Quantile Conformal Prediction

• Quantile conformal prediction is a framework that combines conditional quantile estimation with conformal calibration to achieve finite-sample marginal coverage.
• It adapts quantile regression to compute data-dependent prediction intervals, addressing heteroscedasticity and handling complex calibration scenarios such as group and spatial adjustments.
• Advanced variants extend the method to multivariate contexts, privacy-sensitive settings, distribution shifts, and online prediction, ensuring robust uncertainty quantification.

Quantile conformal prediction denotes conformal procedures in which a quantile model, a quantile-regression calibration rule, or a generalized quantile of nonconformity scores determines the prediction set threshold. In its canonical scalar form, it combines conditional quantile estimation with split conformal calibration: the quantile model supplies input-dependent lower and upper bounds, while the conformal step restores finite-sample marginal coverage under exchangeability. Across the literature, the same quantile-based logic is extended to group-conditional calibration, localized spatial and temporal calibration, multivariate score distributions, differentially private thresholding, distribution shift, missing covariates, and online multi-level prediction sets (Romano et al., 2019, Duchi, 28 Feb 2025, Thurin et al., 31 Jan 2025).

1. Core construction and the nested-set formulation

A canonical construction is Conformalized Quantile Regression (CQR). In split conformal form, one fits lower and upper conditional quantile estimators,

$$\hat q_{\alpha/2}(x), \qquad \hat q_{1-\alpha/2}(x),$$

on a training split, then computes calibration scores

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$

If there are $n$ calibration points, the conformal correction is the $\left\lceil (n+1)(1-\alpha)\right\rceil$-th smallest score, and the prediction interval is

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$

This construction preserves the heteroscedastic shape learned by quantile regression and adds a single global slack sufficient for marginal validity (Romano et al., 2019).

The same construction admits a nested-set interpretation. Starting from

$$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$

one defines the induced score

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$

Calibration then selects the empirical $(1-\alpha)$-quantile of the $r_i$’s and returns $C(x)=\mathcal{F}_{Q_{1-\alpha}}(x)$. This formulation makes explicit that quantile conformal prediction is a calibration problem for a nested family of candidate sets, not merely a residual-thresholding heuristic (Gupta et al., 2019).

Several later methods retain this basic architecture but modify the quantile-estimation stage. A non-crossing variant jointly estimates $$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$0 with the ReLU penalty

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$1

then conformalizes via

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$2

The penalty suppresses quantile crossing without changing the population target because the true ordered quantiles already satisfy the constraint (Tang et al., 2022).

2. Validity notions, conditional ambitions, and impossibility results

The fundamental guarantee in standard quantile conformal prediction is marginal validity. Under exchangeability of the calibration points and the test point,

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$3

with no requirement that the quantile regression model be correct, homoscedastic, or well specified (Romano et al., 2019). This guarantee is finite-sample and distribution-free, but it is unconditional over the covariates.

A persistent misconception is that quantile conformal prediction provides exact conditional coverage. The cited works state the opposite: exact distribution-free conditional coverage is impossible in broad nonparametric settings without trivial or infinite-length sets, so standard CQR is best understood as combining exact marginal validity with conditional-like adaptivity inherited from the quantile model (Romano et al., 2019, Duchi, 28 Feb 2025).

Recent work sharpens the intermediate notion of sample-conditional validity. For the split-conformal threshold set

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$4

Vovk’s result is stated as

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$5

The same work studies covariate-dependent thresholds of the form

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$6

where $$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$7 is obtained by held-out quantile regression,

$$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$8

This yields approximate weighted or group-conditional coverage at essentially minimax-optimal rates, up to logarithmic factors (Duchi, 28 Feb 2025).

Group-conditional calibration is an especially concrete instance. In crop and weed classification, Adaptive Prediction Sets (APS) are calibrated separately by group, or equivalently through quantile regression on group indicators. With APS conformity scores $$E_i=\max\Big\{\hat q_{\alpha/2}(X_i)-Y_i,\ Y_i-\hat q_{1-\alpha/2}(X_i),\ 0\Big\}.$$9, the group-specific threshold is modeled by a regression such as

$n$0

This reformulates group-wise conformal calibration as a single quantile-regression problem over metadata-defined strata (Melki et al., 2023).

3. Learning better quantiles inside the conformal pipeline

A major strand of work keeps the conformal wrapper fixed but improves the quantile learner. The non-crossing deep quantile method already noted enforces ordered lower and upper quantiles and derives non-asymptotic excess-risk bounds at the optimal nonparametric rate for the Hölder class, with polynomial rather than exponential dependence on input dimension in the prefactor (Tang et al., 2022).

A second strand targets the calibration objective itself. “Colorful Pinball Conformal Prediction” introduces the Mean Squared Conditional Error,

$n$1

and derives a density-weighted surrogate for quantile regression: $n$2 The weight is estimated by finite differences from auxiliary quantiles,

$n$3

using a three-headed network with monotone Softplus offsets. The conformal stage is then applied to residuals $n$4, producing

$n$5

This directly refines the quantile engine underlying conformal prediction rather than altering the final rank argument (Chen et al., 30 Dec 2025).

A third line modifies the meaning of the quantile target. In calibrated quantile prediction for Growth-at-Risk, conformal prediction is used to estimate a one-sided lower quantile rather than a two-sided interval. With

$n$6

the conformalized lower quantile is

$n$7

and the central guarantee is

$n$8

This turns conformal calibration into a direct tool for quantile estimation, especially in tail-risk settings such as GaR and VaR (Bogani et al., 2024).

An alternative distributional route begins from an estimated conditional CDF rather than separate lower and upper quantiles. Distributional conformal prediction defines PIT ranks

$n$9

or their estimated analogues, and conformalizes $\left\lceil (n+1)(1-\alpha)\right\rceil$0, with baseline score $\left\lceil (n+1)(1-\alpha)\right\rceil$1. Because ranks are Uniform$\left\lceil (n+1)(1-\alpha)\right\rceil$2 and independent of $\left\lceil (n+1)(1-\alpha)\right\rceil$3 under the true conditional CDF, the method targets approximate conditional validity under consistent conditional-distribution estimation and admits a shape-adjusted score

$\left\lceil (n+1)(1-\alpha)\right\rceil$4

for shorter intervals under skewness (Chernozhukov et al., 2019).

4. Localization, structure, and adaptive calibration

Standard CQR uses one global conformal slack, and later work repeatedly identifies this as the main source of residual non-adaptivity. An “improved conformalized quantile regression” method replaces the single global correction with cluster-specific conformal corrections. Features are weighted by permutation importance,

$\left\lceil (n+1)(1-\alpha)\right\rceil$5

clustered by k-means, and each cluster receives its own calibration quantile $\left\lceil (n+1)(1-\alpha)\right\rceil$6, yielding

$\left\lceil (n+1)(1-\alpha)\right\rceil$7

The purpose is to approximate group-balanced or region-specific uncertainty under heteroscedasticity (Sousa et al., 2022).

Density-Calibrated Conformal Quantile Regression (CQR-d) makes the local/global trade-off continuous rather than discrete. It computes the global score quantile $\left\lceil (n+1)(1-\alpha)\right\rceil$8, a local $\left\lceil (n+1)(1-\alpha)\right\rceil$9-nearest-neighbor quantile $$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$0, a density proxy

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$1

and weights

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$2

The combined correction is

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$3

with $$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$4 chosen numerically on the calibration set. The resulting interval,

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$5

has guaranteed marginal coverage at level $$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$6, where $$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$7 is the numerical optimization tolerance (Lu, 2024).

Localized quantile regression also appears in domain-specific forms. Localized Spatial Conformal Prediction (LSCP) fits residual quantiles from neighborhood features,

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$8

and returns

$$C(x)=\Big[\hat q_{\alpha/2}(x)-\hat q,\ \hat q_{1-\alpha/2}(x)+\hat q\Big].$$9

with $$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$0 chosen to minimize interval width. Its theory replaces i.i.d. assumptions with stationarity and spatial mixing and establishes finite-sample bounds on the conditional coverage gap together with asymptotic conditional coverage (Jiang et al., 2024).

Temporal Quantile Adjustment (TQA) alters not the score but the queried calibration quantile over time. It sets

$$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$1

using a history-dependent rank predictor and then constructs the usual conformal interval at the adjusted level. The method targets simultaneous cross-sectional and longitudinal performance: standard conformal is recovered when $$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$2, whereas TQA uses within-series temporal information to widen or narrow the queried quantile (Lin et al., 2022).

A related localized view appears in Conformalized Unconditional Quantile Regression (CUQR), where the output is both a prediction interval and a relevance subgroup $$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$3, and the guarantee is localized to that subgroup: $$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$4 Here the nested candidate intervals are built from recentered influence functions of residual quantiles rather than from conditional quantile regression (Alaa et al., 2023).

5. Multivariate, vector-valued, and generalized quantiles

Quantile conformal prediction is not restricted to scalar scores. Optimal Transport-based Conformal Prediction replaces the usual scalar nonconformity score with a multivariate score $$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$5 and defines vector ranks through Monge–Kantorovich transport. With reference vectors

$$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$6

the discrete assignment

$$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$7

induces vector ranks $$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$8 and the quantile region

$$\mathcal{F}_t(x)=\big[\widehat q_{\alpha/2}(x)-t,\ \widehat q_{1-\alpha/2}(x)+t\big],$$9

The conformal set is then the preimage

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$0

Under exchangeability and $$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$1,

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$2

The method yields flexible, potentially non-convex prediction regions in multi-output regression and multiclass classification (Thurin et al., 31 Jan 2025).

A semiparametric multivariate construction models the joint CDF of vector scores

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$3

through empirical marginals and a nonparametric vine copula,

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$4

The relevant threshold is the generalized multivariate quantile

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$5

and the paper derives an influence-function correction,

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$6

to improve the plug-in estimate of the joint $$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$7 quantile. The resulting conformal set is asymptotically exact rather than finitely exact (Park et al., 2024).

Distributional conformal prediction provides a scalar-output but generalized-quantile perspective. By conformalizing estimated PIT ranks rather than residuals, it produces intervals from conditional distribution models such as quantile regression and distribution regression, and can recover asymmetric “shape-adjusted” intervals of the form

$$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$8

This suggests that quantile conformal prediction includes not only residual-quantile calibration but also calibration of rank-based surrogates for the conditional distribution (Chernozhukov et al., 2019).

6. Privacy, distribution shift, missing data, and online quantile tracking

In differentially private conformal prediction, the calibration quantile itself becomes the protected object. P-COQS computes the split-conformal threshold with a noisy binary search. If $$r(x,y)=\inf\{t:\, y\in \mathcal{F}_t(x)\} =\max\big\{\widehat q_{\alpha/2}(x)-y,\ y-\widehat q_{1-\alpha/2}(x)\big\}.$$9, then the ordinary threshold $(1-\alpha)$0 is replaced by a DP estimate $(1-\alpha)$1 obtained by repeated noisy counts,

$(1-\alpha)$2

The resulting prediction set

$(1-\alpha)$3

satisfies the approximate finite-sample bound

$(1-\alpha)$4

The privacy–coverage trade-off appears explicitly through the quantile rank error $(1-\alpha)$5 (Romanus et al., 15 Jul 2025).

Under distribution shift, Weighted Quantile Loss-scaled Conformal Prediction (WQLCP) modifies the calibration quantile with reconstruction-loss-based weights,

$(1-\alpha)$6

and computes the weighted threshold

$(1-\alpha)$7

This extends score scaling by a VAE reconstruction loss into a weighted quantile-calibration rule intended to reflect a shifted test distribution (Alijani et al., 26 May 2025).

Missing covariates create a different failure mode: standard conformal prediction on imputed data remains marginally valid because symmetric imputation preserves exchangeability, but coverage can vary strongly across missingness patterns. Missing Data Augmentation (MDA) addresses this by recalibrating with artificially re-masked calibration points and targets mask-conditional validity,

$(1-\alpha)$8

In the CQR version, the interval keeps the usual form

$(1-\alpha)$9

but the calibration set is modified to mimic the test mask exactly or through a nested masking rule (Zaffran et al., 2023).

In online settings, quantile conformal prediction becomes a quantile-tracking problem. For a single level,

$r_i$0

while recent work tracks an entire monotone vector $r_i$1 simultaneously so that the prediction sets are nested across coverage levels. The proposed exponentiated-gradient and projected-gradient algorithms enforce $r_i$2, thereby coupling online conformal prediction to non-crossing quantile estimation over the full risk spectrum (Rivera et al., 12 May 2026).

7. Applications and empirical regularities

Quantile conformal prediction has been used both as a methodological template and as an application-specific workflow. In Svalbard temperature forecasting, the pipeline combines quantile gradient boosting at $r_i$3 with adaptive conformal prediction. After forming residuals

$r_i$4

an AR(1) model,

$r_i$5

is used to whiten the residuals, and quantile random forests estimate case-specific lower and upper score quantiles. The final adaptive interval is

$r_i$6

This is explicitly described as very close in spirit to quantile conformal prediction or conformalized quantile regression (Berk, 28 Oct 2025).

In macro-financial risk analysis, conformalized lower quantiles materially improve calibration. The Growth-at-Risk study reports that conformalization improves calibration especially at extremal quantiles and formalizes the lower-tail statement

$r_i$7

which is directly interpretable as a calibrated tail-risk guarantee under exchangeability (Bogani et al., 2024).

In agricultural image classification, group-specific APS thresholds estimated by quantile regression stabilize coverage across environmental strata defined by metadata such as location and sky conditions. The paper contrasts this with marginal APS, whose overall coverage is near target but whose subgroup coverage varies substantially (Melki et al., 2023).

Across the broader literature, a consistent empirical pattern is reported. Standard CQR usually achieves coverage close to nominal and often shorter intervals than ordinary split conformal in heteroscedastic settings (Romano et al., 2019). Non-crossing and density-aware variants seek sharper intervals or better conditional behavior without abandoning the final conformal step (Tang et al., 2022, Lu, 2024, Chen et al., 30 Dec 2025). Localized spatial, temporal, and subgroup methods report tighter or more uniform calibration within their target structures, at the cost of added assumptions, additional modeling, or approximate rather than exact finite-sample guarantees (Jiang et al., 2024, Lin et al., 2022, Alaa et al., 2023). Multivariate generalizations replace scalar score thresholds by vector quantile regions, and constrained regimes such as privacy, shift, missingness, and online prediction modify the quantile-calibration stage itself rather than the basic conformal logic (Thurin et al., 31 Jan 2025, Romanus et al., 15 Jul 2025, Alijani et al., 26 May 2025, Zaffran et al., 2023, Rivera et al., 12 May 2026).

A plausible implication is that “quantile conformal prediction” is best understood not as a single algorithm, but as a family of conformal procedures whose defining operation is the estimation, calibration, or generalization of a quantile threshold. What remains invariant is the conformal role of that threshold: it is the interface between an adaptive score model and a coverage guarantee.