Conformal Prediction for Time-series Forecasting with Change Points
Authors: Sophia Huiwen Sun
Rose Yu
Venue: NeurIPS 2025 poster ()
Keywords: conformal prediction
time series
uncertainty quantification
Primary area: probabilistic_methods
TL;DR: conformal prediction for time series using switching dynamics systems

Abstract: Conformal prediction has been explored as a general and efficient way to provide uncertainty quantification for time series. However, current methods struggle to handle time series data with change points — sudden shifts in the underlying data-generating process. In this paper, we propose a novel Conformal Prediction for Time-series with Change points (CPTC) algorithm, addressing this gap by integrating a model to predict the underlying state with online conformal prediction to model uncertainties in non-stationary time series. We prove CPTC's validity and improved adaptivity in the time series setting under minimum assumptions, and demonstrate CPTC's practical effectiveness on 6 synthetic and real-world datasets, showing improved validity and adaptivity compared to state-of-the-art baselines.

Conformal Prediction for Time-series Forecasting with Change Points
Sophia Sun, Rose Yu
Computer Science and Engineering University of California, San Diego
Abstract
Conformal prediction has been explored as a general and efficient way to provide uncertainty quantification for time series. However, current methods struggle handle time series data with change points - sudden shifts in the underlying data-generating process. In this paper, propose a novel Conformal Prediction for Time-series with Change points (CPTC) algorithm, addressing this gap by integrating a model predict the underlying state with online conformal prediction to model uncertainties in non-stationary time series. We prove CPTC's validity and improved adaptivity in the time series setting under minimum assumptions, and demonstrate CPTC's practical effectiveness on 6 synthetic and real-world datasets, showing improved validity and adaptivity compared state-of-th--ar baselines. Our code is available at https:: //gi thub. com com/Rose Sese/RoseSSS-- Lab/CPTO
1 Introduction
Uncertainty Quantification (UQ) a key building block of reliable machine learning systems. Conformal prediction (CP) is popular distribution-free UQ method that provides finite sample coverage guarantees without placing assumptions on the underlying data generating process [50] 32] Because of its simplicity and generality, conformal prediction has wide adoption in many tasks, ranging from healthcare S robotics [34), to validating the factuality of generative models [17].
Existing CP algorithms for time series forecasting mostly adopt an online learning framework 44 where the prediction interval adapts to distribution changes in data reactively. The online CP algorithms achieve asymptotic marginal coverage guarantee (the miscoverage rate converges to specified fraction as the time horizon goes to infinity). As a result, the algorithms inevitably exhibit miscoverage when distribution shifts occur, and then have to over- or under-cover in later timesteps to compensate. This behavior is evident in the ACI and CP baselines in Figurel T This is undesirable in practice as these under-covered periods may ineur lot of risk.
This paper builds upon the observation that in real-world scenarios, distribution shifts in time-series often predictable. Take for example the task of forecasting electricity demands: we know that the hidden dynamics may differ between day and night, or during weekdays and weekends. The same logic applies to traffic forecasting and product demand forecasting, where there are patterns in the shifts between "surge times" and "normal times". In this work, we study time series data that exhibit this type of abrupt shifts, change points, in the underlying generative process.
State Space Models (SSMs) [28 1Q are powerful tool for modeling time series data, as they provide structured framework to capture the underlying temporal dependencies through latent states. In partieular, Switching Dynamical Systems (SDS) I a type of SSM with an additional set of latent variables (known switches) that represent the operating mode active at the current timestep. SDS introduces the inductive biases that allow SSMs to switch between a discrete set of dynamics, which can be learned from training data. It is shown to be flexible, robust, and interpretable representation of time series with varying dynamics exhibiting change points 9 [6]. SDS is deployed not only for
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Prediction Intevals Baselines
Prediction Intevals
C.1
True
Undercoverage
Irue Pred
Fast Adaptation
AC
:50
200
253
300
55
400
450
150
50 50-Step Mov ng Average Coverage
Remains Valid
Drops to invalidity
-0.5
230
Figure 1: Comparison of the prediction intervals obtained by our algorithm CPTC (purple) against online conformal prediction baselines on synthetic data. The vertical dashed line marks the distribution shifts; ideal behavior is consistent coverage at the horizontal dashed line in the final panel. Bottom right panel shows that CPTC achieves fast adaptation and remains valid when change points occur.
forecasting as examples given above, but also on a wide range of applications including neuroscience [25 41 30), engineering [45] and sports analytics (58).
energy 29 20 and traffic [59, 15
all would benefit from robust and calibrated uncertainty quantification.
Our contributions are:
We propose new algorithm Conformal Prediction for Time series with Change points (CPTC) that utilizes state transition predictions to improve uncertainty quantification for time-series forecasts. Leveraging properties of a SDS model, CPTC consolidates multiple future forecasts to adaptively adjust its prediction intervals when underlying dynamics shift.
We prove that CPTC achieves asymptotic valid coverage, without assumptions on the data generation process or state transition model accuracy. When predicted state transitions align well with distribution shifts, CPTC can anticipate uncertainty and adapt faster.
We show strong empirical results on 3 synthetic and real-world datasets. Compared to state-of-the-art CP baselines, CPTC achieves more robust coverage with comparable prediction intervals sharpness (example in Figure T]. and is computationally light.
2 Related Work
Probabilistic Forecasting for Time-Series with Change Points. Probabilistic forecasting has become central to modern time-series analysis by modeling a distribution over future outcomes rather than single-point forecast [13. Classic approaches often adopt Bayesian or approximate Bayesian
strategies for uncertainty quantification: for instance, Bayesian Neural Networks (BNNs) 52 54 [1
or neural process models [22] 46 use neural networks to parameterize the underlying stochastic
process. Other methods, such as DeepAR 42), PatchTST [39], or Temporal Fusion Transformer [33 generate probabilistic forecasts directly by minimizing pinball loss (i.e., quantile loss), thereby learning predictive distributions.
key challenge arises when change points occur i..., abrupt distribution shifts in the time series [38]. Change point detection and segmentation have been studied extensively (beyond our scope, see 3 for a recent survey). Modeling time series with change points typically features state-space models (SSM), and in particular switching dynamical systems (SDS) [7 9 37 These SDS approaches use variational inference to fit neural parameterizations of transitions and emission distributions. However, existing probabilistic forecasters often lack strict calibration guarantees and can be overconfident in practice 331, and many require specialized architecture tuning for different tasks, limiting their applicability to real-world UQ challenges.
Conformal Prediction for time-series. Conformal prediction (CP) (50) has emerged as leading framework for UQ due to its algorithmic simplicity, broad applicability, and finite-sample coverage guarantees; see 7 for comprehensive introduction. For time-series forecasting, some works focus on joint coverage over fixed-length horizons 44
others assume stable underlying dynamics uses known covariate shifts (by propensity weighting) to achieve theoretical coverage, but can fail in settings where the shift mechanism is unknown, which is typical for time series forecasting. (26) introduces a multi-model ensemble CP framework, optimizing model selection within a pool experts for combined coverage. CP has also
and adapt only to shifts in residual distributions [56]
been extended for change-poing detection in time series via conformal martingales [5] 49

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Recent works on online conformal prediction develop online strategies that guarantee marginal coverage for adversarial or nonstationary data streams. ACI [23], AgACI [60], DIACI [24]. and Conformal PID control 4 leverage online optimization and have a "reaction" period when distribution
shifts. Multivalid Prediction (MVP) [2 provides group coverage guarantees and learns calibration threshold online, although it can require longer calibration windows. Closer to our work, SPCI [57] learns an autocorrelative estimation of the residual's quantiles through Quantile Random Forest; whereas HopCPT [ condition their conformal interval on similar parts of the time series using a Modern Hopfield Network. The drawback of these two methods is that the regression models trained during inference time, which (1) requires long cold start window to learn meaningful associations and (2) is computationally expensive. Our contribution is a CP algorithm that is computationally light during inference time, and can leverage state prediction capabilities within the SDS model to anticipate and adapt quickly to distribution changes.
3 Background
3.1 Conformal Prediction
We briefly review the algorithm and guarantees for conformal prediction, and refer readers to [7] for a thorough introduction. In this paper CP refers to split conformal prediction (we consider full conformal prediction out of scope, because its computationally complexity renders it nonviable for deep learning settings). The goal of conformal prediction is to produce a valid prediction interval (Def. 3.1) for any underlying prediction model.
Definition 3.1 (Validity). Given a new data point (ar, y), where is the prediction target and æ is the covariates, and a desired confidence - a E (0, 1), the prediction interval (1=*(x) subset of y containing probable outputs given The region I ri-a is valid if
Ply e(yEr-=<(x) 2 11--a
Let D X' x V. Conformal prediction splits the dataset into proper training set Dtrain and a calibration set Dont. A prediction model f X + y is trained on Drrain. We use nonconformity score function A: x + R2o to quantify how well a data sample from calibration conforms to the training dataset. Typically, we choose a metric of disagreement between the prediction and the true label as the non-conformity score, such as the Euclidean distance:
be a dataset whose data points (x* yy) are sampled from distribution on
A(r,v) ca d(y. f(x)) c 11y-f(x)]
(2)
Let S = {A(a*, (x)) denote the set of nonconformity scores of all samples in the calibration set Deat. During inference time given new data =+* the conformal prediction interval constructed as Eqn 3 where Q is the empirical quantile function.
[1=a(""+1) = (v A(x*+1 .) S Q² U
We say a sample is covered if the true value lies in the prediction interval y E (=-o(z))1) Conformal prediction is guaranteed to produce valid prediction intervals (50) if the calibration data and test data are exchangeable (in a dataset [xx,, y ) size n, any of its n! permutations are equally probable).
Conformal prediction has been extended to the online setting. More precisely, the algorithm observes (x1,y1), (I2, y2), ... sequentially, and needs to construct prediction set I!_- for ye time step t. Without making any distributional assumptions on the data generation process, online CP algorithms achieve asymptotic guarantee as Equ where 1.. the indicator function:
T
lim T
Σ 1 (v & ( =
(4)
T
1=1
We refer readers 6 for summary of theoretical results in this setting.
3.2 Switching Dynamical Systems for Modeling Time-series with Change Points
Switching dynamics systems (SDS), or mixed dynamies systems, provide a powerful framework for modeling time series with change points, where the underlying dynamics shift between distinct
3

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Parumeters
Discrete
21
24
0.5
switching states
E (L,
466
-10
Continuous states
81
82
83
84
8t E R
O E Rd
Observations data
01
02
04
181 55
(observation dimension)
Figure 2: (Left) Generative model of the SDS dynamics as Eqn Shaded circles represent observed variables; hollow circles are latent variables. (Right) Example to illustrate notation. We show Lorenz attractor, a canonical nonlinear dynamical system, approximated by a linear SDS (35].
regimes. SDS assumes the system consists of a total of K different base dynamics. It explicitly incorporates discrete switching states to represent the abrupt shifts among base dynamics. We denote the discrete switching states at timestep / as 2, E Σ, 2 = {1...., K} C -+, to index one of K base dynamical systems, and represent the observed sequence 01, ................................ or 01:T for conciseness.
There have been many different variations and implementations of SDS. Classic SDS have memory- less state transitions, i.c. p(zy|2t-1) 1 is parameterized by stochastic transition matrix A E R This results in an "open loop" system and the state duration follows a geometric distribution. Recent works have sought to fix this issue, for example by closing the loop using recurrent dependencies on 8t. [36), or modeling explicit durations [9] Our generative model is illustrated in figure 2 for generality, factor out any non-Markovian covariates model parameters 0. The joint distribution can be factorized as
P(OI:T 81:7, 21:T) II P(oo)ss) Ze, 2,, (2,,
where p(s) |2| (((1) is the initial state prior. The dynamical systems have continuous state transition p(z; St-1,.511..0 0), and the emission p(or|st) can be linear nonlinear (e.g. parameterized by a neural network). Moreover, the SDS framework naturally supports forecasting both the next
observation 01+1 via P(oc+1| St+1) and the next switching state 24+1 via P(*++ St, 21,
4 Conformal Prediction for Time series with Change points (CPTC)
We introduce CPTC, a novel conformal prediction algorithm that utilizes the state prediction from SDS to improve uncertainty quantification for time series forecasting.
Problem Setting We are given sequence observations or E IRd for all potentially with change points. To be consistent with the standard notation in conformal prediction literature, let X C R" denote the input feature domain and y S Rd the target observation space. = .1, K} Z indexes one of K underlying states. The input features can include past observations and other environmental covariates. In our implementations,
k with ie being the look back window.
We consider the online conformal prediction setting where we have a sequence of data points from X' y: (zo, wo), (I1, (,7,)) yIT, We have trained state predictor for ((21(200)) where 2, E the true (latent switching) state time f, state-conditioned forecaster f X' > (a probabilistic SDS model can serve both purposes), and are given desired confidence level 1 At each time step f, we have seen (r 31), (I22 3/2) 1/)...., (x,-1, 3/t-1) )) and the next input feature vector adaptive, time-dependent prediction interval (r(xx)) C y for the unseen true target yt. which we will learn at the next time step. Our goal is to produce prediction intervals with correct empirical coverage, i.c.
Tr. We want to produce
P( E (T())))) 2 a for any 2 0
4.1 Algorithm
Existing online conformal prediction algorithms often rely on assumptions about how distribution would shift in time series. This infeasible for data with change points due to unknown shift mechanism. The main contribution our algorithm makes towards addressing this issue is anticipating this type of abrupt shifts. Our CPTC algorithm outlined pseudo-code algorithm [ In this section, we discuss the general procedure and key components of the algorithm.

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Algorithm 1: Conformal Prediction for Time series with Change points (CPTC)
Input: nonconformity score function A, probabilistic state model (o:t) with prior ((20),
forecaster model f x Z - y. confidence level a, learning rate
Output: Prediction intervals (xx2) for
1 for & E 2 do
Initialize S- 0, 02,00 <-
3 end for
4 if exists warm start data { vo)l2- then
for 1-w,,00 do Sz, f (AAT,, where ~ p(2= z/<0a); ) // warm start scores
end for 6
end if
9 for & [1,7] do
do for 2 [2,( + {y: A(x,,)) S Q1-os1 (.))) specefc CP
11
12
end for
13
(z() + Aggregate the x2, by Equ 10 or Eqn
Output: (1))))
Sample state &, ((21 // state-specific coverage target tracking Update 0k2,1+1 1 4- as,t + 77 . (a erri), where err, (3, & [+()))]:
Update scores S2 + Ss. U (A(z)))))
end for
Switching Behavior and SDS Integration. We explicitly model the abrupt shift dynamics with the switching dynamical systems (SDS) to anticipate the change. In our formulation, the latent state 2, indicates which dynamical regime (out of K possible regimes) is currently active time Given the SDS formulation (e.g. Eqn we can factor the state transition probability from the conformal prediction process and do calibration for each the K dynamics separately. The goal in Eqn 6 can therefore be written as:
Σ P(y [x(x:)] = 2) P( 2|20.)
(7)
State Prediction. The SDS model provides with the state predictor p(2e 224). In practice, SDS models [9] (36] have existing 2, forecasts that we can extract. For other model architectures, one can modify them output an extra covariate prediction for the state, or add auxiliary model to classify or cluster the input space into different regimes. In practice, 2, doesn't necessarily have to come from model, and can be any discrete contextual variable such as weekday/weeeenddhhlidyy or day/night.
Initialization and Warm Start, For each switching state Z, we initialize (1) mode-specific set of nonconformity scores S₂ and (2) the specifpe confidence level 0=,0 a. there are w steps of warm-start data { (xx, yt ) -
initialize the sore sets S. by sampling 2; f(2z 2 2/2r0:) for I t -........ 0, and inserting A(Ta, Ye) into Sz,- Depending on the size of the warm-start window and desired properties, practitioners can adjust the warm- saam-ttarting approach. For example, using the entire warm-start start window S = {A(Iz, yı) (u)He 12_ for all E Z will result in better stability but less adaptability modes. The approach does not change the algorithm's 's coverage guarantees.
State-Specific Prediction Intervals. At every time step & E []. CPTC creates state-specific conformal prediction intervals for all states with nonzero probability. This allows us to generate predictions tailored to the current anticipated dynamics. Specifically, for every state E Z where p(=)20o) O, first obtain the point prediction through forecaster model f(z,, 2), and then the state-specific prediction interval I (xx) by calibrating on nonconformity scores set S. to the adaptive confidence level of 1 0z,t
Online Conformal Prediction. The exchangeability assumption does not apply to the online setting of our work. If we naively calibrate with online data, when the data distribution changes, the coverage probability may deviate from the target level 1 - Therefore, we follow online CP methods such as ACI (23) address this problem by continuously updating 0s,; to track a surrogate miscoverage rate a e as defined in eqn (for each state in our setting), an internal estimate of the target coverage.
a = sup E [0 (a)( < aa), Me,(a)) p(A(zt, w1) Q*-)(S)))
(8)
5

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In eqn 8 M2.t measures the probability that the true label falls outside the prediction set I I, when the predicted state is z. We track a over time by updating the estimate a., with online optimization. In our implementation, we use the simple update from ACI (Fqn 9 though more sophisticated approaches such as may be used its place without affecting the overall algorithm.
022 f ar + 7 (a erri), where erTr = Ly,. $ (xx)))
0 is the step size hyperparameter, and & Z, & p(== 2 2/<o:e) is the state time t sampled to be updated. The update rule increases ag if coverage is too conservative and decreases otherwise, ensuring that for each 2, az,t converges to value that maintains long-term coverage.
Here y
Aggregation. We aggregate the prediction intervals for each state (2,((1)) into one final set (xx)) using the weighted level set, as illustrated in Equ I0
(aee) {u= Σ p( = 1 (T,))))))))))))))))))))))))))))) aa) sE2
(10)
Solution to the constraint in Equ i0 the minimal set that achieves the marginal coverage. can be obtained through discretizing Y' into a fine grid and calculating probability mass at each point, which becomes computationally expensive as the grid resolution or dimensionality increases. For faster computation and in implementation, we approximate the weighted level-set in Eqn T0 taking the union of intervals of the most probable states as in Equ 11 The two aggregation strategies achieve similar results on all our datasets (Appendix (B.4).
T (x:) x (2.(),, where Z = arg min S s.t. Σp(== = = 21) 2 1 1-a a. (11) SC2
#ES
Modularity and the role of the state model. Our algorithm can be viewed as running multiple instances of online conformal inference (lines 14-16 of algorithm[]], one for each underlying state, and adaptively aggregating the confidence intervals when the underlying mode is switching. This differs from the setting where the temporal correlation between residuals needs to be learned online (24, 12 57 [T] the state model allows us to leverage training data for such information. Our algorithm is modular- the state model, forecaster model, and the online adaptive conformal prediction algorithms all operate independently of each other. For example, the state-specific online coverage tracking (line 15 of algorithm I can be replaced with other online conformal prediction variants for data-specific applications. Our theoretical guarantees are agnostic to the forecaster models used.
4.2 Theoretical analysis
In this section, we discuss theoretical guarantees of the CPTC algorithm. In particular, establish finite-sample validity under exchangeability. asymptotic coverage in dynamic (potentially nonstation- ary) settings, robustness to imperfect state classification, and faster adaptation distribution shifts that align with model-preeicted state changes. See Appendix A for proofs and theoretical details.
Finite-sample validity under exchangeability. We start by showing that when data exchangeable (e.g. independent and identically distributed), CPTC achieves finite sample validity. The proof is standard based on showing the marginal coverage of split conformal prediction.
Proposition 4.1 Prinite-saplle validity under exchangeability). If the data (at, v1). t 2 1 are exchangeable, prediction intervals obtained via Algorithm 7 satisfy P(ye E (,(x,)) 2 1 a.
Asymptotic validity without assumptions. Theorem 4.2 ensures that CPTC provides reliable uncertainty quantification in the long run without assumptions on time-series stationarity or accurate state predictions. (We do assume that the distribution of states and state predictions exhibit stable long-term behavior in Assumption [] We achieve our desiderata of Eqn [ on average as T grows asymptotically, consistent with the results of other online conformal prediction works.
Theorem 4.2 (Asymptotic validity of CPTC). For any sample size T > 1, the CPTC algorithm (with weighted average aggregation) achieves:
T
K
11(1- (1- -Ty
E [err]] - a S
Σ
la 0%
T ==1
Where:

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em, 1|2 $ [xx))),, the miscoverage rate.
T2 {L {I,...;T) [): =]. the time steps the predicted state is
0 the optimal miscoverage target for timesteps Tz.
C is a miscoverage constant in Lemma A.1 (23).
Note the RHS of Eqn 4.2 decays at rate of (T)) and satisfies limr- Ei=I B(ern] 0.
Robustness to imperfect state prediction. In wealdworld upplications, the state predictor p(2 (200:))
that plays a large role in our algorithm is often imperfect. The CPTC algorithm accommodates these scenarios by tracking a for each predicted mode 2 with adaptive online updates. Theorem 4.3 quantifies the effect imperfect state prediction by showing that the miscoverage rate is bounded by the product of the misclassification rate and state-specific deviations.
Theorem 4.3 Sriite-Sample Miscoverage Bound with Imperfect State Predictions). For any sample size T 2 1, the CPTC algorithm ensures that:
T Elerre] - 6- SG.7
T
Where e P(â + 2x) is the error rate of the state predictions and 62. the miscoverage deviation from a for any predicted stute defined in Lemma A. For all 82, - 0 as [T] + co.
Theorem demonstrates that CPTC can maintain asymptotic valid coverage even when state predictions are imperfect, with the bound tightening state predictions improve or the number of observations per state increases. In practice, improving the accuracy of the state prediction model directly enhances the coverage performance of CPTC. However, even with imperfect state predictions, the algorithm's adaptive calibration mechanism ensures that the overall coverage remains close to the desired level
experiments show in Section 5.2
Why CPTC? Faster convergence under distribution shifts. The strength the CPTC algorithm is its fast adaptation in dynamic environments with distributional changes. When distribution shift aligns with a predicted state transition, the structure CPTC allows the confidence level az,t to converge to the target error rate faster compared to purely online methods.
Consider a data stream with a distribution shift occurring at time Lsint- Let a denote optimal target error rate for mode j after the shift, and assume that the predicted state &, correctly reflects the shift: laii and a² = oz for >> Laint For conciseness of notation in Theorem 4.4 let d5. and ACILT denote the miscoverage rate starting from lshi for CPTC and ACI respectively.
Theorem 4.4 (Miscoverage Ratio under StaeeCCiincidet Distribution Shift). Under state shift from to j at time step thifh, coinciding with predicted transition, the CPTC algorithm achieves faster convergence the new target of compared to non-state-awar algorithm ACI at a ratio of:
6G SACITT
[ai 11 agl
<
| aaj
The numerator [4 - a91 0 as co, when in-state adaptation has converged; but the denominator [0tua-1 ajl remains greater than zero regardless of the length of the time series. This result highlights the advantage segmenting nonconformity scores by state: the state-specife azzt aligns the target miscoverage updates with distribution shifts, allowing CPTC to quickly adjust its prediction intervals. Such accelerated adaptation allows for shorter miscoverage periods and is critical in woal-world applications where rapid responses to changing conditions are necessary.
5 Experiments
Baselines. We selected the following baseline methods. RED-SDS (Recurrent Explicit Duration
Switching Dynamical System) 9 is a state-of-the-arr Bayesian model that learns state transitions (implementation details in Appendix B.I. We also use REDSDS the base predictor for mode switching for our method the experiments. CP is generalization of conformal prediction to the online setting, also known as Online Sequential Split Conformal Prediction in [60). We choose ACI
7

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23 to represent online CP algorithms leveraging online optimization. SPCI [57] and HopCPT [[]] are state-of-the-ae art CP algorithms that learns adaptive predictive intervals by quantile regression and Modern Hopfield Networks learned from the time series respectively.
Metrics. We evaluate calibration and sharpness for each method. For calibration, we report the empirical coverage on the test set. Coverage should be as close to the desired confidence level as possible. Coverage
- 1 (yv E (x+))
For sharpness, we report the average width or area of the Prediction Intervals (PI). The measure should be as small as possible while being valid (coverage maintains above the specified confidence
level). Width - > (xx))
Datasets. The three synthetic datasets are designed with increasing randomness in mode changes, challenging the adaptivity of CPTC. Bouncing Ball is comprised of univariate time series encoding the height of ball bouncing between two walls with constant velocity and elastic collisions, following The two switching states are going up/down, each associated with a different level of Gaussian noise added to observation (Bouncing Ball obs), or the underlying dynamics (Bouncing Ball dyn) which induces uncertainty in the phase as well. 3-mode system is a switching linear dynamical system with 3 switching states, where each mode samples from Poisson distribution for duration.
[8 9
For real-world datasets, the Electricity and Traffic datasets from 19 have hourly frequency and exhibit seasonality both in terms of the time series itself and volatility. The honey bee trajectory dataset [40] is the most complex, composed of 4-dimensional trajectories with length averaging to 900 frames, where the bees' dance can be decomposed into "left turn", "right turn" and "waggle".
Illustrations of the datasets and detailed description window for synthetic and real datasets are 50 and 100 time steps respectively (excluding bees, whose 15). The datasets represent diverse set of scenarios to demonstrate the robustness and versatility of CPTC, capturing the complexity and variability of practical time-series forecasting problems.
be found in Appendix The warm-star
5.1 Analysis of Coverage and Sharpness Results
Results for uncertainty quantification in table 4 show that we achieve significantly better validity on all datasets compared to baselines. RED-SDS's intervals fail due to lack of calibration, common pitfall of probabilistic methods (shown also in (57)). CP's under-coverage shows that the data are not exchangeable. In the presence of change points, ACT's coverage fluctuates dramatically and did not converging within the horizon (more visualizations in Appendix B.5). For SPCI and HopCPT, which also guarantees asymptotic validity, the invalidity is likely because they are designed for large datasets (T > 10, 000 for HopCPT), and did not have enough data to react or learn useful residual patterns for the relatively short prediction horizon of our datasets (T 200 for synthetic datasets, 300 for electricity and traffic, and 60 for bees). Figure 3 provides a qualitative example of how the prediction intervals compare across different methods. Our method (purple shaded regions) does not over-cover during nonvolatile hours, nor under-cover during busy ones, as does ACI.
Table 1: Performance on synthetic and real-world datasets with target confidence - a = 0.9 (for horizon T' 200 for synthetic datasets, 300 for electricity and traffic, and 60 for bee, mean + standard deviation of the test samples). Methods that are invalid (coverage below 90%) are grayed out. Our method achieves a high level of calibration (coverage close to 90%) consistently.
RED-SDS
CP
ACI
SPCI
HopCPT
Ours
Bouncing Ball obs.
Cow Width
16.90
1296
87.67
6.6
89.82 1.795
87.13 4.77
79.45 1
90.15 1.199
1.60
007
10.54
1.9
3.85 2.000
2.82
4:041
1.98 #090
3.71 = 0:
Bouncing
Cov
13.20
11.50
86.45
1225
89.38
2.5
89.16
81.12
90.47 #230
Ball dyn.
Width
1.37
0.00
11.49
4:499
2.27
# 1.65
2.26 6:028
2.95
1:122
1.76 a.071
3-Mode System
Cov
93.02
+ 3.0
63.98 5.000
89.90
2:2
84.90 541
90.85
94.96 à 1:15
Width
2.08
0.16
1.46
0:0
6.40
3.70
6:8
9.07
1.23
2.45 0.72
Traffic
Cov
23.15
1098
87.06 1m1
90.01
037
84.38
2447
88.91
3.3
92.38 1:1.44
Width
0.05
21.91
1042
27.81
+ 54.0
5.32
1.95
7.50
10.05
7.92
239
Electricity
Cov
62.85
13,67
85.69
6.7
89.79
0.8
84.10
2:77
86,50
2.66
91.22
1:29
Width
162.67
511.33
366.05
# 228045
45.74
279.1
228.14
1207.5
155.71
130443
139.75
à 6:622444
Dancing
Cov
84.92
+ 6,84
79.86
= 2073
86.25
4:10
79.20
= 0.30
72.11
= 1.84
92.64
4:1.9
Bees
Width
0.25 oot
1.65
4.79
1.77
1.06
0:051
0.79 0:3
8

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Input
Fiput
Prediction
Predicton
REnSnS
100
Input
Input
Predictoo
Predicton
ACI
SPCI
Input
Input
Prediction
Preduction
HapCPT
Figure 3: Visualization of prediction intervals on the Electricity hourly demand dataset. The red and blue bars in the bottom reflects the underlying switching state of day and night. Our method (purple) adapts to different levels of volatility between day and night, and achieves stabler coverage over time, whereas ACI (yellow) over-covers during the night and under-covers at change points.
5.2 Ablation Studies
Robustness p(2) accuracy. We conduct ablation studies on the synthetic datasets of which we have ground truth underlying states. As shown in table it is evident that increasing the error the state prediction does not affect the coverage performance. This verifies the robustness of our algorithm and theoretical results (Theorem 4.2 and 4.3) Table 3 further shows that the error of state prediction reflects on the width of the prediction intervals: while remaining the same level of coverage, the better the state predictions, the sharper the intervals.
Table 2: Coverage of CPTC for using ground truth (GT) labels and with various levels of injected noise, on synthetic data with 7 = 200. We can see that coverage performance does not change significantly with error in state prediction.
Table 3: Interval width under the same setting. While the coverage guarantee is the same, the more accurate the state prediction is, the sharper the intervals.
Ground truth state labels
w/ 20% errer
GT 50%
20%
emr
BB obs.
90.31 1.77
90.27 1.30
90.13 1.40
BB ebs.
3.61 +08
3.75 =00
3.94 +
BB dyn.
90,34 0.0
90.32 0.066
90.33
1.6
dyn.
2000
2.15
200
200
mode
92.67
92.70
92.53
3-mode
2.04 #067
2,19 #0.0
2.27 +000
Additional ablation studies presented in Appendix [8.]] On Aggregation Methods (by discretiza- tion as Eqn TO or union as Eqn [TI we found that on our benchmark datasets, union is a close proximation of the true objective, and CPTC achieves similar width and coverage subset. For Long Horizon Forecasting where 2 10, 000, our method still achieves valid and stable coverage, but have wider PI width compared SPCI an HopCPT. They achieve this by frequently re-training their models online, which CPTC does not and is less computationally demanding.
sampled
6 Conclusion and Discussion
In this paper, we introduced Conformal Prediction for Time-series with Change points (CPTC), novel conformal prediction algorithm for uncertainty quantification in non-stationary time series. By leveraging state information in switching dynamics systems models, CPTC offers improvements in adaptivity over existing conformal prediction methods. Our theoretical guarantees ensure validity without assumptions on time-series stationarity or accurate state predictions. Empirical results corroborate our theory and demonstrate the effectiveness of CPTC across diverse synthetic and real-world datasets, achieving robust coverage under distributional shifts with comparable sharpness compared to state-of-ther -frtt-arrt baselines. The adaptivity advantage is most pronounced in shorter time series, when basclines require more data react or learn temporal correlations.
Limitations of CPTC include (1) slower convergence rate in scenarios with frequent or unpredictable state transitions, as the algorithm requires sufficient observations within each state to achieve optimal calibration and sharpness, (2) wider prediction interval width when applied very long time series compared methods specialized for such scenarios (e.g. HopCPT), and (3) requirement of state prediction model, making our method primarily applicable SDS models and discrete SSMss. Future work could explore extending the framework to continuous states and the theoretical implications thereof, and studying the decision theoretic properties of using conformal prediction for safety-critical applications requiring real-time adaptation.

---

Acknowledgments and Disclosure of Funding
This work was supported in part by the U.S. Army Research Office under Amn--ECAE award 1IIN-07-R-0005- the U.S. Department Of Energy, Office of Science, IARPA HAYSTAC Program, and NSF Grants #2205093, #2146343, #2134274, CDC-RFA-FT-23-0066 DARPA AIE FoundSci and DARPA YFA.
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A Theoretical Results and Proofs
A.1 Setup and Notation
Let 2₁ E 2 = (1....,K K denote the unobserved discrete mode, 2t E X denote the continuous state, and y/2 E the observation at time t. We assume have access to probabilistic model p(zz) (xt-)) that estimates the mode 2z.
For target coverage rate 1 1 let (X,) be the prediction set at time t. We define the miscoverage indicator M, as follows, and aim to bound the overall miscoverage rate over T time steps:
T
M,
erTy,
where
erri 1(M (x())),
A.2 Proofs: Lemma
In practical scenarios, the state prediction model may not be perfectly accurate. We now analyze the impact of imperfect state predictions on the coverage guarantees of the CPTC algorithm.
We derive the following lemma from the convergence result of Proposition 4.1 in [23] with two additional assumptions also according to [23]: (1) the existence of single fixed optimal target [0, 1J. and (2) [[ .M (a+). These are reasonable assumptions that allow to quantify distribution shifts and are necessary for analyzing adaptation behavior (whereas proposition 4. itself a worst-case upper bound.) We refer the readers to the original paper for the complete derivation.
Lemma A.1 (Adaptive Conformal Inference Miscoverage Bound Within Predicted States). Let == (1.......... T}
)} denote the set of times when the predicted state For any and predicted state 2 2, the Adaptive Conformal Inference (ACI) al algorithm ensures that there exists a constant åz, such that:
Elerr:] S 62.7
T
LE
Explicitly,
1 (1 cy)|T=
S.T
la a"
1
where
of is the optimal miscoverage target for timesteps T
y is the slep size
c the miscoverage constant.
Note that & decays rate of O(T) and satisfies limy, 82, 0.
A.3 Proof of Theorem 4.2
Theorem
4.2 Asymptotic validity CPTC ). For any sample size T 1. without placing any assumptions, the CPTC algorithm ensures that:
T
K
1 (1
17.1
Σ
Elern]]
S
T
ax O
T
T
which decays at rate of O(T') and satisfies
lim T
T M
E(err]] - a
To prove theorem 4.2
we assume that both the distribution of the underlying modes and the predictions is stationary (Assumption
14

---

Assumption (Stationary Distribution of States), The sequence of true states {2, } has stationary distribution T22). and the sequence of predicted states {2r} also has a stationary distribution ø (2). Specifically:
E 2,
lim p(= = = T()), lim p( = = #=),, t-000
This assumption is well-justified in practical scenarios where the sequence of observations and predictions arises from processes that, while non-stationary over short intervals, exhibit stable long-term behavior. In time-series dynamic environments, regularities in data generation or prediction models often lead empirically observed stationarity in distributions. While restrictive, this assumption aligns with prior work in sequential and online prediction frameworks, where stationarity assumptions are standard to derive thcoretical guarantees.
Let T. {t E (I....,7) & 2) represent the set of timesteps when the predicted state is z. For the total number of timesteps T, the overall miscoverage error can be expressed as:
Proof of Theorem 4.2
1
Σ
B
Blrr
Elerr:]
T
T
#=1
TE
K
-
T
E(err,
T
LET
From Lemma A.1 for any predicted state 2, the miscoverage within Tg satisfies:
1
Σ
Elerr]] a S 82,7.
T=
TET.
where:
(1- oy)
GC.
a a.l-
|
Substituting this bound into the overall miscoverage error, obtain:
E errr
Ta 1
Expanding 82,T, the overall bound becomes:
1
1 S
(1 - (T)
E(err:]
aa
T
7
By assumption as T' + oc, the size of [Tgl for each state z grows proportionally to T' The term 1 - (1 c7) approaches 1, while |/ converges to the relative proportion of time spent in
state 2, denoted as p.. Hence, as T + co, the overall bound
i ΣT= Ejerr:] a +
7
Thus, the CPTC algorithm ensures that the overall enror converges to the target coverage level a at rate OTT), satisfying:
T
lim T-x
Ejerr]] - a = 0.
T
1=1
15

---

A.4 Proof of Theorem
Theorem 4.3 Sinie-Sample Miscoverage Bound with Imperfect State Predictions). For any sample
size T 2 1, the CPTC algorithm ensures that:
T
E elerr a S e. -.-max &,,7
T
where:
erri = (Y; $ (x)))) the miscoverage rate.
{t (I....,7) T) = 2), the set of times when the predicted state
P(é; * 2 the misclassification rate the state predictions.
62.r is the deviation from a within any predicted state as in Lemma A.1
Proof of Theorem
4.3 First we will partition time into correct vs. incorrect predictions. Define:
I T
C {t = 21); I {t be 2 2), and
Calibrated expected coverage correctly predicted states C. Since there are no distribution shifts within each state, have the same expected error from calibration data. Therefore,
E eerr
El [err.]] - a = 0.
Bounded deviation on incorrectly predicted states I. On I, we have [lerr, al s 8e. Therefore.
[[err]] at
E(err:] al
Elerry al S 0 [Z] 82.
|I] s2.
Since [T
dividing by T yields
E(err, -al I Ed
Combine achieve the overall bound. By the triangle inequality, we know that
T
1
Ejer, a <
[[erre] a).
T
T
Hence, from the previous step,
1
E Jerry
<
E
T
Since on the predicted state & could vary among different values, we take dgr S max, 8g. Thus,
err,
e max 82.
T
16

---

A.5 Corollaries
Theorem 4.3 indicates that the overall miscoverage rate deviates from the desired level a by at most max, Ss.T. The misclassifications rate contributes directly to the deviation, and the calibration error 5g.7 within each predicted state decreases as more data becomes available. As T increases and max; SGTT 0, the deviation is primarily governed by the misclassification rate e.
Corollary A.2 (Zero miscoverage rate under Accurate State Prediction). predicted state probabili- ties are accurate 2z (1:1) = p(z)|=1a-1), then e O, therefore E[errı] = a for all T, achieving optimal performance of online conformal prediction.
Corollary A.3 (Asymptotic calibration under eventually correct State Prediction). If the predicted state probabilities p(2) T1:t-1) converge in probability to the true state distribution p(zi (11.-1) 1) as T + oc. With T 0, - 0 as well. Hence, the overall miscoverage rate converges asymptotically to 0.
A.6 Proof of Theorem
Consider a data stream with a distribution shift occurring at time Laift- Let az denote the optimal target error rate for mode j after the shift, assume that the predicted state &, correctly reflects the shift: tain and a aj for fort > tsint- For conciseness of notation in Theorem 4.4 let 65.7 and ÓACI,T denote the miscoverage rate starting from fauit for CPTC and ACI respectively.
Theorem (4.4 Miscoverage Ratio under Cate-Coinn Distribution Shift). Under a state shift from i at time step whin, coinciding with predicted transition, the CPTC algorithm achieves faster convergence the new target oz compared non-state-aware algorithm ACI ratio of:
55TT ACITT
10i 03 a3]
Proof of Theorem 4.4 We want compare the convergence behavior of CPTC and non-segmented online conformal prediction algorithm (e.g.. ACI) when distribution shift occurs at time fshin and this shift coincides with predicted state transition from to j.
We assume that State prediction coincident shift, meaning a = a for I fahin and a = az for t Lhift
From Lemma A.T and Theorem we have bounds on these deviations. Taking the ratio of the bounds, we have:
8j.T
C.
ay.hmin Aaibinn11 11 az1
|aj 03
ÒACIT
c At - 095 |ataia a3)
We elaborate the implication of this result. The ratio indicates that the CPTC achieves lower miscoverage rate when the initial target 0r for state j in CPTC is closer to the new optimal target af than the global target 0t -1 used by ACI. This is because CPTC maintains separate targets for each state, allowing it to better track statc-specific optimal targets. The base model takes around hours of training time on the GPU
17

---

B Experiment Details and additional results
B.1 Hyperparameters for the base forecasting model
We train our RED-SDS on the synthetic datasets, and use the model checkpoints provided by the authors for the real-world datasets. The model architecture consists of discrete switching component with K E [] categories and a continuous state space with dimensionality d₂ {2, 4 For training, we employ the ELBOv2 objective with a learning rate 1 E [5 x 10 3, 7 x 10 warmup steps of 1000, and gradient clipping at 10.0. The model uses batch size B E {32, 50] and is trained for E (20, 000, 30, }00} steps. The continuous transition and emission models are parameterized by nonlinear MLPs with hidden dimensions h = 32, while the inference network uses either a bidirectional RNN or transformer with embedding dimensions de = For real-world datasets, apply target transformation and Jacobian correction, while synthetic datasets use raw observations. The model's s capacity is controlled through weight decay À = 10 and MLP hidden the component. For forecasting. the model is trained to
dimensions h E {8, 32, 64) depending forecast a window of t = 50 time steps for synthetic datasets (bouncing ball) and a windows for real-world datasets (electricity, traffic) specified by their respective metadata. The model generates N = 100 (Iz, yt, 24 ) triplet trajectories from the trained model by Monte Carlo sampling, and use the samples calculate prediction quantiles for table 1, in accordance to the original paper.s. We segment all datasets by 70/10/20 train/validation/test split. Conforaml prediction results are reported the test set only.
B.2 Computational Resources
All experiments are done on a server machine with an Nvidia A100 GPU, with some data processing and analyties performed on Apple Macbook Pro laptop computer with M1 chip. As rumning time vary greatly depending on implementation and hardware, we provide a rough range of algorithm runtime. Training the RED-SDS base model seed) takes 2-5 hours for our datasets, and inference take around 15-20 minutes for all datasets. Among the conformal prediction baselines, SPCI is the most computationally demanding, taking 8-10 hours for inference on our benchmarks. HopCPT follows, taking around hour. CP, ACI, and our method SPCI do not require extensive inference-time computation, completing all inference within 5 minutes.
B.3 Dataset descriptions, continued
All datasets are visualized in F with their underlying states color coded.
Electricity and Traffic. The Electricity and Traffic dataset are from the UCI machine learning repository [19] and we use the train/test split from GluonTS (2). The electricity dataset consists of hourly electricity consumption data for 370 customers, each with 5,833 observations. The traffic dataset consists of hourly occupancy rates from 963 road sensors, each with 4,000 data points.
Dancing Bees. The honey bee trajectory dataset is from 40] Honey bees convey information about the location and distance to a food source through a dance carried out inside the hive. This dance consists of three distinct phases: "left turn", "right turn", and "waggle". The duration and orientation during the waggle phase represent the distance and direction to the food source. Experiments are conducted on 6 trajectories, with lengths of 1058, 1 125, 1054, 757, 609, and 814 frames, respectively. In this paper, the prediction interval is on the first two dimensions (the coordinates of the bee) only, but all dimensions are used for prediction.
left
waggle
right turn
Figure 4: Dancer bees are tracked by appearance based tracker from video sequences. The tracked bee is shown in green rectangle in the left figure above. The right figure shows a stylized bee dance through which bees talk to the other bees about the orientation and distance to the food sources.
18

---

Bounc ng Ball Ooservation
120
Bouncing Ball Dynamics
15
2.5
40
63
80
130
170
3-made system
Dancing Bees
Electricity
300
20
40
60
80
100
320
Traffic
Mode
Model
Faconstruction
20
60
80
100
Time
Figure 5: Examples datasets and prediction results. Dotted line represents prediction whereas solid lines represents ground truth; background colors represent the different operating modes segmented by our base predictor RED-SDS.
19

---

B.4 Additional Experiments
Additional Baselines We add comparison to three baselines that handles distribution shift in time series data. They are DIACI [5), ECI [55 and L-ARC [61].
We would like to emphasize that these baselines reactive. The question we try to address with CPTC is what happens when we can anticipate distribution shift? CPTC designed such that greatly improves coverage compared to purely reactive methods, while still maintaining the ability to adapt to unknown shifts. To our knowledge, there are no other models that explore the same problem; CPTC provides practitioners with simple and sound solution, when they have the ability (such as via a state space model) to predict shifts.
Table 4: Performance on synthetic and real-world datasets with target confidence 1 - a 0.9 (for horizon 200 for synthetic datasets, 300 for electricity and traffic, and 60 for bee, mean + standard deviation of the test samples). Methods that are invalid (coverage below 90%) are grayed out. Our method achieves a high level calibration (coverage is close to 90%) consistently.
DIACI
ECI-cutoff
L-ARC
Ours
Bouncing Ball obs.
Cov
92.45
4 1.51
49.80
4.944
80.98 3,65
90.15 a 1.39
Width
1.06
+ 0,40
6,54
#91
1.41 #031
3.71 0.988
Bouncing
Cov
92.54
55.71
#.00
81,66
+
90.47
#
Ball dyn.
Width
1.18
7.53 3.16
1.47 0.21
1.76
3-Mode System
Cov
89.21
+ 1.25
59.27
4:3.4
80.54
4 4.02
94.96
# 196
Width
1 0.03
1.027
0.25
0.07
2.45 1.7.2
Traffic
Cov Width
87.68
1 1.28
68.83
1.3.10
82.89
1. 2.43
92.38
1. 1:24
1.92
12.53
1.5.0
90.35
21.44
7.92
2.98
Electricity
Cov Width
95.63
# 2,77
65.00
9.9.9
82,00
4: 3,31
91,22
# 1.29
193.25
1 80.19
106.25
159.82
83,42
20.53
139.75
1 620.44
Dancing Bees
Cov
84.52
4 2.98
45,32
1. 8.49
70.52
4. 5,33
92.64
4: 3.39
Width
0.03
+ 0.01
0.08
= 0.00
0.22
:: 0.09
0.79
+ 0.27
Longer horizon Data Result for the same set of experiments performed on longer horizon (T 10, 000 for synthetic datasets amd = 4, 000 for Electricity and Traffic datasets) is presented in 5 We added the setting where the added noise to the bouncing ball dataset is varying over the course of the time series, which is more challenging than the original settings. In the longer horizon case, both SPCI and HopCPT achieve their state of the art performance. Our method remains valid yet achieves less sharp interval width.
CPTC offers distinct advantages over SPCI and HopCPT in data- lata-limited regimes and applications requiring rapid deployment. While SPCI and HopCPT achieve excellent long-term performance on extensive time series (T > 10,000) by learning residual patterns through quantile regression and Hopfield networks respectively, they require substantial data to train these models effectively during inference. Our results demonstrate that CPTC significantly outperforms these methods on shorter horizons (T < 300), where insufficient data prevents SPCI and HopCPT from learning meaningful temporal correlations or residual patterns.
20

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Table Longer time horizon (T 10, 000 for synthetic datasets amd T 4, 000 for Electricity and Traffic datasets). Performance in synthetic and real-world datasets with target confidence --a = 0.9. Methods that are invalid (coverage below 90%) are grayed out. Our method achieves a high level of calibration (coverage is close to 90%) consistently.
RED-SDS
ACI
SPCI
HopCPT
Ours
Bouncing Ball
Cov
18.53 7,19
65.28 5a9
89.25
1.40
90.05
90.02 =00
90.53
0:2.3
obs.
Width
1.65 a.1
2.65 8:0
3.53
à oM
1.95 0.12
1.99 a o.0
2.25 2:054
Bouncing Ball
Cov Width
15.12 15.00
42.88
6.5
87.95 Los
90.04
# 0.04
90.01 00
90.05 + 0:004
dyn.
1.67
0.95
3.10
3.24
3.07
3.18
3-Mode
Cov
68.52
65.24
89.75
90.05
90.02
90.08
System
Width
2.56
0.021
2.33 #:083
3.74
+ 0.6
2.67 2.034
2.49 *
3.12 #226
Bouncing Ball
Cov Width
10.83
2253
40.25
89.60
90.02
90.04
90.10
obs, varying
2.10
0.45
1.73 9.991
3,70
4 0:9
2.11 0:03
2.14 4:075
2.41 1:122
Bouncing Ball
Cov Width
14.25
16:
41.73
#:132
88.05
1.00
90.05
= 0.00
90.02
= 0.44
90.08
+ 0.0
dyn. varying
0.12
0.055
3.13
4: 1.14
2.92 0:2.1
3:05 4:0.9
0:032
Truffic
Cov Width
87.14
10.1
70.52
230
89.15
0:3
90.01
+ 0.00
90.03
0003
90.02
4 000
2.91
à: 1.02
0.43
a 0.25
55.10
à 74.15
4.81
2:44
7.84 10.33
8.55
a: 14.11
Electricity
Cov Width
76.10
13.52
69.53
3221
89.05
0.333
90.02
= 0.00
90.00
000
90.43 4 0.0
91.30
62.10
31.40
162.02
161.70
173.38
Aggregation methods. In this study we compare the two different aggregation methods. First is the case where we chose the minimum size, i.e. (x+) Eqn 10 Algorithmically, we discretize the output space y into intervals of length 0.02, and calculate probability mass for each interval. We refer to this method as min. Aggregation by union numbers (the same as in the main text using Eqn I1] are referred to union. Results are present in table where the performance are very similar. When there are two modes (bouncing ball cases) the prediction intervals are almost identical.
Table Ablation Study on two different Union (union, eqn 11 and grid-diicrretizatonn (min, eqn 10
BB obs.
BB dyn.
3-mode system
traffic
electricity
bees
Coverage (union)
90.53
90.44
93.35
92.76
92.62
93.43
Coverage (min)
90.03
90.32
91.50
91.87
89.82
90.28
Width (union)
3.73
1.72
2.45
7.94
166.88
0.79
Width (min)
3.62
1.65
2.41
7.95
153.57
0.77
Discussion on Misspecified Regimes. We designed the number of latent states to be hyperpa- rameter given by the user. Our method ensures that the validity guarantee still holds even if the state cardinality is incorrect (theorem 4.2), as incorrect state cardinality can be formulated as case of incorrect classification. Determining state space cardinality is well-studied problem (see for example [14)), and we consider it to be beyond the scope of our paper.
Regarding the effect on performance, our ablation studies (Section 5.2, Tables 2 and 3) demonstrate that while the validity (coverage) of CPTC is robust to errors in state prediction, the sharpness (width of prediction intervals) can be affected. Table 3 explicitly shows that "the more accurate the state prediction is, the sharper the intervals." This implies that selecting a more appropriate number of regimes that better captures the true underlying dynamies would lead to narrower and more efficient prediction intervals.
B.5 Qualitative Results
We provide additional visualizations to present qualitative comparisons CPTC compared to baseline models. Figures 6 7 show show consistent results with the analysis in section 5, where CPTC shows stronger adaptivity and validity for uncertainty quantification time series with these nonstationary dynamics.
21

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10
10
10
10
RED-SDS
CP
AC1
SPCI
CPTC
40
20
40
60
20
40
60
20
10
60
20
40
00
10
10
RED-SDS
ACI
SPCI
CPTC
40
20
60
20
40
64
20
40
60
20
50
10
10
40
20
20
Figure 6: Qualitative comparison of the prediction interval on the bouncing ball w/ dynamics noise dataset. The dashed line is the predicted state whereas the solid gray line is the observation. We see that our CPTC algorithm can increase the interval in anticipation of an uncertain state change (top panel), is stabler when the prediction accuracy compared to ACI and SPCI (middle panel), and is adaptive to inaccurate predictions (botom pancl).
10
10
10
5
25
50
0
25
50
25
50
25
50
25
50
10
10
10
10
10
0
RED-SDS
0
25
50
0
25
50
25
50
25
50
25
50
10
10
10
10
10
RED SDS
CP
0
ACI
SPCI
CPTC
25
50
25
50
Figure 7: Visualization of prediction intervals on the Bouncing Ball Obs dataset.
22

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10
10
10
10
10
5
MP3 10
-5
RED-SDS
-5
CP
-5
ACI
-5
SPCI
-5
CPTC
25
50
50
25
25
50
10
10
10
10
10
5
-5
RED-SDS
CP
-5
ACI
SPCI
-5
CPTC
25
25
50
1.0
10
10
10
5
RED-SDS
CP
AC1
SPCI
CPTC
-5
-5
25
Figure 8: Visualization prediction intervals on the 3 mode system datasets.
B.6 Runtime analysis
We provide runtime analysis comparing the aggregation methods in Equations 10 and 11 and with baselines.
Notation
T: Total timesteps, K: Number of states, M: State prediction model inference complexity
Score sets S₂ maintained in sorted order (quantile computation in (1), insertion Olog 7))
Complexity Per Timestep
1. Computing quantile Q' ('=")(s!)) for each state, totaling O(K).
2. State Prediction: computing (x))(x))) and sampling a state, totaling (
3. Aggregation
Eq. 10, Weighted Level-Set: O(K [1/6]d) where ô is discretization resolution
Eq. 11, Union: O(K log K') for sorting states by probability
4. Updates: Ollogt) t) for inserting new score into sorted set.
Therefore, the total complexity of CPTC with horizon T is: O(T (M + K [1 log T) if we use the Weighted Level-Set as aggregation in Eq. 10, and O(T (M + K log K) + T' log if use the more efficient Union aggregation in Eq. 11. In practice, both M (inference cost) and (number of discrete states) are small. For applications where K (1)) and M (1), CPTC achieves near-linear scaling O(T log T), the same ACI (see table below).
If we use the more precise weighted level set aggregation method, the runtime largely depends on the discretization resolution 6, output dimension d, and the number of states K. We remark the level set approach scales poorly if the resolution, dimension, or number of states are high.
Method ACI
Time Complexity
Remarks
OT log T')
Single score set, no state awareness
SPCI/ HopCPT
(T log T)
Re-train regression model every time step
CPTC (Union)
( log T)
State-aware, pre-trained predictor
23

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30


Reviews:
[{"review_id": "2EcsQlw8bA", "reviewer": "Reviewer_pacc", "summary": "This paper proposes a new approach to conformal prediction for time series data with change points. The proposed method leverages switching dynamical systems to model different underlying dynamics in online environments. For each state, a separate prediction interval is constructed, and a final prediction set is formed by aggregating these state-specific intervals based on the predicted state distribution.", "questions": "1-\tCould the authors provide the computational complexity of the proposed method and compare it with that of previous approaches?\n\n2-\tCould the authors clarify how the warm-start data is selected? Additionally, is there any empirical evaluation of its impact on coverage and prediction set size?\n\n3-\tCould the authors clarify how the state transition model is constructed? Specifically, is the transition matrix learned offline, and is the probability of switching between states kept fixed during all test-time steps?\n\n4-\tThe proposed method creates state-specific prediction intervals for each latent state and then aggregates them. I would kindly ask the authors to compare their approach with prior methods that also generate multiple prediction sets and then select or aggregate among them. For example, ACI’s sensitivity to learning rate has been addressed in [1], and other methods such as [2] construct prediction sets for each learning model and then select or combine them. \n\n[1] Gibbs, I. and Candès, E.J., 2024. Conformal inference for online prediction with arbitrary distribution shifts. Journal of Machine Learning Research, 25(162), pp.1-36.\n\n[2] Hajihashemi, E. and Shen, Y., 2024. Multi-model ensemble conformal prediction in dynamic environments. arXiv preprint arXiv:2411.03678.", "limitations": "Limitations are mentioned in the conclusion", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:  The paper presents a novel idea with solid theoretical guarantees. The motivation is clearly stated, and the paper is well-written and easy to follow.\n\nWeakness: Please check the Questions.", "was_revised": true, "final_justification": "The idea proposed in this paper is novel. During the rebuttal, the authors addressed most of my concerns, such as providing the computational complexity and comparing the proposed method with previous approaches; therefore, I have decided to increase my confidence in the score. I suggest that the authors include the computational complexity in the final version of the manuscript.", "rebuttal": ""}, {"review_id": "XTQ2uraIs9", "reviewer": "Reviewer_Nmg7", "summary": "The manuscript describes an algorithm for a conformal prediction to quantify uncertainty for time series data, whose data generating distribution changes over time. The proposed method relies on a pre-trained switching state model to identify the hidden state at each time step and build a conformal prediction model as a mixture distribution.", "questions": "1. The probabilistic model shown in (5) is inconsistent with the rest of the paper.\n  - the model in (5) is a generative model that leans the distribution. However, rest of the paper, the baseline model looks like a deterministic forecast model.\n  - in most of the places, the conformal prediction is computed based on $p(z_t|x_t)$, which is inconsistent with the probabilistic model in (5). Moreover, $p(z_t|x_t)$ indicates the latent state is determined by the data in the current time step, $x_t$. However, in the time series problems, in general, given only one time step value, $x_t$, the system state cannot be uniquely determined. Did you mean $p(z_t|x_0,\\cdots,x_t)$?\n\n2. There are two aggregation schemes Eqns (10) and (11). And the authors ague that both methods results in similar prediction interval based on observations on a few data sets. However, in all of their experiments, the number of latent state is very small, 2 or 3. So, it is unclear if the behavior can be extrapolated to a larger number of latent states.\n\n3. Similar to Q2, what's the effects of the size of the latent state? How do you know what's going to be the right number of latent states for each data set?\n\n4. What is $E[err_t]$ in Theorem 4.2? Expectation over what distribution?\n\n5. In Appendix B, it is mentioned that the forecast was made with a \"deterministic latent states\" ($z_t$) and \"stochastic systems states\" ($x_t$). This seems not consistent with the probabilistic model assumed in the paper. What is \"deterministic latent states\"? Can you prove that it gives a correct Monte Carlo estimate of the given probabilistic model?\n\n6. What's the effect of the data set size?\n\n7. It looks like the proposed method works for univariate time series. Can it be used for a multi-variate time series? Then, how to consider the covariance?\n\n8. The organization can be improved. In particular, online CP is introduced in section 3.1. But here the authors only describe the theoretical guarantee without outlining the core method. And in section 4.1 the online CP is described together with the new features. This organization is a little bit confusing. I would suggest the authors to move the description of the online CP to section 3.1 and only focus on novel features in section 4.1.", "limitations": "See Questions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength:\nThe paper is straightforward and easy to read the core idea and the proposal to tackle the challenges. The proposed method is model-agnostic as long as the baseline model provides a discrete latent space. \n\nWeakness:\nThe experiments are weak, and there are many places that need clarification / further explanation. See the questions below", "was_revised": true, "final_justification": "I believe that the manuscript is interesting enough and the proposed method can be useful for time series prediction models, many of which are deterministic. The authors addressed most of my concerns. I am still not confident about the experiments and hyperparameter studies, but the study is worth reporting to the community.", "rebuttal": ""}, {"review_id": "mJogL3MsKS", "reviewer": "Reviewer_hm7E", "summary": "This paper presents a novel method (CPTC) that integrates switching dynamical systems (SDS) with online conformal prediction to handle uncertainty quantification in non-stationary time series with change points. The approach is theoretically grounded and empirically validated, addressing a significant gap in existing CP methods.", "questions": "1. In table 1, why is the coverage of ACI invalid? It seems that both theoretical and empirical results in other papers (see \\[1] and \\[2]) show that ACI will achieve long-term coverage despite fluctuations and inefficient intervals. Does this correlates to the choice of learning rates $\\gamma$? The authors should dicuss it in depth.\n\n2. In the experimental section, most of the baseline models used for comparison are relatively outdated. It is recommended that the authors compare their model with more up-to-date online CP methods tackling changepoints, such as \\[2] and \\[3]. \n\n3. CPTC assumes the system consists of several underlying states, such as day/night. However, in the area of conformal prediction, it seems more appropriate to use group-conditional-based methods like \\[4] and the baseline should add such methods. How do the authors comment about this?\n\n[1] Gibbs I, Candes E. Adaptive conformal inference under distribution shift, NIPS 2021;\n\n[2] Wu J, Hu D, Bao Y, et al. Error-quantified conformal inference for time series, ICLR 2025;\n\n[3] Angelopoulos A N, Barber R, Bates S. Online conformal prediction with decaying step sizes, ICML 2024;\n\n[4] Zecchin M, Simeone O. Localized Adaptive Risk Control, NIPS 2024.", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n 1. A new method of conformal prediction for time series with changepoints, leveraging an SDS model.  \n 2. The mathematical proofs are easy to follow and the notations are clear.\n 3. Many online CP methods similar to ACI can be incorporated into the framework.\n 4. The authors demonstrate the practical utility of their methods with an extensive series of experiments.\n\n**Weaknesses:**\n\n 1. The CPTC algorithm proposed in this paper simply integrates an SDS model into the online CP framework. The switching scenarios are also relatively limited, and the authors could consider the switching mechanism's effect on online CP in greater depth.\n 2. Incomplete experiments. See questions.", "was_revised": true, "final_justification": "Most of my concerns have been addressed. Since this work is relatively derivative, so I give rate 4 (Borderline accept).", "rebuttal": ""}, {"review_id": "c0Q3JaCef3", "reviewer": "Reviewer_3vb3", "summary": "The author introduced CPTC method, which is a conformal prediction method for time series that is based on switching dynamical systems. Supposing that the time series can be classified into different states, the CPTC method split the dataset into subsets and combine the prediction set with the state predictor. The method is able to quickly respond to the change points in the time series and generate valid confidence interval for each state. In theory, the authors prove the asymptotic validity under the assumption that the true state distribution follows an underlying stationary distribution. In experiment, the authors compare the method with several common bayesian or CP methods to show that their method is able to achieve valid coverage in the change point setting while the other methods cannot.", "questions": "1. Could authors explain when the assumption for the time series holds?\n\n2. Could authors explain how to select the number of possible regimes in predicting the states? Would that affect the performance of the method?\n\n3. If the state predictor works very poorly, will the performance of the method be similar to the performance of ACI?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. Writing is clear\n2. Have solid experiments, compare with several common benchmark methods on 3 synthetic datasets and 3 real datasets\n3. Show the method is asymptotic valid under certain assumption in theory, also show the method is robust and can quickly react to change point\n4. Have novelty in method\n\n\nWeakness:\n1. The method assumes that the time series is switching between some true states, and requires a state predictor to build the prediction interval.\nNeed more discussion about when the assumption holds and what to do when the assumption fails to hold.", "was_revised": false, "final_justification": "", "rebuttal": ""}]

Author rebuttal:

Meta-review: This paper introduces a new conformal prediction algorithm (CPTC) for time series with change points, leveraging switching dynamical systems to anticipate distribution shifts. The reviewers generally found the idea novel and well-motivated, with solid theoretical guarantees and clear presentation. While some raised concerns about the scope of experiments and clarifications, the authors have addressed these during the rebuttal with additional baselines, runtime analysis, and ablation studies. Overall, the contributions are timely and relevant, and the paper is a worthwhile addition to the literature.
Decision: Accept (poster)