Kernel-based Optimally Weighted Conformal Time-Series Prediction
Authors: Jonghyeok Lee
Chen Xu
Yao Xie
Venue: ICLR 2025 Poster (2025)
Keywords: Conformal prediction
Nonparametric kernel regression
Time series
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: A sequential conformal prediction method for time-series through nonparametric kernel regression, with rigorous theoretical analysis and strong empirical performances.

Abstract: Conformal prediction has been a popular distribution-free framework for uncertainty quantification. In this work, we present a novel conformal prediction method for time-series, which we call Kernel-based Optimally Weighted Conformal Prediction Intervals ($\texttt{KOWCPI}$). Specifically, $\texttt{KOWCPI}$ adapts the classic Reweighted Nadaraya-Watson (RNW) estimator for quantile regression on dependent data and learns optimal data-adaptive weights. Theoretically, we tackle the challenge of establishing a conditional coverage guarantee for non-exchangeable data under strong mixing conditions on the non-conformity scores. We demonstrate the superior performance of $\texttt{KOWCPI}$ on real time-series against state-of-the-art methods, where $\texttt{KOWCPI}$ achieves narrower confidence intervals without losing coverage.

Published as a conference paper at ICLR 2025
KERNEL-BASED OPTIMALLY WEIGHTED CONFORMAL TIME-SERIES PREDICTION
Jonghyeok Lee, Chen Xu & Yao Xie
H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332, USA (jlee4177, cx9711) Bgatech. edu, yao. yac. edu
ABSTRACT
In this work, we present a novel conformal prediction method for time-series, which call Kernel-based Optimally Weighted Conformal Prediction Intervals (KOWCP I). Specifically, KOWCPI I adapts the classic Reweighted Nadaraya- Iadaraya-Waason Watson (RNW) estimator for quantile regression on dependent data and learns optimal data-adaptive weights. Theoretically, we tackle the challenge of establishing a conditional coverage guarantee for non-exchangeable data under strong mixing conditions the non-conformity scores. We demonstrate the superior performance KOWCPI on real time-series against state-of-the-ar methods, where KOWCP I achieves narrower confidence intervals without losing coverage.
INTRODUCTION
Conformal prediction, originated in Vovk et al. (1999 2005). offers a robust framework explicitly
designed for reliable and distribution-fres uncertainty quantification. Conformal prediction has become increasingly recognized and adopted within the domains machine learning and statistics
(Lei al. 2013 Lei & Wasserman, 2014 Kim al. 2020; Angelopoulos & Bates 2023). Assuming nothing beyond the exchangeability of data, conformal prediction excels in generating valid prediction
sets under any given significance level, irrespective of the underlying data distribution and model assumptions. This capability makes it particularly valuable for uncertainty quantification in settings characterized by diverse and complex models.
Going beyond the exchangeability assumption has been research challenge, particularly as many real-world datasets (such as time-series data) are inherently -on-exchangeable. Tibshirani al. (2019) addresses situations where a feature distribution shifts between training and test data and restores valid coverage through weighted quantiles based on the likelihood ratio of the distributions. More recently, BBabber et al. (2023) bounds the coverage gap using the total variation distance between training and test data points and minimizes this gap using pre-specified data- inttaindeeendeen weights. However, it remains open to how to appropriately optimize the weights.
To advance conformal prediction for time series, we extend the prior sequential predictive approach (Xu & Xie 2023a b) by incorporating nonparametric kernel regression into the quantile regression method on non-conformity scores. A key challenge of adapting this method to time-series data lies in selecting optimal weights accommodate the dependent structure of the data. To ensure valid coverage of prediction sets, it is crucial to select weights inside the quantile estimator so that it closely approximates the true quantile of non-conformity scores.
Sadtraya--aason In this paper, we introduce KOWCPI, which utilizes the Reweighted Nadaraya- Watson estimator (Hall al. 1999) to facilitate the selection of deta-dependent optimal weights. This approach anticipates that adaptive weights will enhance the robustness of uncertainty quantification, particularly when the assumption of exchangeability is compromised. Our method also addresses the weight selection issue in the weighted quantile method presented by Barber et al.] (2023), as KOWCP I allows for the calculation of weights in a data-driven manner without prior knowledge about the data.
In summary, our main contributions are:
"Correspondence: www.xxi@@sscgggtcchh..

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We propose KOWCP I, a sequential time-series conformal prediction method that performs nonparametric kernel quantile regression on non-conformity scores. In particular, KOWCPI learns optimal data-driven weights used in the conditional quantiles.
We prove the asymptotic conditional coverage guarantee of KOWCPI based on the classical theory of nonparametric regression. We further obtain the marginal coverage gap of KOWCPI using the general result for the weights on quantile for non- non--ecaangeable data.
We demonstrate the effectiveness of KOWCPI I on real time-series data against state-of-the-an baselines. Specifically, KOWCPI can achieve the narrowest width of prediction intervals without losing marginal and approximate conditional (i.e., rolling) coverage empirically.
1.1 LITERATURE
RNW quantile regression In Hail et al. (1999), the Reweighted Nadaraya-Watse Watson (RNW, often referred to as Weighted or Adjusted Madaraya-Watsone estimator was suggested as a method to estimate the conditional distribution function from time-series data. This estimator extends the renowned Nadaraya- Watson estimator (Nadaraya 1964 Watson 1964) by introducing an additional adjustment the weights, thus combining the favorable bias properties of the local linear estimator with the benefit of being a distribution function by itself like the original Nadaraya- Watson estimator (Hall et al. 1999 Yu & Jones, 1998). The theory of the regression quantile with the RNW estimator has been further developed by Cai (2002). Furthermore, Cai (2002) and [Salha (2006) demonstrated that the RNW estimator is consistent under strongly mixing conditions, which are commonly observed in time-series data. In this work, we adaptively utilize the RNW estimator within the conformal prediction framework to construct sequential prediction intervals for time-series data, leveraging its data-driven weights for quantile estimation and the weighted conformal approach.
Conformal prediction with weighted quantiles Approaches using quantile regression instead of empirical quantiles in conformal prediction have been widespread (Romano et al. 2019 Kivaranovic jet al. 2020 Gibbs et al., 2023). These methods utilize various quantile regression techniques to construct conformal prediction intervals, and the convergence to the oracle prediction width can be shown under the consistency of the quantile regression function (Sesia & Candès) 2020). Another recent work by Guan (203)) uses kernel weighting based on the distance between the test point and data to perform localized conformal prediction, which further discusses the selection of kernels and bandwidths. Recent work in this direction of utilizing the weighted quantiles, including Lee jet al. (002), continues to be vibrant. As we will discuss later, our approach leverages techniques in classical non-parametric statistics when constructing the weights.
Time-series conformal prediction There is a growing body of research on time-series conformal prediction (Xu & Xie 222Tbb Gibbs & Candès (02T). Various applications include financial markets
(iibss & Candès 1222)) 1). anomaly detection (Xu & Xie 2021a), and geological classification (Xu &
Xic|2022), In particular, Gibbs Candès 2221 2024) sequentially construct prediction intervals
by updating the significance level based the mis-coverage rate. This approach has become a major methodology for handling online, non-exchangeable data, leading to several subsequent developments of adaptively updating the single-parameter threshold that determines the prediction sets at each time step (Feldman et al. [022 Auer el al. 20233 [Bhttnaarr et al. 2023 Zaffran et al. 2022 Angelopoulos et al. [0023] Yang et al. 2024] Angelopoulos et al. (024). On the other hand, Xu & Xie (2023b); Xu et al. 2024) take a slightly different approach by conducting sequential quantile regression using non-conformity scores. Our study aims to integrate non- -onaamamttiic kernel estimation for sequential quantile regression, addressing the weight selection issues identified by Barber et al. (0023). Additionally, our research aligns with Guan 2023), particularly in utilizing a dissimilarity measure between the test point and the past data.
PROBLEM SETUP
We begin by assuming that the observations of the random sequence (X,, Y) E = obtained sequentially. Notably, X, may represent exogenous variables that aid in predicting Y. the historical values of Y, itself, or a combination of both. (In Appendix A we expand our discussion to include cases where the response Y, is multivariate.) A key aspect of our setup is that the data
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non-exchangeable and exhibit dependencies, which are typical in time-series data where temporal or sequential dependencies influence predictive dynamics.
IR trained on a separate dataset or on past data. This predictor maps feature variable in IRd to a scalar point prediction for Y, Given a user-specified significance level a E (0, 1), we use the initial observations to construct prediction intervals C2 (X,) for Y, in sequential manner from t = T + 1 onwards.
Suppose we are given a pre-specified point predictor f R fR
Two key types of coverage targeted by prediction intervals marginal coverage and conditional coverage. Marginal coverage is defined as
PPY:, E cx,((x)) 21-a.
which ensures that the true value Y, falls within the interval Či- least 100(1 a)% of the time, averaged over all instances. On the other hand, conditional coverage is defined as
IP(Y, E C (Xx) Xx)) 2 a, which is a stronger guarantee ensuring that given each value of predictor X,, the true value Y, falls within the interval
(2)
at least 100(1 - a3% of the time.
METHOD
In this section, we introduce proposed method, KOWCPI (Kernel-based Optimally Weighted Conformal Prediction Intervals), which embodies our approach to enhancing prediction accuracy and robustness in the face of time-series data. We delve into the methodology and algorithm of KOWCPT in-depth, highlighting how the Reweighted Nadaraya- -atrayyn-aatoon Watson (RNW) estimator integrates with our
predictive framework.
Consider prediction for univariate time series, Y1, ...... We have predictors Xx given to us time f ... which can depend on the past observations (Y,-1, Yr- and possibly other exogeneous time series Zz. Given a pre-trained algorithm f, we also have sequence of non-conformity scores indicating the accuracy of the prediction:
& Y, f(X.), t= 1,2.....
We denote the collection of past T non-conformity scores at time <>> T
We construct the prediction interval Ch. (X,) with significance level time t T as follows: C [f(x) + 9a- (Et). f(X)) + -i-aass (E,),,
8 argmin (()+)+)(EE)) - (x(E.))
(c(0.o)
Here, t is a quantile regression algorithm that returns an estimate the S-quantile of the residuals, which will explain through this section. We consider asymmetrical confidence intervals to ensure the tightest possible coverage.
3.1 REWEIGHTED NADARAYA- WATSON ESTIMATOR
The Reweighted Nadaraya-Wasse Watson (RNW) estimator is general and popular method for quantile regression for time series. Observe (X,, Yi). I...., n, where Y, IR, and the predictors X; can be p-dimensional. The goal is predict the quantile P(Ý < b)8), b I. given test point X using training samples. The RNW estimator introduces adjustment weights on the predictors to ensure consistent estimation. We define the probability-lik- adjustment weights (x( = 1,..., n, by maximizing the empirical log- likeelhodd Σm log (X)), subject to (X) 2 0, and
p(())= =
(4)
(X) X)Ka(X, - -)=
(5)
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The RNW estimate of the conditional CDF P(Y b|X) is defined follows:
F(b[X) Σ W(X s.),
(6)
where the final weights W; are given by
Px( XX
(X( Σ (X) -X) i= 1=1....,
computed as a weighted average of the adjustment weights p. based on the similarity between X to each sample X; measured by the kernel function K: R + R. Here, (u) h h>"K(h-lu) for E RR. Any reasonable choice of kernel function is possible; however, to ensure the validity of our theoretical results discussed Section g the kernel should be nonnegative, bounded, continuous, and possess compact support. An example is K(u) k[], where k: R R the Epanechnikov kernel.
The primary computational burden of the RNW estimator lies in calculating the adjustment weights Ps. However, as Lemma 3.1 shows, this reduces to simple one-dimensional convex minimization problem, ensuring that the RNW estimator is not computationally intensive. This simplification significantly alleviates the overall computational complexity. Furthermore, Lemma 3.1 serves as a starting point for the proof of the asymptotic conditional coverage property of our algorithm, which will be addressed in Appendix B.I
Lemma 3.1 ((Hall et al. 1999 Cai 2001 The adjustment weights P,(X), 1,... for the
RNW estimator are given as
= 1 [1 (( -X81) -x), x)]=²
(x(
(8)
ni
where [X] denotes the first element of a vector X, and X E TR. is the minimizer of:
= Σ log(1- + A((X. (X, -
(9)
3.2 RNW FOR CONFORMAL PREDICTION
To perform the quantile regression for prediction interval construction at time T + 1, a sliding window approach, breaking the past T residuals ε441 = (ér:...f)) into n == (T w) overlapping segments of length w. We construct the predictors and responses to fit the RNW estimator as follows:
y, = E X, = (Eit E.),
1,...., n.
With RNW estimator F fitted on ((8,, Y.)) the conditional 8 8-quantile estimator Q is defined as
2x(x) = inf R: F(glX) $3.
(10)
After time t = T + 1, we update & by removing the oldest residual and adding the newest one, then repeat the process (see Algorithm D. Section 4 we prove that due to consistency of Qs. KOWCPI I achieves asymptotic conditional coverage despite the significant temporal dependence introduced by using overlapping segments of residuals.
Bandwidth selection. Estimating the theoretically optimal bandwidth that minimizes the asymptotic mean-squared error requires additional derivative estimation, which significantly complicates the problem. Consequently, similar to general con-paramttric models, one can use cross-validation select the bandwidth. However, gross-validation can be computationally burdensome and may deteriorate under dependent data (Fan et al. 1995). Therefore, we adapt the nonpparametrie AIC (Cail Tiwari 2000), used for bandwidth selection local linear estimators. This method is applicable because the RNW estimator belongs to the class of linear smoother 2002). Let S be linear
smooth operator, with the (i, j-th element given by Sij WX,,) (Hastie 1990). Recognizing that
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82
Feow-2
% m Reew
time
Zeour-2
8 (feupp-2
Ro-1 (Boom Rpoa)
Histogram of (6:1-1
PACF
81 (mm-)))
1.0
0.5
0.0
126
20
40
Figure 1: Illustration of KOWCPI, sequential conformal prediction method. In the absence of exchangeability the data, as indicated by the empirical distribution residuals and the PACF plot, it is critical to consider the sequentially dependent structure of the data. In KOWCPI non-conformity score blocks are updated sequentially using a sliding window, which provides prediction intervals for future scores through nonparametric quantile regression.
Algorithm Kernel-based Optimally Weighted Conformal Prediction Intervals (KOWCPI) I)
Require: Training data (X,, Y,), & I..... T, pre- preitrained point predictor f. target significance
level E (0, 1), window length w, non-conformity score block count n. = T W.
1: Compute prediction residuals for the training data: & Yt f(X.), =1, I....,
2: for 7 1,7 + do
3: Update residual history & (..
4: Break & into overlapping segments: X;
...... 1.
5: Compute À that minimizes L(:; Xn+1) in equation
6: Derive adjustment weights p. i l...., n, and calculate final weights (X(x++1) in
equation] 7
7: Using Y = É 1,...,n, n, compute the quantile estimator (Xx for
B E [0, a).
Determine 3 argmin (1- (Xn+1) -Qq(
8:
9: Return prediction interval (X,) [f(x,) + Q3 (Xnt1), f(Xi) + Q1-a- (Xn+z)].
10: Obtain new residual é Y, f(X.).
end for
the degree of freedom of the RNW smoother can be given as tr(SST). choose the bandwidth that minimizes
(hCo(h) = log(RSS) + + TTSST
(11)
(tr(SST)
where RSS is the residual sum of squares.
Window length selection. To select the window length w, cross-validation can employed. In experiments, we chose w with the smallest average width that achieves target coverage in the validation set. Another approach to use a weighted sum of the average under-coverage rate and the average width obtained for a given as the criterion. We note that the performance is generally less sensitive to the choice of w across broader range compared to the bandwidth Additionally, in Appendix E we introduce adaptive window selection approach that enables to be determined a data-driven manner, eliminating the need for hyperparameter tuning with minimal loss in performance.

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4 THEORY
In this section, we introduce the theoretical properties of the RNW estimator, a quantile regression method use, and demonstrate in Theorem that our KOWCP asymptotically displays conditional coverage under the strong mixing of residuals. It turns out that the asymptotic conditional coverage gap can be derived from known results in the context of kernel quantile regression.
4.1 MARGINAL COVERAGE
We begin by bounding the marginal coverage gap of the KOWCPI method. The following result shows the coverage gap using our weights, compared with the oracle weights; the results are established using a similar strategy in Tibshirani et al. (2019 Lemma 3):
Proposition 4.1 (Non-asymptotic marginal coverage gap). Denote by P the joint density of (YI...., 1n+1). Then, we have
P ((YY+1 (9(X++1)) - (1- S (B)( W + IEW++1) + 2EA((n+1) (12)
where A A(Xu+1) is the discrete gap defined in equation 17 and W is the vector of oracle weights with each entry defined as
W = Eno(n +1)= P(Y to(++1))
= 1, 1...., 1, (oracle weights)
(13)
Eo P(Yo))) ................................ (n))))
and a is permutation on fl.....n n 1}.
The implication of Proposition 4.1 that
The "under-coverage" depends on the l di-distance between the learned optimal weights and oracle-optimal weights (that depends on the true joint distribution of data).
Note that the oracle weights W cannot be evaluated, because in principle, it requires considering the (n + 1)! possible shuffled observed residuals and their joint distributions.
The form of the oracle weights W," from equation 13 offers an intuitive basis for algorithm development: can practically estimate the weights through quantile regression, utilizing previously observed non-conformity scores.
4.2 CONDITIONAL COVERAGE
In this section, we derive the asymptotic conditional coverage property of KOWCPI. For this, we introduce the assumptions necessary for the consistency of the RNW estimator. To account for the dependency in the data, we assume the strong mixing of the residual process.
A stationary stochastic process [V be strongly mixing (a-mixing) if mixing coefficient (T) defined as
probability space with probability measure P is said
a(T)
sup A B
IP(An (P(A)P(B))
satisfies (TT) + as + x., where My -00 S s 5 oo, denotes a o-algebra generated by (Vs, Vot1s..., V/}. The mixing coefficient (()) quantifies the asymptotic independence between the past and future the sequence {Vi)-=---
Assumption 4.2 (Mixing of the process). The stationary process (V₂ (X,, strongly mixing with the mixing coefficient a(T) (2++)) for some 8 > 0.
We highlight that strong mixing assumptions apply the residuals, which is far less restrictive condition than assuming the original time series itself is strongly mixing. Even when the original time series departs significantly from stationarity, the unobserved noises may still retain stationarity and strong mixing properties. For instance, in vector auto-regressive model with a time-dependent drift, the noises are drawn from the identical distribution without serial correlation.

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Furthermore, the strong mixing property is widely regarded relatively weak condition and is com- monly met by many time series models, making it typical assumption in time-series analysis (Cail 2002). For instance, both linear autoregressive models and the broader class of bilinear models satisty strong mixing conditions with exponentially decaying mixing coefficients under mild assumptions. Similarly, ARCH processes and nonlinear additive autoregressive models with exogenous variables are recognized for their stationary and strong mixing behavior (Masry & Tjøstheim 1995: 1997).
Due stationarity, the conditional CDF of the realized residual does not depend on the index i; thus, denote
F(b(ž) PY;; < b|X; it)
as the conditional CDF of the random variable Y, given X, I. In addition, we introduce the following notations:
Let (() be the marginal density of X, at ā. (Note that due stationarity, we can have common marginal density.)
Let 91,2, i 2 denote the joint density of X1 and X
The following assumptions 4.3 4.5)
common in nonparametric statistics, essential for attaining desirable properties such as the consistency of an estimator ( [Tsybakov[200%)
Assumption 4.3 (Smoothness of the conditional CDF and densities), For fixed ÿ E R and 5 e IR.,
(i) 0 < F() 1.
(ii) F is twice continuously partially differentiable with respect c.
(iii) g > and g(-) is continuous at ã.
(iv) There exists M 0 such that [g,,; g(")g(v)) M for all u, v and i
Regarding Assumption 4.3 we would like to remark that there is a negative result: without additional assumptions about the distribution, it is impossible to construct finite-length prediction intervals that satisfy conditional coverage (Lei & Wasserman| 2014: Vovk 2012).
Assumption 4.4 (Regularity of the kernel function). The kernel K R - R is nonnegative, bounded, continuous, and compactly supported density function satisfying
(i) JR uK uK(u) (u)du= O,
(ii) Jgw un T "(u)du u2l I for some 112 E (0, oc),
(ii) JRw. (²(u)du- 1/ and fg.. un (2(uddu 22l for some vo, 1/2 E (0, oc).
Assumptions 4.4
and (ii are standard conditions (Wand & Jones 1994) that require K to be "symmetric" in sense that that the weighting scheme relies solely the distance between the observation and the test point. For example, if K isometric, i.e., K(u) = k([[u[]) for some univariate kernel function k: R - R. it can satisfy these conditions using widely adopted kernels such as the Epanechnikov kernel. Assumption 4.5 (Bandwidth selection). As n + co, the bandwidth h satisfies
h 0, and =1+2/1)) -> 0o.
We note that Assumption 4.5 met when selecting the (theoretically) optimal bandwidth h* -1/(w+4)) which minimizes the asymptotic mean squared error (AMSE) of the RNW estimator, provided that 5 > 1/2.
We prove the following proposition following a similar strategy as Salha 2006 technical details:
by fixing several
Proposition 4.6 (Consistency of the RNW estimator). Under Assumptions 4.2 4.5 given arbitrary t and jJ, as nl + XC,
F(ī F(j(z) ([ (((
(14)
where D2 F)) denotes the Hessian of F F(j)ū) with respect to T.
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This proposition implies pointwise convergence in probability of the RNW estimator, and since it is the weighted empirical CDF, this pointwise convergence implies uniform convergence in probability (Tucker, [967] p. 127-128). Consequently, obtain the consistency of the conditional quantile estimator in equation T0 to the true conditional quantile given as
Qa(ā) inf y
E R F 2) 8}.
Corollary 4.7. Under Assumptions 4.2 4.5
for every 3 E (0, 1) and ã, as nt + oc,
Q5(5) - Q2(ū) in probability.
(15)
As direct consequence of Corollary 4.7 teed by the consistency of the quantile estimator used in our sequential algorithm.
the asymptotic conditional coverage of KCWCPI is guaran-
Corollary 4.8 (Asymptotic conditional coverage guarantee). Under Assumptions (0, oc,
4.2
4.5 for any
P(Y, E Cr (x)(()) + (1 - a) in probabiliy.
(16)
Thus, employing quantile regression using the RNW estimator for prediction residuals derived from the time-series data of continuous random variables, assuming strong mixing of these residuals, KOWCP I can achieve approximate conditional coverage with a sufficient number of residuals utilized.
To further specify the rate of convergence, define the discrete gap
A(X)= == sup [F(Or(8))8) - - max (XY)., $E(0..)
(17)
=I......
introduced by the quantile estimator being the generalized inverse distribution function.
Theorem 4.9 (Conditional coverage gap). Under Assumptions 4.2 4.5 for any (0, 1) and xı, as + 00,
P Ci X. 21) -(1- (1-a) ai) S OOp( + (nh")) 2/2) + 2-2((nn4)) (18)
where In+1 is the realization of Xn+1 given X; Zt.
Given that the adjustment weights p: (z) uniformly concentrate 1/n (Steikert 2014), one can see that the conditional coverage gap tends to zero, although its precise rate remains an open question.
EXPERIMENTS
In this section, compare the performance of KOWCPI against state-of-the-ar conformal prediction baselines using real time-series data. Additional experimental results, not included in this section, using both real and synthetic data, are provided in Appendices can consistently reach valid coverage with the narrowest prediction intervals.
We aim to show that KOWCPI
Dataset. We consider three real time series from different domains. The first ELEC2 data set (clectric) (Harries 1999) tracks electricity usage and pricing in the states of New South Wales and Victoria in Australia for every 30 minutes over a 2.5-year period in 1996-1999, The second renewable energy data (solar) (Zhang et al.) 2021) are from the National Solar Radiation Database and contain hourly solar radiation data (measured in GHI) from Atlanta in 2018. The third wind speed data (wind) (Zhu et al. 2021) are collected at wind farms operated by MISO in the US. The wind speed record was updated every 15 minutes over one-week period September 2020.
Baselines. We consider Sequential Predictive Conformal Inference (SPCI) (Xu Xie 2023b). Ensem- ble Prediction Interval (EnbPI) (Xu Xie 2023a), Adaptive Conformal Inference (ACI) (Gibbs & Candès 2021), Aggregated ACI (AgACI) (Zaffran al.) 2022). Fully Adaptive Conformal Inference
(FACI) Gibbs Candes 2024). Scale-Free Online Gradient Descent (SF-OGD) (Oraoona & Pál
2018 Bhatnagar al. (202)). Strongly Adaptive Online Conformal Prediction (SAOCP) (Bhatnagar
ct al. 2023), and vanilla Split Conformal Prediction (SCP) (Vovk et al.) 2005). Additionally, we
included comparison where weights were derived from the original Nadaraya- Watson estimator
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Table 1: Empirical marginal coverage and average width across three real time-series datasets by different methods. The target coverage is 1 - a = 0.9. The values in the bracket are standard deviation across five independent trials.
Flectric
Wind
Solar
Coverage
Width
Coverage
Width
Coverage
Width
KONCPI I
0.90 (2.3e-3)
0.22 (1.5e-3)
0,91 (2.8e-3)
2.41 (3.2e-2)
0.90 (1.2e-3)
48.8 (9.4e- I)
Plain NW
0.89 (1.7e-3)
0.31 (2.2e-3)
0.95 (7.4e-3)
3.58 (1.0e-1)
0.41 (2.7e-3)
20.1 (1.8e+0)
SPCI
0.90 (1.1e-3)
0.29 (1.9e-3)
0.94 (1.0e-2)
2.61 (2.1e-2)
0,92 (1.7e-3)
84.2 (1.7e+0)
EnbPI
0.93 (3.4c-3)
0.36 (2.7c-3)
0.92 (2.3c-3)
5.25 (4.3c-2)
0.87 (1.1c-3)
106.0 (2.3c+0)
ACI
0.89 (0.0c-0)
0,32 (2.0c-3)
0.88 (0.0c-0)
8.26 (2.8c-2)
0.89 (1.0c3)
143.9 (2.3e-1)
FACI
0.89 (2.5e-3)
0.28 (1.2c-3)
0.91 (3.2c-3)
7.77 (1.7e-1)
0.89 (0.0c-0)
141.9 (6,4c-1)
AgACI
0.91 (3.1e-3)
0.30 (2.3e-3)
0.88 (1.1e-2)
7.54 (1.2e-1)
(2.4e-3)
144.6 (1.4e+0)
SF-OGD
0.79 (5.8e-4)
0.25 (1.0e-3)
0.11 (2.6e-3)
0.29 (7.0e-4)
0.00 (0.0e-0)
0.50 (0.0e-0)
SAOCP
0.93 (6.1e-3)
0.33 (2.4c-3)
0.76 (1.1e-2)
4.00 (4.5e-2)
0.64 (1.9e-3
33.5 (7.3e-2)
SCP
0.87 (2.8e-3)
0.30 (5.9c-4
0.86 (3.2e-3)
8.20 (1.5e22
0.89 (1.0e-3)
142.0 (3.8e-1)
(Plain NW). For the implementation of ACI-related methods, we utilized the R package AdaptiveCon- formal mntpps://g http///ttubbb.. used the Python code from http:////gg.tthb..ocm/aaaaeeeeea
For SPCI and EnbPI,
Setup and evaluation metrics. In all comparisons, we use the random forest as the base point predictor with the number of trees 10. Every dataset is split in a 7:1:2 ratio for training the point predictor, tuning the window length w and bandwidth h, and constructing prediction intervals, respectively. The window length for each dataset fixed and determined through cross-validation, while the bandwidth is selected by minimizing the nonparametric AIC, as detailed in equation I1
Besides examining marginal coverage and widths of prediction intervals on test data, we also focus on rolling coverage, which is helpful in showing approximate conditional coverage at specific time indices. Given a rolling window size 0, rolling coverage RC, time t is defined as RC, - D 1[Y,-411 E (X,+))) m7
Results. The empirical marginal coverage and width results for all methods are summarized in Table [. The results indicate that KOWCP I consistently achieves the 90% target coverage and maintains the smallest average width compared to the alternative state-of-the-ar methods. While all methods, except SF-OGD, SAOCF, and Plain NW nearly achieve marginal coverage under target 1 a = 0.9, KOWCPI produces the narrowest average width on all datasets. In terms of rolling results, we show in Figures and and 2c that the coverage of KOWCPI intervals consistently centers around 90% throughout the entire test phase. Additionally, Figure 2b shows that KOWCP I intervals are also significantly narrower with a smaller variance than the baselines.
Lastly, Figure 2d depicts the weights W (in log scale) assigned by the RNW estimator the first time index of the test data. Notably, the most recent set of non-conformity scores (in terms of time indices) is assigned the heaviest weights, which aligns intuitively as these are the most similar to the first test datum in terms of temporal proximity. We believe that this heavy weighting of recent residuals contributes significantly to KOWCP I's performance. Datasets where KOWCPI I demonstrates significant superiority, such as the Solar dataset, typically exhibit active volatility changes. In these cases, KOWCPI adapts quickly to changing conditions by leveraging the high weights assigned to recent residuals. For instance, in Figure A.3 which visualizes the performance of KOWCPI, SPCI, and ACI on the Solar dataset, KOWCPI I dynamically adjusts its interval widths to reflect whether is in high or low-volatility region. This adaptive behavior allows KOWCPI I to avoid over-coverage and maintain narrower average widths compared to methods like SPCI and ACI, which produce intervals with relatively constant widths across all regions. At the same time, we acknowledge that such fast-adapting behavior, avoiding conservative intervals, can occasionally lead to brief coverage failures in some regions due to the aggressive adaptation to rapidly changing conditions.
In Appendix C.I] datasets in terms of rolling results. See Appendices C.2 and D for additional experimental results
we show additional comparisons of KOWCPI against the baselines on the other two

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KONOP
10801
PAi kW
H|-owso
222-
A31
00:0
001
0.65
0.70
0.75
0:80
0:85
0.90
0.80
1.50
0.8
0.2
0/
06
1.0
nolling coverage
with
(a) Rolling coverage (boxplot)
(b) Width of prediction intervals (boxplot)
1.0
100
1010
-10
Dovergg
KOWCH
Ran NN
SPC
b10/1
661
0.8
(ghC1
3F:00
SADCP
-00
SCP
325
350
305
425
450
475
500
Time
(c) Rolling coverage over prediction time indices
(d) log W.) by the time lag
Figure 2: Comparison of empirical rolling coverage and width on the electric dataset by different methods in (a) rolling coverage: (b) widths of intervals, (c) rolling coverage over time, and (d) an instance of computed final weights. The target coverage is 90%. In (a), the red dotted line is the target coverage and (b), the blue dotted line the median width of KOWCII.
using a wider variety of real and synthetic datasets, where we consistently observe the coverage validity of KOWCPI I while yielding the shortest intervals on average.
CONCLUSION
In this paper, we introduced KOWCP I, a method to sequentially construct prediction intervals for time-series data. By incorporating the classical Reweighted Nadaraya-Watson estimator into the weighted conformal prediction framework, KOWCPI I effectively adapts to the dependent structure of time-series data by utilizing data-driven adaptive weights. Our theoretical contributions include providing theoretical guarantees for the asymptotic conditional coverage of KOWCP I under strong mixing conditions and bounding the marginal and conditional coverage gaps. Empirical validation on real-world time-series datasets demonstrated the effectiveness of KOWCP I compared to state-of-ther art methods, achieving narrower prediction intervals without compromising empirical coverage.
Future work could explore adaptive window selection, where the size of the non-conformity score batch is adjusted dynamically to capture shifts in the underlying distribution. A preliminary implemen- tation of this approach is discussed in Appendix E showcasing its potential to improve flexibility and adaptability in practice. Additionally, the natural compatibility of kernel regression with multivariate data can be leveraged expand the utility of KOWCP for multivariate time-series data, as detailed in Appendix[A] A There is also potential for improving theoretical guarantees and practical performance by designing alternative non-conformity scores.
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A MULTIVARIATE TIME SERIES
In the main text, our discussion has centered cases where the response variables Y, are scalars. Here, we explore the natural extension of our methodology to handle scenarios with multivariate responses. This extension requires defining multivariate quantiles, introducing multivariate version of the RNW estimator for estimating these quantiles (Salha 2006), and adapting our KOWCPI method for multivariate responses.
Multivariate conditional quantiles Consider a strongly mixing stationary process ((X;, Y;) which is a realization of random variable (X, ) E IRP x R². Following Abdous & Theodorescu 1992), we first define pseudo-norm function || I1-1,,,a IR + R for E (0,1) 1),1)a as
|( U1 + (2a
lua (2xx 1)
|ell,,a =
2
for R², where l1-112 is the Euclidean norm R Let
20 = B[| 0 0||2,a |2
X = 2].
Definition A.1 (Multivariate conditional quantile (Abdous & Theodorescu 1992)). variate conditional a-quantile (3) for a E (0, 1) as
Define multi-
0g (i) argmin H. (0.
(A.1)
DER
Remark A.2 (Compatibility with univariate quantile function). For scalar Ý E IR, its conditional quantile given X - is
0g (z)
argmin E ||I- 0112,0 X = ai]
BER
argmin E || 01 + (2a - 1)(8 -0) X = x)
PER
argmin B(( -) () 1(² s 0)) X =
PER
Qa (8),
for any a (0, 1). Thus, Definition A.1 is consistent with the univariate case.
Multivariate RNW estimator for multivariate responses
Following the definition equation we obtain the RNW estimator
DE W 8 (
F
(A.2)
Σ W (x=2
where, according to equation 7 and equation g
K(u)
W(u) 1 Au AwnK(w) (w)
for u = up) Now, let Wa(u)= = h (W-PWWh-a uu), and define an estimator for Ha (f, x)) as
En W. (&. =) (1D: 0112,a ||
(0, = / (1b 0 01|,, |, F(d)la))))))))
Σi Wn(X, - z)
and consequently the RNW conditional quantile estimator 0₂ as
(G) = argmin (0, = argmin
Wx 2) 0122, - 11/1l2,.)
(A.3)
DERP
DERP
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Multivariate KOWCPI
Suppose sequentially observing (X,, Yi) R t 1,
Based on the construction the multivariate version of the RNW estimator, we can extend our KOWCPI I approach to multivariate responses in the same manner described in Algorithm [ with multivariate residuals
é Y--Y ER,
as non-conformity scores. This adaptation allows for the application of our methodology broader range of data scenarios involving dependent data with multivariate response variables, which were similarly studied in (Xu et al. 2024 Sun & Yu 2024 Stankevičiuté al. (021).
B PROOFS
The following lemma is adapted from the proof of Lemma of Tibshirani et al. (2019); however, do not assume exchangeability.
Lemma B.1 (Weights quantile for non-exchangeable data). Given sequence of random variables
(V...., Va+13 with joint density P and a sequence of observations (U1,-.., Pat1). Define the event
E ((V Vn+13 = (ui,..., 1n
Then we have for i 1, ,,...,n+1,,
P(V++1 = EE Eoia(n+))=d ((Uo(I)...... (o(n=1)
E [0,
Eo P(Vo(I)..... vo(++1))
Note that when the residuals are exchangeable, W 1/(n + 1), as also observed in Tibshirani et al. (2019). Now prove Proposition 4.1
Proof of Proposition 4.1 The proof assumes that Y,, for 1,.., are almost surely distinct. However, the proof remains valid, albeit with more complex notations involving multisets, if this not the case. Denote by Quantile, (Q) the --quantile of the distribution Q on R and by 8, the point
mass distribution at a E R. Define the event E {{),, Pn+1J = {fI,.... 1n Then, by the tower property, have
P(Y++1 E (XT+1)
E(P(Yr+1 E (Xx4)) E))
E
Quantilog-
Σ Wrdn.
< Yot1 I Quantile -
Σ Wion.
E
Quantileg
E
Pv
Σ Wron.
SV S Quantilee-or --
Σ Wedv.
where pw sn+ W 8v,, and in the last line, we have used the result from Lemma B.
Y ~ PW"
Denote the weighted empirical distributions based on W W(X as
pW
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This gives the marginal coverage gap
P(Yn+1 (Xn+1)) - (1 aa)
S E
Pv~ww
Quantileg-
Σ
Widn
S S Quantile 1-003
Wgdw:
P Vv>p
Quantileg-
Wrdes
SV 1 a0+
Wid.. ) (1 a)
+ P. V~PW
Quantile
Wio.,
SV S Quantile a00
- Eld Eldry [PW" _P)]]
+ F F ( @2--++ (Xa+1) Xnt1) - a1 8*) +E|F (X () -B8
$ Bl(" Jmn WIl1 EW" n+1 + 2E max
1
Wi(nn+)). i=I....n
where we denote by drv(..-) the total variation distance between probability measures, and the second inequality is due the definition of the total variation distance.
B. PROOF OF ASYMPTOTIC CONDITIONAL COVERAGE OF KOWCP I (COROLLARY 4.8 AND
THEOREM 4.9
In deriving the asymptotic conditional coverage property of KOWCPI, the consistency of the RNW estimator plays crucial role. Therefore, we first introduce the proof of Proposition (46) which discusses the consistency of the CDF estimator. Corollary 4.7 which states the consistency of the quantile estimator, is natural consequence of Proposition and leads us to the proof for our main results, Corollary 4.8 and Theorem 4.9 Proof of Proposition 4.6 adopts the similar strategy as /alha (2006) and Cai 2002).
To prove Proposition 4.6, essential to first understand the nature of the adjustment weight p: Thus, Lemma 3.1 is not only crucial for the practical implementation of the RNW estimator but also indispensable in the proof process of Proposition 4.6
Proof of Lemma 3.1 For display purposes, denote [X] X1. By equation 5
have that
Σ
(a) 12) =.
(A.4)
Let
. (2))
(og + A1
1 Σ p1) - X2
(i( (8 - KX, -),
i=1
where \1, A2 E R are the Lagrange multipliers. From OCC/Op;(2) 0 for 1,.... n. we get
p (+) X + X2 3 Kn ( Xx G) 0,
Since (i)'s sum up to in equation
letting - -22 \ا, we have
[1 XXX T1) - R2
p
+ X(X,1 - 21 ) (X, -
Using equation 4 lggain with equation
A.4 this gives
1+ * - KX - 2)] +))-
Σ
(1) 1,, - 23]
=
==1
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and therefore equation 8 hholds. With equation 5 this gives
(X, K1 (X, 3) aI(():
0 1 A(X,1 K1)
By
Note that ((y=) 0y²
2 O, implying that L(:;āt) is indeed convex function.
Lemma B.2. Under the assumptions of Proposition 4.6
define
0glä)
08, 122
c(ī) ( P22
Then,
== 1t c( + (1)) -,(h)²)
(A.S)
Proof. Let
- - S. - ,X, 8).
Then, by Assumption 4.4
S; is bounded above by some constant Cj. Let
S
(S.)*
for k 1,2. Then, from equation A.4
we have
0 = (1+ AS -s 2 |
Σ
5 +
AS,)
ST >
ST
which gives
ST
|I| v
2
C1
Using the Taylor expansion (Wand & Jones
1994). we obtain that
ESL
(u1 K1) E)g(u)du
R
h I un (())((( + hu)ddu
R
h f un m(((u) g(5) +
0g(
du + o(12)
új
Oīg
1=1
n² [gg(ū)
op(1)))
03, 122
where the last equation comes from Assumptions 4.4 and derive that
11I
With similar argument, we can
/ (ui 2 ())( h-w+2
ESS
w+2 {g(#)u2 (h))
using Assumption 4.4 (ii) Therefore, we obtain equation A.5
Decomposing F() +) F())2) in bias and variance terms, we get
F(īāă) F(j
Σ (5 (z) (1 (X: () < 9) F()
En p: (x, - i))
(A.6)
EL (i, 8)5. + En Di (X. (X. F(
E Kz) -
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where 6, 1( - 3) F)) Note that E[6,] due to the tower property. Now, let
(z) b, (X,; = [1 ht (x, (1)
-i]] -
Then, by Lemma B.2 have that
Pt( = b, (1
(A.7)
Define the approximations for the terms in the decomposition presented in equation A.6
J1 = n 11/2//122 12
b, - a),
J2 F F(1E)) ((B)), 8),
J3 = (X) -X,
so that
F(j(a) F( ((n
1/ + J2)J (1 (p()))
(A.8)
Therefore, we will derive Proposition 4.6 by controlling the terms J1, J/2 and J3.
Lemma B.3, Under the assumptions of Proposition 4.6
J1 Op())
(A.9)
Proof. Let
& = hw/2 (x),, Z), so that J1 =n 111122 & Since E(d, = 0, we have that IB(G.) (((((E/(.)) 0, and thus
EJ1 = 0.
(A.10)
Also, due to the stationarity of X,, we have that
Var(Ji) EE?
Σ
i
Cov(EI, Ex).
(A.11)
2=2
By Assumption 4.4 through expansion, we have
have that lim bz(
1, which gives B(b,) 1 o1(1). Therefore,
Eº, = h
E (3) 2)5? x:]]
= hh" (T) -)) F(
= =I(3) (( F(.-))(()))))) (z)
op
=
where in the third line the convolution operator. To control the second term in the right-hand side of equation A.1l we borrow the idea of Masry (986). Choose d, O(h 17373) and decompose
x
1-1 n
Cov(EL, Et)
Σ
dn
i- n
Co) +
(1- i
Cov(EI,
n
i=d., +1
We have that [b:(#)5bl S C2 for some constant Cz. By Assumption 4.3 (iv), we obtain
| / EiEigia(u, v)dudo - I. Sig(u)du {
[Cov(Er,&)) = 1 JRn
Eag(v)du
< C2 [ [ K(u)K(v)loLAAE hu, hw) g(E hu)g ho))ddud
< <CMM
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so that
dn
1 - 0p(1).
n
By Assumption 4.4 we have [[X; 2 Kn (X; - I) - C3. so that |s,l S Ch Ch-w/2 Then, by Theorem 17.2.1 of
Ioragimov al. (197I). have that (Cov(E), s Ch -aaiaa(- 1).
Thus, we get
i-1
Cov(f1, S Ch w Σ
a(i) S Ch d 1-1- = o
n
i2
Therefore, we obtain
J1 = EJ1 + Op (Var(Ji)) = =p((1).
(A.12)
Lemma B.4. Under the assumptions of Proposition 4.6
1 J2
2 h² u2tr ( + oz²))
(A.13)
J3 ( (1).
(A.14)
Proof. By Assumption 1, equation 4 and equation
5 we have through expansion that
J2 = (2m) 1 Σ (X, - I) (1h
Since
B[ =5]
-1103)())E(XX-0(((0 (3) x)))
we have
J/2 = 1 p Dz + (1²))
(A.15)
Finally, by applying the expansion argument routinely, get
J3 g(ō) + (1).
(A.16)
L
Proof of Proposition
4.6 Cai 2002 Salha 2006 Combining Lemmas B.3 and
B.4 with equa-
tion A.8. Assumption 4.5 gives the result.
Proof of Corollary 4.7 Cai 2002) Given ã. Proposition 4.6 implies uniform convergence of F(+ā)
to F( F(I ()) in probability Tucker 1967 p. 127-1228) since F(- ④ is a CDF That is,
F F()la)) 0 in probability.
sup jER
Given E > 0, let = 8(E) = min{3 F(Qu(#) - = [)) F(Qx(() + +e(²) - B}. Note that > 0 due to the uniqueness the quantile. We have
P ((0s(()) -((ss()-QQ((t) Qo(i)) > <P ((Qa(()) 3/ >)
Fū P(TTT) > 6)
SP
sup
jek
The uniform convergence of F(-(ū) in probability gives the result.
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Proof of Corollary 4.8
From the definition of Č7_1 in equation 3 have
P Yv C (x.) (Pn+1 E [Qx- (8n+1); Q1-0+0 (Xn-1)) (xt1)
(Q1-ato- (Xn+1) Xn+1) - (Xnt((ttt))
By Theorem 4.7 we have the consistency of 23 for all 3 e (0,1). 1). On that, the continuous mapping theorem and Assumption 4.3 gives
F Qi-ato (Xn+1) (
(Xx+tt {x+1)
F Q1- (Xn+z) (Xntz)({ntt)) F (Qx- (a-(Xx+t) Xn+1)
=1-0 a,
where the convergence is in probability.
Proof of Theorem
4.9
From the definition of Õ2_ in equation 3 we have
P
Y E (x) Xt = - (1-a)
P (iu+1 E
[2-- (i), 1--at- (8)
Xn+1 = (1
(1-a) (1--a)
F (2- (i)
Xnt1 = 8) F (70 (ū) X Xn- =τ) = -11-a)
P
@ -0 (i) 8at1 8) -(1 a 8t) + | ((2x.(88) Xat1= = = -8
F (Q1-ato- (7)
[n+1 = F - (8)x = =t)
F (ox- (z) Xx+1 = = f ( ( = =x)
< (²) + O Op((mh")-1/2 + 1)²),
where the last inequality comes from equation A.8 and the definition of the discrete gap A.
C ADDITIONAL REAL DATA EXPERIMENTS
C.I WIND/SOLAR DATA EXPERIMENT RESULTS
detailed description of the results for the solar and wind datasets introduced We provide Iigures/A.IJang and A.2 illustrate the rolling coverage and interval width results for the in Section 5 Figures A
solar and wind datasets, respectively. As described in Section KOWCP consistently achieves the narrowest intervals while maintaining valid coverage. For qualitative explanations, we also include Figure A.3 that demonstrates the performance of KOWCPI, I, SPCI, and ACI on the Solar dataset.
C.2 AAPL DAILY STOCK PRICE
We compare KOWCP1 I with baseline methods using Apple's daily closing stock price data from from January 2020, December 12, 2022. This publicly available dataset can be accessed on Kaggle (https / /www. www.wa.gggl..mmm com wwt.sattaaaesa/aaaaaaaaa..a stweet///pptttt..m tmmt momoohyyoooey.... stock mirkkmman market (akee-ddaaa) data). The goal is to construct confidence intervals for the daily closing prices. The first 80% of the data is used for training, with the remaining 20% reserved for evaluation. We can observe from Table that KOWCPI attains the narrowest interval.
D SYNTHETIC DATA ANALYSIS
D.1 HETEROSKEDASTICC MIXTURE MODEL
To evaluate the robustness KOWCPI under heteroskedastic conditions, we conduct simulations using heteroskedastic mixture model. This model incorporates AR(1) component, GARCH(1,1)
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Pax
CO
A0
000
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00000
10000
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0.000
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2.55
0.59
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(a) Rolling coverage (boxplot)
(b) Width of prediction intervals (boxplot)
1.0
San
1621
66-
S-303
SACCP
Time
(c) Rolling coverage over prediction time indices
Figure A.1: Rolling coverage and width comparison the solar dataset by different methods.
Table A.1: Empirical marginal coverage and interval widths for confidence intervals of AAPL closing prices, with target coverage of 90%. Standard deviations calculated under three independent trials.
Coverage
Width
KOMCPI
0.912 (2.3e-3)
15.74 (1.2e-1) 1)
SPCI
0.952 (3.3e-3)
19.39 (2.3e-1)
EnbPI
0.912 (8.7c-3)
38.67 (1.5e-1)
ACI
0.871 (1.7e-3)
44.84 (8.7c-2)
FACI
0.891 (5.2e-3)
43.93 (1.7e-1)
AgACI
0.878 (1.1e-2)
43.19 (2.6e-1)
SAOCP
0.619 (7.1e-4)
17.90 (4.3e-2)
SCP
0.796 (2.0e-3)
36.60 (1.0e-1)
structure for time-varying variance, and an additional small Gaussian noise term. The model is defined
Y, 0.8Y.- oztr + Ee.
of 0.1 + 0.3Y + 0.60, fid N(O, & dd N.0.0.13). 4
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Pax
Pix M3
A0
A0
00000
10:00
243
043
6.12
0.55
(a) Rolling coverage (boxplot)
(b) Width of prediction intervals (boxplot)
1.0
66-
Time
(c) Rolling coverage over prediction time indices
Figure A.2: Rolling coverage and width comparison the wind dataset by different methods.
This mixture model generates irregular, large volatility bursts, as evidenced in the simulated sam- ple paths. Such extreme variations make conformal prediction challenging, they require rapid adaptation to maintain valid coverage while avoiding overly wide intervals.
We simulate five independent paths of the model and evaluate the performance of KOWCPI against baseline methods, with target coverage of 90%. Unlike methods such as SPCI and SAOCP, which often overreact volatility changes by producing excessively wide intervals and struggle to recover quickly after burst, KOWCPI effectively adapts to these changes using its adaptive weighting mechanism. The results for each sample path summarized in Table A.2
D.2 NONSTATIONARY TIME SERIES
We also consider model with strong seasonality, clearly representing non-stationarity:
Y log(t') sin
(13 X2I + 188
x.1 + 18 1-8T8x, X, 3/11 +
12
where = mt,(t, (,, 12) introduces a seasonal component with 12-period cycle, and X, II- Y,-]] represents features of lagged values. The noise term follows AR(I) process, given by & 0.6 + & with & ~ N(0, 1). Table A.3 demonstrates that KOWCPI performs best in general among methods that achieve valid coverage of 90%. As KOWCP actively leverages reweighting to assign higher weights to recent residuals, it demonstrates the ability to quickly adapt to changes in volatility.
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200
True Y and prediction
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100
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-50
0
20
40
60
80
100
Y
Č(x, a)
(a) KORCP
250
True Y and prediction
200
150
100
-50
-100
20
40
60
80
100
Y
Y
Č(X, a))
(b) SPCI
200
True and prediction
150
100
50
-50
-100
60
Y
Č(X,
ACI
Figure A.3: Comparison of prediction intervals generated by KOWCPI, SPCI, and ACI on the Solar dataset.
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Table A.2: Empirical marginal coverage and interval widths from simulations using hetcroskedastic mixture model, with target coverage of 90%. Here, C and W denote the empirical marginal coverage and average interval width, respectively.
Path
Path
Path 3
Path 4
Path 5
C
W
C
W
C
C
W
W
KORCPI
0.91
4.61
0.89
5.57
0.90
11.44
0.93
8.84e3
0.92
23.09
SPCI
0.99
8.68
0.89
5.85
0.97
18.46
0.90
8.53e3
1.00
99.22
EabPI
0.93
5.47
0.87
5.56
0.91
15.60
0.91
9.14e3
0.96
74.62
ACI
0.93
5.21
0.92
6.99
0.92
14.11
0.89
1.79c4
0.95
27.84
FACI
0.92
5.11
0.92
7.10
0.92
13.75
0.89
1.88c4
0.92
24.28
AgACI
0.93
5.20
0.91
6.83
0.93
13.17
0.88
1,58c4
0.92
25.83
SAOCP
0.79
3.58
0.82
4.62
0.72
7.39
0
36.1
0.67
10.87
SCP
0.93
5.17
0.91
6.73
0.93
14.19
0.84
1.06e4
0.97
29.23
Table A.3: Empirical marginal coverage and interval widths from nonstationary time-series simula- tions, with target coverage of 90%. Standard deviations are calculated under five independent trials.
Coverage
Width
KOWCP
0,90 (1.2c-3)
11.41 (2.3e-2)
SPCI
0.91 (2.7e-3)
11.73 (3.1c-2)
EnbPI
0.86 (1.2e-2)
10.45 (1.8e-2)
ACI
0.90 (1.le33)
12.57 (8.7e-3)
FACI
0.90 (4.1ee3)
12.65 (1.2e-2)
AgACI
0,90 (2.20-3)
12.71 (1.4e-2
SAOCP
08.499.cc4
8.89 (3.2e-3)
SCP
0.90 (2.8e-3)
12.50 (4.1e-2)
ADAPTIVE WINDOW LENGTH SELECTION
In this section, we explore an adaptive selection of w, where is no longer treated a fixed hyperparameter but is instead dynamically adjusted for each time step. In KOWCPI, I, the window length w originally serves as a hyperparameter that requires tuning. To alleviate the burden of manual tuning and introduce a more data-driven approach, we implement adaptive selection process for w based on two-sample test on the residual distributions.
At each time step t, we compare the distributions two blocks of residuals using the two-sample Kolmogorov-Smimoo test: One block contains the most recent w residuals (c.-1 E. and another block consists of the residuals immediately preceding, (Er-ar11,... 2m We then select the smallest w for which the p-value drops below, e.g., 0.01.
While this is simple preliminary approach, it allows for data-driven and adaptive selection of w without requiring additional hyperparameter tuning. Through experiments on the real data, we have confirmed that this method achieves comparable performance to values pre-selected by cross-validation (See Table A.4). [ggure/A.4iillusttatee how the chosen window size changes over time the Wind dataset.
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Table A.4: Comparison of KOWCP I on real datasets using pre-fixed window lengths selected by cross-validation versus adaptive window selection based the two-sample KS test. Target coverage is 90%, and standard deviation is derived across five independent trials.
Electric
Wind
Solar
Coverage
Width
Coverage
Width
Coverage
Width
Fixed
0.90 (2.3e-3)
0.23 (1.5e-3)
0.91 (2.8e-3)
2.41 (3.2e-2)
0.90 32) 0.90 (1.3e-3)
48.8 (9,4e-1)
Adaptive
0.92 (3.0e-3)
0.22 (1.3e-3)
0.90 (4.4e-3)
2.44 (2.7e-2)
50.6 (1.1c+0)
Optimal window size time
Optimal window size
12
10-
60
100
120
140
Test set time index
Figure A.4: Dynamic adjustment of the window size (w) for each prediction step on the Wind dataset, using the adaptive selection process based on the two-sample KS test.
25


Reviews:
[{"review_id": "QY9BBS9rn7", "reviewer": "Reviewer_E4WM", "summary": "The authors introduce KOWCPI to address the difficulties that arise from employing conformal prediction in the time series setting. In order for standard conformal prediction procedures to have valid statistical coverage guarantees the data needs to be exchangeable. However, this requirement is violated in time series data. KOWCPI uses the Reweighted Nadaraya-Watson estimator for quantile regression in the presence of dependent data to learn the distribution of residuals. Using this information, intervals can be constructed that have been theoretically shown to provide conditional coverage and empirically have been found to be narrower than other baselines while maintaining desired coverage.", "questions": "- Is quantile crossing ever an issue in $\\hat Q_\\beta$?\n- The form of the intervals is $\\hat C^\\alpha_{t-1} = [\\hat f(X_T) + \\hat Q_{\\beta^*}(\\tilde X_{n+1}) , \\hat f(x) + \\hat Q_{1 - \\alpha + \\beta^*}(\\tilde X_{n+1})]$. Are there cases where both quantile estimates are positive/negative and possibly leading to invalid intervals?\n- Are there ever discontinuities over time? (ie the quantile estimators dramatically jump around)\n- Are there any diagnostics that can be used to assess the plausibility of the assumptions? \n- How robust is this method to violation of assumptions?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Authors provide a conditional coverage guarantee (much more meaningful than marginal coverage guarantee)\n- Proofs are included and assumptions are clearly stated\n- Strong coverage and efficiency results on intervals", "weaknesses": "- Grammatical errors/typos scattered throughout\n- Notation is rather dense. More plain language accompanying the mathematical formalism would help the reader\n- Presentation of Algorithm 1 seems to be missing some pieces. In the required line should it be $(X_t, Y_t), t=1, 2, ..., T$? Variables $n, w$ are referenced but not explained earlier in the box. \n- Table 1 is a little bit confusing at first glance. Typically bold indicates \"best\" but in this case it only shows what the authors' method is\n- 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": "jQfI4PFRmM", "reviewer": "Reviewer_LFjW", "summary": "This article studies how to use the conformal prediction framework to establish prediction intervals for time series data.\nThe key point of the method is: to estimate the conditional distribution of future residuals using the Reweighted Nadaraya-Watson (RNW) method. Then establish the prediction interval for the future response with mean function and estimated distribution of residual. The experiments indicate that the method produces shorter and more efficient prediction intervals while meeting coverage requirements.", "questions": "-  $T$ represents the number of past residuals used to give the estimation of residual at the next time point. Can the authors give particular analyses related to $T$, such as a sensitivity analysis showing how prediction interval performance changes with different values of $T$, or a discussion of how $T$ was chosen for each dataset? \n\n- The method involves two important parameters: the bandwidth $h$ and the window length $w$. While the paper discusses the settings for these parameters, it is unclear whether the experimental results are sensitive to them. Additional experimental results may be needed to clarify this.\n\n- When the dimension of $\\tilde{X}$ is large, the results from kernel estimation may not be robust. In this case, is it possible to train a function to compress $\\tilde{X}$, aiming to extract effective information and reduce dimensionality for $\\tilde{X}$?\n\n- In classic time series models like the ARMA-GARCH model, can the choice of $w$ or $h$ be discussed more precisely in theoretical parts as a corollary?\n\n- Can the authors provide some experimental results on non-stationary time series?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This article applies the conformal prediction and the RNW method to establish a prediction interval. Therefore, the proposed method is independent of specific time series models and has a broader range of applications.\n\n- Instead of establishing prediction intervals using the quantiles $Q_{\\alpha/2}$ and $Q_{1-\\alpha/2}$. The method seeks a $\\beta$ such that the distance between the quantiles $Q_{\\alpha/2+\\beta}$ and $Q_{1-\\alpha/2+\\beta}$ is minimized to form the prediction intervals. This approach is more effective for certain residual distributions.\n\n- The authors consider time-series structures. In conditional coverage properties, the authors loosen the stationary condition only on strongly mixing residuals.\n\n- The experiments indicate that the method produces shorter and more efficient prediction intervals while meeting coverage requirements.", "weaknesses": "- The authors make a natural integration of the conformal prediction with the RNW method. However, the innovation of this approach is not particularly strong. The theoretical results primarily focus on the asymptotic accuracy of distribution estimation.\n\n- When the dependence in the time series is strong, the dimension of $\\tilde{X}$ may be large. In this case, when the data is limited, kernel estimates may not yield reliable results. Also, large $w$ leads to a low convergence rate of coverage. So it is better to include experimental results showing how coverage and interval width as $w$ increases for different sample sizes and datasets with varying degrees of temporal dependence.\n\n-  The method establishes prediction intervals for stationary time series. Can the method be extended to handle local stationarity? Have authors considered techniques like differencing or detrending to handle non-stationarity? Additionally, please provide a more explicit discussion of the limitations of the proposed method for non-stationary data.", "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": "wHmtiVBkXk", "reviewer": "Reviewer_75Ps", "summary": "This paper proposes a conformal prediction method for time-series data using the Reweighted Nadaraya-Watson estimator. The authors provide non-asymptotic marginal coverage gap in Proposition 4.1 and conditional coverage guarantees under strong mixing conditions in Theorem 4.9. Its efficiency is demonstrated through experiments on three real-world datasets.", "questions": "1. Regarding the theoretical results, one of my concerns is could the author provide theoretical guarantees for other types of dependent time-series beyond strong mixing assumption or explain the difficulties under other dependent conditions?  \n\n2. Related works and experiments: \n- Could the author provide more details about the benchmarks and related works to help readers unfamiliar with online conformal prediction understand why KOWCPI outperforms other methods? For example, providing a brief summary of the key differences between KOWCPI and each baseline method, highlighting why these differences might lead to improved performance may be helpful.  You can also include a table to compare the main features of each method if possible.\n- Besides, the comparison methods in experiments seem to be incomplete. For example, the conformal PID in [1], the BCI in [2] and the decay-OGD in [3] should be considered. \n\n4. Optimal choice of window length $\\omega$: \n - Is it possible to design some strategy to determine optimal $\\omega$ values that adapt to real-time changes in data volatility or distribution patterns, thereby further enhancing the KOWCPI? For example, using change point detection algorithms or online learning techniques to dynamically adjust the window length.\n\nReferences:\n\n[1] Angelopoulos A, Candes E, Tibshirani R J. Conformal PID control for time series prediction[J]. NeurIPS, 2023, 36.\n\n[2] Yang Z, Candès E, Lei L. Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series[J]. arXiv, 2024.\n\n[3] Angelopoulos A, Barber R F, Bates S. Online conformal prediction with decaying step sizes. ICML, 2024", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written and easy-to-follow.\n2. The considered problem is important and well-motivated.\n3. The proposed KOWCPI is efficient in various real datasets.", "weaknesses": "1. Regarding the experiments, the author only provides real datasets to demonstrate the efficiency of marginal and conditional coverage without offering information about dependency. However, I believe it would be beneficial to include additional synthetic experiments to explore different dependency structures and more comprehensively assess the scope and validity of the methods.  \n\n2. While a theoretical guarantee is provided, the bound heavily depends on the estimation of weights and the strong mixing assumptions.", "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": "W2uCXaxhko", "reviewer": "Reviewer_WMCc", "summary": "The paper presents a method to construct prediction intervals for time-series data, its main contribution being using RNW estimators for quantile regression. The paper builds upon a large body of work on the topic and offers a new algorithm under mild assumptions.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is very well-written and easy to follow. \n- The algorithm that leverages the RNW estimator is innovative. \n- Theory section (as far as I have understood) is correct and well-argued.\n- The experiment comparison is comprehensive (includes almost all the baselines in the space. thanks for doing all this work).", "weaknesses": "**1. Differentiation from SPCI.**\n\nOne concern I have is that this approach feels very similar to SPCI both in terms of algorithm (using an estimator to model residuals) and guarantee / proof technique (relies on the convergence property of the estimator). In terms of experiment results, KOWCPI also does not significantly outperform SPCI (in figure 2(c), there is a significant chunk of time where SPCI is more valid). \n\nCan you add some more explanation on how your method is different from SPCI, in terms of theoretical guarantees, assumptions on data, and performance? It will not only improve clarity, but also help readers in practice know which method to choose for their specific applications. \n\n**2. Experiment results & analysis**\n\nI would strongly encourage the authors to add more details to their experiment results, including: \n\n- Add standard deviations to table 1. \n- Explain the results in table 1. What are the rationale of highlighting the first row? You should also highlight other methods that achieve the same coverage, and explain  why lower width was not highlighted. \n- Add experiments for one or more values of $\\alpha$ to demonstrate consistent performance.\n- Provide some qualitative examples to show why/how the KOWCPI confidence bands are better. \n- Elaborate on figure 2(d), perhaps adding a further study/analysis on weights. The remark in line 494-496 is not very convincing when only one example is provided. \n\nIf these concerns are addressed, I'm happy to increase my score.", "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)