.7–4.11 in the experiments (or provide diagnostics/proxies).\n3. I'm not sure about the key reason that the flow-based model yields substantially smaller prediction sets than baselines. What properties of flows drive the gains, and why alternatives like quantile regression or transformer encoders cannot achieve similar improvements?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The integration of flow matching with conformal prediction is original and conceptually appealing.\n2. The method provides solid coverage guarantees, both marginal and conditional.\n3. Experimental results show significant improvements over baselines.\n4. The paper is well written and easy to follow, with clear motivation and presentation.", "weaknesses": "1. While flow matching may help model the distribution of prediction residuals, it can be computationally expensive.\n2. Results may be sensitive to hyperparameters, and extensive hyperparameter search can further increase computational cost.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)