Conformal Prediction with Cellwise Outliers: A Detect-then-Impute Approach
Authors: Qian Peng
Yajie Bao
Haojie Ren
Zhaojun Wang
Changliang Zou
Venue: ICML 2025 poster (2025)
Keywords: Conformal prediction
Cellwise outliers
Detection-imputation method
Nonexchangeability
Primary area: general_machine_learning
TL;DR: This paper proposes a general detect-then-impute conformal prediction framework  to address the cellwise outliers in test data.

Abstract: Conformal prediction is a powerful tool for constructing prediction intervals for black-box models, providing a finite sample coverage guarantee for exchangeable data. However, this exchangeability is compromised when some entries of the test feature are contaminated, such as in the case of cellwise outliers. To address this issue, this paper introduces a novel framework called *detect-then-impute conformal prediction*. This framework first employs an outlier detection procedure on the test feature and then utilizes an imputation method to fill in those cells identified as outliers. To quantify the uncertainty in the processed test feature, we adaptively apply the detection and imputation procedures to the calibration set, thereby constructing exchangeable features for the conformal prediction interval of the test label. We develop two practical algorithms, $\texttt{PDI-CP}$ and $\texttt{JDI-CP}$, and provide a distribution-free coverage analysis under some commonly used detection and imputation procedures. Notably, $\texttt{JDI-CP}$ achieves a finite sample $1-2\alpha$ coverage guarantee. Numerical experiments on both synthetic and real datasets demonstrate that our proposed algorithms exhibit robust coverage properties and comparable efficiency to the oracle baseline.

Conformal Prediction with Cellwise Outliers: A Detect-then-Impute Approach
Qian Peng' Yajie Bao' Haojie Ren 2 Zhaojun Wang' Changliang Zou'
Abstract
prediction models in risk-sensitive applications hinges on valid uncertainty quantification. Conformal prediction (CP, Vovk et al. (2005)) offers a flexible and robust frame- work for uncertainty quantification of arbitrary machine learning models by constructing prediction interval (PI). To be specific, suppose we have collected an i.idd la- beled dataset Di = {(X,, Yi) C Rd x R where (X,, Yi) P = Px Prix. Given the test data Xn+1. CP issues PI CC (Xn+1) satisfying the marginal cover- age P(Y,+1 E COP (X,+1)) 2 1 a: if the test data and labeled data are exchangeable. However, the distribution of some variables in test data may deviate from that of the training data due to measurement error, natural variation, or feature contamination. This issue, referred to as cell- wise outliers (Alqallaf et al., 2009), can result in inaccurate predictions and flawed decisions. For example, consider a hospital that seeks use the patient's biomarkers pre- dict incidence rate based on a model trained on data from recovered patients. For newly admitted patient, some of whose biomarkers may differ from those in the training set. Conducting robust uncertainty quantification for machine learning models in the presence of cellwise outliers remains an underexplored area.
Conformal prediction is a powerful tool for con- structing prediction intervals for black-box mod- els. providing a finite sample coverage guarantee for exchangeable data. However, this exchange- ability is compromised when some entries of the test feature are contaminated, such as in the case of cellwise outliers. To address this issue, this paper introduces a novel framework called detect- then-impute conformal prediction. This frame- work first employs an outlier detection procedure on the test feature and then utilizes an imputation method to fill in those cells identified outliers. To quantify the uncertainty in the processed test feature, we adaptively apply the detection and im- putation procedures to the calibration set, thereby constructing exchangeable features for the con- formal prediction interval of the test label. We develop two practical algorithms, PDI-CP and JDI-CP, and provide a distribution-ffee coverage analysis under some commonly used detection and imputation procedures. Notably, JD JDP-CP achieves a finite sample - 2a coverage guar- antee. Numerical experiments on both synthetic and real datasets demonstrate that our proposed algorithms exhibit robust coverage properties and comparable efficiency the oracle baseline.
In this case, observe contaminated test feature Xn+1 with entries
Xntiss je [d] O
Xntis
(I)
Znttij j jEO O²
1. Introduction
where O C [d] denotes coordinates of cellwise outliers in X and Zn41 ~ Pz is arbitrarily distributed outliers and independent of {(4,, Y)) Our goal is to build PI for test label Y,+1 with a target coverage level 1 when only Xx+1 is observed. However, the classical CP method is incapable of providing a valid PI as test data (X,-1. Yn+1) is not exchangeable with the labeled data Dr. Furthermore, the distribution shift from the labeled data to the test data is unknown and difficult to estimate without ad- ditional distribution structure assumptions, which makes the weighted conformal prediction (WCP) method (Tibshi- rani et al., 2019) that requires consistent likelihood ratio estimation unsuitable in this context as well. Therefore, a more principled CP approach is needed to address cellwise outliers test data.
As the volume and complexity of data expand, machine learning models and algorithms have become essential tools for enhancing dicisioonmaking processes across var- ious domains, including autonomous driving (Min et al., 2024), wing design (Zhang et al., 2024), and disease di- agnosis (Shang et al., 2024). Ensuring the reliability of
'School of Statistics and Data Sciences, LPMC, KLMDASR and LEBPS. Nankai University, Tianjin, China Schooo of Mathe- matical Sciences. Shanghai Jiao Tong University, Shanghai, China. Correspondence Yajie Bao <yajicbao@@ mankai.cdu.cm Hao- jie Ren <haojicren @ @jtu.edu.cn>..
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Conformal Prediction with Cellwise Outliers
To deal with cellwise outliers, our framework starts from the common approach, detect then impute (Raymaekers & Rousseeuw, 2021; 2024b). We implement detection method to identify the locations of outliers in Xn+1 and mask those entries, followed by using an imputation method to impute the masked entries. After this procedure, we obtain the processed feature X (see Section 3.1 for de- tails). The next step is to construct the PI using the labeled data Dj and X- However, X and X2 for i E [n] remain nonexchangeable because the detection and imputa- tion procedures cannot exactly recover the uncontaminated feature Xn+1- To tackle this challenge, our key strategy involves adaptively applying detection and imputation pro- cedures to the labeled features X; for E [m], aiming to produce exchangeable copies of the processed test feature X Subsequently, we utilize the split conformal predic- tion (SCP) method (Papadopoulos et al., 2002; Vovk al., 2005) to construct the Pl based on the residuals computed from the processed labeled data. Building on this concept, we first propose an oracle detection innttttttitttttiionn conformal prediction (ODI-CP) method and prove that it can achieve a finite sample coverage property. Following this, we develop a proxy detection-imputatinn CP (PDI-CP) method based on the data-driven approximation to CDI-CP. To further have finite sample coverage guarantee, we propose an- other algorithm joint iitection-imputationn CP (JD (JDI-CP) to construct Jackknife+ type PI (Barber et al., 2021).
GMM method to detect cellwise outliers in heterogeneous populations. Inspired by these detection methods, Liu et al. (2022) proposed the BiRD algorithm to control the false discovery rate in detection. Besides detection, there are various cellwise robust methods available for cases includ- ing discriminant analysis (Aerts & Wilms, 2017), principal component analysis (Hubert et al., 2019), regression models (Öllerer et al., 2016; Filzmoser et al.. 2020), cluster analy- sis (Garefa- Preeca--scuderr et al., 2021), location and covariance estimators (Raymaekers & Rousseeuw, 2024a) and so on. To the best of our knowledge, this paper is the first work considering the predictive inference problem with cellwise outliers.
Conformal prediction without exchangeability. Our paper is related to the line of work on the CP approaches catered to nonexchangeable data. Chernozhukov et al. (2018) ex- tended CP to time series data based on block-permutation. Tibshirani et al. (2019) proposed the weighted conformal prediction (WCP) framework to deal with the covariate shift between labeled data and test data. Further, Yang et al. (2024) developed doubly robust approach to construct Pis satisfying approximate marginal coverage under covariate shift. Podkopaev & Ramdas (2021) studied the conformal PI under the label shift case. For arbitrary distribution shift issues, Barber et al. (2023) developed a robust weighted CP method with deterministic weights that could provide ap- proximately marginal coverage under gradual changes in the data distribution. Cauchois et al. (2024) studied construct- ing distributional robust PI for the test data sampled from a distribution in f-divergence ball around the population of labeled data. This paper focuses discussing another special case of nonexchangeability caused by cellwise out- liers, which cannot be addressed by the methods proposed so far.
Our contributions have three folds: (1) we propose novel CP framework to efficiently address the issue of cellwise outliers in test data, which be wrapped around any well-known detection and imputation procedure; (2) for arbitrarily cellwise outliers, we establish coverage error bounds for PDI-CP and finite sample 1 - 2a coverage property for JDI-CP, and all results are distribution-fies (3) experiments on synthetic data and real data show that the proposed method has a robust coverage control and a comparable performance with oracle approaches.
Predictive inference with missing data. Another topic closely related to our paper is the construction of PI in the presence of missing values in features. The entries identified as cellwise outliers by the detection procedure are masked, transforming the primary problem into predictive inference with missing data. Zaffran et al. (2023; 2024) examined a setting where missing values may occur in both labeled and test features, proposing to perform SCP after imputing the missing entries in both sets. This approach could achieve finite-sample marginal coverage if the random missing patterns in the labeled and test sets exchangeabl.. However, this assumption does not hold in our context, as the masked entries identified by the detection procedure lack exchangeability due to cellwise outliers in the test feature. In addition, Lee et al. (2024) proposed the propensity score discretized conformal prediction (pro-CP) to construct Pls with missing values in outcomes. This is different from the scenario considered in this paper.
2. Related work
Cellwise outliers. In the context of handling extensive datasets, it is increasingly common for only a few individual entries (cells) to show anomalies, called cellwise outliers, first published by Alqallaf et al. (2009). Building upon this concept, Van Aelst et al. (2011) proposed the earliest cell- wise detection method and Roussceuw & Bossche (2018) pi- oneered the Detect Deviating Cells (DDC) algorithm, which predicts the value for each cell and identifies outliers by the significant deviations of predicted values from the original values. Thereafter, Raymackers & Roussceuw (2021) pro- posed the cellHandler method, which integrates all relations in variables to identify complex adversarial cellwise outliers. Along this line, Zaccaria et al. (2024) proposed the cell-
2

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Conformal Prediction with Cellwise Outliers
Conformal inference for outlier detection. In recent years, conformal inference has emerged as an important tool in the field of outlier detection at the intersection of statistics and machine learning. Due to the space limit, we list some relevant works (Guan& & Tibshirani, 2022; Zhang et al., 2022; Bates et al., 2023b; Marandon et al., 2024; Bashari et al., 2024; Liang et al., 2024). However, the above works aim to apply conformal inference to test the presence of outliers, while this paper focuses on the uncertainty quantification of prediction and the construction of Pl when some entries of the observed test feature are contaminated.
(2001)) and Multivariate Imputation by Chained Equations (MICE, Van Buuren et al. (1999)).
Hereafter, we abbreviate the detection and imputation pro- cedure as DI, and denote the processed test feature as
X XDI = (Xn+1, On+1 = D(X,+1). (2)
3.2. Oracle detection-imputation
Because DI cannot exactly recover the ground truth feature Xa+1, i.e., R+1 # Xn+1- Therefore, we must consider the recovery uncertainty from DI when constructing the PI. We begin with an oracle procedure by assuming O is known and D satisfies the following sure detection assump- tion. Later we will discuss the necessity of this assumption for informative coverage in Section 3.3.
3. Problem setup and oracle approach
In this section, we present problem formulation and moti- vation of our method from an oracle perspective. Before that, we list some useful notations. We split the labeled data D, into the training set 'D2- = {(X,, (()) yeT and the cali- bration set D. = {(X,, Y;) iimgg where 1 < no < n. The prediction model ja()) : Rd + R is fitted on the training set. Given real numbers {ra}m_ and a E 1), we de- note 95 (fri)==)) and it (Er))=z) as the [a(n+1)] and [(1 - (n + 1h smallest value in {r:) respectively. Given x E IRd, we write the l-th coordinate of as Zz.
Assumption 3.1 (Sure detection). We call the detection pro- cedure D has the sure detection property, if all the cellwise outliers are detected, namely O C E = D(X++)) for test point Xn+1
Assumption 3.2 (Isolated detection). Given the score func- tion 8, and threshold Ty. the detection indicator 1 (2) S Ty } depends only on I for j E [d].
Denote Õnt1 (X) as the detection subset by apply- ing D to the uncontaminated feature X which cannot be obtained in practice. Assumption 3.2 says that the de- tection indicator of each coordinate is independent of each other, which is used to build the connection between Ön+1 and Õn+1, and we will also provide coverage results when this assumption does not hold in Section 5. Next lemma shows that the randomness of Ön+1 completely originates from Ön+1-
3.1. Detection and imputation procedures
This subsection introduces the detection (D) and imputation (I) procedures to clean the test feature for constructing PI.
A generic detection procedure (D) includes two ingredi- ents: cellwise score function (y):): IR- + R and threshold T IR for j E [d]. which are both fitted on the training set D., Given any E Rdd we write the output of detec- tion procedure as D(x)= = [j E [d] 8y(25) > } For example, Z-score (Curtis et al., 2016) is a detection method based on standardization, which has score 2j := (x,)) = (,) where Hi and oy are the mean and standard deviation of the j-th coordinate. Typically, the entry xy is considered as an outlier when [zyl > 3. Other more phisticated detection methods include DDC (Rousseeuw & Bossche, 2018), one-class SVM classifier (Bates et al., 2023a), etc.
Lemma 3.3. Under Assumptions 3.1 and 3.2, it holds that Ou+1 =On+1 O and
XDI n+1 = O0
(3)
Denote Ô, D(X,) for i .o..... n as outputs by ap- plying D in calibration features. Since s,(-) and Tj are independent of calibration set D. and (Xx+1, Yn-1) we know that (Ō, U O² D are exchangeable. Define the oracle delection-impuuattonn (ODI) features
We define the imputation procedure (I) as a function of feature I E IE and entry subset O _ [d], then the imputed value for coordinate j O is given by
(4) By comparing (3) and (4), we have X = XPr_ Let R = |Y, x(X;)) ) be residuals computed by ODI features. We introduce the PI defined as
x = I(X, O(XX,) i = no
n n +1. 1.
[xx, Oi1 = Tjj
1
((rihe))) j jeo'
where d5)(): Rdabe - IR with dais = d0 is imputation function for the j-th coordinate, and is also fitted on the training set D, The output of most popular imputation methods can be formalized in the above form, such as Mean Imputation, k-Nearest Neighbour (kNN, Troyanskaya et al.
CODI (xn+1) = ((X +
(5)
and call this oracle method as ODI-CP. For simplicity, we use the absolute residual score to construct PI here, but
3

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Conformal Prediction with Cellwise Outliers
our method supports various non-conformity scores as well, such as normalized residual (Lei et al., 2018), conformalized quantile regression (CQR) (Romano et al., 2019), et al. The following proposition provides finite sample coverage for O-P.
4.1. Proxy detection-imputation
Since O is impossible to know in practice, the best proxy we have access to is the output On+1 of detection procedure D. By replacing O' with Ox+1 in (4), we have the prexy detection- imputation (PDI) features
Proposition 3.4. Suppose the detection procedure D satis- fies Assumptions 3.1 and 3.2, then
X: = I [(,,, Ó. U i ,., n.
(7)
Denote the residuals in calibration set as {R. == |V; ((X.))) no and we construct the PI for Ya+1 as:
P (Yota CoDi (Xnte)) 2 21- al-a a.
(6)
( (PDI(Xn+1) = ((X = Ta [R
(8)
The role of data-driven mask O in ODI features (4) to keep the exchangeability with the processed test feature (3) masked by On+1- This idea is similar to the operations the calibration set in selective conformal prediction litera- ture (Bao et al., 2024b;a; Jin & Ren, 2025), where they aim to construct PI for selected test data and consider perform- ing a similar selection procedure on the calibration set to satisfy the post-selection exchangeable condition.
Algorithm CP
Input: Calibration set ((X,, Vi))=nn test feature Xa+1,
prediction model fi, detection procedure D, imputation procedure I, miscoverage level a.
1: On+1 D(X,+1)
2: X I(X 1( On+1)
3: for i -Ro...., do
3.3. Necessity of sure detection property
4: Ö, + D(X;)
5:
X, + I(X,, O, UŌ
The next theorem gives a negative result for the case when the sure detection property of D does not hold.
6:
Ra + IY; - -x;-)(X,)
7: end for
Theorem 3.5. If X contains cellwise outliers, given any PI taking the form C(XDI)) = ((XDL1) + In and satisfying marginal coverage P( E ((xD+))) 2 1
CPPI (PPt((n+1) + p(X + qd ({R:))=no))
Output: (PDI(Xn+H)
a. For arbitrary M > O, there exists prediction model û. distributions P and Py such that for (Xn+1. Yu+1) ~ P
We call the construction stated in (8) as PDI-CP, and sum- marize its implementation in Algorithm Next, we analyze the coverage property of PDI-CP.
and Zn+1 ~ Pz. P(gn 2 M) 2 1 - a. In other words,
limm--e E [((XPL = cc.
Definition 4.1 -eesssitiiviy) The -i-ssiiityy of predic-
tion model j is sup) x->1) [A(x) - ((x²)) S Sja.
This theorem indicates that if outliers persist within the pro- cessed feature 1 the further PI will fail to provide meaningful coverage guarantee in distribution-fee and model-free fashion. Therefore, the sure detection property introduced in Assumption 3.1 is reasonable and necessary for predictive inference tasks with the existence of cellwise outliers. The proof of Theorem 3.5 and a similar negative re- sult under CQR score are deferred to Appendix C.2. In fact, Assumption 3.1 can be satisfied in practice if we choose relatively smaller detection thresholds (T, )= in D proce- dure. In fact, similar sure detection condition also appears in Wasserman & Roeder (2009); Meinshausen et al. (2009); Liu et al. (2022).
The ii-eniittvvtyy measures the stability of a prediction model to its input, which is similar to that in differential pri- vacy literature (Dwork et al., 2014). For example, for a lin- ear regression model p(x)) 8 + b, whose fi-seniitivity is given by Sp = [|,8]]1 Definition 4.2 (Mean Imputation). The imputation function is (,(+) = Σ = T Eme X1; for j E [d]. i.e, the cellwise sample mean obtained from the training set.
The following theorem provides the coverage property of PDI-CP under the Mean Imputation rule.
Theorem 4.3. Suppose the detection procedure D satisfies Assumptions 3.1 and 3.2, and ja has the E--sensitivity Sp. in Definition 4.1. Let E, := maxjeld [Xaj Zzl. i
4. Detect-then-imuute conformal prediction
Ro...., ........ n.n + 1 be error of Mean Imputation in Definition 4.2. We have
In this section, we develop two practical algorithms named PDI-CP and JDI-CP to construct the PI, and the corre- sponding coverage property is analyzed. The PDI-CP is based on a proxy of the ODI features in the previous sec- tion, and the JDI-CP is a Jackknife+ type construction to guarantee finite sample coverage.
P f Yn+1 E (PDI 2 21-a
FFR (at (R) + +Sp.E, E,, S5, \O'1)
- FFR (at (IR - Sp E, 10 \ O01I0= )

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Conformal Prediction with Cellwise Outliers
where Fg- is the distribution function of ODI-CP residuals (R;) (
Algorithm 2 JDI-CP
Input: Same as Algorithm 1.
1: On+1 D(X++1)
According to the representation, the coverage gap of PDI-CP depends on the error of Mean Imputation and the number of false discoveries 1rnt1 \ O*, which repre- sents the number of inliers incorrectly identified as outliers by the detection procedure D. In an ideal scenario where On+1 = O³, PDI-CP can achieve the finite sample cover- age as ODI-CP. The proof for Theorem 4.3 is deferred to Appendix C.3.
2: for i -o..-.., do
3:
O + D(x;)
4: X t I[X, O O+)1)
5: X I[(x+1,, O Un+1))
R f 1 (1--(Xxtt))
end for
8: Qa t7 (1A(X Rí )=n
9: Ôt -
qo ((ū(X 1+++)= + Rí R
10: (JDI(X,+1 x -1Q 10,22ā1
4.2. Joint detectioniimputation
Output: (JDI(Xx+1)
The coverage gaps of PDI-CP are caused by the nonex- changeability between PDI feature X; and processed test feature XD This subsection proposes a robust approach to constructing exchangeable features by applying a mod- ified DI procedure on test feature Xn-1, which achieves 1 2a finite sample coverage in the worst case.
to the intrinsic limit of Jackknife+ type method (Barber et al., 2021), the finite sample result in Theorem 4.4 is still important in practical tasks for two reasons: (1) Ön+1 may contain many false discoveries which leads to large coverage gap for PDI-CP; (2) the coverage property of JDI-CP holds for arbitrary imputation rules. In Appendix B.2, we provide an alternative version of JDI-CP, which achieves a finite sample the same conditions in Theorem 4.4. However, it is quite conservative compared with the original version because it uses Ui=.. Ó U On+1 1 to mask calibration features and test feature. In our experiments, JDI-CP has a robust coverage control without adjusting the quantile level in (9).
We begin with a careful inspection of pairwise exchange- ability in test feature and calibration feature after DI. Un- der Lemma 3.3, we can rewrite PDI feature (7) as X; = I(X,, O, UUUGEE Ón+1 U O*). Since Ó, and Õut1 are ex- changeable, it is easy to verify that I(X,, Ô, 10
coverage guarantee under
and I(Xn+1, O On+1 U UO)) are also exchangeable. Using Lemma 3.3 again, the latter one is identical to I(X 1,, O₂ U Ón+1). In light of this observation, have (JDI) fea-
the following pairwise joint tures: for i no. : 11
We notice that Jackknife+ technique is also used in CP- MDA-Nested algorithm in Zaffran et al. (2023; 2024) to build PI with missing values in features, and similar 1 1- 2a coverage property is proved. However, the algorithmic de- signs of CP-MDA-Nestedd and JDI-CP are very different. The former uses the observed and exchangeable missing pat- terns to perform nested masking for each pair of calibration and test features, while JDI-CP uses outputs of D (i.e., O and O.+1) 1), which are not exchangeable.
X 1(x,+1, Un4H)), x = I(X,, U,U
Notice that, X is the same as the PDI feature (7), and we use new notation here to emphasize the pairwise relation.
With the pairwise JDI features, we leverage the Jackknife+ technique proposed by Barber et al. (2021) to construct the PI for Y
To end this section, we use Figure to illustrate the connec- tion and difference of the proposed methods.
(JDI(Xnt)) (1A(X Rs ) to ((0(( x+1) + Rr p 1
5. Coverage analysis beyond isolated detection
where Ri = IY - as the residuals in (8). The method presented above is called JDI-CP, and we summarize the implementation in Algorithm 2.
is the same
The coverage results in the previous section are based on the isolated detection in Assumption 3.2, now we discuss the theoretical properties of our method when the detection score relies on other coordinates, e.g., DDC detection pro- cedure (Roussccuw & Bossche, 2018). Now we rewrite the detection score (z,,, = (x,;; where æ-j is the subvector of by dropping Ty. Denote
Theorem 4.4. Suppose the detection procedure D satisfies Assumptions 3.1 and 3.2, then
P PfYut E ('DI(( 2 1 2o.
T = (j E [d] \ 0- 3) Snt1, > T
The proof of Theorem 4.4 is given in Appendix C.4. Even though JDI-CP cannot achieve the target level a due
Jns = (j E [d] \ O" =8 Xn+1,-z) > T.).
5

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Conformal Prediction with Cellwise Outliers
Calibration feature
Test feature
Pl type
Coverage rate
ODI-CP
xi = 1,,
X =1((n+1,
SCP
Proxy mask calibration features
PDI-CP
X, 1(X, O; Ūn+1)
X (Xx+110 mn+1)
SCP
- a a
Joint mask in test feature
JDI-CP
x I(X, Ox 0 U v
Xn1 1(xx11, U Ō
Jackknife+
- 1
Figure 1. An illustration of the relationship of ODI-CP, PDI-CP PDI-CP
and JDI-CP. Ó; = D(x)) and Õn+1 = D(8=+1).
The two sets above represent the false discoveries of D applied on contaminated feature Xx+1 and clean feature Xn+1 except for j E O", respectively. Without Assumption 3.2, the ODI features {X; and X are no longer exchangeable, and the coverage guarantee of ODI-CP and JDI-CP will be broken. The next theorem characterizes the coverage gaps of our algorithms through Tn+1 and Tn+1. Theorem 5.1. Suppose the detection procedure D satisfies Assumptions 3.1, and ji has the l fe-seivity Sp in Defini- tion 4.1. Under the Mean Imputation in Definition 4.2, (1) PDI-D- CP satisfies that
We use the DDC method as example to demonstrate the relation between T and Tx+1 (see Appendix D for details of DDC method). A commonly used similarity metric of two sets is Jaccard similarity (Murphy, 1996):
Jaccard(Tn+ TT [ N TT+1ll
E [0, 1]
(T U
with a value closer indicating Tn+1 and Jn- are more similar. Figure 2 visualizes Jaccard Tu+1) and the number of occurrences of Tn+1 Tn+ 0 during 500 trials, each with 100 test points. It can be seen that T1 and Tx+1 are almost the same except in a few trials. In addition, the frequency of Tn+1 Ta+1 =
Yat1 E CCPI 21-2)21--1 21-a
reaches above 0.6.
PR- (1t ((R)) Fi + Ant1) Fn- ((R))mm)))
FR To ((R) + Spa E. 1T-TT+22)
Jaccard
Empty
Not Empty
0.980 0.982 0.994 0.966 0.988 1.000
80 100
-FR fat [R] Sp Ei (Ta+ )=mm (10)
Jaccard Similarity
Number of empty
where An-1 Sp. [Tn+1 L TT+1l 1|- En+1 and A denotes the symmetric difference.
60
40
(2) JDI satisfies that
P PfYYti Yoti E C (XPt((n+1)) 2 -2x
- [[r( + 24n -F
100
200
300
400
500
- -[FY(QT)) Fy -Fv(Q.)FFv(Q; -24n1)) 21n+1)
Figure 2. illustration of the similarity of T and To (left axis, red cross marks) and the number of occurrences of empty and nonempty sets (right axis, blue and gray dots).
where Fy is the distribution function Y, Q7 and Q are defined Algorithm 2 where X7+*1 and X a+1 are replaced by I(X,, O, U O U fn+1 and I(X U. f respectively.
6. Simulation
The proof of Theorem 5.1 is given in Appendix C.5. If the sets Ta+1 and Tn+1 are infinitely close, then the error Ap+1 + ( and (JDD((X.4) still has 1 2x coverage guarantee. Further, if [T.+1l = |Tx+1| + 0, it means the detection method can accurately find the cellwise out- liers without false discoveries, then (PDI (Xn,+) ) can also achieve the target a coverage.
We write N o²) for the normal distribution with mean i and variance o² SN o² a.) for the skewed normal with skewness parameter cx, L(ke) for the l-distribution with k degrees of freedom, and Bern(p) for the Bernoulli distri- bution with success probability p. Given any & E Rd define f(z) = E(Y|X, = z) and 7) =Y, f(X.). We consider three data generation settings Lei et al. (2018):
6

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Conformal Prediction with Cellwise Outliers
Setting A (linear, homoscedastic, light-tailed))/(())
SCP Baseline t PDI
2, (X1))=1 i.i.d. N(0, 1); (O. 1 X2. Setting B (nonlinear, homoscedastic, heavy-tailed): f(z) is an additive function of B-splines of 21,..., Zdd
WCP
t ODI
[ JDI
1.00
Coverage
(Xi))]=1 i.dd. N(0, 1); 7 1(2); m X,,
Setting C (linear, correlated features, heteroskedas- tic, heavy-tailed); f(x) = 3 x; {Xuly=1 s.d.d
1), + (0, + (ern()..). and then given autocorrelation by redefining sequen- tially each Xis to be a convex combination of Xij, Xis-1,.... Xx.U-3)A11 standard deviation 1 + 2 (x.))* / /E (I)()))))
1(2), with variable
Length
Let be the probability that cell in the test feature is contaminated. Then we obtain the test feature with entries
5.0
X Xn+1j> with probability 1-c
Xntis=
(11)
Znti,s
with probability
Figure 3. Simulation results of six methods. D is DDC, I is Mean
where the noise Zn+1. s.i.d. N(, o) and μ,, a are randomly sampled from (1, 10). All simulation results the fol- lowing are averages over 200 trials with 200 labeled data and 100 test data. The nominal coverage level is set to be 1- 90% and d 15. For comparison, also conduct following methods (see Appendix 1.1 for specific forms):
Imputation, 0.1 and coverage level 1 a 90%.
/ /xi.0.95 The red line is the target
DDC. Here consider two procedures: the one-class SVM classifier method (Bates et al., 2023a) with тj 0.2 and the celIMCD estimate method (Raymaekers & Rousseeuw, 2024a) with = -Xxi,0.99 where Tj determined to control the FDR (false discovery rate). The former satisfies Assumption 3.2 while the latter does not. According to Figure 4, method still has the coverage guarantee when employing other detection procedures. The empirical TPR (true positive rate) and FDR of detection methods are given in Appendix E.2.
1. Basel line mask the entries of {X,]==m Xx+1 by O', then compute residuals on the calibration set and construct SCP interval, which can be seen as an optimal oracle approach to construct split conformal PI in the cellwise outlier case;
2. SCP: directly compute residuals on the calibration set and contaminated test feature, then construct SCP in- terval without performing DI;
3. WCP: first estimate the likelihood ratio between labeled data and test data using random forests approach, then construct WCP interval.
6.2. Combinations with other imputation methods
Besides Mean Imputation, we also conduct experiments under the other two imputation methods: k-Nearest Neigh- bour (kNN, Troyanskaya et (2001)) and Multivariate Imputation by Chained Equations (MICE, Van Buuren et al. (1999)). Figure 5 shows that if we replace Mean Imputation with kNN or MICE in DI, then our method is still able to achieve target 1 a coverage.
Figure presents the empirical coverage and length of six methods. As expected, SCP and WCP fail reach the target coverage level. Notably, when the contamination probability is high, e.g. 0.2, WCP even outputs PI with infinite width. Meanwhile, our proposed method can achieve 1 a target coverage and return PIs with almost the same lengths as Basel 1 ine method. Since SCP and WCP cannot pro- vide valid coverage control, we regard Baseline as the benchmark for the subsequent experiments.
6.3. Performance under different contaminated ratios
From (11),
the probability that at least cell of X is contaminated by Zn+1 is 1-11-1e6 which grows very quickly
and d increase. For example, in our setting d d= 15, c = 0.05 suffices to have over 46% of contaminated test data on average, while = 0.2 can achieve over 97% of contaminated test data. Here we explore the effect of
6.1. Combinations with other detection methods
This experiment is to verify the validity of our methods under other plausible cellwise detection methods besides

---

Conformal Prediction with Cellwise Outliers
PO1
Bascline (0.1
0.95
Coverage
Longth
Coverage
3.00
Length
10
0.00
0.80
(a) One-class SVM
(a) KNN
Banelne l PDI
Basekine DDI c.J01
0.83
Coverage
Longih
0.80
0.83
0.86
0.80
0.50
(b) celIMCD
(b) MICE
Figure Simulation results Basel ne and our methods with different D procedures. is Mean Imputation and 0.1.
Figure 5. Simulation results Base inne and our methods with different I procedures. D is DDC, 0.1 and v X1.0.95.
Beseline O2I e FDI g= 0.15
contamination levels our method. We set D DDC,
=
0.1
==
1.00
and the parameter p detection threshold Tj = v xt.p as (0.95, 0.93, 0.9) corresponding to 6 E 0.11, 0.2) to
Coverage
0.35
control FDR. According Figure our method can achieve very tight coverage control under each contamination level. Other supplementary simulation results refer to Appendix E.3.
0.80
Length
15.0
6.4. Performance under different detection thresholds
We conduct this experiment by varying the DDC detection threshold / Xř.p (adjusting p). As the threshold decreases, the number discoveries increases, leading to a higher FDR and higher TPR. Table shows that our method has ro- bust coverage performance when the threshold changes and achieve approximate control of coverage when TPR<1.
Figure 6. Simulation results of Basel i ne and methods when E (0.1, 0.15, D is DDC and is Mean Imputation.
CQR to construct Pls. We compare the Basel ine with method under Setting A with different contaminated ratios. Figure shows that the Pīs constructed by our method with CQR score can still approximately achieve the target coverage rate.
Table 1. Empirical TPR and FDR of DDC with different thresholds under Setting A when 0.1.
p
0.99
0.9
0.7
0.5
Baseline ODI E PDI (3)
1.00
TPR
0.987
0.992
0.995
1
FDR
0,035
0,340
0.669
0.793
Coverage
Length
PDI coverage
0.902
0.901
0.907
0.909
0.50
IDI coverage
0.895
0.899
0.904
0.904
0.85
0.80
0.1
0.15
0.1
0.15
0.2
6.5. Performance with CQR score
To demonstrate that our method can be combined with other non conforrfryy scores, we conduct an experiment using
Figure 7. Simulation results Baselin and methods when Pl is constructed by CQR. D is DDC and is Mcan Imputation.
8

---

Conformal Prediction with Cellwise Outliers
7. Application real data
0.6
7.1. Airfoil data
Coverage
0:00
Lengih
We apply the proposed method to the airfoil dataset from the UCI Machine Learning Repository (Dua & Graff, 2019),
where the response Y and covariates X (with 5 dimensions) described in Appendix E.4. We select 1000 labeled data and 500 test data in 100 trials. Since it is unknown which cells are outliers in reality, we artificially introduce outliers with = 0.02 to construct test features with both genuine and artificial cellwise outliers. The details of the experiment are presented Appendix E.4. Figure shows that our method can achieve 1 a coverage while SCP and WCP fail to reach the target coverage.
SGP
Pb.
Figure 10. Experimental results on riboflavin dataset. D is DDC and is Mean Imputation.
found in Appendix E.6. Figure 10 shows that methods maintain coverage above the target -- 90% while WCP fails to provide meaningful Pls, that is, the PIs constructed by WCP may have infinite length.
Coverage
Lenghh
8. Conclusion
0.66
This paper proposes
setect-then-immute conformal prediction framework to address the cellwise outliers in the test feature. We develop two efficient algorithms PDI-C2 and JDI-CP to construct prediction intervals, which can be wrapped around arbitrary mainstream detection and impu- tation procedures. In particular, JDI-CP achieves a finite sample 2a coverage guarantee under the sure detection property of the detection procedure. We conduct extensive experiments to illustrate the robustness and efficiency of our algorithms in both synthetic data and real applications.
0.00
SGP
WCP Baeelin
PP1
JDI
SGP
WCP Gasseine
051
JDI
Figure 8. Experimental results on airfoil dataset. D is DDC and
is Mcan Imputation.
7.2. Wind direction data
Another example involves the hourly wind direction data from meteorological station in the Central-West West region
of Brazil (https:/tempo.inmeg) www.l//mmpo.imm..gooo.. www.bb/Tabcaassssosss//AA00
A Acknowledgements
In 100 trials, randomly select 1000 labeled data and 500 test data, and add artificial outliers with 0.02 to test features. The variable list and data processing steps are presented in Appendix E.5. From Figure 9, we conclude that our method outperforms SCP and WCP actual cases where cellwise outliers are present in the test feature.
We would like to thank the anonymous reviewers and area chair for their helpful comments. Changliang Zou was supported by the National Key R&D Program of China (Grant Nos. 2022YFA1003800 202 1003703), and the National Natural Science Foundation of China (Grant No. 12231011). Haojie Ren was supported by the National Key R&D Program of China (Grant No. (012200)))2200,,, and the National Natural Science Foundation of China (Grant No. 1247 1262). Young Elite Scientists Sponsorship Pro- gram by CAST and Shanghai Jiao Tong University 2030 Initiative. Yajie Bao was supported by the Postdoctoral Fel- lowship Program of CPSF (Grant No. GZC20251996 and the fellowship from CPSF (Grant No. 2025M773046).
0.80
Coverage
0.66
Lenghh
0.80
SCP WCP Baseline GEI
P31
JD1
SCP
WCP
Baseline
OC1
PDI J0
Figure 9. Experimental results on wind direction dataset. D is DDC and I is Mean Imputation.
Impact Statement
This paper presents the work to advance the field of confor- mal prediction, which promotes the reliability of machine learning. The tools introduced in this study can be utilized across a wide range of machine learning applications, includ- ing those with societal implications, but they do not directly implicate any specific ethical or impact-related concerns, so there is nothing we feel must specifically highlighted.
7.3. Riboflavin data
To further demonstrate robustness, we test our method on the gene expression dataset for riboflavin production provided by DSM (Kaiseraugst, Switzerland), which was offered by Bühlmann Mandozzi (2014) and confirmed to have cellwise outliers by Liu et al. (2022). The details can be

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Conformal Prediction with Cellwise Outliers
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Conformal Prediction with Cellwise Outliers
A. Notation table
Table 2. The mathematical notations in this paper.
Notation
Meaning
Comments
X; Y Xa+1 Zatz
The ith instance
Data model
The label corresponding to X;
Data model
The test feature with cellwise outliers The instance of outliers with arbitrary distribution
Data model
Data model
X atl X a
The test feature processed by DI The jth coordinate of the ith instance The vector x (æ1, 22, ...., ..., x22" IRd
Data model
Data model
Data model
The æth coordinate of
Data model
X;
The ODI feature
Data model
X;
The PDI feature
Data model
The JDI feature
Data model
X
The JDI feature which is the same as the PDI feature X,
Data model
R
The residual computed by ODI features The residual computed by PDI features
Algorithm output
R4
Algorithm output
Di
The labeled dataset
Set
D.
The training dataset
Set
D.
The calibration dataset
Set
O Ó
The coordinates of cellwise outliers in X
Set
The detected coordinates by applying D X; The detected coordinates by applying D to Xn+1
Set
Ou+1
Set
T
The false discoveries of D applied on X except for j E O The false discoveries of D applied on X,+1 except for j E O
Set
T
Set
The prediction model
Function
The [a(n + 1)] quantile function
Function
(
The [[1 (n + 1)] quantile function
Function
The cellwise score function of jth coordinate
Function
2)()
The imputation function of jth coordinate
Function
((x)
The output of detection procedure
Function
I(x,
The output of imputation procedure
Function
Fr
The distribution function of ODI-CP residuals {1R Jimno
Function
The critical point of split
Algorithm parameter
a
The target coverage level
Algorithm parameter
Tj
The detection threshold of jth coordinate
Algorithm parameter
N(,,
The normal distribution with mean and variance o² The skewed normal with skewness parameter a The I-distribution with & degrees of freedom The Bernoulli distribution with success probability
Distribution model
SN(H, ((k) Bern(p)
Distribution model
Distribution model
Distribution model
B. More methods to construct PI
B.1. Naive combination of DI and CP
consider method by naively combining the DI procedure with split conformal prediction, which abbreviated as Naive -e.ve-DI. The processed features of Naive-DI are defined as
X Naive-ul I(X,, On-1), no, ...
12

---

Conformal Prediction with Cellwise Outliers
and the Pl for n=
Maliiee DI (Xn+1) = ji( Xllvvie SI) d2 (()) M M(X DI) D
(B.1)
However, Naive-DI constructed in this way fails to achieve 1 - coverage in several cases. As an example, we consider the coverage of PIs constructed by CDI-CP, PDI-CP and Naive-DI under Setting A when e = {0.1, 0.15, 0.2},
T - -/x,,.. and Z that when the gap between Zu+1 and the clean data is not very large, and the detection threshold Tj is small enough detect more outliers, the Na i ve-DI method fails 1 coverage regardless of the level of contaminated probability. In comparison, method has a better coverage in these cases, which is due the consideration of the effect of {O, ) when masking. This empirical evidence confirms that simply combining DI with CP (without our proposed modifications) cannot guarantee valid coverage.
i.i.d. ~ N(,, a) where are randomly sampled from 0.1.. From Figure 11, it can be seen
ODI-CP E PDI-CP
Naive-DI
1.00
4.5
0.95
Coverage
Length
4.0
0.90
3.5
0.85
0.80
3.0
0.1
0.15
0.2
0.15
0.2
Figure 11. Simulation results of ODI-CC, and Naive-DI under Setting A when e (0.1,0.15,0.2) 0.2). is DDC, I is Mean
Imputation.
B.2. An alternative version of JDI
we take into account the cellwise outliers contained in all calibration features, we can provide the following conservative version of JDI, whose features and are listed below.
Conservative-JDI (C-JDI) features:
X-CJDII = I(X, O
i == no. ......;..
t=
and the PI for Y,+1:
(C-IDI (C-IDI(X M ((CCPII) + ot (LIY - A(XC-IDI )
(B.2)
We introduce Theorem B.I to demonstrate the coverage guarantee of C-JDI. Theorem B.l. Suppose the detection procedure D satisfies Assumptions 3.1 and 3.2, then
PP (YYt E &C-JDI (X 2 a
Proof. Notice that, Lemma 3.3 shows that Ö = Ó.+1 U O², which implies that U δ,U0- - - Ô U Given the unordered set of yn+1 the joint mask Un+1 Ô, O fixed. Hence we know (XC-JDIaat) J= are
exchangeable. Using the standard proof of split conformal prediction (Lei et al., 2018; Tibshirani et al., 2019), we can prove the result.
13

---

Conformal Prediction with Cellwise Outliers
We present an example for comparison of PDI-CP, JDI-CP, and C-JDI in three settings, where 0.05, Tj and Z the those in Appendix B.1. From Figure 12, which is consistent with simulation results. Although the C-JDI method can also achieve the target coverage, its PI is wider and coverage rate is looser than PDI-CP and JDI-CP, indicating that the PIs constructed by C-JDI are more conservative. Taking into account both Appendix B.1 and B.2, our method outperforms Na i ve-Di I and C-JDI, which able to build tight PIs that satisfy the target coverage.
that method can achieve the target coverage rate,
PDI-CP
JDI-CP
C-JDI
1.00
20
0.95
Coverage
Length
0.90
10
0.85
0.80
5-
C
B
Setting
Setting
Figure 12. Simulation results of PDI, JDI and C-JDI. D is DDC, Is Mean Imputation and 0.05.
C. Technical Proofs
In this section, we fix the training set {(x,, Te and only consider the randomness from calibration set {(x,, Y:) 1=ng and test data (1,+1, 11 1.+1).
C.1. Proof of Lemma 3.3 and Proposition 3.4
Proof. Firstly, we prove Lemma 3.3. Notice that for any E [d] \ O*, we have X = Xat15- It follows that
Õu+1 = O" { O" 18 > T/}
o Or :0 (X > Tj}
-j' (,(Xn+15) > U [d] \ \0 :85(Xn+1,) Ty}
O² O''Uân+1
where the first equality holds due to Assumption 3.1; and the second equality holds due to Assumption 3.2. From the definition, we know
Xutl.ji ((( ( [[ O On+1U0")),
i O UO
I( 10')
j E Ô n+ O
Thus we have
X I(X,+1, On41) I(X 0 1Ix
(C.I)
which is exchangeable X; I[X,, for no. ...... n.
Next, we prove Proposition 3.4. Recalling that R2- |Y: A (Xi) for no..... n. Denote R YY41 ((XPL)
By the exchangeability of (R:) ) im we can guarantee
n+1
P (Y CODI (x01(X P R S dt
(C.2)
14

---

Conformal Prediction with Cellwise Outliers
C.2. Proof of Theorem 3.5
Proof. We consider the case where d 2. Suppose is a two-dimensional standard normal random variable with observations, the corresponding labels are Y; X i X where X,,,, Uniform 1]) for i E [n]. Suppose
j a linear regression model (xx) 31 + f222 where 32 # The test point Xn+1 (X=+1,1 Xa+1,2) where Xn+1,2 = M+1
B2 1[3 2 1] + M-8,+2 1 < 1 for some large positive value M.
32
Now we consider the case X still contains cellwise outlier after DI, that is D(Xn+1) 1 and X Xa+1- When
31 > 1, notice that
Ya+1 - 10H1-A(XPH)) A(X [11 31) Xn+12 - M 1 M.
(C.3)
By the form of Pl, it holds that
1 - a = P( E C(XP4)) = P[ - ( $.) S (gn 2 M).
When 31 notice that
[Y++1 - ((Xx+t)) = 1(1-)/))) + Xx+112 MM 2 811 2 M,
and (C.3) still holds.
Theorem C.1. If X contains cellwise outliers, given any PI taking the form -
- ( Qn. fuP (up(Xpt)) x+++)++
dn) where flo and fup are the lower and upper quantile regression models, and ân the quantile of the empirical distribution of CQR score computed on the calibration set. Suppose C(XPI) 1+ satisfies marginal coverage ((Yn+1 CC(DD))) 2 1-a.
For arbitrary M > 0, there exists fio and fuP, and distributions P and Pz such that for (Xut1, Ynt1) ~ P and 7n+1 ~ Pz, P(în 2 M) 2 - a.. In other words, lim [0(XDL)) oc.
Proof. Following the proof of Theorem 3.5 in Appendix B.2, the labels are Y; = Xi,1 + X1,2 where X,1,X12 Uniform([0, 1]) for i [n + 1]. Suppose fl)(z) 81 + 3/222 where 8 + O, and the test point X
where the outlier is given by
M+1
M+2
M
8 +
Zat12 =
1(3 21
1 100 < g < 11+
1 { < 0}
g
for some large positive value M. If D-DI n+1
still contains Z and CXPPI m+1) covers the true label, we have
max (X R - Yo+1, Yu+1 - jup (X foa (X Yat1 2 M,
which means P( M) PP E 2 1--
C.3. Proof of Theorem 4.3
".,..., Raa the Mean Imputation values of X, for Proof. According to Definition 4.2, we denote by C. = (.......,......
i ... , Denote
= 11)(())))) TH i= no...., E [d],
(C.4)
and
Å diag( i = no. .o.....14 1,
(C.5)
A diag(( diag(fôn41.)ee)) ))))))
(C.6)
A Ap-1 diag(ton+1.bbeld)
15

---

Conformal Prediction with Cellwise Outliers
where 8nt1.- 1 ((,((X)+15) S Tyy and On+1s = 1(j [d] O*). Then the difference between CDT feature and PDI
feature be written
X;
x A,Án+1 + - A,
|a: (1- Á,(Á - 1 Ái An+1)a
+ (1- A,n
= Ant)) - -
(C.7)
Denote by Dig = j - أyا the Mean Imputation error of each entry for = no. .... n. Invoking & es-senittiiity of p. we
have
R2 R&] = - -- -( IV, x(X;)1l
- - S Sp 1 |x - Xil1
Spp Σ 8 5et1aa [x F.3]
= Sp Σ
(5nt61 - 8mt1,s) Eii
jE(d](O
S Sp max E.g
Σ
(8n+1. - 8nt1aa)
%Eld]
jE(d).o
= Sp. E, 1O \0%
(C.8)
where E, == ma Big and the third equality holds due to Assumption 3.1.
By the construction of ČODI ( in (5), we have
P (Yat E (X {|1n+1 10Y+111(XX7))) S at ((
=P =P (Ri-1 S Va (nn:)
(C.9)
and the coverage gap between CODI(En+1) and (PDI(Xn+1) is
p Yn+1 E (XPi(X)+)) PP (Xn+1 E (iD((()+))
{ S att (( - P PR S 1
P {min (ot (f dt ((RT))=no)) < R2+ S max (( 9t (1 ((
S P da (IR - Sp E, 1nn-1 \0 10 ) Rh+1 St (IR) + Sp Es 1On+1 1 0 11))=_m)))) }
= FR- (at [IR] + Sp. E. 10 \ (01)%= ))
FR- gtt (IR) - Sa E, 1 \ O*1H=no))
(C.10)
where FR- is the distribution function of ODI-CP resiuuals 3.4, we can obtain the coverage property of PDI-CP
{ RR Ji Combined with Assumption 3.2 and Proposition
n1
IP { Yn+t1 E (PPI((n+1))
Fn- (at (RR; + Sp E, -1041 \0'))=)))) Fr- (at (IR - Sp E. E..00 )O'lB=n))) []] (C.11)
C.4. Proof of Proposition 4.4
Proof. We follow the proof in Barber et al. (2021) to show the result. We abbreviate the set of integers (no...., n) as (no, na] in this subsection. Notice that under Assumptions 3.1 and 3.2, (X:+1, Y.) U (X Yn41) are exchangeable. For
i, j [no, n + 1] with * j. denote X = I(X,, G U*)). Then we assert that ((X)) # ((X)) for i i j.
16

---

Conformal Prediction with Cellwise Outliers
Define matrix of residuals, D [R( ( with entries
+00,
p= q
Dm (1ptto-1 p ((
(C.12)
p
for p.q + 2 no]. i... the off-diagonal entries represent the residual for the (p + no 1)-th point which processed by DI with mask Op U U 9 U
Define comparison matrix, A E {0, ((( with entries
Apg = 1(Dpa > Dyph,
and App 1 means that the residual of Xp+no-1 is larger than that of Xg+ which processed with the same mask Optno-1 U UO, UOg-no-1 O and we say that data point (p + no - 1) "wins" (q + no 1).
Define set S(A) C [n + nal of strange points
< S(A) p p + 2-nol:
Σ
Apg 2 (1 a) + 2 na)
(C.13)
i.., p E S(A) if it holds that, when we compare the residual Dpu of the (p no 1) point against residual Dap of the (g 1o - 1)-th point, the residual Dm is the larger one for a sufficiently high fraction of these comparisons. Note that each strange point E S(A) can "lose" against at most (n + 2 - mo) - other strange points, this is because point (p + 72 - 1) must "win" against least (1 an + no) points in total because it is strange, and point (p + no 1) cannot win against itself as the definition of Dpq-
Denote by (A)] the number of strange points. The key realization is now that, if we think about grouping each pair of strange points by the losing point, then we see that there are at most
8 (a(n + no) - 1)
pairs of strange points. So we have
8(8 1) S s (a(nn + 2- 2n)) 1),
(C.14)
which simplifies to S 2n(n + 2- - < 2n(n + no)
Since the data points {(X/, (.)) x.) are exchangeable and the fitting algorithm is not affected by DI, it follows that A d IAAIIT for any (n + no) (n + no) permutation matrix II. In particular, for any index q E [n + 2 no). suppose we take II to be any permutation matrix with II 22 = (i.e., corresponding to permutation mapping + to + no 1), then we have
n + 2 no E S(A) - q E S(TATT)
and therefore
(n + 2 S(A)) = =P(a S(TTATT)) P(g P S(A))
In other words, we compare an arbitrary calibration point g + no where q E In + nu] versus the test point m 1, these two points are equally likely be strange, by the exchangeability of the data. So we calculate
1 2
Pfn + 2 - no E E(A)) =
1
Σ
Plg E S(A)}
+ 2 D10
T=1
E 2 1[g E S(A)}
E[S(A))]
< 2o,
(C.15)
n 2 Ro
n + ng
n2 2 no
according to (C.14). By the construction of JJDI (ODI(X(+)) (8n+1) in (9), we have the following equivalence relation
Yat1 & (DD((xn+1) + > at (fū(X + +R: )=))))
17

---

Conformal Prediction with Cellwise Outliers
or Yn+1 < Pa (fu(En+t) - -R, [Ro)==o))
(C.16)
In either case, we have
+22 no) -
Σ 1fY++1 & ((Xx+))) + Ri}
t=
Σ 1( - ((x+tt) < Vu+1 ixx n(*++))
1=0
a+1-11 no
n+1
1(Dgat2-n < D
A
(C.17)
and therefore n + 2-2nn 2ng E ((A) according to (C.13). Combined with (C.15), we have
P[Ya+1 & (. (Xx+(())) $ P(n + no E S(A) I 20.
(C.18)
C.5. Proof of Theorem 5.1
C.5.1. PROOF OF PDT-CP
Proof. Under Assumption 3.1, denote
XD1 I(xx+1, O¹ U Tnt)),
(C.19)
Xi+1 I((x+11, 0¹ U Tn+1).
(C.20)
The difference between X DI n and X is
X
X
(AT (A A n+1 1 + (AT An+1 1n=))
= A (A a+1 - ( - -mn-1),
(C.21)
where
ÄT R+1 = diag diag (1TT-1))) = diag({1() $ Fr+i Hhet))
AT
diag([1(j & In+1)
#+1
and Øn+1 = (21...., ....... Ta)
T is the Mean Imputation value of Xx+1-
Denote Rn+1 = Yn+1 ((X
1), and R = |Y +(Xi++)) is residual computed by ODI feature Xit1- Invoking
l1 ei-eeniitvv of ft we have
R
R S Sp |X - x
= Sp Es 18at1s - 87 51 |X - Fzy
Sp Ejeld 18 1 ST 8 [xnt1g Zz]
S Sp ATTTIAAIATTTO Ent1
= Ant1,
(C.22)
where En+1 == maxjeld [Xn+1J - F, is the same the detinition in Theorem 4.3. Notice that X is exchangeable with ODI features {Xxx yi under Assumption 3.1, then the coverage gap of ODI-CP
p (Ytts E (OD((En+1)) (1 xx)
18

---

Conformal Prediction with Cellwise Outliers
|P Rn+1 0o ((R))en (1 a1-
S PP Ra+1 Aut1 S la (RR) }} IP { S ( )
Fn- (4% (1 (mng)) ) + Ant1) - Fi- (0 (1 12 )i=
(C.23)
where FR- is the distribution function of ODI-CP residuals { Similar to the proof of Theorem 4.3, obtain the coverage of PDI-CP:
P (r (PDI(Xn+1)) 21-a a- (Fii- (6t (1 Ii li-no) Ant1) - FR- (4. ( li=ng?)]
FFr d (IR) + Sp Fij F.g.ITn+t13 11117++11i==m)) FR- (at (R Sp. F. T 1 ))
C.5.2. PROOF OF JDI-CP
Proof. Under Assumption 3.1, denote
X = I(X,, (Tat), i= no......: n2
(C.24)
X n+1 I(X 0,, Ufa41), i= n0: nt.
(C.25)
According to Proposition 4.4, we have
p P(Y E (UDi(Xn+1)] 21 -20,
(C.26)
where
CJDI(AAH = [T ((A(X; - - --xxxxx*+++)))) ) da (fa(( + |V, ((x:+))) )
(C.27)
Note that if Assumption 3.2 is satisfied, X and X equal to X,+1 and X Based on this, characterize the differences between features and obtain the coverage property of CJDI (Xnt())+))) beyond Assumption 3.2. First, notice that the difference between X x and X X
X n+1
X n+1
Á,A - (T+1) + A A
= (AT at1 AT+1) Int1).
(C.28)
and the difference between X and X is
X X AT - + A A4+1);
-x;- 21).
(C.29)
Invoking l--sensitivity of R. we have
|A(* +1) 2(x;+ )l s Saa 1 - Xi+1ll
=Sp E 18ntiol 18 87 82t1sl 1.1111 - 3zl
=Sp - 18 8T Shunol 1 Zzl
SSa (Tnt1 .En+1
(C.30)
and
|| )(X++)) 1 A(X+++))| < SA TT Anail Enat.
(C.31)
19

---

Conformal Prediction with Cellwise Outliers
Notice that the upper bound in (C.30) and (C.31) is exactly An-1 in (C.22). Combining with (C.26), we can obtain the coverage gap:
P Yo+i (X P Ynti E C (X
IP (1 n+1) [Y, A(X++))) )i Yt1 S 90 (1 +Y;
-P (fA(()
Y A(X7 M|
Y21 S % (fp( + IY ((x,+))))
|Pa ((1(X)+1) - Ant1 (|Y; -xx;+
+ A )i=no) < Y4t1 S
(1ū(X x+1) +n+ (|V: i((;+t))
+ An+1) }=
-TP (P (L ((i(X x+1) -Y,, (( ) < Yn1 S 30 (LA + 14 ((x
H
|P -2n < - 0: +2n++1
P P(O nn S Q:
FQx + +an+1) - -Q Fy (Qa) - -Fr - -2nn+1),
(C.32)
where
Q_ = do (1A(Xx++)) - -V -x(xxx++))
(C.33)
Q+ (TA((( -
(C.34)
D. DDC Method
The DDC method is the most widely used method to detect cellwise outliers. We display the process of DDC in Figure 13, and the details of the robust functions (highlighted orange) can be found in Rousseeuw Bossche (2018).
Input
Output
x
standerdization
Z
univariate
U
① bivariate relations
flagging outliers
21
M11
2sd
|ryll >
Xa1
Znd
Myt
Mad
estimated
2
2
values
For each column of Xy:
① carn = robCorr, (Mgaam)
byn = robSlope, (uyl un)
When |a:l > t:
For each column / of X
m, robLoc; (Xty)
11 [h: coT Ia 0.5]
8j robScale, (xiy -mmm)mm))
NA
② ž G (b Uey H,) II
Zzy-21 roboicale Ciyy -4)
(xy mm)/8,
By 2, robSlope; (z 12/))
Figure 13. implementation of DDC detection method.
E. Additional experiment results
E.1. Three benchmark methods in simulation
Now we present the specific construction forms of the three methods used for comparison in Section
Baselines mask the entries of {X,) U Xx by O", and obtain the processed features
x Base
I(X;, 0*), i = .10...., . I(8x+1,0)) O²) I(Xat1, 0*),
X Base
n+1
20

---

Conformal Prediction with Cellwise Outliers
then construct SCP interval
Basse (Xn+1)= = jx(X ( + d ((1% ( Base M)
SCP: directly compute residuals R₂ = 1 -
(1yy-((X,)) fi(X;)) on the calibration set, then construct SCP interval
CSCP (Xn+1) (=( + dot (1
(E.1)
WC2: consider the weights of residuals
(Xi)
pf (a:)
no....;
n= w (X() + w(x)
w(x)
P Ph+1(1) (x) Clono w(Xi) w(x)
where the likelihood ratio w(x) dP, ( with X,+1 Px can be estimated by Random Forests (more generally, any classifier that outputs estimated probabilities of class membership) as mentioned in Tibshirani al. (2019), then construct WCP interval
" EDr (X + P (X
OWCP (Xn+1) (((x+11) = dt
(E.2)
1=
E.2. TPR and FDR for detection methods In Table 3-5, summarize the empirical TPR and FDR obtained through the application of the detection methods in the simulation.
Table 3. Empirical TPR and FDR of DDC under three seltings when 0.15, 0.1), where the noise Zu+1,5 is 10.
TPR
FDR
Setting
0.2
0.15
0.1
0.2
0.15
0.082
0.113
0.179
B
1
0.067
0.108
0.232
C
1
1
0.080
0.116
0.182
Table Empirical TPR and FDR of DDC under three settings when c {0.2, 0.15, 0.13. randomly sampled from (000 10).
where the noise Znti, s.d N(,, 0), u.o
TPR
FDR
Setting
0.2
0.15
0.1
0.2
0.15
0.1
A
0.991
0.990
0.990
0.084
0.115
0.186
0.990
0.990
0.990
0.078
0.116
0.236
C
0.990
0.990
0.990
0.085
0.116
0.182
Table Empirical TPR and FDR of one-class SVM classifier and cellMCD estimate methods under three settings when 0.1, where the noise Zn+1i
S.i.d. N(u, oi) and μ,, a are randomly sampled from (0, 10).
TPR
FDR
Setting
SVM
MCD
SVM
MCD
A
0.988
0.990
0.026
0.173
0.988
0.989
0.023
0.172
0.988
0.989
0.027
0.175
21

---

Conformal Prediction with Cellwise Outliers
E.3. Supplementary simulation results for 6.3
In addition to Mean Imputation, we present the empirical coverage and length of Baseline and our method under different contamination probabilities when I is kNN or MICE in Figure 14 and 15, respectively. The parameters the same those in 6.3.
Baseline @ ODI @ PDI JDI
0.1
= 0.15
= 0.2
1.00
0.95
Coverage
0.90
0.85
0.80
20.0
Length
10.0
5.0
A
Figure 14. Simulation results of Basel ine and our methods when E (0.1, 0.2]. D is DDC and I ANN.
Baseline @ ODI a PDI JDI
0.1
= 0.15
c 0.2
1.00
0.95
Coverage
0.90
0.85
0.80
20.0
15.0
Length
10.0
5.0
Figure 15, Simulation results Basel ine and our methods when E (0.1, 0.1,0.0,5,02) D is DDC and is MICE.
22

---

Conformal Prediction with Cellwise Outliers
E.4. Supplement of airfoil dataset study
Table Available variables airfoil dataset.
Variable
Unit dB
Scaled sound pressure level of NASA airfoils (target)
Frequency
Hz
Attack-angle
deg
Chord-lenghee
m
Free-stream--elocity
m/s
muettoo-sieedisslleeeeettttthhii
m
Creating training data, test data, and covariate shift: We repeated an experiment for 200 trials, and for each trial we randomly partition the data {(X,, H=) into two equally sized subsets D, and De, and construct test set Drest containing cellwise outliers with the following steps.
Log the first and fifth features of X across all the data.
According to the contamination probability e = 0.02, randomly select some entries to assign outlier 50.
Use DI to handle covariates, where Tj is set to / Vx,,0.95 to control FDR.
The set O used in Baseline and ODI-CP is the coordinates of those artificially added noises.
E.5. Supplement of wind direction dataset study
Table 7. Available variables in the wind direction dataset.
Variable
Unit
Wind direction (target)
rad
Cosine of wind direction in the previous hour Sine of wind direction in the previous hour Atmospheric pressure in the previous hour Air temperature (dry bulb) in the previous hour
dimensionless
dimensionless
mB
CC
Dew point temperature in the previous hour Relative humidity in the previous hour Wind gust in the previous hour Wind speed in the previous hour
*C
%
m/s
m/s
Additional details are referred to E.4, but the test set Drest is constructed with the following steps.
According the contamination probability e 0.02, randomly select some entries to assign outlier 100.
Use DI to handle covariates, where Tj is set to X2,0095 to control FDR.
The set O* used in Basel lane and ODI-CP is the coordinates of those artificially added noises.
E.6. Supplement of riboflavin dataset study
Table 8, Available variables in the riboflavin dataset.
Variable
Comments
Logarithm of riboflavin production rate
target
log-transformed gene expression levels
p = 4088 (co)variables
23

---

Conformal Prediction with Cellwise Outliers
Additional details are referred to E.4 without introducing artificial outliers.
Use DI to handle covariates, where 7j is set to V Vxi,0.95 to control FDR.
The set O* is unknown, so the experiments of Baseline and ODI-CP are not conducted.
There may be cellwise outliers in Dj and De.
24


Reviews:
[{"review_id": "Enmpl7PudP", "reviewer": "Reviewer_vdr2", "summary": "When some entries of the test features are contaminated, the paper introduces a detect-then-impute conformal prediction framework. This framework first applies an outlier detection procedure to identify contaminated entries in the test features and then uses an imputation method to fill in the identified outliers. Moreover, the authors apply the detection and imputation procedures to the calibration set, ensuring the construction of exchangeable features for the conformal prediction interval of the test label. Two practical algorithms including PDI-CP and JDI-CP are proposed, with lower bounds on marginal coverage probability established under certain conditions. Numerical experiments on both synthetic and real datasets are provided to demonstrate the performance of the proposed algorithms.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">**Q1**: The validity of Assumption 3.1 (sure detection) depends on quality of detection, which may be restrictive in practice.\n\nThanks for your valuable question! We acknowledge the imperfection of Assumption 3.1 and make the following explanations.\n- Sure detection/screening conditions are commonly used in existing works (Wasserman&Roeder,2009; Liu et al.,2022) to ensure all relevant variables are retained. We adopt a similar condition in Assumption 3.1 for model-free coverage guarantee.\n- Assumption 3.1 is essential for meaningful prediction intervals (PIs): if outliers persist in $\\check{X}\\_{n+1}^{\\rm{DI}}$, Theorem 3.5 shows PIs can become infinitely wide in expectation.\n- Violating Assumption 3.1 impacts our method by introducing a total variation coverage gap: if outliers persist in the test point $\\tilde{X}\\_{n+1}$ after detection, Lemma 3.3 fails due to the exchangeability between $(\\check{X}\\_{n+1}^{\\rm{DI}},Y_{n+1})$ and $\\\\{(\\check{X}\\_i^*,Y_{i})\\\\}\\_{i=n_0}^n$ is broken, where $\\\\{\\check{X}\\_i^*\\\\}\\_{i=n_0}^n$ are the ODI features. At this point, there is a total variation coverage gap of the PI:$$\\mathbb{P}\\\\{Y_{n+1}\\in\\hat{C}^{\\rm{ODI}}(\\tilde{X}\\_{n+1})\\\\}=\\mathbb{P}\\left\\\\{|\\hat{\\mu}(\\check{X}\\_{n+1}^{\\rm{DI}})-Y_{n+1}|\\le\\hat{q}\\_{\\alpha}^+(\\\\{R_i^*\\\\}\\_{i=n_0}^n)\\right\\\\}\\ge 1-\\alpha-\\frac{1}{n-n_0+1}\\sum_{i=n_0}^n d_{\\rm{TV}}(\\check{X}\\_{n+1}^{\\rm{DI}},\\check{X}\\_i^*).$$The impact is mild if $d_{\\rm{TV}}(\\check{X}_{n+1}^{\\rm{DI}}, \\check{X}_i^*)$ is small; similar bounds apply to PDI-CP and JDI-CP.\n- In practice, Assumption 3.1 (TPR=1) can be satisfied when the detection threshold is small; our method still maintains target coverage in simulations and real examples even if some outliers were not completely detected. For your convenience, we also added a new experiment under different DDC detection thresholds $\\sqrt{\\chi\\_{1,p}^{2}}$ (varying $p$) based on Setting C in Appendix D.2, which shows Assumption 3.1 is exactly satisfied (TPR=1) at $p=0.5$. When TPR<1, our method can still achieve approximate control of coverage.\n|$p$|0.5|0.7|0.9|0.99|\n|-|-|-|-|-|\n|TPR|1|0.995|0.992|0.987|\n|FDR|0.793|0.669|0.340|0.035|\n|PDI coverage|0.909|0.907|0.901|0.902|\n|JDI coverage|0.904|0.904|0.899|0.895|\n\n[1] Wasserman, L. and Roeder, K. High dimensional variable selection. The Annals of Statistics, 2009.\n\n[2] Liu, Y., Ren, H., Guo, X., Zhou, Q., and Zou, C. Cellwise outlier detection with false discovery rate control. Canadian Journal of Statistics, 2022.\n>**Q2**: Would the conclusion in Theorem 3.5 hold with other adaptive prediction sets like conformalized quantile regression?\n\nThank you for your question! We prove a similar result for the CQR score, and add new simulation results based on CQR.\n- The PI constructed from CQR is $\\hat{C}(X)=[\\hat{f}^{lo}(X)-\\hat{q}\\_n,\\hat{f}^{up}(X)+\\hat{q}\\_n]$, where $\\hat{f}^{lo}$ and $\\hat{f}^{up}$ are the lower and upper quantile regression models, and $\\hat{q}\\_n$ is the quantile of empirical distribution of CQR computed on the calibration set.\n- Following the proof of Theorem 3.5 in Appendix B.2, $Y_i=X_{i,1}+X_{i,2}$ where $X_{i,1},X_{i,2}\\sim\\rm{Uniform}([0,1])$ for $i\\in[n+1]$. Suppose $\\hat{f}^{lo}(x)=\\beta_1^{lo}x_1+\\beta_2^{lo}x_2$ where $\\beta_2^{lo}\\neq0$, and the test point $\\tilde{X}\\_{n+1}=(X_{n+1,1},Z_{n+1,2})^{\\top}$ where$$Z_{n+1,2}=\\frac{M+1}{\\beta_2^{lo}} \\mathbb{1}\\\\{\\beta_1^{lo}\\geq 1\\\\}+\\frac{M+2}{\\beta_2^{lo}}\\mathbb{1}\\\\{0<\\beta_1^{lo}<1\\\\}+\\frac{M-\\beta_1^{lo}+2}{\\beta_2^{lo}}\\mathbb{1}\\\\{\\beta_1^{lo}\\leq 0\\\\}$$for some large positive value $M$. If $\\check{X}\\_{n+1}^{\\rm{DI}}$ still contains $Z_{n+1,2}$ and $\\hat{C}(\\tilde{X}\\_{n+1})$ covers the true label, we have$$\\max\\\\{\\hat{f}^{lo}(\\check{X}\\_{n+1}^{\\rm{DI}})-Y_{n+1},Y_{n+1}-\\hat{f}^{up}(\\check{X}\\_{n+1}^{\\rm{DI}})\\\\}\\geq\\hat{f}^{lo}(\\check{X}\\_{n+1}^{\\rm{DI}})-Y_{n+1}\\geq M,$$which means $\\mathbb{P}(\\hat{q}\\_n\\geq M)\\geq\\mathbb{P}(Y_{n+1}\\in\\hat{C}(\\tilde{X}\\_{n+1}))\\geq1-\\alpha$.\n- We also conduct an experiment using CQR to construct PIs, where the Baseline used in our simulation can be considered as the optimal method for constructing split conformal PI for cellwise outlier, which masks calibration and test features by $\\cal{O}^*$. This experiment will be added in future revisions.\n||Baseline|ODI|PDI|JDI|\n|-|-|-|-|-|\n|Coverage| 0.905|0.902|0.900|0.885|\n|Length|4.366|4.291|4.282|5.544|\n\n>**Q3**: How does the contamination rate affect the length of the prediction sets?\n\n- Figure 6 in Section 6.3 shows the coverage and length across cell contamination probabilities with DDC and Mean Imputation, while Figures 12-13 in Appendix D.4 display results for kNN and MICE imputation.\n- Results indicate the length of our method remains stable across varying contamination rates, whereas that of Baseline increases with higher contamination. Figure 3 demonstrates our method's competitive length and target coverage, surpassing the classical WCP method.\n\nHope these explanations can ease your doubts!", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "The paper evaluates the proposed method on synthetic data and two real datasets and compares its performance against several relevant baseline methods.", "theoretical_claims": "I checked the theoretical proofs of Lemma 3.3 and Proposition 3.4 and believe they are correct.", "experimental_designs_or_analyses": "I have reviewed all experimental parts in the paper.", "relation_to_broader_scientific_literature": "The paper proposes a framework to construct prediction sets with marginal coverage guarantees when the test input feature is contaminated.", "essential_references_not_discussed": "No critical references appear missing.", "other_strengths_and_weaknesses": "No.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. The validity of Assumption 3.1 appears to depend heavily on the quality of the detection rule, which may be restrictive in practice. In Theorem 3.5, the negative result is derived using a specific form of the prediction set. Would the conclusion change if other adaptive prediction sets, such as those based on conformalized quantile regression (CQR), were used instead?\n\n2. How does the contamination rate in the test features affect the length of the prediction sets?", "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": "xLVIIkR3Ya", "reviewer": "Reviewer_8B4L", "summary": "This paper proposes a DI-CP framework to handle cellwise outliers in conformal prediction. The key idea is first to detect outliers in the test feature vector and then impute them before applying conformal prediction. To maintain exchangeability, a similar detection-imputation process is used to calibration samples. \nThe authors propose two methods: 1, PDI-CP (Proxy Detection-Imputation CP), which applies detection and imputation separately. 2, JDI-CP (Joint Detection-Imputation CP), which modifies detection rules to ensure theoretical coverage guarantees.\nThe paper provides theoretical guarantees, including a finite-sample coverage bound for JDI-CP and empirical results showing that DI-CP performs robustly under contamination.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">**Q1**: Assumption 3.1 is impractical. How does coverage degrade when detection is imperfect?\n\nThank you for your insightful question!\n\n- We acknowledge the imperfection of Assumption 3.1 and have obtained a new coverage gap bound in total variation: if there are still outliers in test point $\\tilde{X}\\_{n+1}$ after the detection procedure, Lemma 3.3 will not hold because the exchangeability between $(\\check{X}\\_{n+1}^{\\rm{DI}},Y_{n+1})$ and $\\\\{(\\check{X}\\_i^*,Y_{i})\\\\}\\_{i=n_0}^n$ is broken, where $\\\\{\\check{X}\\_i^*\\\\}\\_{i=n_0}^n$ are the ODI features. At this point, there is a total variation coverage gap of the prediction interval (PI):$$\\mathbb{P}\\\\{Y_{n+1}\\in\\hat{C}^{\\rm{ODI}}(\\tilde{X}\\_{n+1})\\\\}=\\mathbb{P}\\left\\\\{|\\hat{\\mu}(\\check{X}\\_{n+1}^{\\rm{DI}})-Y_{n+1}|\\le\\hat{q}\\_{\\alpha}^+(\\\\{R_i^*\\\\}\\_{i=n_0}^n)\\right\\\\}\\ge 1-\\alpha-\\frac{1}{n-n_0+1}\\sum_{i=n_0}^n d_{\\rm{TV}}(\\check{X}\\_{n+1}^{\\rm{DI}},\\check{X}\\_i^*).$$Notice that, we can still have a approximate coverage control if $d_{\\mathrm{TV}}(\\check{X}_{n+1}^{\\mathrm{DI}}, \\check{X}_i^*)$ is small. A similar bound can be obtained for PDI-CP and JDI-CP and we will add these results in the revision.\n- In addition, we add a new experiment to show the influence of imperfect detection. Please see **Q6** for details.\n\n>**Q2**: If choosing a larger detection threshold ensures Assumption 3.1, doesn’t this break exchangeability? How does this affect empirical coverage?\n\n- According to the construction, we know that false positives do not break the data exchangeability in ODI-CP and JDI-CP. Since we use $\\tilde{\\cal{O}}\\_{n+1}$ to approximate $\\cal{O}^*$ in PDI-CP, it will break the exchangeability between processed test data and calibration data because $\\tilde{\\cal{O}}\\_{n+1}$ depends only on test data. As we stated in Theorem 4.3, the coverage gap is affected by the number of false discoveries $|\\tilde{\\cal{O}}\\_{n+1}\\setminus\\cal{O}^*|$.\n- In Appendix D.3, we summarized the empirical false discovery rate (FDR) and true positive rate (TPR) of the detection methods in our simulation. We will include this discussion in future revision, using the table from **Q6** as an illustrative example. As the threshold decreases, the number of discoveries increases, leading to a higher FDR. Notably, the table demonstrates the empirical coverage of our method remains robust to variations in FDR.\n\n>**Q3**: Theorem 4.3 and 4.4 assume imputation doesn't change the residual distribution.\n\nWe apologize for any confusion. Here, $F_{R}$ denotes the distribution function of the ODI-CP residuals $\\\\{R_i^*\\\\}\\_{i=n_0}^n$, **not** the residuals from the raw calibration data before the DI procedure. Thus, our method doesn't assume \"imputation preserves the residual distribution.\" We appreciate your attention and will clarify this notation in the revision.\n>**Q4**: Have you tested other imputation strategies? How coverage varies under different imputation and detection methods?\n\n- In addition to Mean Imputation, we evaluated two other imputation methods in Section 6.2: **kNN** and **MICE**. Figure 5 shows our method maintains robust empirical coverage across all imputation strategies.\n- Our analysis in Sections 6.1-6.2 compares coverage and length of PI across various detection and imputation methods. Figures 4-5 show our method consistently maintains robust coverage control across all configurations with stable interval lengths.\n\n>**Q5**: Direct-ODI and Direct-PDI contradict the main method’s justification.\n\nWe apologize for any ambiguity and would like to clarify:\n- Direct-ODI and Direct-PDI in Appendix A.1 are naive combinations of DI with CP, which are **not** our proposed methods.\n- Figure 8 shows Direct-PDI fails to maintain proper coverage while our PDI-CP successfully achieves target coverage. This empirical evidence confirms simply combining DI with CP (without our proposed modifications) cannot guarantee valid coverage.\n\nWe appreciate your careful review and will revise Appendix A.1 for clarity.\n>**Q6**: The paper does not study sensitivity to different detection thresholds and imputation choices.\n\nRegarding the sensitivity of our method: \n1. **Imputation methods**: Figure 5 in Section 6.2 demonstrates our method maintains robust performance across different imputation choices. \n2. **Detection thresholds**: We conduct additional experiments by varying the DDC detection threshold $\\sqrt{\\chi_{1,p}^2}$ (adjusting $p$) based on Setting C in Appendix D.2. As shown in the table, our method demonstrates robust coverage performance when threshold changes.\n|$p$|0.5|0.7|0.9|0.99|\n|-|-|-|-|-|\n|FNR |0|0.005|0.008|0.013|\n|FDR|0.793|0.669|0.340|0.035|\n|PDI coverage|0.909|0.907|0.901|0.902|\n|JDI coverage|0.904|0.904|0.899|0.895|\n\nPlease let us know if you would like any additional details.\n>**Q7**: Could you provide a math notation table?\n\nThanks for your helpful advice. We will add a math notation table to the appendix of the revision.\n\nHope these explanations are acceptable for you!", "claims_and_evidence": "This paper presents a novel and relevant problem in conformal prediction, but the theoretical guarantees depend on overly strong assumptions about detection accuracy. The method may fail if detection is imperfect or if too many inliers are misclassified as outliers.", "methods_and_evaluation_criteria": "There exist two problems: \n\n1. Strong assumptions on detection accuracy (Assumption 3.1 is unrealistic).\nThe method assumes perfect outlier detection. This is impractical because most real-world detection methods have false negatives.\nIf detection is imperfect, theoretical guarantees do not hold, making the results less applicable to real-world data.\n\n2. Excessive false positives break the method.\nThe authors claim that choosing a larger detection threshold guarantees Assumption 3.1. However, this results in too many false positives.\nExcessive imputation may change the calibration distribution, violating exchangeability assumptions (Lemma 3.3 fails).\nThere is no analysis of how misclassified inliers affect conformal validity.\n\n3. Appendix A.1 considers Direct-ODI and Direct-PDI, which do not modify calibration samples. This setting contradicts the main method’s justification that calibration samples must be processed to maintain exchangeability.", "theoretical_claims": "1. Theoretical results assume that imputation does not introduce significant bias. However, imputation methods systematically shift feature distributions, leading to biased conformity scores.", "experimental_designs_or_analyses": "Empirical evaluation lacks robustness analysis.\nThe paper does not study sensitivity to different detection thresholds and imputation choices.", "relation_to_broader_scientific_literature": "This paper reviews the literature of cellwise outliers, conformal prediction without exchangeability, predictive inference with missing data, and conformal inference for outlier detection. It also lists the key difference of settings or tasks for works in the related area.", "essential_references_not_discussed": "I think authors have achieved a fairly complete literature review.", "other_strengths_and_weaknesses": "Strength:\n\nNovel problem setting: The problem of cellwise outliers in conformal prediction is important and underexplored. The paper provides a structured approach to address it.\n\nExchangeability considerations: The idea of applying detection and imputation jointly to calibration and test samples is a novel extension that ensures that conformal prediction remains valid despite outliers.", "other_comments_or_suggestions": "There are a lot of math notations proposed in this paper. Could the author provide the math notation table in the appendix?", "questions_for_authors": "1. The method assumes perfect outlier detection. How does coverage degrade when detection is imperfect? \n\n2. If choosing a larger detection threshold ensures Assumption 3.1, doesn’t this break exchangeability by misclassifying inliers as outliers? How does this affect empirical coverage?\n\n3. Theorem 4.3 and Theorem 4.4 assume that imputation does not change the residual distribution. How do results hold when imputations introduce bias?\n\n4.  How does the method perform when detection is imperfect?\n\n5. Have you tested other imputation strategies?\n\n6. Can you provide a sensitivity study showing how conformal coverage varies under different imputation and detection methods?", "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": "ra5N5dcLYz", "reviewer": "Reviewer_xkXt", "summary": "This paper addresses conformal prediction with feature-wise outliers in the test sample. It assumes access to a detection oracle satisfying the sure detection and isolated detection assumption, and impute the values of the outlier features. After the detection and imputation procedure, split conformal prediction and Jackknife+ are applied, respectively, resulting in two kinds of prediction sets. A distribution-free finite sample coverage guarantee is proved for the latter. Beyond isolated detection, the coverage guarantee worsens with the difference of two detection set, which is empirically shown small. Experiments on synthetic and real datasets with synthetic perturbations show valid coverage and controllable sizes of the prediction set.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">**Q1**:Concerns focus on Assumptions 3.1 and 3.2, assuming detection with FNR=0. Assumption 3.2, crucial for finite length prediction interval, also excludes joint feature space outliers. Figure 2 shows FDR is around 0.4 , indicating unmet assumptions.\n\nThanks for your valuable questions! We acknowledge the imperfection of Assumption 3.1 and explain as follows.\n\n- Assumption 3.1(FNR=0), also used in Liu et al.(2022) and Wasserman&Roeder(2009), can be met by choosing a small detection threshold in practice and is theoretically essential for model-free coverage guarantee. When Assumption 3.1 doesn't hold, we derived a new bound showing the coverage gap depends on the total variation between processed calibration and test data (see Reviewer **vdr2**’s Q1 due to space limit). We also added experiments by varying DDC detection thresholds $\\sqrt{\\chi_{1,p}^{2}}$ (adjusting $p$) based on Setting C in Appendix D.2. When FNR$\\neq$0, our method can still approximately control the coverage.\n|$p$|0.5|0.7|0.9|0.99|\n|-|-|-|-|-|\n|FNR |0|0.005|0.008|0.013|\n|FDR|0.793|0.669|0.340|0.035|\n|PDI coverage|0.909|0.907|0.901|0.902|\n|JDI coverage|0.904|0.904|0.899|0.895|\n\n- We'd like to clarify the primary purpose of Assumption 3.2 is to ensure exchangeability of $\\\\{\\hat{\\cal{O}}\\_i\\cup\\mathcal{O}^*\\\\}\\_{i=n_0}^{n+1}$(see Lemma 3.3), not for finite length prediction intervals (PIs). In Section 5, we also provide theoretical coverage results when Assumption 3.2 doesn’t hold. In simulations, we also used detection methods that unsatisfy Assumption 3.2, such as **DDC** in Figure 3 and **cellMCD** in Figure 4(b), which don't lead to infinitely wide PIs.\n- Actually, Assumption 3.1 requires FNR equal to zero, while the frequency of $\\tilde{\\cal{T}}\\_{n+1}=\\hat{\\cal{T}}\\_{n+1}=\\varnothing$ in Figure 2 means the false discovery (positive) rate (FDR) is zero.\n\n>**Q2**:Why weighted conformal prediction is less competitive?\n\nWCP requires precise likelihood ratio estimation, but arbitrary cellwise outliers in our setting render this impossible. Figures 3 and 11 highlight this limitation, showing WCP fails to meet target coverage.\n>**Q3**:Localized conformal prediction as a special form of WCP is not discussed.\n\nThanks for your advice and we apologize for overlooking these references.\n- LCP controls conditional coverage under i.i.d. test and calibration data, differing from our focus on distribution shift caused by cellwise outliers in test data, which LCP cannot cope with. We’ll add a detailed discussion of LCP and these references in the revision.\n- We also compare our method with baseLCP, calLCP (Guan,2023), and RLCP (Hore&Barber, 2023) through experiments. Clean data follows Section 5.1.1 of Hore&Barber(2023), and cellwise outliers are generated via Equation (10) in our paper with $\\epsilon=0.03$. Results show LCP methods fail to provide informative PIs.\n||Bandwidth|0.1|0.2|0.4|0.8|1.6|\n|-|-|-|-|-|-|-|\n|baseLCP|Coverage|0.90|0.90|0.90|0.90|0.95|\n||Infinite PI%|0.90|0.90|0.90|0.90|0.81|\n|calLCP|Coverage|0.94|0.94|0.94|0.94|0.93|\n||Infinite PI%|0.88|0.88|0.88|0.87|0.66|\n|RLCP|Coverage|0.90|0.90|0.90|0.90|0.90|\n||Infinite PI%|0.90|0.90|0.90|0.86|0.48|\n|PDI|Coverage|0.90|||||\n|| Infinite PI\\%|0|||||\n|JDI|Coverage|0.89|||||\n||Infinite PI\\%|0|||||\n\n>**Q4**:The form of non-conformity score is restricted to absolute deviation.\n\nFor simplicity, we use the absolute residual score, but our method supports various non-conformity scores like CQR score. Please refer to Reviewer **vdr2**'s Q2 for a discussion about CQR.\n>**Q5**:Theorem 4.3's coverage lower bound may be vacuous with large calibration size.\n\nAfter checking our proof, we found the expansion to $\\max_{i= n_0,\\ldots,n}E_i$ is unnecessary and the last inequality in (B.10) (Appendix B.3) should be removed. The modified coverage gap is given by:$$\\mathbb{P}\\left\\\\{\\hat{q}\\_{\\alpha}^+(\\\\{R_i^*-S_{\\hat{\\mu}}\\cdot E_i\\cdot|\\tilde{\\cal{O}}\\_{n+1}\\setminus\\mathcal{O}^*|\\\\}\\_{i=n_0}^n)<R_{n+1}^*\\leq\\hat{q}\\_{\\alpha}^+(\\\\{R_i^*+S_{\\hat{\\mu}}\\cdot E_i\\cdot|\\tilde{\\cal{O}}\\_{n+1}\\setminus\\cal{O}^*|\\\\}\\_{i=n_0}^n)\\right\\\\},$$which will not be vacuous for large calibration size.\n\nThank you for catching this key technical nuance!\n>**Q6**:Dataset with real outlier will be more convincing.\n\nTo further demonstrate robustness, we test our method on a riboflavin gene expression dataset(Liu et al.,2022) with confirmed cellwise outliers. Our method maintain coverage above the target $1-\\alpha=0.9$ while LCP methods ($h=0.1$) fail to provide meaningful PIs.\n||SCP|WCP|baseLCP|calLCP|RLCP|PDI|JDI|\n|-|-|-|-|-|-|-|-|\n|Coverage|0.83|0.85|0.90|0.96|0.90|0.93|0.95|\n|Length|1.82|Inf|Inf|Inf|Inf|3.08|3.29|\n\n>**Q7**:Are other contamination models worth considering?\n\nThe Tukey-Huber Contamination Model generates casewise outliers, assuming most samples follow a target distribution $F$, while the others follow an arbitrary distribution $H$:$$X\\sim(1-\\epsilon)F+\\epsilon H,$$where $\\epsilon\\in[0,1)$ is the contamination ratio.", "claims_and_evidence": "The claim lacks support in L48 which states weighted conformal prediction is unsuitable for CP with outliers because distribution shift cannot be estimated. For example, localized conformal prediction reweigh the calibration samples according to their distance to the test sample, without estimating the distribution shift. The WCP is clearly applicable, since the experiments also report Tibshirani et al.'s WCP method. There could be more discussion why weighted conformal prediction is less competitive in this setting.", "methods_and_evaluation_criteria": "1. The major concern is over the strength of assumption 3.1 and 3.2. They assume oracle access to an outlier detector that has zero false negative rate. And assumption 3.2, which is necessary for finite length prediction sets, additionally excludes outliers in the joint space of features. Figure 2 has shown that the false discovery rate is around 0.4, which indicated that the assumptions are not approximately satisfied.\n\n2. The form of non-conformity score is restricted to absolute deviation.", "theoretical_claims": "For theorem 4.3, the coverage lower bound can be vacuous for large calibration size since max E_i is non decreasing.\n\nI have not checked the proofs.", "experimental_designs_or_analyses": "1. The experiments show that the proposed method is insensitive to different detection and imputation methods, which indicates the flexibility of the design.\n\n2. Results with more recent weighted conformal prediction methods will consolidate the claim, such as localized conformal prediction.\n\n3. Datasets with real outliers will be more convincing. The current results on real datasets introduce artificial perturbations.", "relation_to_broader_scientific_literature": "The most significant contribution of this paper is the first to consider conformal prediction with cellwise outliers, and proposes an algorithm with valid distribution-free coverage guarantee for finite samples (Theorem 4.4).", "essential_references_not_discussed": "Related works on weighted conformal prediction are listed but not fully addressed on why they are not applicable with outliers. Additionally, localized conformal prediction as a special form of WCP is not discussed. They are free of estimation of distribution shift which is claimed in the paper as a reason for incompetence.\n\n[1] Leying Guan. Localized conformal prediction: A generalized inference framework for conformal prediction. Biometrika, 110(1):33–50, 2023.\n\n[2] Rohan Hore and Rina Foygel Barber. Conformal prediction with local weights: randomization enables local guarantees. arXiv preprint arXiv:2310.07850, 2023.", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "NA", "questions_for_authors": "Are there contamination models other than Equation 1 that are worth consideration?", "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": "2ug06tTGRU", "reviewer": "Reviewer_92Px", "summary": "Conformal Prediction (CP) provides prediction intervals with guaranteed coverage for black-box models under exchangeability assumptions. However, cellwise outliers isolated contaminated entries in test features break this exchangeability, leading to unreliable PI. This paper addresses this challenge by introducing a detect-then-Impute conformal prediction framework to robustly handle cellwise outliers.  \nThis paper proposed two novel algorithms of conformal prediction, PDI-CP and JDI-CP, and provided a distribution-free coverage analysis under detection and imputation procedures. This paper establishes coverage error bounds for PDI-CP and JDI-CP achieves a finite sample 1 − 2α coverage guarantee. In section 6 and section 7, author experiments on synthetic and real datasets to demonstrate that proposed algorithms are robust and efficient.\n\n\n=============\n\nI thank the reviewer for the responses which clarified my concerns. \nI have read them and keep my score unchanged.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">**Q1**: Author should highlight the purpose of this paper instead of giving the notation in introduction.\n\nWe sincerely appreciate this constructive critique of our manuscript's organization. In response to your suggestion, we will comprehensively restructure the introduction in the future revision to:\n\n1. **Problem Focus**  \n   - Lead with the critical challenge of cellwise contamination in conformal prediction  \n   - Highlight gaps in existing methods' ability to handle this data corruption  \n\n2. **Solution Framework**  \n   - Clearly articulate our methodological innovations  \n   - Emphasize how we address previously unsolved problems  \n\n3. **Presentation Strategy**  \n   - Defer technical notation to methodology sections  \n   - Maintain narrative flow while preserving rigor  \n\nThese adjustments will prioritize readability and better highlight the scientific significance of our approach. \n>**Q2**: Assumption 3.1 is unrealistic in practice.\n\nWe appreciate your inquiry regarding Assumption 3.1 and we acknowledge its imperfection.\n- We'd like to stress that Assumption 3.1 is theoretically necessary for model-free and distribution-free coverage guarantees and can be satisfied by choosing a small detection threshold in practice.\n- Additionally, we also present a total variation coverage gap of our method when Assumption 3.1 is violated. If the total variation between processed calibration data and test data is small, the coverage can still be approximately controlled. Please refer to Reviewer **vdr2**'s Q1 for details and an additional experiment about Assumption 3.1 (due to the space limit).\n\nWe would be happy to discuss this further if additional clarification would be helpful.\n>**Q3**: Assumption 3.2 may not hold for correlated features.\n\nThanks for your question and we’d like to make the following explanations. For correlated features:\n- Assumption 3.2 can be satisfied if we use **one-class SVM** method to learn the score $s_j$ for each coordinate.\n- In simulations, we also used detection methods do not satisfy Assumption 3.2, such as **DDC** in Figure 3 and **cellMCD** in Figure 4(b).\n-  In Section 5, we provided theoretical results for our method when Assumption 3.2 does not hold.\n\n>**Q4**: A new experiment with setting linear, homoscedastic, light-tailed could be added.\n\nThank you for your advice. In fact, we have included the experimental result of the setting you suggested in Appendix D.2 but didn't mention it in the main text due to the space limitation. We apologize for any possible reading incompleteness and will add this setting to the main text when we have an additional page in the revision.\n>**Q5**: JDI-CP requires O(n) pairwise operations, which could be inefficient for large datasets.\n\nWe appreciate this insightful observation about our method's computational aspects.  \n1. **JDI-CP Design Considerations**:\n   - Primary objective: Finite-sample coverage guarantee\n   - Trade-off: Achieves robustness (demonstrated in Figure 7) at some computational cost\n   - Reference: Similar trade-offs exist in Barber et al. (2021)\n2. **Future Directions**:\n   - Actively researching more efficient implementations\n   - Will optimize computational performance while maintaining theoretical guarantees\n\n[1]Barber R F, Candes E J, Ramdas A, et al. Predictive inference with the jackknife+. The Annals of Statistics, 2021.\n>**Q6**: This paper seems simply combining DI and CP. Author cites many theories which are proved in prior work like theorem 3.5. Focus too much on detection instead of CP.\n\nThanks for your insightful comments on the contribution of our approach. We’d like to provide some clarifications to highlight the contribution of our work.\n- Firstly, our approach fundamentally differs from naive DI+CP combinations. As discussed and shown in Appendix A.1, such a direct combination cannot achieve coverage control. Therefore, we proposed new techniques to adaptively deploy DI on calibration data and proposed PDI-CP and JDI-CP.\n- Regarding Theorem 3.5, this is a negative result we obtained to illustrate the necessity of Assumption 3.1, and we have not found similar results in other papers. Additionally, we have provided a new counterexample under the CQR score, please refer to Reviewer **vdr2**'s Q2.\n- Finally, we believe that detection is essential to cope with cellwise outliers in predictive inference tasks. The classic CP methods under exchangeability assumption have been extensively studied. Here, we are concerned with the nonexchangeable problem caused by cellwise outliers, which is challenging and has not been studied before. The detection and imputation steps are used to identify and remove those outliers, so we can construct informative conformal prediction intervals. The detection is key to constructing exchangeable processed features in ODI-CP and JDI-CP, which rebuilds exchangeability between calibration data and test data and enables coverage control.\n \nHope these explanations are acceptable for you!", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "Theorems are correct under assumptions, but the assumptions are restrictive. Assumption 3.1 is critical for ODI-CP but unrealistic in practice. Assumption 3.2 may not hold for correlated features.", "experimental_designs_or_analyses": "The experiment is reasonable overall, but a new experiment with setting linear, homoscedastic, Light-tailed could be added in simulation.", "relation_to_broader_scientific_literature": "The key contributions of the paper are positioned at the intersection of cellwise outlier detection, conformal prediction, and missing data imputation, building on and extending prior work in these areas.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "**Weakness**\n1. **Computational Cost**: JDI-CP requires O(n) pairwise operations, which could be inefficient for large datasets.\n2. **Logic**: The logic of the article introduction is confusing and should highlight the research issues of the article. Author should highlight the purpose of this paper is using DI method to solve cellwise outlier issues instead of giving the notation in introduction.\n3. **Innovation**: This paper seems simply combining DI and CP, I don’t see any theoretical innovation. Author cites many theories which are proved in prior work like theorem 3.5 and focus too much on detection outlier instead of CP.\n4. **Application**: Assumption 3.1 assume all the cellwise outliers are detected, but it sounds impossible in real world.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "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:

Meta-review: This paper studies a scenario in conformal prediction when some entries of the test features are contaminated, which leads to a violation of the exchangeability assumption that is critical for the correct coverage probability. The proposed approach is first to apply an outlier detection method to identify contaminated entries in the test features and then impute these outliers before applying conformal prediction. 

Two practical algorithms are proposed, with lower bounds on marginal coverage probability established under certain conditions. Numerical experiments on both synthetic and real datasets are provided to demonstrate the performance of the proposed algorithms. A bonus is that the same approach can also be applied to the calibration set. Since the calibration data presumably came from the same source as the training data, it would seem that this approach needs to be extendable to the setting when the training set contains outliers, or realistically, a worse scenario when both the training and calibration data are contaminated. It is important that the authors address this question in the final version. 

The problem is highly relevant but one reviewer thinks the theoretical guarantees depend on overly strong assumptions about detection accuracy. The method may fail if detection is imperfect or if too many inliers are misclassified as outliers.
Decision: Accept (poster)