ResCP: Reservoir Conformal Prediction for Time Series Forecasting
Authors: Roberto Neglia
Andrea Cini
Michael M. Bronstein
Filippo Maria Bianchi
Venue: ICLR 2026 Poster (2026)
Keywords: Conformal prediction
Time series
Uncertainty quantification
Primary area: learning on time series and dynamical systems

Abstract: Conformal prediction offers a powerful framework for building distribution-free prediction intervals for exchangeable data. 
    Existing methods that extend conformal prediction to sequential data rely on fitting a relatively complex model to capture temporal dependencies. 
    However, these methods can fail if the sample size is small and often require expensive retraining when the underlying data distribution changes. 
    To overcome these limitations, we propose Reservoir Conformal Prediction (ResCP), a novel training-free conformal prediction method for time series. 
    Our approach leverages the efficiency and representation learning capabilities of reservoir computing to dynamically reweight conformity scores. 
    In particular, we compute similarity scores among reservoir states and use them to adaptively reweight the observed residuals at each step. With this approach, ResCP enables us to account for local temporal dynamics when modeling the error distribution without compromising computational scalability. We prove that, under reasonable assumptions, ResCP achieves asymptotic conditional coverage, and we empirically demonstrate its effectiveness across diverse forecasting tasks.

Published as a conference paper at ICLR 2026
RESCP: RESERVOIR CONFORMAL PREDICTION FOR TIME SERIES FORECASTING
Roberto Neglia' UIT the Arctic University of Norway
Andrea Cini'
IDSIA USI-SUPSI, Università della Svizzera italiana Swiss National Science Foundation Postdoctoral Fellow
Michael M. Bronstein University of Oxford AITHYRA
Filippo Maria Bianchi
UiT the Aretic University of Norway NORCE Norwegian Research Centre AS
ABSTRACT
Conformal prediction (CP) offers a powerful framework to build distribution-free prediction intervals for exchangeable data. Existing methods that extend CP to se- quential data rely fitting a relatively complex model to capture temporal depen- dencies. However, these methods can fail if the sample size small and often re- quire expensive retraining when the underlying data distribution changes. To over- come these limitations, we propose Reservoir Conformal Prediction (RESCP), a novel training- frree CP for time series. Our approach leverages the efficiency and representation capabilities of Reservoir Computing to dynamically reweight con- formity scores. In particular, we compute similarity scores across reservoir states and use them to adaptively reweight the observed residuals. RESCP enables us to account for local temporal dynamics when modeling the eтror distribution with- out compromising computational scalability. We prove that, under reasonable as- sumptions, REsCP achieves asymptotic conditional coverage, and empirically demonstrate its effectiveness across diverse forecasting tasks.
INTRODUCTION
Despite deep learning often achieving state-of-the-arr results in time series forecasting, widely used methods do not provide a way to quantify the uncertainty of the predictions (Benidis et al., 2022), which is crucial for their adoption in risk-sensitive scenarios, such healthcare (Makridakis al., 2019), load forecasting (Hong & Fan, 2016), and weather forecasting (Palmer & Hagedorn, 2006). Moreover, many cases, prediction intervals (PIs) must not only be reliable but also fast compute. Existing uncertainty quantification approaches for time series often rely on strong distributional assumptions (Benidis et al., 2022; Salinas et al., 2020), which do not always fit real-world data. Moreover, they often require long and expensive training procedures, as well as modifications to the underlying forecasting model, which limits their applicability to large datasets. recently gained attention in uncertainty quantification is CP (Vovk et al., 2005; Angelopoulos et al., 2023). CP assumes exchangeability between the data used to build the PIs and the test data points, meaning that the joint distribution of the associated sequence of random variables does not change when indices are permuted. Clearly, this does not hold for time series, as the presence of temporal dependencies and, possibly, non-stationarity violates the assumption. Moreover, temporal dynamics can be heterogemeous and often result in heteroskedastic errors. This requires mechanisms to make PIs locally adaptive (Lei et al., 2018; Guan, 2022).
framework that
To extend CP to time series data, a popular approach is to reweight the observed residuals to handle non-exchangeability and/or temporal dependencies (Barber et al., 2023; Tibshirani al., 2019; Auer et al., 2023). In particular, Auer et al. (2023) proposes an effective reweighting mechanism based on learned model and soft-attention operators, which, however, suffers from the high computational cost typical of Transformer-lin architectures. Other methods build PIs by directly learning the
"Equal contribution. Correspondance to: roberto.negi i ia@uit. no; andrea ci innuii.... ch.

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quantile function of the error distribution (Jensen et al., 2022; Xu & Xie, 2023a; Lee et al., 2024; 2025; Cini et al., 2025). This, however, often results in high sample complexity and might require frequent model updates to adapt to changes in the data distribution.
To address these limitations, we propose Reservoir Conformal Prediction (RESCP), a novel CP method that leverages reservoir, i.e., randomized recurrent neural network (Jaeger, 2001; Lukoševičius & Jaeger, 2009), driven by the observed prediction residuals to efficiently compute data-dependent weights adaptively each time step. To capture locally similar dynamics, the weights are derived from the similarity between current and past states of the reservoir, which is implemented as an echo state network (ESN) (Jaeger, 2001; Gallicchio Scardapane, 2020). This enables RESCP to model local errors through weighted empirical distribution of past residuals, where residuals associated with similar temporal dynamics receive higher weights. The ESN does not require any training and is initialized to yield stable dynamics and informative representations of the input time series each time step. This makes our approach extremely scalable and easy to implement. RESCP can be applied on top of any point forecasting model, since it only requires residuals from disjoint calibration set.
Our main contributions are summarized as follows.
We provide the first assessment of reservoir computing for CP in time series analysis and show that it is valid and scalable alternative existing methods.
We introduce RESCP, novel, scalable, and theoretically grounded tool for distribution-fres uncertainty quantification in sequential prediction tasks.
We prove that RESCP can achieve asymptotic conditional coverage under reasonable assump- tions on the data-gen aata-generating process.
We introduce variants of RESCP to handle distribution shifts and leverage exogenous variables.
We evaluate RESCP on data from real-world applications and show its robustness and effectiveness by comparing it to sfate-of-the-art baselines.
BACKGROUND AND RELATED WORK
We consider forecasting setting where the goal is to predict future values of time series based on past observations. Let {y, ] 1 be the sequence of scalar target observations at each time step t. We denote by V1.r the entire history of observations up to time T. Moreover, let {u, [ur be the sequence of exogenous variables at each time step of dimension D In this context, we are also given forecasting model ÿt- s(y Wat, Ut-W:: u:: that produces point forecasts of H-steps- ahead, with H 2 1, given window W 2 1 of past observations. The model f can be any kind of forecasting model, e.g., a recurrent neural network (RNN). The objective is to construct valid PIs that reflect the uncertainty in the forecasts. Ideally, PIs should be conditioned on the current state of the system, i.e., we want to achieve conditional coverage:
P y E C7 ( USG ugg)
2 -
where Ö7 H. is the prediction interval for the forecast ÿc+H at significance level a.
2.1 CONFORMAL PREDICTION TIME SERIES FORECASTING
CP (Vovk al., 2005; Angelopoulos al., 2023) is framework for building valid distribution- free Pls, possibly with finite samples. CP provides a way to quantify uncertainty in predictions by leveraging the empirical quantiles of conformity scores, which in regression setting define as the prediction residuals
(2) These scores measure the discrepancy between the model predictions and the observed values. The CP framework we consider in this work is split CP (SCP) (Vovk et al., 2005), which constructs valid Pls host-hoc by leveraging disjoint calibration set to estimate the distribution of prediction errors. In its standard formulation, CP treats all calibration points symmetrically when building Pls. In heterogeneous settings, this lead overly conservative intervals conformal scores
F: V: ÿt-
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often heteroskedastic. To address heteroskedasticity, CP can be made locally adaptive by pro- viding PIs whose width can adaptively shrink and inflate at different regions of the input space (Lei et al., 2018). In this context, Guan (2022) proposed localized CP (LCP), which reweights con- formity scores according to the similarity between samples in feature space, given a localization function. Hore & Barber (2023) introduced a randomized version of I.P,, with approximate condi- tional coverage guarantees and more robust under covariate shifts. Although this reweighting breaks exchangeability, validity is recovered through finite-sample correction of the target coverage level. Other approaches rely on calibrating estimates of the spread of the data (Lei & Wasserman, 2013) or quantile regression (Romano et al., 2019; Jensen et al., 2022; Feldman et al., 2023), but often require ad-hoc base predictors (i.e., they cannot be applied on top of generic point predictors).
Conformal prediction for time series Under the assumption of data exchangeability, CP meth- ods provide, in finite samples, valid intervals given specified level of confidence (Angelopoulos et al., 2023). However, in settings where exchangeability does not hold, standard CP methods can fail in providing valid coverage (Barber et al., 2023; Tibshirani al., 2019). This is the case of time series data where temporal dependencies break the exchangeability assumption. Barber et al. (2023) show that reweighting the available residuals (in data-independent fashion) can handle non- exchangeability and distribution shifts, and introduce NexCP, a weighting scheme that exponentially decays the past residuals over time. Barber & Tibshirani (2025) provides a unified view of several weighted CP methods as approaches to condition uncertainty estimates on the available informa- tion on the target data point. Other recent approaches, instead, fit a model to account for temporal dependencies and rely on asymptotic guarantees under some assumptions on the data mata-gennrating process (Xu & Xie, 2023a; Auer et al., 2023; Lee et al., 2025). For example, SPCI (Xu Xie, 2023a) uses a quantile random forest trained at each step on the most recent scores. Conversely, HopCPT leverages a Hopfield network to learn a data-dependent reweighting scheme based on soft attention (Auer et al., 2023). Lee et al. (2025) use reweighted form quantile regression over the past nonconformity scores and derive the weights from the kernel function. Cini et al. (2025) proposes instead use graph neural networks (Bacciu et al., 2020; Bronstein et al., 2021) to additionally condition uncertainty estimates on correlated time series. As already mentioned, we focus on methods that estimate uncertainty on the prediction of pre-trained model; we refer to Benidis et al. (2022) for a discussion of probabilistic forecasting architectures.
Watson estimator to per-
2.2 ECHO STATE NETWORKS
ESNs are among the most popular reservoir computing (RC) methods (Jaeger, 2001; Lukoševičius & Jaeger, 2009). In particular, ESNs are RNNs that are randomly initialized to ensure the stability and meaningfulness of the representations and left untrained. An ESN encodes input sequences into nonlinear, high-dimensional state representations through the recurrent component, called reservoir, which extrapolates from the input a rich pool of dynamical features. The state update equation of the reservoir is
h = (1 - 1) 1 + la (Wzz, + Whh+11+b).
(3)
where æ, denotes the input at time I, W. RR Dw.D= and Wn BRxxDs are fixed random weight matrices, be E RDa is a random bias vector, h, E RR is the reservoir state, 1 E (0, 1] is the leak rate which controls how much of the current state is retained at each update, and a is a nonlin- ear activation, typically the hyperbolic tangent. When Wa and W₁, are properly initialized (sec,e.g., Lukoševičius 2012), reservoir states ht provide expressive embeddings of the past inputs æ5,,, which can be processed by a trainable readout to perform downstream tasks such as time series classifica- tion or forecasting (Bianchi et al., 2020). Since embeddings already model nonlinear dynamics, the readout is usually implemented as a simple linear layer which can be trained efficiently.
Echo state property The reservoir dynamics should be neither too contractive nor chaotic. In the first case, the reservoir produces representations that are not sufficiently expressive. In the latter, the reservoir is unstable and responds inconsistently to nearly identical input sequences. ESNs are usually configured to ensure the Echo State Property (ESP), necessary condition for global asymp- totic stability under which the reservoir state asymptotically forgets its initial conditions and the past inputs. Formally, for any initial states ho, h1 and input sequence &I,T, by calling J'v Ta(ao, a1:r) the final state of the reservoir initialized with hg and being fed with ær.r, then the ESP is defined as
[| .IT) - [n(h6 mirr)) 0 T
(4)
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Calihration
Test
Reservoir
T T4 8
hIT
Similadit
ga/2
THT
Samp1 kg
Figure I: Let (xt and {re){=1 be the time series and residuals of the calibration respectively. Each sequence Ei:t generates the state of the reservoir hu- The last state by the query state for the prediction of (T+ We compute similarity scores between hy and the calibration states T- which used reweight with ( associated residuals {r: (r+)7= by resampling them. Quantiles are computed from the sampled residuals and used to build the Pl for ÿT+
The ESP can be achieved by properly initializing W₁ and by setting its spectral radius p(W,) < 1 (Yildiz al., 2012; Gallicchio & Scardapane, 2020).
RESERVOIR CONFORMAL PREDICTION
We propose novel approach for computing CP intervals for time series forecasting based the representations generated by RC to model temporal dependencies. Our method leverages CP to provide accurate uncertainty estimates while maintaining the computational efficiency of RC.
3.1 RESERVOIR SAMPLING CONFORMAL PREDICTION
We construct PIs from calibration set consisting of a time series of residuals of length T. In particular, we aim at reweighting the residuals based on similarities with the local dynamics the target time step. We leverage the reservoir capability of embedding the observed dynamics zgg into its state hG, An overview the whole procedure depicted in Figure In general, the input æ; can include any set of endogenous and exogenous variables available at time step t. In the following, unless differently stated, we consider 2 to be the prediction residual (x, = r, obtained by training generic forecasting model on disjoint training set); the discussion on how effectively incorporate exogenous variables is deferred to Section 3.2.
Given ESN with dynamics defined in Equation 3 and hyperparameters 0, the time series 21d = (#1,..... 2x) is embedded into a sequence reservoir states which encode observed dy-
namies each time step
h (21:). h E P...
(5)
In particular, compute the PT for the H-step-ahead prediction Ht+H, take query state the current state hy (since we do not have access to states hsr) and compute similarity scores between h, and the states in the calibration set. Note that in most scenarios, the calibration set can be
iteratively updated as new observations become available (i.e., we can include in the calibration set all the observations up the current time step t). Weights are computed
zy (hs) = SIM(h,, hw),
15 5T H
(6)
(1(
2r H (h))
{w) (hy).......... WT WT- (ba)) SOFTMAX
(7)
where SIM is similarity score (e.g., dot product) and is temperature hyperparameter. Our goal is to approximate the conditional cumulative distribution function (CDF) of the residual at time ++ H with
T-
F(r h.) =
Σ
w. (he, (r++# S r) s P(r++ Sr xgr)
(8)
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where w, (he) is weight proportional to the similarity of h2 to h, and such that E, (g(hi) 1. Quantiles of the conditional error distribution can then be estimated as:
(b,) == in E R F(r ha) 2 3).
In practice instead of computing the weighted quantile in Eq. 9 directly, we approximate it through Monte Carlo sampling akin to Auer al. (2023). More specifically, we sample residuals accordingly to the weights in Eq. 7 and compute the standard empirical 3-quantile on the sampled residuals:
1++H (he).
22/2
dl-a/2 4++11
Q1-a/2(tee)
(10)
PIs with the desired confidence level are obtained as:
(b++H) [ + 20/2 UC+H + 90+H
al-a/2]
(11)
To account for skewed error distributions, we follow the approach Xu & Xie (2023b) and refine the interval in 11 by selecting the level 3* that minimizes its width:
3 arg min @11-+t ( BE(0,a]
T7 (ŪT+H) = (iTHH + Q (hz), UT+H H + 11-a+0 (hi)
(12)
This can provide narrower Pls, at the cost of searching for the optimal level B (Xu & Xie, 2023b).
Discussion Essentially, RESCP is local CP method for time series where local similarity is gauged by relying on reservoir states. By sampling more residuals associated with similar states, we condition the estimates on specific dynamics of interest in the time series. This allows us build lo- cally adaptive intervals. While Guan (2022) provides asymptotic guarantees for standard local LCP for i.i.d. data, such results cannot be applied in the context of time series. In the following section, we discuss the conditions that allow RESCP to provide (asymptotically) valid intervals. Notably, this will require assumptions on the nature of the process generating the data and on the reservoir dynamics. Finally, note that with finite samples, even with i.i.d. observation, data-dependent weights might introduce a bias that should be accounted for (Guan, 2022). In practice, we found that tun- ing RESCP hyperparameters on a validation set was sufficient to provide accurate coverage in most scenarios (see Sec. 4).
3.1.1 THEORETICAL ANALYSIS
We analyze the theoretical properties of RESCP, and show in Corollary 3.7 that asymptotically guarantees coverage under some regularity assumptions. As already mentioned, guarantees of stan- dard CP methods break down for time series unless one relies on highly unrealistic assumptions (e.g., exchangeable or i..dd. observations) (Barber et al., 2023). Moreover, even assuming exchangeabil- ity, impossible to construct finite-length PIs with distribution- free conditional coverage guar- antees (Lei & Wasserman, 2013; Vovk, 2012). We start by introducing the assumptions needed to prove the consistency of the weighted empirical CDF in Eq.
Assumption 3.1 (Time-invarian and mixing process). We assume that the process (Z, (zt, Tt+ n1 )2 is lime-invariant and strongly mixing (a-mixing) with coefficient o<(k)) +
as
Intuitively, Assumption 3.1 implies that the system's 's dynamics do not change over time and that "forgets" its initial conditions and past structure, thereby allowing observations that are far apart in time to be treated as if they were independent.
Assumption 3.2 (ESP and Lipschitz properties in ESN). Ler æsi = (1,,, denote the history. Define the fading- memory metric for 7 E (0,1) by
dim ( xe a'go)
* | 1-1- 1 2/-241112-
Then the ESN state map ESNg ESt H b4 E Dr. well-defined and causal with fading memory (i.e., the output only depends the past and present inputs), and there exist constant Lx > such that for all histories æse, x'zr
[[SNN((Rs)) - ESNo (2'sea S T.x den den (2gge, alge)
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Assumption 3.2 tells us that reservoir has stable and contractive dynamics: past imputs influence the state in a controlled, exponentially decaying way, and small perturbations are eventually washed out from the reservoir's state. These assumptions commonly used in RC (Grigoryeva & Ortega, 2019; Gallicchio & Micheli, 2011).
Assumption 3.3 (Continuity of the conditional law), Let P(re+ E R (gge) denote the conditional law of T14H given the past. Then for every r E IR the map
Est F(r xg(e)
is continuous in dgt with respect to dia i.e.
(1 a'gi) 00 = sup P( agge)= - F(r a's:)l -
Intuitively, Assumption 3.3 ensures that if two histories are similar (in the aading-memove sense), then the distributions of the future residuals conditioned on these histories are also close. This is reasonable assumption to enable learning.
Definition 3.4 (Effective sample size). Let be the number of available calibration samples, the effective sample size (Kong el al., 1994; Liu, 2008) is defined in our setting as
Σ (wumi)y
mn
(13)
Note that at high temperature, weights are approximately uniform (w. (n) ~ }) and (w,, (w, = n.
hence m, R n, which means that the method collapses to vanilla SCP as all calibration points are treated symmetrically. If instead the temperature is too low, then all the mass concentrates on only one point, meaning that Σ, (w: (wo(")²
and hence my, 1.
Assumption 3.5 (Softmax weighting scheme). Let n be the number of available calibration sam- ples, and let T, r(n) be the lemperature parameter of the SOFTMAX of equation 7, my the associated effective sample size, and w(a)) the output weights. Assume:
(i) Tn is configured to slowly decrease as n - oc, so that mn + 00;
(ii) for every 8 > O,
Σ
(he)))))) (hr)
0.
h. h 26
Condition (ii) states that for sufficiently small temperature Tre the softmax normalization should concentrate mass on those calibration points whose states lie in shrinking neighborhood of h Conversely, we also need these points within the shrinking neighborhood of h, to increase as the size of the calibration set increases. This can be seen bias-variance tradeoff. In practical terms, the temperature Tn must be set to a small enough value to localize weights at points similar to the query state (reducing bias), but the same time it should be large enough to guarantee a good effective sample size. This requirement translates into Assumption 3.5 (i) which preseribes that the temperature has shrink reasonably slow rate: one that allows the number of effective neighbors m. to diverge as n grows.
h2) denote the conditional weighted empirical CDF in Eq. 8 with calibration data {(x., )) and Fa(r h2) = P( Sr l Under Assumptions 3.1-3.5, we have for any query state h E RR,
Theorem 3.6 (Consistency of the weighted empirical CDF). Let Pal
sup Fr h2) - -F(r h2) FER
0.
The proof can be found in App. A. Theorem 3.6 implies the consistency of the empirical condi- tional quantile estimator (h+). defined in Equation 9, with respect the true conditional quantile Qo(hs). Finally, it is trivial to show that the asymptotic coverage of RESCP is guaranteed. Corollary 3.7 (Asymptotic conditional coverage guarantee). Under Assumptions 3.1-3.5, for any E (0, 1) and for n - xc,
P (vtth (Be44) h1)
} (1 - -a).
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Remark In the theoretical analysis, we model the distribution of future residuals conditioned on the state h4, i..., on the reservoir state at time t. This is clearly weaker than conditioning on the entire history zge. Recalling Eq. 8, whether or not F(r he) is a good approximation of F(r (gg)) entirely depends on the ability of the ESN to encode all relevant information from the past in its state. In particular, if the state representation captures all the relevant information, then RESCP provides asymptotic conditional coverage given such history. Conversely, in the extreme case where every sequence is mapped by the reservoir into the same uninformative state, RESCP would simply provide marginal coverage, as one would expect. This introduces an additional bias-variance tradeoff. For example, the reservoir mapping all trajectories to similar states would result in small variance, but would have small discriminative power (large bias). We refer the reader to the rich body of literature on the expressiveness of ESNs, particularly regarding their universal approximation properties (Grigoryeva & Ortega, 201 8a;b; Li & Yang, 2025) and their effectiveness in extracting meaningful representations from time series data (Bianchi et al., 2020). In particular, we refer to (Lukoševičius, 2012) for practical guidelines on how to set up ESNs effectively.
3.1.2 TIME-DEPENDENT WEIGIITS
RESCP, by default, compares the query state with any time step from the calibration set. Note that state representations do not include any positional encoding that accounts for how far back in time a certain sample is, treating recent and distant residuals alike. This is not problematic for time- invariant and stable processes, which we assumed in 3.1.1 to provide coverage guarantees. To deal, instead, with distribution shifts, we 1) update the calibration set over time, keeping its size N fixed using a first-in-frsttouu approach, and 2) make the weights time-dependent, following an approach similar to NexCP (Barber et al., 2023). In particular, we condition the weights on the distance between time steps as
wy (h,,,t) = (t, (he),
(14)
where 7 + R is discount function that maps the distance between time steps (t,, i) to decay factor. The discount schedule can be chosen in different ways, e.g., exponential like NexCP (Barber et al., 2023) or linear. In our settings, we observed that a linear decay-i.e., ((, t, a)) == 1/A(t, i)- allowed to keep PIs up- date without reducing too much the effective sample size. This is partic- ularly important if assumption 3.1 is violated and the underlying dynamics change over time. It is worth noting that RESCP is non-parametric approach without learnable parameters that is much more robust to non-stationarity compared to approaches that train model (Auer et al., 2023; Cini al., 2025), which must necessarily be updated when time-invariance is lost.
3.2 RESERVOIR CONFORMAL QUANTILE REGRESSION
Usually CP methods operate directly conformal scores. However, in some case it might be benefi- cial to consider exogenous inputs (covariates). Adding exogenous variables as input to the reservoir can affect the internal dynamics of the network and, thus, its states. Depending on the characteris- ties of specific exogenous variables, they can harm the effectiveness the reservoir in localizing prediction w.r.t. the dynamics of the target variable. Since the network is not trained, adjusting to the relevance and characteristics of the covariates is not possible with the completely unsupervised approach discussed in Sec. 3.1. As an alternative RESCP these scenarios, we consider a variant of our original approach, called Reservoir Conformal Quantile Regression (RESCQR), which can account for exogenous inputs by relying on quantile regression. In practice, we use a linear readout to map the state at each time step to set of quantiles of interest. Specifically, we use states hIT-- and residuals T-+1-T in the calibration set to fit linear model (hx; quantile predictor. To do so, we train the readout to minimize the pinball loss for the target quantiles {S1...., B3x) } of the conformity scores:
(1 (1- (1- (hi) -)),
Qhe 2r Qo (he) <r
La. (h),)))),
Bi(r Qa, (h.)),
We can then construct PIs as discussed in Sec. 3.1. As we will show in Sec. 4, RESCQR pro- vides a practical methods approach that can work well in scenarios where the availability of enough calibration data allows for training the readout and informative exogenous variables are available.
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Table 1: Performance comparison for a 0.1. ACov values color-coded for undercoverage
cases:
yellow (1-2%), orange (2-4%), red (>4%). The top three Winkler scores for each scenario
are highlighted: bold+underlined (Ist), underlined (2nd), dot-underlined (3rd).
Learning
Non-learning
Metrie
SPCI
HopCPT
CP-QRNN
RESCQR
SCP
NexCP
RESCP
ACov
0.05
-1.6441.88
-0.2622092
-1.102091
0.37
1.46
0.74
Transf RNN
PI-Width
10.4
60.49:2.10
55.74
59.99
79.15
89.56
22.25
Winkler
151.50.11.8 112.4619.44
78.42.10.0
82.76
171.41
164.96
104, 104.244077
Solar
ACov
-0.16:10.66
1.32
1.37=#2.99
16.21
0.35
1.46
+.09
PI-Width
71.36#3.21
61.49:1..4
55.70
59.56=11.9
79.03
89.21
63.34+111
Winkler
152.85+10.7
107.59
77.61:0.22
82.16:0322
169.64
163.34
103,13+0.58
RNN ARIMA
ACov
0.51
2.88
-0.41=204t
-2.03
0.16
1.76
0.68
PI-Width
91.71+1.18
143.32
59.17=0.41
66.19
124.19
137.53
77.17+2207
Winkler
148.86+40.4
173.49 45.15
+..34
85.38
215.77
207.58
+
ACov
-1.7340..7
-5.18
-1886=2.16
-1.21
-0.32
0.00
-0.70
PI-Width
67.93+144
68.47
61.71
65.53
67.99
69.79
65,96++2.50
Winkler
124.51+225
140,50+-43.41
104.03
105.43
126.41
124.10
106.07
Beijing
Transf
ACov
-0.98
11605
-1.07
-1.43
-0.41
0.03
-0.49
PI-Width
69.96:14.53
61.76::14.39
1.66
64.41
67.46
69.64
64.06++174
Winkler
125.41+1.43
140.30
102.81
105.97
126.63
124.35
103.64 103.64
Transf RNN ARIMA
ACov
-0.23++0.0
+1.37
-1.54
-1.42
-0,24
-0.16
0.63
PI-Width
14.68+121
67.78
6..80
66.01
75.72
76.45
70.43+00.6
Winkler
130.59
122.48+-4.36
101.84
107.20+121
135.07
132.03
108.75
ACov
2.98::005
2.75
-1.07
3.18
2.29
1.64
1.13
PI-Width
0.024110007 0.040410.000
0.0341-0.00
0.0383+00008
0.0444
0.0405
0.0210
Winkler
0.0287+00007 0.0482++0.00
0.0461
=0.000
0.0464
0,0517
0.0492
0.0264
Exchange
ACov
4.44:10.3 2.98+0007
-0.57=1.58
0.82+1.58
4.57
3.25
1.46:10.18
PI-Width
0.0255 0.0399+0.0001
0.0037=000016
0.0365
0.0544
0.0509
0.0229 10,000
Winkler
0.0300:1..0ms 0.0479+t0.001
0.0480=00000
0.0475
0.0620
0.0602
0.0294:uumm
ARIMA
ACov
3.4940041
2.07=:0.18
-1.22
0.68#1.5%
3.08
2.13
0.38 11.41
PI-Width
0.0242::0006 0.0379+t0000
0.0330
0.035 00000
0.0387
0.0356
0.0207
Winkler
0.0289
0.0456:20.000
0.0455=0.000
0.045520000
0.0462
0.0447
0.0268 100001
ARIMA Transf RNN
ÁCov
-0.78
-2.18200.00
-12.37 #8.98
-18.86=7.44
-0,99
-0.33
1,56
PI-Width
8.994068
18.90-10.0
15.86=1.99
15.23
19.63
20.15
9.61
Winkler
14.27=0.19
27.56++0.00
32.61
34.61
27.60
26,83
12.91
ACEA
ACov
-1.41
-2.51
-13.35
-26.92.4768
-5.52
-0.45
3.54
PI-Width
9.10+0223
18.29
14.82+2.02
13.20
16.53
20.20
10.10
Winkler
14.58+0.36
27.53
33.47
39.98
29.24
27.47
12.90
ÁCov
1.41+00.0
+3.58
-2..35
-27.10
-0.75
-0.40
5.02
PI-Width
12.46+0.35
34.84+t0.00
18.16#2.48
17.39
38.13
36.08
13.63
Winkler
17.36#0.13
44.69+:0.00
53,89
48.49
45.99
43,70
16.21
4 EXPERIMENTS
We compare the performances of RESCP against state-of-the-art conformal prediction baselines on time series data coming from several applications.
Datasets and baselines We evaluated our method across four datasets. 1) The Solar dataset comes from the US National Solar Radiation Database (Sengupta al., 2018). We used the dataset contain- ing 50 time series from different locations over period of 3 years, as done in previous work (Auer et al., 2023). 2) The Beijing dataset contains air quality measurements taken over a period of years from 12 locations in the city of Beijing, China (Zhang et al., 2017). 3) The Exchange dataset consists of a collection of the daily exchange rates of eight countries from 1990 to 2016 (Lai et al., 2017). 4) ACEA contains electricity consumption data coming from the backbone of the energy supply network in the city of Rome (Bianchi et al., 2015). More details are reported in App. B. We compare RESCP and its RESCQR variant against the competitors presented in Sec. 2: 1) vanilla
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Table 2: Runtime (in seconds) using the RNN point forecasting baseline.
Learning
Non-learning
Dataset
SPC1
HopCPT
CP-QRNN RESCQR
SCP
NexCP RESCP
Solar
1039.8+2 4574.641355
172.4
82.1
18
66
12.9
Beijing
351.412.1
1838.51202.4
81.613
46421
34.6114
Exchange
50.6205
31820
36.9112
16409
2
2
6,5105
ACEA
227.6158
2262.829189
95.3195
56.6129
70.5
SCP (Vovk et al., 2005), 2) NexCP (Barber et al., 2023), 3) SPCI (Xu Xie, 2023a), and 4) RNN with a multi-dimensional
HopCPT (Auer al., 2023). We also include 5) CP-QRNN, output trained to perform quantile regression on the calibration set using the pinball loss. The archi- tecture is analogous
the model called CORNN introduced in Cini et al. (2025) (more details in App. C). For RESCP, we use time dene-depnddent weights (with a linear decay schedule) and the cosine similarity between reservoir states as the similarity score.
Experimental setup and evaluation metries our experiments, we adopted a 40% 40%%/40%/52 /20% split for training, calibration, and testing sets, respectively. As base models, we consider three different point predictors: a simple RNN with gated recurrent cells (Cho et al., 2014), a decoder-only Transformer (Vaswani et al., 2017), and an ARIMA model (Box & Jenkins, 1970). After training, we evaluated each model on the calibration set and saved the residuals, which we then used all the baselines compute the Pls. As evaluation metries, between the specified confidence level - a and the achieved coverage on the test set, the width of the Pls, and the Winkler score (Winkler, 1972), which penalizes the PI width whenever the observed value falls outside the computed interval, with the penalty scaled proportionally to the magnitude of the deviation. Model selection for all methods done by minimizing the Winkler score over a validation set, except for HopCPT, which follows custom procedure (Auer et al., 2023). More details on model selection for our methods are reported App. C. For RESCP, when approximating the quantiles via Monte Carlo sampling, we set the number of observations used for calibration as the sample size. mentioned in Sec. 3.1, we account for skewed distributions by building PIs as in Eq. 12. The optimal is chosen from 100 linearly spaced values between 0 to
considered the ACov, i.e., the difference
4.1 RESULTS
Results across the datasets and base models are summarized in Tab. 1, where CP methods grouped based on whether they rely on learning or not. Additional results for different miscov- erage levels can be found in App. (see Tab. 5 and Tab. 6). RESCP achieves competitive perfor- mance across all settings and datasets, both in terms of coverage and Winkler score, while remaining highly scalable. Notably, RESCPs outperforms HopCPT almost all scenarios. This is partieu- larly remarkable since both methods use weighted empirical distributions to model uncertainty, but HopCPT requires training an attention-based architecture end-to-end, while RESCPs does not per- form any form of training. Moreover, similarly to NexCP and differently from most of the other methods, RESCP provides approximately valid coverage across all scenarios, but with large im- provements in PI width, which is reduced up to 60%. The CP- CP-QRNN baseline achieves strong
Fechospe
Brstip ResOQR Sestz" CI-QUIX
Obterved Coverae
2.00
Mixa
8.3
8.40
0.00
8.30
000
1.50 Nounical Coverage
0.50
1114
8.70
600
1.00
100
Classagg
Clooerage
Nuninal Choerage
Figure 2: Calibration curves of RESCP, RESCQR, CP-QRNN and NexCP over all datasets with the RNN baseline.

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Published as a conference paper at ICLR 2026
Table 3: Ablation of RESCP. RESCP without decay, RESCP without sliding window, and the combination of the two, using the RNN base model. Best Winkler scores are shown in bold.
Metric
RESCP
No decay
No window
No window, no decay
Solar
ACov
+.74
-0.10+026
0.89=0.20
1.70
PI-Width
62.25 #0.75
60.59
59.814007 +0,7
60.90
Winkler
104.24.0.79
19
104.70..20
104.46.10.0
Beijing
ACov
-0.70
-1.644161
0.
1.6421088
PI-Width
65.96=2.50
64.0421.8
64.91 13:36
69.53=1102
Winkler
106.07
1 a1
97.99
98.25
ACEA Exch.
ACov
1.13
2.01
4.19
4.20
PI-Width
0.02 1000001
0.0219
0.0249
0.0254.10.003
Winkler
0.0264
0.0269
0.0284220001
0.0291 000
ACov
1.56
2.79
5.34
4.96
PI-Width
9.61 1::26
10.15
11.88
12.15
Winkler =2.21
13.41
14.80
15.25
performance on the large datasets with informative exogenous variables, outperforming all methods in Solar, but fails in achieving good results in the smaller datasets, such as ACEA and Exchange. Moreover, on those datasets, trainable methods obtain worse performance, likely due to distribution changes that require the trained models to be updated over time. RESCQR achieves competitive performance against the competitors, including more complex models such as CP- QRNN in Solar, while being more scalable. All other baselines obtain good coverage in most settings, but produce prediction intervals that are much more conservative. The runtime of each CP method reported in Tab. 2 and shows the advantage of RESCP in terms of scalability against methods that require fitting a model. Moreover, as RESCP does not need centralized training on a GPU: it can be easily scaled in distributed setting.
Additional experiments and ablation studies Fig. 2 presents calibration curves for RESCP and RESCQR and the CP-QRNN and NexCP baselines, across all datasets and using RNN as the base point predictor: The curves show that RESCP provides accurate estimates at all the considered coverage levels. Note that while NexCP is well calibrated, it produces much wider intervals (see Tab. 1, Tab. 5 and Tab, 6). Additional results are provided in App. D, including an experiment of RESCP on non-stationary synthetic dataset. RESCP. In particular, we evaluated the impact of 1) removing deeedeepndennt weights (No decay), 2) using all the available samples for calibration rather than only the most recent ones (No window), and 3) the combination of the previous two ablations (No window, no decay). The results clearly show the impeat of the proposed designs. More comprehensive ablation results for all base models are presented in App. E, which also include an ablation study the use of exogenous variables. Finally, App. reports additional results on the sensitivity RESCP to different hyperparameter configurations.
Tab. 3, we also report an ablation study of
5 CONCLUSION AND FUTURE WORKS
We introduced Reservoir Conformal Prediction (RESCP), a simple, fast, and effective method for uncertainty quantification in time series forecasting that provides accurate uncertainty estimation across several tasks. We showed that RESCP is principled and theoretically sound. In partieu- lar, assuming strong mixing conditions and reasonable regularity conditions, we demonstrated that RESCP provides asymptotically valid conditional coverage guarantees. Results on diverse suite of benchmarks show that RESCP achieves state-of-the-art performance while being more scalable and computationally efficient than its competitors. There are several directions for future work. In particular, it would be interesting to explore extensions to modeling joint distributions in multistep forecasting (Sun & Yu, 2024), treat multidimensional time series (Feldman et al., 2023; Xu et al., 2024), and spatiotemporal data (Cini et al., 2025).
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ACKNOWLEDGMENTS
This work is supported by the Research Council of Norway through RELAY: Relational Deep Learn- ing for Energy Analytics (project no, 345017) and through its Centre of Excellence Integreat The Norwegian Centre for knowledge-dree driven machine learning (project no. 332645), the Swiss National Science Foundation grant no. 225351 (Relational Deep Learning for Reliable Time Series Forecast- ing at Scale), the EPSRC Turing Al World-Leading Research Fellowship No. EP/X040062/1 and EPSRC Al Hub No. EP/Y028872/1. The authors wish to thank Nvidia Corporation for donating some the GPUs used in this project and Carlo Abate for the help in checking the proof.
REFERENCES
Anastasios N Angelopoulos, Stephen Bates, et al. Conformal prediction: gentle introduction. Foundations and Trends@ in Machine Learning, 16(4):494-591, 2023.
Andreas Auer, Martin Gauch, Daniel Klotz, and Sepp Hochreiter. Conformal prediction for time series with modern hopfield networks. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.), Advances in Neural Information Processing Systems, volume 36, pp. 56027- 56074. Curran Associates, Inc., 2023.
Davide Bacciu, Federico Errica, Alessio Micheli, and Marco Podda. A gentle introduction to deep learning for graphs. Neural Networks, 129:203-221, 2020.
Rina Foygel Barber and Ryan J Tibshirani. Unifying different theories of conformal prediction. arXiv preprint arXiv:2504. 02292, 2025.
Rina Foygel Barber, Emmanuel J. Candès, Aaditya Ramdas, and Ryan J. Tibshirani. Conformal prediction beyond exchangeability. The Annals of Statistics, 51(2):816 - 845, 2023. doi: 10. 1214/23-A0S2276.
Konstantinos Benidis, Syama Sundar Rangapuram, Valentin Flunkert, Yuyang Wang. Danielle Mad- dix, Caner Turkmen, Jan Gasthaus, Michael Bohlke-Schneider, David Salinas, Lorenzo Stella, François-Xate Xavier Aubet, Laurent Callot, and Tim Januschowski. Deep Leaminng for Time Series Forecasting: Tutorial and Literature Survey, ACM Comput. Surv., 55(6), dec 2022. ISSN 0360- 0300. doi: 10. 145/3333333338. URL https:
/ //ooi 10. 2ttp///.1.1113333333
Filippo Maria Bianchi, Enrico De Santis, Antonello Rizzi, and Alireza Sadeghian. Short-term elec- tric load forecasting using echo state networks and pea decomposition. IEEE Access, 3:1931- 1943, 2015. doi: 10.1 109//CSSS
Filippo Maria Bianchi, Simone Scardapane, Sigurd Løkse, and Robert Jenssen. Reservoir computing approaches for representation and classification of multivariate time series. IEEE Transactions Neural Networks and Learning Systems, 2020.
George EP Box and Gwilym M Jenkins. Time Series Analysis: Forecasting and Control. Holden- Day, 1970.
Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint 11XX::22104.3478 2021.
Andrea Ceni and Claudio Gallicchio. Edge of stability echo state network. IEEE Trans. Neural Netw. Learn. Syst., 36(4):7555-756) April 2025.
Kyunghyun Cho, Bart van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv:1406.100) 1078 2014.
arXiv preprint
Andrea Cini and Ivan Marisca. Torch Spatiotemporal, 2022. URL httttt://gtthhttt com/ cannSpatiotepporall
Andrea Cini, Alexander Jenkins, Danilo Mandic, Cesare Alippi, and Filippo Maria Bianchi. Re- lational Conformal Prediction for Correlated Time Series. International Conference on Machine Learning, 2025.
11

---

Published as a conference paper at ICLR 2026
Jonathan Dong, Erik Börve, Mushegh Rafayelyan, and Michael Unser. Asymptotic stability in reser- voir computing. In 2022 International Joint Conference on Neural Networks (ICCNN), volume 148.34, pp. 01-08. IEEE, July 2022.
Paul Doukhan. Mixing: Properties and Examples. Information Systems Development: Reflec- tions. Challenges and New Directions N=85. Springer, 1 edition, 1994. ISBN 1461226422; 9780387942148; 0387942149; 9781461226420.
William Falcon and The PyTorch Lightningg team. PyTorch Lightning,, 3 2019. URL https:
///gttub.cc./ ///ithubb.com//
Shai Feldman, Stephen Bates, and Yaniv Romano. Calibrated multiple-output quantile regression
with representation learning. Journal of Machine Learning Research, 24(24):1-48, 2023.
Claudio Gallicchio and Alessio Micheli. Architectural and markovian factors of echo state networks.
Neural Networks, 24(5):440-44 456, June 2011. ISSN 6093-6080. doi: 10. 1016/...eumet.2011111 002. ht
ht dx doi org/10..//.. 100.101.. 1016/1161 neunet. 2011. 02 002.
Claudio Gallicchio and Simone Scardapane. Deep Randomized Neural Networks, pp. 43-68. Springer International Publishing, Cham, 2020. ISBN 978-3-030-43888-88 doi: 10.1007/ 778-3-030-43883-8 3. URL https:tt////.. //coi////////// meeeiiorg/110..
Lyudmila Grigoryeva and Juan-Pablo Ortega. Echo state networks are universal. Neural Networks, 108:495-508, 2018a. ISSN 0893-6080.
doi: htpps://oi.orr/10.001//....mm
2018.08.025. URL /tLhrrsss//..sseeeeeeee www.a//eeeerett.oee/eeeeeeeeeeee $0893608018302511,
Lyudmila Grigoryeva and Juan-Pablo Ortega. Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems. Journal of Machine Learning Research, 19(1):892-931, January 2018b. ISSN 1532-4435.
Lyudmila Grigoryeva and Juan-Pablo Ortega. Differentiable reservoir computing. Journal of Machine Learning Research, 20(119): 2019. htllhtttp//mm..oot ottpppppppp//p///7/7
1099150.000 html.
Leying Guan. Localized conformal prediction: a generalized inference framework for conformal prediction. Biometrika, 110(1):33-50, 07 2022. ISSN 1464-3510. doi: 10. 10.1093/hiomet/asacc///
Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant. Ar- ray programming with NumPy. Nature, 58577825)3357-36,, September 2020, doi: 10. 1031038/ s41586-028 020-0049-22222 URL https: /tte:://doo.ooo www.i..org/ooo ott.e/.og1/10......1 s41586416666 020- 2649-2.
Tao Hong and Shu Fan. Probabilistic electric load forecasting: A tutorial review. International
Journal of Forecasting, 32(3):914-938, 2016. ISSN 0169-2070.
doi: http::/doiioon org//0. www.coooo///o.ooo ccom/sccence//
1016//.iforccas.2055.1..110
URL httt::://ww..cienettt
aaticile/p11/00112270055000
Rohan Hore and Rina Foygel Barber. Conformal prediction with local weights: randomization enables local guarantees. arXiv preprint arXiv:2310.07850, 2023.
Herbert Jaeger. The "echo state" approach to analysing and training recurrent neural net- works. GMD Report 148, GMD German National Research Institute for Computer
Science, 2001. URL http://// www. faculty. jacobbss.yccssso...o.oo.ooooosse .df.
jacobs-ccooooooee wwwiiesiiiiiiieeeee wew.iit.ddee.jjeee de/hjaager/
Vilde Jensen, Filippo Maria Bianchi, and Stian Normann Anfinsen. Ensemble conformalized quan- tile regression for probabilistic time series forecasting. IEEE Transactions on Neural Networks and Learning Systems, 35(7):9014-9025, 2022.
12

---

Published as a conference paper at ICLR 2026
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun (eds.), ICLR (Poster), 2015.
Augustine Kong, Jun S. Liu, and Wing Hung Wong. Sequential imputations and bayesian missing data problems. Journal the American Statistical Association, 89(425):77-288, 1994. ISSN 01621459, 1537274X. URL http:////// //www jstor. wtt.oo...rggsaaaee/2222222
Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long- and short-term temporal patterns with deep neural networks. arXiv preprint arXiv: 703.07015, 2017.
Jonghyeok Lee, Chen Xu, and Yao Xie. Kemne--based optimally weighted conformal time-series prediction. The Thirteenth International Conference on Learning Representations, 2025, URL https: /ttp////peeeeeeeeeeeeeeeeeee net foo.mmidd=oP7rr.oooo
Junghwan Lee, Chen Xu, and Yao Xie. Transformer conformal prediction for time series. arXiv preprint arXiv:2406.05332, 2024.
Jing Lei and Larry Wasserman. Distribution-free prediction bands for non-parametric regression. Journal of the Royal Statistical Society Series B: Statistical Methodology, 76(1): 07 2013. ISSN 1369-7412. doi: 10.1111/ssb.12221. URL https: petttps:/doi..rgggg/ 10t.1//00.00.10.00rss....... 12021.
Jing Lei, Max G'Sell, Alessandro Rinaldo, Ryan J Tibshirani, and Larry Wasserman. Distribution- free predictive inference for regression. Journal of the American Statistical Association, 113 (523):1094-1111, 2018.
Zhen Li and Yunfei Yang. Universality and approximation bounds for echo state networks with random weights. IEEE Transactions on Neural Networks and Learning Systems, 36(2):2720- 2732, 2025. doi: 10.109//NNIS.20033333111
Jun S. Liu. Monte Carlo Strategies in Scientific Computing. Springer Publishing Company, Incor- porated, 2008 ISBN 0387763694.
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In International Conference on Learning Representations, 2019. URL
wtt.rr/ewwweeetnmnmmmm forum?
Mantas Lukoševičius. A practical guide to applying echo state networks. In Neural Networks:
Tricks of the Trade: Second Edition, pp. 659-686. Springer, 2012.
Mantas Lukoševičius and Herbert Jaeger. Reservoir computing approaches to recurrent neural network training- Computer Science Review, 3(3):177-149, 2009. ISSN 1574-0137. doi:
URL https: /www. sciencedirect. com/
Spyros Makridakis, Richard Kirkham, Ann Wakefield, Maria Papadaki, Joanne Kirkham, and Lisa Long. Forecasting, uncertainty and risk; perspectives on clinical decision-making in pre- ventive and curative medicine. International Journal of Forecasting, 35(2):659-666 2019. ISSN 0169-2070. doi: /ttpe//ncee/rttiieee////////--// sciencedirect. com/
URL https://www/
Tim Palmer and Renate Hagedorn (eds.). Predictability of Weather and Climate, Cambridge Uni-
versity Press, Cambridge, United Kingdom, 2006.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang. Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high- performance deep learning library. In Advances in Neural Information Processing Systems 32, pp. 0024-8033. Curran Associates, Inc., 2019. URL http: wwwppspppppppaaaaaaap http://ppppps......ppppe cc ctt /appr// 2015-pytorch-ale an o pdf.
13

---

Published as a conference paper at ICLR 2026
Yaniv Romano, Evan Patterson, and Emmanuel Candes. Conformalized quantile regression. Advances in neural information processing systems, 32, 2019.
David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. DeepAR: Probabilistic forecasting with autoregressive recurrent networks. International Journal of Forecasting, 36(3):
1181-1191, 2020.
Manajit Sengupta, Yu Xie, Anthony Lopez, Aron Habte, Galen Maclaurin, and James Shelby. The national solar radiation data base (nsrdb). Renewable and Sustainable Energy Reviews, 89:51-60, 2018. ISSN 1364-0321. doi: http://ooo.oogg////ooo/d/eeeeees
URL https: htt:://wwwww //www.
sci sciencediress com/ wwm//ccinneen http:///eeeeeeerrrooo//////////2
Sophia Huiwen Sun and Rose Yu. Copula conformal prediction for multi-step time series prediction.
In The Twelfth International Conference on Learning Representations, 2024. URL https:
/openzeview wwt. http://oooooomndoooooooo
The pandas development team. pandas-dew/pandas: Pandas, February 2020. Utthttss://doi.
org/ 00/000.0000 5281 zenodo .2010do3300101113.. 3509134.
Ryan Tibshirani, Rina Foygel Barber, Emmanuel Candes, and Aaditya Ramdas. Conformal predic- tion under covariate shift. In H. Wallach, H. Larochelle, A. Beygelzimer, d'Alché- Buc. E. Fox, and R. Garnett (eds.). Advances in Neural Information Processing Systems, volume 32. Cur- ran Associates, Inc., 2019. URL https:ttp..... httpttp::///pcceeeeee http///pooeddi/eeoeeeee/ ngs. neuri wttpippp..ccpppeee paappparrflleeee paper/2019/ fileilllll e f221 ee 7227777726252555675555
Guido Van Rossum and Fred L. Drake. Python Reference Manual. CreateSpace, Scotts Valley, CA, 2009. ISBN 1441412697.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Eukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NeurIPS), volume 30, 2017.
Vladimir Vovk. Conditional validity of inductive conformal predictors. Steven C. H. Hoi and Wray Buntine (eds.), Proceedings of the Asian Conference on Machine Learning, volume 25 of Proceedings of Machine Learning Research, pp. 475 490, Singapore Management University, Singapore, 04-06 Nov 2012. PMLR. URL h /ttp///poceenn...ppee/////e mlr press/2es//25/ vovk12. html.
Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world, volume 29. Springer, 2005.
Wes McKinney. Data Structures for Statistical Computing in Python. In Stéfan van der Walt and Jarrod Millman (eds.), Proceedings of the 9th Python in Science Conference, pp. 56 - 61, 2010. doi: 10.25080/Majora
001A88/MMMaora2921119220000 9wff9922-l9000 00aa
Robert I.. Winkler. A decision-theoreticc approach to interval estimation. Journal of the American Statistical Association, 177331) 1972.
Chen Xu and Yao Xie. Sequential predictive conformal inference for time series. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scar- lett (eds.). Proceedings of the 40th International Conference Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp. 38707-38727. PMLR, 23-29 Jul 2023a.
Chen Xu and Yao Xie. Conformal prediction for time series. IEEE Transactions Pattern Analysis and Machine Intelligence, 11555-111555118877 2023b. doi: 10. WW9/TPAMM.202333222333
Chen Xu, Hanyang Jiang, and Yao Xie. Conformal prediction for milti-dimeesionall time series by ellipsoidal sets. In Forty-first International Conference on Machine Learning, 2024. URL https: www./eeeeeveeeeeeeeeeeeeeeeeeeee ew net forum? d=uN3 9P8b.
Izzet B Yildiz, Herbert Jaeger, and Stefan J Kiebel. Re-visiting the echo state property. Neural networks, 35:1-9, 2012.
Shuyi Zhang, Bin Guo, Anlan Dong, Jing He, Ziping Xu, and Song Xi Chen. Cautionary tales on air-quality improvement in beijing. Proc. Math. Phys. Eng. Sci., 473(2205):20174457, Septembe 2017.
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APPENDIX
A PROOFS FOR THE ASYMPTOTIC CONDITIONAL COVERAGE
We start the proof with the following lemma.
Lemma A.1 (Regularity of the conditional CDF), Under Assumptions 3.2-3.3, the conditional CDF
F(r h4) is continuous in h2 E Rdd,, uniformly
Proof. By Assumption 3.2, the map æςτ + h4 is Lipschitz with respect dan- By Assumption 3.3, the map Øşt H F(r (gg)) is continuous. Therefore the composition Zgt + he ++ F(r hi) is continuous in h4.
We then continue with the proof of Theorem 3.6, which we start by proving point wise convergence.
Proof of Theorem 3.6. Define
Fa(r he) = Σ
w (n) (hr) hh))
i=1
Decompose
Fr h.) - F(r h.) (F h.) Fr h.)) + h,) F(r h.))
(1)
(IT)
We handle the two terms separately and show both going uniformly to in probability.
Term (I):
Fa(r Fr h4) = Σ w.
(n) (hi) 5 -r)-F(r h.)) = Σ w (n) (h.))
i=1
with Y;(r) = 1(r++ S rr - F(r h2)) We can see that (r) h:] = 0, since
E[I rr h4] = 1- P(ra < h4) 0- P(r. h.) P(r: hi) = F(r h.).
Therefore, E (S. = 0. Moreover, [Y((r) < 1. To prove that (r) مم 0, leverage
Chebyshev's inequality
(([SA(r))>c) Var(S,
2²
and then expand the variance as
Var(S,((r) = Var
Σ
w
(1)
Y;
= Σ (w: (m) 2 + Σ
wi:h wi w/ Cov(Y,, Yy) (15)
1=1
in9
(w (n) 2 + Σ w
w (1)
(ov Y,)l
Starting by the variances term, since [Y]] S 1, we have Var(Y) S 1, thus the first term of Eq. 15 is bounded Σm= (w (a) Var((() < Σi (w, (2)² + 0 thanks to Assumption 3.5 (i). We now move to the covariances. From Lemma of Doukhan (1994), we know that since the process generating (æ,, TH) is mixing by Assumption 3.1, for bounded variables Y,, Y, that are function of this process the following holds
[Cov(Y,, Y2)1 < 4a(li - jl),
where akk) is the mixing coefficient. Hence, in case,
Σ z
w (m) w (n) Cov(Y, Y,)1 < 4 Σ
(n) m0 a(li (o((i-1) -j)).
w
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We can split the sum in two terms
4
Σ
w "!"/
w a(li
j1)
+4
Σ
w. uo (r) alli
a(li - jl)
パー
such that for all lags k > e (this is always possible).
K the mixing coefficient is arbitrary small, i.e. for any > akk( <
()) =
max a(k)
Σ
wi
(n) (n)
iskSk K
(1) (n)
S Amaz
W; w
(n) (m) Witk
2A
k=1
Using the inequality 2ab < a² + +b::
K k
K
nー k
n k
2.mmzz
w
w. i4t
(n)
S
Amar
w
)2
(1)
+
(w)
itk
}2
1=
1=1
1=
22
i=1
2A K
(we
(2)
i=1
2Amoz K
- 0.
mn.
For the second term
(**) Σ
w W0 aali jl)
(n) (10
< Σ
w (n)
< 6
where e is arbitrarily small.
Term (II):
(r h2) F(r ha) = Σ f=1
(n) (F.) h.) - F(r h.))
Because weights are convex combination (nonnegative, sum to 1),
Fx(r he) - F(r ht)l
w
(n) (kr)) h.) F(r ho)
Now Lemma A.1. For any 0 we can choose 5 0 that l|h, - h2 5 iimllies
sup; ((r h2) F(r h4] We can split the sum into indices with ||: h.ll 8 and l|;
hill 2 8:
Σ w (n) (hx) F(r h2)) - F(r h2)l = Σ
w ( (ho)) h.) F(r hal
alll: co
Σ
w (m)) (h:))) h2) F(r hz)
+: 1h-11115
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The first term is smaller than
wi (d) (h:) F(r h.) F(r ha) <e ;: 1
Σ
Σ
(m) (ha) <
which we can make arbitrarily small. The second term goes to probability under Assump- tions 3.5-(i) and 3.5-(i)): the temperature goes to zero, the weight concentrates observations h; close to feg and the weight of points not in the neighborhood of he vanishes
Σ
"
Th) (kt) h.) F(r h.) 2
Σ
W (n))
(h+)
(h+)
0.
12-000
k:l 22
s:1|
126
Uniform convergence: Consider points
The proof here analogous to proofs the CeveenkoCCanter theorem.
0c = ro <n < <TK-1 < TK = 00
such that for all e
F(TR h.) F(re-1 he) S
This implies
(r h2) - F(r h.)l max ISkSK
(rk: h2) - F(rx he)l +
sup
Since we already proven pointwise convergence, we have that
(r he) F(rx hc)l
max
n-+00
and since e can be made arbitrary small
Palr he) F(r hi) O.
sup
FR
Proof of Corollary 3.7. Starting from
IP (yc+H E Ö (f++H) ne) = -P(re E (Rape(te):00aa//p m.)
we have
PP (rLHH* [Qaj2(hi), Q1-a2((()))()))) h.) =F (Q1-a/2(th)) [ -F hi)
and since Theorem 3.6 gives us the consistency of the quantile estimator to the true conditional quantile, have that
F' (Q1-a/2(tto) mi) F (Qa/a(te)) ms) 2=00
1 a.
B DATASETS AND POINT PREDICTORS
B. DATASETS
In the experiments that we performed, we considered four datasets coming from different real-world scenarios and application domains.
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Solar data The solar data comes from the US National Solar Radiation Database (Sengupta al., 2018). We worked with the dataset of 50 time series generated by Auer et al. (2023), each consisting of 52,609 half- halflyy measurements. These were resampled to obtain 26.304 hourly observations. Solar radiation (dhi, Diffuse Horizontal Irradiance) was designated as the target variable, and the dataset includes eight additional environmental features:
dni, Direct Normal Irradiance: amount of solar radiation received per unit area from the sun's s direct beam on a surface kept perpendieular (normal) to the sun's rays.
dew point: the temperature at which air becomes saturated with water vapor, causing condensation (dew) to form.
a r.temperature the ambient temperature of the air.
wind.speed: the rate at which air is moving horizontally.
the depth of water in column of the atmosphere all its water vapor were condensed and precipitated.
totalapreciiitallee
relati humi di ty: the percentage ratio the current water vapor in the air to the maximum amount the air hold at that temperature.
solar_zeni taaggle:: the angle between the sun's rays and the vertical direction at given location.
surface a albedo the fraction of incoming solar radiation reflected by the Earth's sur- face.
Air Quality data The dataset was first introduced in Zhang et al. (2017) and comprises 35,064 measurements from 12 locations in Beijing, China. It includes two air-quality measurements (PM10 and PM2.5); for our experiments, we used PM10 as the target and excluded PM2.5 from the feature set. The dataset also contains ten additional environmental variables:
502: sulfur dioxide.
NO2: nitrogen dioxide.
CO: carbon monoxide.
Og ozone.
TEMP: ambient air temperature.
PRES: atmospheric pressure.
DEWP: dew point temperature.
RAIN: amount of precipitation.
wd: the direction from which the wind is blowing. This was encoded following the method in Auer et al. (2023).
WSPM: horizontal wind speed.
Exchange rate data The collection of the daily exchange rates eight foreign countries, includ- ing Australia, Britain, Canada, Switzerland, China, Japan, New Zealand, and Singapore. The data span 1990-2016, yielding 7,588 samples.
ACEA data The data was first introduced in Bianchi et al. (2015). The time series represents electricity load provided by ACEA (Azienda Comunale Energia e Ambiente), the company that manages the electric and hydraulic distribution in the city Rome, Italy. It was sampled every 10 minutes and contains 137.376 observations.
B.2 POINT PREDICTORS
For each dataset, we trained three distinct point-forecasting models that span different modelling paradigms. First, RNN built with gated recurrent unit (GRU) cells and single layer with hidden state size of 32. Second, decoder-only Transformer configured with hidden size 32, a feed- forward size of 64, two attention heads, three stacked layers, and dropout rate of 0.1. Third, an
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Autoregressive Integrated Moving Average (ARIMA) model of order (3, 1).. The first two models were trained and configured similarly to Cini et al. (2025), i.c. by minimizing the mean absolute error (MAE) loss with the Adam optimizer for 200 epochs and batch size of 32.
For each dataset, every model was trained independently using 40% of that dataset's samples as the training set. The first 25% of the calibration split is used as a validation set for early stopping.
C HYPERPARAMETERS AND EXPERIMENTAL SETUP
C.I EVALUATION METRICS
Given a prediction interval (+)) = Dt + +. De + 4: a2/2 computed for desired confidence level a and true observation Yt. we evaluated our method against the baselines using three funda- mental metrics in uncertainty quantification.
Coverage gap
The coverage gap is defined as
(1(y= (
ACov 100
and indicates the difference between the target coverage 1 and the observed one.
Prediction interval width This metric quantifies the width of the prediction intervals, i.e., how uncertain the model is about prediction i/
PI-Width = d. 11-a/2 a//2
q:
Winkler score This is combined metric that adds to the PI-Wh a penalty if the actual value Y falls outside the PI. The penalty is proportional by a factor to the degree error of the PI, i.e. how far the actual value is from the closest bound of the interval. The Winkler score is defined
( 1-a/2 / + 100002 Va)
if Vv < sa/2
W=
(g. 2/2
20/22
if 20/2 S S 4t
11a/22
(q) 1-a/2 2a/2 (yr /2/2
qt
if ye qi
1-a/2
C.2 ESN HYPERPARAMETERS
If properly configured, ESNs provide rich and stable representations of the temporal depen- dencies in the data. The main hyperparameters that we considered are: the reservoir size, which represents the dimension Dr of the high-dimensional space where reservoir's 's states evolve (the "ca- pacity" of the reservoir); the spectral radius p(W,) of the state update weight matrix W,, which governs the stability of the internal dynamics of the reservoir; the leak rate, which controls the proportion of information from the previous reservoir state that is preserved and added to the new state after the non-linearity; and the input scaling, which regulates the magnitude of the input signal before it is projected into the reservoir, and thereby control the degree of non- non-lincyryty that will be applied to the state update.
Other hyperparameters that define how the ESN behaves are the connectivity of the reservoirs, which denotes the proportion of non-zero connections in the state update weight matrix and thereby de- termines the sparsity structure of the reservoir, the amount and magnitude of noise injected during state update, and the activation function for the non-linearity. Sometimes, also the topology of reservoir can lead to meaningful dynamical properties (a ring-like arrangement of the reservoir's internal connection as opposed to random connections).
C.3 CP-QRNN ARCHITECTURE
The architecture of CP-QRNN is identical to the model called CORNN introduced by Cini et al. (2025). It consists in a encoder-decoder architecture, in which the encoder is implemented as multi- ple GRU layers, and the decoder is multilayer perceptron (MLP) that maps the learned representa- tions to the predictions of the quantiles. The model trained end-to-end by minimizing the pinball loss.
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Table 4: Best hyperparameter values found with the grid search described in the previous section for each dataset and base model combination.
Spectral Radius
Leak Rate
Input Scaling RESCQR
Temperature RESCP
Sliding Window
Model
RESCP RESCQR
RESCP RESCQR
RESCP
RESCP
Solar
RNN
0.9
0.9
0.75
1.0
0.7
0.1
0.1
3900
Transformer
0.9
0.9
0.8
1.0
0.4
0.1
0.1
7600
ARIMA
1.0
0.95
0.8
0.9
0.2
0.1
0.1
1500
Beijing
RNN
1.3
0.99
0,95
1.0
0.25
0.1
0.1
3200
Transformer
1.0
0.9
0.8
1.0
0.25
0.1
0.1
10500
ARIMA
1,45
0.99
0.65
1.0
0.75
0.1
0.15
3800
Exch.
RNN
0.95
0.9
0.8
0.9
0.5
0.25
0.1
1000
Transformer
0.99
0.99
0.8
0.9
0.5
0.1
0.1
1000
ARIMA
0.95
0.99
0.8
0.8
0.5
0.1
0.1
1000
ACEA
RNN
0.95
0.99
0.8
0.8
0.25
0.25
0.1
1000
Transformer
0.99
0.99
0.8
0.8
0.25
0.25
0.1
3000
ARIMA
0.95
0.9
0.8
1.0
0.25
0.1
0.1
1000
C.4 MODEL SELECTION
We performed hyperparameter tuning for all combinations of point predictors and datasets on validation set, depending on the model.
RESCP For our method, we used 10% of the calibration set for validation. To perform model grid search over the spectral radius for values in the interval [0.5, 1.5). for the leak rate [0.5, 1] (1 meaning no leak), for the input scaling [0.1,3) 3). for the temperature [0.01, 2] and for the size of the calibration window [100, "all"). where "all" considers all states in the calibration set. The remaining ESN hyperparameters where kept fixed (reservoir size 512, connectivity 0.2, tanh activation function, no injected noise during state update, random connections). Given the high scalability of the method, performing such a grid search was very fast. The same was done for RESCQR, but only on the ESN hyperparameters (reservoir's size, spectral radius, leak rate and input scaling). In this setting, to train the readout we used the Adam optimizer (Kingma & Ba, 2015) with an initial learning rate of 0.003, which was reduced by 50% every time the loss did not decrease for 10 epochs in row. Each epoch consisted of bini-batches of 64 samples per batch.
selection,
NexCP The search was made for the parameter used to control the decay rate for the weights. The search was over the values (0.999, 0.99, 0.95, 0.9).
SPCI Since SPCI (Xu & Xie, 2023a) required fitting quantile random forest at each time step, the computational demand to use the official implementation was limiting. Therefore, we used the implementation provided by Auer et al. (2023), and followed the same protocol for the training. SPCI was using fixed window of 100 time steps, which corresponds to the longest window considered in Xu Xie (2023a).
HopCPT We followed the same model selection procedure and hyperparameter search the orig- inal paper (Auer et al., 2023). The model was trained for 3,000 epochs in each experiment, validating every 5 epochs on 50% of the calibration data. Adam W (Loshchilov & Hutter, 2019) with standard parameters (3, 0.9, B2 0.999, 8 0.01) was used to optimize the model. A batch size of time series was used. The model with smallest PI- Width and non-negative ACov was selected. no models achieved non-negative ACov, then the one with the highest ACov was chosen.
Also for this baseline, followed the same model selection procedure as in the original paper (Cini et al., 2025). The number of neurons and the number of GRU layers was tuned with small grid search 10% of the calibration data.
CP-QRNN
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Table 5: Performance comparison for 0.05. ACov values color-coded for undercoverage
cases: yellow (1-2%), orange (2-4%), red (>4%). The top three Winkler scores for each scenario
highlighted: bold+underlined (Ist), underlined (2nd), dot-underlined (3rd).
Learning
Non-learning NexCP
Metric
SPCI
HopCPT
CP-QRNN
RESCQR
SCP
RESCP
Transt RNN
ACov
-0.191006
-0.90
-0.33
-0.38
0.11
0.97
0. 0.182
PI-Width
124.2514.19
82.29
2..14
77.762087
141.23
154.71
86.561008
Winkler
214.33:22.03
149.8811.15
94.73.10.46
108.42.10
239.24
225.85
138,564127
Solar
ACov
-0.400+:026
0.50
1.05
0.06
1.00
+1.29
PI-Width
123.48::5.82
85.55
1.154
76.15
139.52
153.24
62.61
Winkler
215.15433001
133,84
93.31
107.31
236,43
223.17
105 80
RNN ARIMA
ACov
-0.52+0.17
0.75
=0.40=0037
-0.62
0.07
1.21
0.21
PI-Width
124.61+1.77
188.97
70.67
87.34
188.57
202.58
99.68
Winkler
193.31+0.59
26..20
88.30
115.66+40.2
281.61
265.15
+33.
ÁCov
+1.388
-1,06++:1.00
-0.36
-0.58
-0.18
-0.01
-0.60
Pl-Width
97.22
112,10+31.4
84,51+22990
87.50
98.51
100.48
86.14
Winkler
167.81+2.37
183.15
130.62++0.69
170.36
166.04
124.24
Beijing
Transf
ÁCov
-1.27
-0.46:+9.56
-0).43
-1.05
-0.27
-0.03
-0.52
PI-Width
95.82:15.88
100.941:..0
84.24
87.662601
97.75
100.17
82.11 1188
Winkler
167.34.2.11 167.90+2.4
131.27
137.80
171.09
166.74
119.46:0..5
RNN ARIMA
ACov
-0.53:10.36
-1.20:10.45
-1,30=006
-1.13
-0.23
-0.14
-0,29
PI-Width
104.81 94.47+2.13
81.64
15.17
106.81
107.73
94.54
Winkler
175.14
16..8811.21
128.63
136.63
180.18
174.82
154.79.403
ACov
0.58++0.1
1.72
-2.00
-0.52
1.35
1.18
0.54
PI-Width
0.0290++0.000
0.0513
0.0404=00001
0.0479
0.0549
0.0497
0.0260
Winkler
0.0356:00.00
0.0608 +0.0000
0,0568
0,0002
0.0578:t00009
0.0645
0.0604
0.0321
0.0000
Exchange
Transf
ÁCov
1.98
1.81
0.94
-0.01
2.67
1.93
0.67 =0.11
PI-Width
0.0.10:10.008 0.0501+..0000
0.0441
0.0463
0.0646
0.0601
0.0281
Winkler
0.0375-+0.000 0.0600++0.000
0.0585=0.000
0.0595
0.0745
0.0712
0.0355 : 4-00002
RNN ARIMA
ACov
1.01::04
1.39
-1.50
-1.7622063
1.87
1.32
-0.03
PI-Width
0.0293 0.0482+t0.000
0.0404=0.0006
0.0419200016
0.0489
0.0442
0.0258 000000
Winkler
0.0357
0.0576::00000
0.0060
0.0547
0.0585
0.0552
4.0328-1umm
ÁCov
- 1.20
-1.924:0.00
-8.29=1.64
-15.3224457
-0.89
-0.29
0.32
PI-Width
11.132088
23.20:10.00
19.85
18.18 58
24.31
24.48
11.9010.00
Winkler
18.3810.4
32.48:+0.00
39.722466
46. 16.34
32.57
31.23
16.31 16.31.1.20 1:023
ACEA
Transf
ACov
-1.75:1.19
-2.04-1:0.00
-10,72:47.28
-21.13
-3.49
-0.27
1.72:10.06
PI-Width
11.22:048
22.80:10.00
18,44
16.25
21.99
25.34
12.62+10.7
Winkler
18.8440.7
32.54-10.00
41.67:9.46
52.93
34.43
32.01
16.19
ARIMA
ACov
0.92+40.7
-1.874+0.00
117.36
-23.8447..9
-0.13
-0.24
-2. 2.700
PI-Width
14.86+0.40
39.74-4-000
25.26+44444
20.67 +0.67
43.65
41.42
11.68
Winkler
21.83+0.19
49.57-40..0
63.53
67.04++
50.52
48.69
18.01 +0.53
C.5 RESULTING HYPERPARAMETER
The best hyperparameters found with the grid search described in the previous section can be found in Tab. 4.
D ADDITIONAL RESULTS
D.I RESULTS WITH DIFFERENT SIGNIFICANCE LEVELS
In addition the results presented in the main text for miscoverage level a 0.1, we also evaluate the performance of RESCP and RESCQR at significance levels a = 0.05 and a = 0.15. Tables and 6 summarize the results for these additional significance levels across all datasets and base fore- casters and against all the baselines considered in the main text. The results are consistent with those discussed in the main text for 0.1, confirming the effectiveness of RESCP in producing well-
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Table 6: Performance comparison for 0.15. ACov values color-coded for undercoverage
cases: yellow (1-2%), orange (2-4%), red (>4%). The top three Winkler scores for each scenario are highlighted: bold+underlined (Ist), underlined (2nd), dot-underlined (3rd).
Learning
Non-learning
Metrie
SPCI
HopCPT
CP-QRNN
RESCQR
SCP
NexCP
RESCP
ACov
0.72.t023
-1.7740.33
-1.102183
0.10=1m0
0.64
1.82
-0.47
Transt RNN
PI-Width
46.78 46.781081
47. 46.66
46.84
47
47.85
55.73
46.37
Winkler
119.10
96. 0..30
69.15
134.09
130.47
87,444022
Solar
ACov
0.34
2.88
2.54
-3.19
0.60
1.81
0.18
PI-Width
16.56#1.69
46. 46.78
46.47:1.25
47.06=0036
48.72
56.72
39.37:00.11
Winkler
119.81
89, 3..30
68.48
74.44
133.18
129.62
75.69+0.99
RNN ARIMA
ACov
0.87+0043
1.98
-100620.33
-0.43
-0.07
2.29
0.30
PI-Width
1..25
118.82
50.18#0.44
60.41
93.98
102.74
62.24
Winkler
124.75+40.5
171.47+:1132
68,52
87.26+
179.83
174.48
14,54
ÁCov
-0.94+
0.56
-1.31
10.26
-0.48
0.10
+0.98
Pl-Width
55.46+2.03 67.60+118.7
51,21
54.58
52.93
55.07
52.21+2206
Winkler
103.43
112.44+11.23
87,49
90.42-+0.3
104.56
103.16
86.72
Beijing
Transf
ÁCov
-0.69
-0.45
-1.3
-1.68#181
-0.46
0.11
-0.75
PI-Width
54.51
55. 1413.11
50.67
52.98
52.42
54.64
40.31
Winkler
102.7440.5
103.53
87.71
90,97
104.60
103.21
84,47
RNN ARIMA
ACov
0.74:20.0
-3.0840.97
-2.04.0089
-2.0720079
-0.25
-0.09
-0.32
PI-Width
60.45:0885
53.46+1.10
50.58
51.64
60.29
60.73
14.70
Winkler
108.16.4047
107.50
86.46_0955
90.85
112.79
110.66
99.59
ACov
5.13+0.59
3,05
-3.28
2.40
3,06
2.26
1.77
PI-Width
0.0207+0.0004 0.0002
0.0284=0.0009
0.0323
+0.0000
0.0380
0.0351
0.0180
+00000
Winkler
0.0247+00.002 0.0420+t.0000
0.0413 000005
0,0410
0.0452
0.0436
0.0234
Exchange
Transf
ÁCov
6.43::0.6
3.78
0.25=1186
-2.20
6.03
4.08
2.37
PI-Width
0.0221 1:00000
0.0341+:0.000
0.0301
200007
0.0313
0.0483
0.0455
0.0195
Winkler
0.0261 010.004
0.0421
2.0425500000m
0.0422+00008
0.0558
0.0545
0.0262
Transf RNN ARIMA
ACov
6.23 2.96+00.0
-2.26=1.499
-2.04
4.25
2.77
1.03
PI-Width
0:0013:00000 0.0022+t0.000
0.0279=0.006
0.0294100006
0.0330
0.0303
0.0175+0.00 200000
Winkler
0.0253-10.00
0.0399
0.0405
0.0401
0.0404
0.0392 VerDees
0.0234
ÁCov
- 1.841148
-2.43:1:000
-16. 15:9.0
-20.50=5.38
-1.55
-0.46
1.8610.00
PI-Width
7.42+071
15.93+10.00
13.09
13.29-1:31
16.34
17.07
8.06
Winkler
12.33
24.51
28.71 1:377
30.47=2.40
24.66
24.15
11.14:0.11
ACEA
ACov
-2.58:0078
-2.98:100.00
-17.58=10.86
-29.97
-5.76
-0.56
4.20
PI-Width
4.34
15.26:10.00
11.98
10.99
13.66
16.73
8.58
Winkler
12.6940.10
24.40:00.0
29.59
33.91
25.63
24.52
11.42:1020
ARIMA
ACov
0.39
-4.8140.00
-36.21
-34.84
-1.27
-0.72
1.16+1.10
PI-Width
10.92
30.89+-0.00
14.51
13.39#191
33.68
32.72
8.31+ +-0.22
Winkler
15.51+0.12
41.82++..0
4..54
44.95
43.00
40.68
11.66
calibrated and accurate PIs across different significance levels, datasets, and base point predictors.
D.2 CALIBRATION CURVES
To better evaluate the calibration performance of RESCP and RESCQR, we plot their calibration curves for all datasets and base models. The results are shown in Fig. 2. The calibration curve plots the target coverage against the observed coverage, where a perfectly calibrated method would follow the diagonal line y = I, which is depicted as a black dashed line in the plots. The target coverages {0.7, 0.75, 0.8, 0.85, 0.9, 0.95), are shown on the æ-axis, and the corresponding observed coverages are shown on the y-axis. Each row of the grid corresponds to dataset, while each column to a base point forecaster. The results confirm the findings discussed in the main text for a = 0.1, showing that RESCP produces well- cell-calibrated PIs across different significance levels, datasets, and base models. RESCQR also generally generates accurate prediction intervals, but its performance poor in some benchmarks such ACEA.
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Transfommer Solar
ARIMA Sodar
0.95
0.30
Observed Coverage
0.85
0.90
0.75
ResOQR
0.70
RNN Beiling
Transformt Beiling
ARIMA Betjing
0.95
0.90
Obsrrved Coveruge
0.85
0.80
0.25
RNN Exchange
Transormer Exchange
ARIMA Exchange
0.95
Obsrrved Coveruge
RNN ACEA
Trandormer ACEA
ARIMA ACEA
Obxervel Cowruge
0.85
0.50
0.75
0.70
0.70
0.25
0.80
0.85
0.50
0.95
0.70
0.75
0.50
0.85
0.00
0.95
0.70
0.75
0.90
0.95
Nominal Coverage
Nominal Coverage
Nominal Coverage
Figure 3: Calibration curves of RESCP and RESCQR over all datasets and baselines.
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Table 7:
Ablation study the AR synthetic dataset for three miscoverage levels (a
{0.05, 0.1, 0.15)).
Metric
RESCP
No decay
No window
No decay, window
NExCP
SCP
0.05
Coverage
94.41
93.50=0.21
91.6220m
88.392017
94.34
86.11
ACov
-0.59
-1,50.0.23
+3.38
-6,611017
-0.66
-8.89
0.1
Coverage
89.75
88. 88.25
84,45
80.85
88.84
78.48
ACov
-0.25
-1.7520.41
-9
-9.154917
-1.16
-11.52
Coverage
84.93+0.54
83.1 =0.35
78,33
73.99
84.14
71.92
ACov
-0.07
-1.892035
-6.67.0.20
-11.01
-0.86
-13.08
D.3 SYNTHETIC DATASET
To further validate the effectiveness of RESCP, we conduct an ablation study on synthetic data where the ground truth is known. We generate 10000 samples from synthetic autoregressive (AR) process of order
V/ 0:30-11 with ~ (0,11
with time- varying coefficients Φi across the calibration and test sets. Specifically, we introduce three change points in the coefficient sequence o between the calibration and test sets. Overall. 6, assumes the values (-0.9, 0.3, 0.5, 0.7). The first change point occurs at the midpoint of the calibration set, while the remaining two are placed at equal intervals over the rest of the series. The same setup as in Sec. is used: data is split into training, calibration, and test sets with proportions 40% /40% 40%//0% /20%, and RNN is employed as the base forecaster. We then compare the performance of RESCP by removing its two key components: (i) the time-dependent weights, (ii) the sliding window of past residuals used for calibration, and (iii) the combination of the two. The results, reported in Tab. 7, demonstrate that both components significantly contribute to the performance of RESCP, with the full model achieving the best results in terms of coverage. This ablation study synthetic data further confirms the robustness and effectiveness RESCP in capturing the underlying dynamics of time series data.
D.4 VISUALIZATION THE ADAPTIVE PIs
To give better intuition of how RESCP adapts the width of the PIs over time in response to changes in the data distribution, we visualize some examples of adaptive PIs produced by RESCP on the ACEA and Solar datasets in Fig. 4 and Fig. 5, respectively. These plots illustrate how RESCP effectively adjusts the uncertainty quantification based on the observed patterns.
E RESCP ABLATION STUDY
E. ABLATION ON RESCP COMPONENTS
In this section, we report the remaining results the ablation study on RESCP components pre- sented in Sec. 4 for the RNN base model, extending them to all base models considered in experiments. The results, summarized in Tab. 8, confirm the findings discussed in the main text. Specifically, we see that both the time-dependent weights and the sliding window of past residuals significantly contribute to the performance of RESCP across all datasets and base models.
E.2 USE OF EXOGENOUS VARIABLES
We further investigate the impact of incorporating exogenous variables into the RESCP framework for the Solar and Beijing datasets. To this end, we conduct experiments comparing the performance of RESCP with and without exogenous inputs using all base models. The results, summarized in Tab. 9, show that the inclusion of exogenous variables in RESCP generally leads to worse perfor- mance in terms of coverage and Winkler score. This supports our choice of excluding exogenous variables from RESCP. Removing exogenous variables from RESCQR, instead, leads to clear de- graded performances on the Solar dataset, while on the Beijing dataset the results are similar (ex- ogenous inputs do not appear particularly relevant to quantify uncertainty in this dataset).
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RtpPP Interval
20
46
Vahue
8
NecCP RISPP
Prodiction interall widhh
10000
2000
38.00
10000
12:00
18:49
Tima
Figure 4: Prediction intervals of RESCP (top) and prediction intervals' width of different methods (bottom) for the ACEA dataset
REsCPP intervala
86-
36-
Value
NeeCP
RESPP
Productioon Interaa withh
100
100
Time
Figure 5: Prediction intervals of RESCP (top) and prediction intervals' width of different methods (bottom) for the Solar dataset
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Table 8: Ablation of RESCP. RESCP without decay, RESCP without sliding window, and the
combination of the two. Best Winkler scores are shown in bold.
Metric
RESCP
No decay
No window
No window, no decay
RNN
ACov
0.74
00.1020.6
2.89
1.70
Pl-Width
62.25
60.59
59.81
60.9010.0
Winkler
104.2440.9
107.4620.49
104.70=a20
104,46:
Solar
Transf
ACov
3.09
2.22
-0.37
0.73
Pl-Width
63.34
58.49 0.882
53.4111044
56.37
Winkler
18
103.35
103.71
101.36::0.3
RNN ARIMA
ACCvv
0.68#005
-1.06=1.27
2.93+0034
2.97
PI-Width
77.17
80.27
82.68:12.29
85.91
Winkler
110.384403
113.09
+1.,41
118,20+:285
ACov
-0.70
-1.64
0.10+136
L644
PI- H1-Width
65.96:1250
64.04_1169
64.91:1336
69.53
Winkler
106.07
114.85
97.99+
98.25
Beijing
RNN ARIMA Transf
ACov
-0.49
-1.53
-1.45
-1.09
PI-Width
64.06:11.74
62.23
62.37:1196
63.94-1201
Winkler
103.64:0.21
112.31 +0.74
111,88+000
1 112.40
ACov
0.63+o22
0.35++023
1.3 1.31
2.36+:041
PI-Width
80
70.90=0.51
72.85
77.26
Winkler
108.75
119.72+0..4
118. +00.14
118.50+t.66
ACCov
1.13
2.01
4.19
4.20
PI-Width
0.0210 0.0219
0.0249 #00002
4.0054
Winkler
0.0264:2.0mzz
=..26920.002
0.028440000
0.0291 1:0000
Exch.
Transf
ACov
1,46
2.36
4.45
4.80 14
P1-Width
0.0229 #0.000
0.0241
0.00002
0.0265 +-0000
0.0267+000000
Winkler
0.02942a.0m1
0.0298
0.0306
0.0307
RNN ARIMA
ACov
0.38
1.61 20.21
3.90+021
3.40
Pl-Width
0,0207
0.0215 000000
0.0245
1.0248+00002
Winkler
0.0268
0.0273
0.0288
0.0293400002
ACov
1,56
2.79
5.34
4.962:03
PI-Width
9.6 9.61
10.15
11,88
12.15+0.7
Winkler
12.91
13.41
14.80
15.25
ACEA
ARIMA Transf
ACov
3.54
3.29
2.97
3.04
PI-Width
10. 10
10.32
9.34+t015
9.45
Winkler
12.90
13,45-20115
12.27:1026
12.49
ACov
5.02#t0.0
5.17
7.68+a22
7.75+004
PI-Width
13.6340.5
13.50=0.32
16.744to72
16.91
Winkler
16.21
16.27=2045
19,24:1064
19.44-1045
F SENSITIVITY ANALYSIS ON ESN HYPERPARAMETER AND TEMPERATURE
After performing model selection and evaluating our model with the resulting hyperparameters, we conducted small sensitivity analysis to determine how each part of the reservoir influenced model performance in RESCP, allowing us to assess the robustness of our chosen hyperparameters and identify the key aspects of the reservoir architecture that contribute most to quantification accuracy. We considered the RNN point forecasting baseline.
Fig. 6 shows how the value of the Winkler score and of the coverage varies as the temperature used in the SOFTMAX X in equation Eq. 7 increases. As can be seen, there is a well-defined optimal range for the temperature in which the Winkler score is minimal and the coverage close to the targeted one. Notice that for small temperature, the method is unable to achieve the nominal coverage. This is the bias-variace tradeoff that we mentioned in Section 1.1.1, more specifically in Assumption 3.5: the temperature should be small enough to localize only the meaningful residuals close to the query one, yet high enough to avoid collapsing the neighborhood to a single or limited set of similar samples.
Fig. 7 shows the behaviour of RESCP depending on the capacity of the reservoir. Apart from the Beijing dataset, the method can achieve the nominal coverage also when the dimensionality of the
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Table Comparison of RESCP and RESCQR with and without exogenous variables for Solar and Beijing datasets. Best Winkler scores (lowest) within each method are shown in bold.
Metric
RESCP
exeg
RESCQR
exeg
RNN
ACov
0.74
1.34
-1.101091
-0.70
PI-Width
62.25
$3.87
49.99
56.71
Winkler
104,24:0.99
105,55
82.76+0.6
88 88.46=042
Solar
Transf
ACov
3.09
3,57
16.2
1.37
PI-Width
63.34
56,96+0227
59.56+1.99
57.84-0.36
Winkler
103.13
101.73++.33
82.16:0.32
88.30=0053
RNN ARIMA
ACov
0.68
-1.122013
2203
-1.03
PI-Width
77.17+220r
78.09
66. 19401
66.85=095
Winkler
110.38
127.79
85.38+
88.19
ACov
-0.70
4..71
-1.21+1165
-0.53
PI-Width
65.9612.50
56.294000
65.53
65. 10.544
Winkler
106.07 +0.47
115.73++0.00
105 -0..43+0085
104.93
Beijing
ARIMA Transf
ACov
+0.49
-1.87
+..43
0.20=
PI-Width
64.0621.74
59.5221022
64.41
66.49_1117
Winkler
103.64:022
112.90
105.97
105.44
ACov
0.63
+0,96
-1.42:+1.15
-0.16
PI-Width
70.43::0.6
66.01
67.33_2228
Winkler
108.75
120.59
107.20
103.97
Solar
Beijing
240
160
220
Winkler 503te
150
088
Winhder score
200
008
140
0.7
Coneragg
180
0.7
130-
160
0.5
140
0.5
110
0.44
0.4
0.01
0.05
0.1
0.25
2.0
0.01
0.05
0.1
0:25
2.0
Temperature
Teuperatuze
Exchhange
ACEA
0.060
0.9
0.9
0.015
222
Winkle wade
0.000
Coverage
20
0.7
Waaket
18
0.6
0.6
10000
14-
0.44
004
0.000
12
0.05
2.0
Temperature
Temperatuze
Winkler scare
Coverage
Figure 6: Sensitivity analysis of the temperature.
reservoir space contained, at the price of higher Winkler score. For the Beijing dataset, bigger reservoir dimensionality is required to reach the target coverage.
The sensitivity analysis the spectral radius can seen in Fig. 8. interesting to see that in the Solar and Beijing datasets, the best performance is achieved even with p(W,) > 1. This is due to the condition being only sufficient, not necessary: while (W,) < 1 guarantees contractive dynamics in the autonomous linearized reservoir (Gallicchio & Scardapane, 2020), driven reservoirs with input scaling, leaky units, and bounded nonlinearities (Dong et al., 2022; Ceni & Gallicchio, 2025).
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Solar
Beijing
105
107.2
107
Whiker 8000
0.91
Waskler Score
0.91
100
Clovrange
106
Chovernge
0.50
00.00
105
-0.50
106
-0.50
0.58
106.6
0.55
62
125
512
125
Hiddeea Size
Hidden Stee
Exchange
ACEA
0.53
0.50
010274
15.0
010272
0.92
0.32
0.070
0.91
Wincktr 8008
14.5
0.90
Winlder
Coremige
Cororigg
0.0968
0.90
0.50
0.0264
13.5
0.0262
0.57
332 64
128
256
512
64
128
25G
612
Ridren Sine
Hidrenn Siop
Winkler score
Coverage
Figure 7: Sensitivity analysis of the reservoir size.
Solar
Beljing
0.0
0.95
120
0.90
112
0.50
Wablee sote
Wialder scare
Coverage
Coocr HUgS
0.88
110
110-
0.80
108
0.90
0.77
106
0.75
0.5
0.75
1.1 Spectral Radins
0.9
0.99
0.5
0.75
0.99
1.1
1.5
Spectral
Exchange
ACEA
0.00
0.95
0.000
0.90
0.90
Walka scone
15
Cooerage
Wakket
0..60
0.008
+0.00
-0.50
0.027
13
0.76
0.75
0.99
1.1
Spectral Radius
Spectzal Racius
Winiske score
Cowerage
Figure 8: Sensitivity analysis of the spectral radius.
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Solar
Beijing
0.95
0.95
116
0.90
0.90
114
11.5
114
0.55
Winddn 803000
Winddr autiee
0..50
Conetagg
0.550
112
110
01773
0.27
108
0.70
110-
0.70
108
0.65
108
0.65
100
0.7
0.7
0.55
Loak Rate
Leak Rate
Exchange
ACEA
0.95
0.05
0.90
0.90
0.002
18-
0.85
0.85
Wilki acuec
0.50
Chricrige
0.50
0.000-
0.77
Willet
16-
0.25
0..70
0..70
0.088
0.65
14-
0.05
0.000
-0.00
0.006
0.5
0.7
0.8
0.9
1.0
0.5
0.7
0.8
0.9
1.0
Look Rate
Leak Rate
Winkder
Coverage
Figure Sensitivity analysis of the leak rate.
Solar
Beijing
170
035
0.53
160
0.92
116-
0..0
Winklct soue
150
0.91
Winkeet scoze
114
0.91
Coricaige
140
0.90
132
0.50
200
00.50
00.50
+0.58
-0.55
108
110
0.57
0.57
100
0.01
0.1
0.25
0.5
1.0
2.0
0.01
0.1
0.25
11.5
1.0
2.0
Ingeat Scaling
Inpot Sealing
Exchange
ACEA
0.008
0.000
24-
22
Conergg
Winkder
0.90
Winkke
0.00
Commnnge
0.000
0..89
0..55
0.028
0.58
0.55
14-
0.026
0.57
0.557
0.25
Inpat Sealing
Inpot Scaling
Winkler score
Coverage
Figure 10: Sensitivity analysis of the input scaling.
In Fig. 9, we observe how varying the leak rate affects the reservoir's performance. In particular, a low value of the leaking rate slows the reservoir's state updates, causing the network to retain past information for longer but reducing its responsiveness to new inputs. We can see how this heavily affects the observed coverage in almost all datasets.
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Finally, the effect of different values of input scaling be found in Fig. 10. High values of input scaling lead the tanh activation function to saturation during state update, yielding loss of sensitivity to imput variations. On the other hand, small input scaling keeps the activation function in the linear part, limiting the richness nonlinear dynamics necessary to capture complex temporal patterns.
G IMPLEMENTATION DETAILS
The code have been written in Python (Van Rossum & Drake, 2009) using the following open-source packages:
Numpy (Harris et al., 2020);
Pandas (The pandas development team, 2020; Wes McKinney, 2010);
Py PyTorch (Paszke et al., 2019);
PyToocch Lightning (Falcon & The PyTorch Lightning team, 2019);
Torch Spatiotemporal (Cini & Marisca, 2022);
reservoir-comput ing (Bianchi al., 2020).
Experiments were conducted on machine equipped with AMD Ryzen 9 7900X CPU and NVIDIA GeForce RTX 4090.
H USE OF LARGE LANGUAGE MODELS
We acknowledge the use of Large Language Models a writing tool for minor edits in single sentences.
I ETHICS STATEMENT
The work presented in this paper is about basic machine learning research and we perform experi- ments on standard, publicly available datasets. The authors have read and adhere the ICLR Code of Ethics and do not foresee any direct ethical concerns or potential for misuse.
J REPRODUCIBILITY STATEMENT
The code to reproduce the experiments presented in this paper will be made publicly available.
30


Reviews:
[{"review_id": "mawJfYMZt0", "reviewer": "Reviewer_WWw1", "summary": "The work proposes a novel method that utilises Reservoir Computing to reweight conformity scores. This helps account for the local temporal dynamics, and under some assumptions, the method can achieve conditional coverage. Empirical evidence for the method's efficacy is provided across various forecasting tasks.", "questions": "1. Line 078: If I understand correctly, y_t is a scalar observation. How would the method incorporate the cases where we have multidimensional time series forecasting?\n\n2. Line 081: If I understand correctly, y_hat_{t+H} is a vector corresponing to H steps of prediciton. Since both times, a small y is used, it becomes notationally confusing to follow the stuff here. \n\n3. In equation 2, shouldn't it be the absolute residuals, since they are used as the nonconformity scores for regression? \n\n4. Line 102: \"To address heteroskedasticity\": does it refer to heteroskedasticity over the values, or time index, or both?\n\n5. Line 191: Is x_t supposed to be the residual, not the history i.e y_{t-1}...\n\n6. Line 196: Is x the time series now instead of y, and not the exogenous variables or residuals?\n\n7. Equation 6: Shouldn't s be upper-bounded by T and not T-H?\n\n8. Equation 11: If we want symmetric intervals, can we work with absolute residuals and adapt them accordingly?\n\n9. Assumption 3.1: Earlier, it was alluded to that under mild assumptions, the coverage is satisfied; however, strongly mixing seems to be a rather strong condition. Would any of the real-world time series even satisfy it?\n\n10. \"Assumption 3.2 tells us that the reservoir state evolves in a stable and predictable manner\"  This also seems a strong assumption, especially if there is a change of distribution. \n\n11. Line 338 \"do not include any positional encoding\": Could having a positional encoding help the algorithm?\n\n12. Regarding Table 1: Firstly, it is interesting that adding exogenous variables is hurting the performance, as apparent from RESCQR. As for RESCP, it is not clear if there are huge advantages; in most cases, across learning and non-learning, the average width is better for Learning methods, whereas many of the learning methods still have better Winkler score.\n\n13. The target confidence is set as high as 0.9, which is okay. However, it is necessary to demonstrate the performance of the proposed method with varying significance levels. A calibration curve may be helpful in determining if the proposed method works effectively in all cases.\n\n14. The theoretical statements are there for conditional coverage, but there is no empirical evidence for the same.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The idea is quite interesting to use Reservoir Computing to adapt the residuals. The background is well-written and thorough. The experiments are also conducted with various datasets, baselines, and models. The experiments are repeated across several runs, giving a good picture of the method's stability.", "weaknesses": "The biggest weakness is the comparison with only one significance level. 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\nThe notations are unclear at times and the presentation could be improved.\n\nSee \"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": "t492SGhK7E", "reviewer": "Reviewer_TvuY", "summary": "The paper studies prediction set construction for time-series data.\nThe proposed method builds on reservoir computing and learns the \nresidual distribution by dynamically weighting the calibration samples.\nThe method is evaluated on real datasets, showing advantages in \ncoverage and computation efficiency.", "questions": "Please refer to the \"Weaknesses\" section.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written and easy to follow. \n2. Integrating conformal prediction with reservoir computing demonstrates the potential to unite statistical efficiency and computational efficiency in predictive analysis of time-series data—achieving the best of both worlds.", "weaknesses": "1. The theoretical results are relatively clean, but depend on a set of assumptions. \nIt would be helpful to evaluate the method in synthetic environments (where the data-generating process is fully known)\nand systematically investigate the method's sensitivity to violations of its assumptions.\n\n2. In the numerical results, it appears that CP-QRNN often achieves strong statistical performance in \nreasonable computing time, compared with ResCQR. I wonder if the authors could provide more discussion on \nthe comparison, to clarify better the advantage of ResCQR and how a practitioner should choose between the methods \nin different scenarios.", "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": "kLhqPfpksd", "reviewer": "Reviewer_SLWv", "summary": "This study proposed a conformal prediction method for time series forecasting using reservoir computing implemented with echo state network.", "questions": "-\tIn 2.2 Echo State Networks, it is stated that when “$W_h$ and $W_x$ are properly initialized, …”, however throughout the paper, I can only find the condition about $W_h$ which is required to set to satisfy the spectral radius < 1. What is “proper initialization” for $W_x$?\n\n-\tIn line 217, “we approximate it through Monte Carlo sampling akin to Auer (2023)…”.  I believe its better to describe how the authors did this since its important things to construct the prediction intervals with some details: what sampling size did the authors used? Did authors used sampling size same for all datasets or different size for datasets?\n\n-\tIn line 225, the authors refined the intervals by selecting $\\beta$ that minimizes width. What are $\\beta$ candidates did authors used to obtain $\\beta^*$ that minimizes interval width?\n\n-\tIn Assumption 3.2, are there any assumptions need for $W_h$ and $W_x$? how did authors derive the bound for ESN based on $d_{\\text{fm}}$\n\n-\tWhat is CP-QRNN baseline? While the authors cited [1], there is no method entitled CP-QRNN in the cited paper. Additionally, the important details of CP-QRNN such as model architecture are missing in the main text and Appendix, which makes difficult to interpret the results since CP-QRNN is the baselines that showed the best performances in the experiments on Beijing and Solar datasets.\n\n-\tWhat exogenous variables are used for the experiments? which methods used exogenous variables?\n\n\n-\tIn sec 4.1, the author stated “CP-QRNN baseline achieves strong performance on the large datasets with informative exogenous variables,… but fails in achieving good results in the smaller datasets”. This claim does not sound convincing since we do not know whether exogenous variables are informative. It could be interpreted that in other two datasets where the proposed methods outperformed other baselines,  the exogenous variables are not informative? The authors need to provide details of exogenous variables or carefully designed experiments using synthetic dataset to support this claim. Additionally, what is the definition of the smaller datasets? Based on the dataset details provided in the appendix, the size of the ACEA dataset is the largest (solar data is 26k, Beijing air quality dataset is 35k, exchange dataset is 7.5k, and ACEA dataset is 137k), while the author stated that ACEA dataset is a small dataset.\n\n-\tIn line 60, the author claimed that “ResCP can be applied on top of any point forecasting model, since it only requires residuals from a disjoint calibration set”. However, isn’t this true for any CP methods? Since technically CP is base prediction model-agnostic.\n\n\n[1] Relational Conformal Prediction for Correlated Time Series, Cini et al., 2025", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "-\tThe study utilized reservoir computing implemented by echo state networks, which has not been studied before in the context of conformal prediction for time series.\n-\tThe study provided an asymptotic conditional coverage guarantee and the theoretical analysis was explained thoroughly.\n-\tDetailed backgrounds and motivation related to the existing studies.", "weaknesses": "-\tIn Assumption 3.1, the author assumed stationarity of $(x_t, r_{t+H})$. I believe the author used residuals for $x_t$ (i.e., $x_t = r_t$) for ResCP, and I think assuming stationarity of residuals (i.e., $(r_t, r_{t+H})$) makes sense. However, in 3.1, the author claimed that “the input $x_t$ can include any set of endogenous and exogenous variables at time step t”.  I am concerned that in this case the assumption does not hold, since in general time series features are not guaranteed to be stationary, which makes conditional coverage somewhat restricted to ResCP with $x_t = r_t$ (since I guess ResCQR utilizes other variables except residuals).\n\n\n-\tThe author proposed and used ad-hoc methods, such as using time-dependent weights described in 3.1.2 and handling exogenous inputs (i.e., ResCQR) described in 3.2, on top of the main proposed method (ResCP). However, with the current experiment design, it is difficult to see what is the performance of the proposed method and what is the improvement due to the ad-hoc methods. Based on my understanding, the authors used time-dependent weights as the default to ResCP and used exogenous variables only for ResCQR. Ablation studies regarding these ad-hoc methods will be helpful for readers to understand and interpret the experimental results. For example, ablation study: ResCP vs. ResCP + time-dependent weights vs. ResCP + exogenous variables vs. ResCP + time-dependent weights  + exogenous variables vs. ResCQR.\n\n\n-\tIn line 328 under Remark section, the author claimed that “good approximation of F entirely depends on the ability of the ESN to encode all relevant information from the past in its state”. I am not sure what it means since ESN does not have trainable parameters and only has echo state property, which says it asymptotically forgets its initial state. Even if the claim is true, then what is the advantage of using ESN over RNNs or other sequence models? Since I believe RNNs have more capacity to learn “all relevant information from the past in its state” to compute better weights to approximate F (even tho it would not feasible to achieve the asymptotical conditional coverage theoretically as in the way the author derived using other sequence models).\n\n\n-\tThe experiment results did not show consistent results supporting the effectiveness of the proposed methods, and the manuscript did not contain enough details about the implementation of the method, experiment setup, to interpret the results (see questions).", "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": "eFTkG9b2ro", "reviewer": "Reviewer_3iSQ", "summary": "This paper introduces ResCP (Reservoir Conformal Prediction), a novel training-free method for constructing conformal prediction intervals in time series forecasting. They use Echo State Networks (ESNs) - randomized recurrent neural networks - to compute data-dependent weights for conformal prediction without requiring expensive model training compared to existing works. The authors provide comprehensive theoretical and experimental support for the CP technique.", "questions": "- Strongly mixing data generation process was not an assumption by SPCI and HopCPT, I believe they both permit distribution shifts or regimes in the data generation process. I also do not think neither weather data (solar and Beijing) nor financial data is strongly mixing? I would love to see a discussion on what kind of data is NOT suitable for ResCP and it performs worse than SPCI for example. \n\n- Condition 3.5 (ii) is quite strong and perhaps need more discussion. How is this clustering property ensured? Is there evidence that the ESN hyperparameters (spectral radius, input scaling, etc.) produce a reservoir that maps similar dynamics to nearby states? Does it place a strong requirement that  the input sequence have strong recurring patterns? The paragraph below assumption 3.5 feels like 3.5(ii) can be a direct result of 3.5(i), which i do not think is the case.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- Excellent presentation. The motivation, algorithm, and results are all very clearly explained and easy to follow. \n\n- The algorithm is strong and fills a needed gap. In 2023 / 2024 there are a bunch of papers that use regression inside the CP algorithm in order to handle patterns in the residual (SPCI and HopCPT are prime examples), although they did achieve better efficiency and are inspiring, the methods are slow and data-hungry, negating some of the advantages of CP itself. Having a high performing training-free (almost non-parametric) alternative is great.\n\n- Strong Empirical Performance. Achieves competitive or superior results across multiple datasets, particularly outperforming HopCPT while being much faster.", "weaknesses": "Not much. The only criticism is perhaps the assumptions for ResCP is quite limiting, making the algorithm to be sensitive to data conditions (assumption 3.1) and hyperparameter choices (assumption 3.5). The sensitivity analysis shows performance varies significantly with these choices.", "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)