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Conformal Prediction for Spatially Dependent Data via Sequential Whitening

Ayush Baran Sen, Arkajyoti Saha

arXiv:2610.10168v1stat.MLcs.LGstat.APstat.ME

TL;DR

Spatial dependence and nonrandom sampling geometry can undermine the exchangeability required for split conformal coverage, while existing spatial methods leave calibration-only predictive information unused. The paper sequentially conditions on fitting and calibration residuals; under stated conditions it restores exact coverage and yields narrower, more stable intervals, with simulations and a PM2.5 application showing practical gains and coverage-risk diagnostics.

  • Problem

    Spatial dependence and nonrandom sampling geometry can undermine score exchangeability, while existing spatial methods leave predictive information in calibration residuals unused.

    Source: §1 · p. 1, §2.2.1 · p. 4

  • Method

    The method sequentially conditions on calibration residuals as well as fitting residuals to predict and standardize calibration and target errors.

    Source: §1 · p. 2, §7 · p. 19

  • Results

    In simulations, whitening produced the narrowest, more stable intervals, and in the air-quality application it was narrowest under random folds and comparable to LSCP under whole-state holdout.

    Source: §5.2 · p. 13, §7 · p. 19

  • Takeaways & Limitations

    The method also provides a diagnostic that identified coverage-loss regions in simulations and high-risk regions in the application.

    Source: §7 · p. 20

    Exact coverage requires an elliptical residual law and correct working covariance, while efficiency and misspecification results treat the estimated covariance as fixed.

    Source: §7 · p. 20

Abstract

from arXiv · show

Split conformal prediction uses prediction errors on held-out (calibration) data to determine how wide the prediction intervals should be. It guarantees distribution-free finite-sample coverage when these errors and the error at the target site are exchangeable. This assumption may fail under spatial dependence and nonrandom sampling geometry. Existing spatial methods use fitting residuals to remove the predictable part of spatial variation from calibration and target errors. However, the spatial variation that only the calibration residuals can predict remains in both the target and calibration errors, reducing the efficiency and stability of the interval. We address this by additionally conditioning on the calibration residuals sequentially, which scales to large networks through nearest-neighbour approximations. Under a correct working covariance and an elliptical residual law, the resulting interval has exact finite-sample coverage under any spatial design, and under further conditions it is asymptotically oracle efficient. We also bound coverage loss under covariance misspecification and develop a diagnostic that identifies regions at risk of undercoverage. In simulated data, our method produces narrower and more stable intervals than global and localized state-of-the-art alternatives. In a national PM2.5 application, it produces narrower intervals within the network and identifies regions at risk of coverage failure.

1 Introduction

Spatial split conformal can lose efficiency, stability, and coverage when dependence and sampling geometry undermine exchangeability. The paper proposes sequential conditioning on fitting and calibration residuals, with exact coverage under stated covariance and residual-law conditions and improved interval performance.

2.1 Conformal prediction for independent data

  • Split conformal achieves finite-sample marginal coverage when calibration and target errors are exchangeable, without requiring a correctly specified mean model or error distribution.The paper’s setup partitions observations into fitting and calibration sets, then uses held-out residuals to calibrate the interval.Source: §1 · p. 1, §2.1 · p. 3

1 Introduction

  • Sequential conditioning uses predictive information in both fitting and calibration residuals, yielding narrower, more stable intervals and restoring score comparability when the working covariance captures residual structure.Under a correct working covariance and elliptical residuals, the method has exact finite-sample coverage under any spatial design; under additional assumptions it is asymptotically as short as the shortest conditionally valid interval.Source: §1 · p. 2

2.2 What changes for spatial data

  • Spatial dependence does not by itself rule out exchangeability: under random location sampling, calibration and target scores can remain exchangeable, whereas nonrandom geometry can make them incomparable.The paper distinguishes spatial dependence from the sampling design’s effect on exchangeability.Source: §2.2 · p. 4, §2.2.1 · p. 4, §2.2.2 · p. 5

2.2.1 When random sampling preserves exchangeability

  • Under random sampling, naive split conformal and GSCP can be valid, but GSCP leaves calibration-only spatial information unused, while naive intervals are unstable and do not adapt to target location.GSCP uses fitting residuals to predict spatial variation and adapts to prediction uncertainty, yet applies the same rule to calibration points and the target.Source: §2.2.1 · p. 4

2.2.2 When geometry breaks exchangeability

  • Under distant calibration, naive conformal and GSCP undercover; localized methods trade coverage and width, and strong localization can produce unbounded intervals.In the two-cluster example, GSCP ranks are bimodal under distant calibration, while strong localization can yield infinite intervals and the bandwidth balancing coverage and width is unknown in practice.Source: §2.2.2 · pp. 5–6, Figure 1 · p. 6

3 Method

The method sequentially predicts and standardizes calibration residuals, then the target residual, using a response-independent order. This is equivalent to triangular whitening; under stated conditions it restores score exchangeability, stabilizes thresholds, and reduces interval widths.

3 Method

  • Sequentially conditioning on calibration residuals uses their predictive information without making each residual predict itself, which would produce a degenerate score.The target is placed last and predicted from both fitting and calibration residuals.Source: §3 · pp. 6–7

3.1 Sequential prediction and standardization

  • Each calibration residual is predicted from fitting residuals and earlier calibration residuals, then its prediction is subtracted and divided by its predictive standard deviation.The target residual is predicted from both fitting and calibration residuals after a response-independent ordering.Source: §3 · p. 7, §3.1 · p. 7

3.2 Equivalence to triangular whitening

  • The sequential prediction and standardization are equivalent to triangular whitening of the ordered residual vector.Source: §3.2 · p. 8

3.3 The whitened conformal interval

  • Under an absolutely continuous elliptical residual law with the working scatter matrix, the whitened interval has finite-sample validity; the result also allows estimated mean and covariance when the conditional residual law meets the stated conditions.The estimation result requires conditional calibration and target residuals to have an absolutely continuous elliptical distribution with working kriging predictors and working conditional covariance.Source: §3.3 · p. 8

3.4 What whitening changes

  • Whitening removes additional calibration-residual correlation, makes distant-calibration GSCP ranks bimodal but whitened ranks nearly uniform, and can reduce interval width when calibration residuals add predictive information.The whitened-to-GSCP width ratio is asymptotically the ratio of their target predictive standard deviations; conditioning on calibration residuals cannot increase prediction variance.Source: §3.4 · pp. 8–9

3.5 Implementation

  • Nearest-neighbour Gaussian-process covariance approximation gives the inverse-Cholesky factor at most K + 1 nonzero entries per row, with linear time and storage.A dense working covariance factorization costs cubic time and quadratic storage and may be prohibitive for moderate to large samples.Source: §3.5 · p. 10

4 Theoretical guarantees

The section establishes asymptotic oracle efficiency under stated conditions, bounds coverage error under covariance misspecification, and gives a local diagnostic for identifying potential undercoverage.

4.1 Efficiency

  • Under a correct working covariance and the theorem’s residual-law conditions, the whitened interval width is asymptotically equal to the shortest conditionally valid oracle width.The oracle and whitened intervals share a predictive standard deviation; their multipliers differ because one uses a population quantile and the other an empirical calibration quantile.Source: §4.1 · p. 10

4.2 Coverage under covariance misspecification

  • Under covariance misspecification, coverage is within 1/(m + 1) of nominal under the exchangeable reference law, with an additional change bounded by the total-variation distance to the actual law.Applying Pinsker’s inequality yields a bound on the coverage error.Source: §4.2 · p. 11

4.3 A local coverage diagnostic

  • The relative predictive scale indicates coverage direction: approximately nominal at 1, conservative below 1, and undercoverage above 1; regional held-out errors estimate it without the true covariance.The diagnostic compares the regional root-mean-square transformed error with the conformal threshold used by the interval.Source: §4.3 · pp. 11–12

5 Simulation study

The simulations compare whitening with naive, global, and localized split conformal across random and nonrandom spatial designs. Whitening generally yields narrower, more stable intervals, while a diagnostic identifies settings at risk of undercoverage.

5 Simulation study

  • The study evaluates three simulation questions across random and nonrandom designs, comparing naive split conformal, GSCP, LSCP, and whitening on 500 independently generated datasets.The questions concern interval efficiency and stability, informative calibration residuals, and coverage failures across environments.Source: §5 · p. 12, §5.1 · pp. 12–13

5.2 Results

  • In all settings, whitening yields the narrowest intervals, 4% and 7% smaller than GSCP and LSCP, respectively, and 14–48% narrower than naive split conformal (35% average).The comparison is reported in Table 1.Source: Table 1 · p. 13, §5.2 · p. 13
  • Whitening also has the lowest CVR in every setting, tied with GSCP at ϕ = 30; its advantage over GSCP and LSCP is largest under strong dependence.Under weak dependence, all methods approach the independent-data stability benchmark.Source: §5.2 · pp. 13–14
  • Across six informative-calibration designs, whitening's width reduction relative to GSCP has correlation 0.98 with available gain; GSCP and LSCP widen as more nearby observations enter calibration.Whitening's width stays nearly unchanged because the total information in labelled observations remains constant.Source: Table 2 · p. 14, §5.2 · p. 14
  • For distant-patch calibration, GSCP undercovers and LSCP reaches nominal coverage only when unbounded intervals count as covered; whitening attains nominal coverage with the narrowest interval.Patch calibration errors are closely correlated with one another but nearly uncorrelated with the target error.Source: §5.2 · p. 15
  • Under block holdout every method undercovers; under varying range every method except LSCP undercovers, as a stationary working covariance cannot capture the abrupt range change.In block holdout, calibration sites have fitting-set neighbours but targets do not, making their predictive variances respond differently to estimated-range error.Source: §5.2 · p. 15
  • With 200 additional labelled observations from target regions, predictions differ from realized coverage by at most 0.006 in four designs where whitening maintains coverage and predict undercoverage within 2 percentage points under block holdout and varying range.Source: §5.2 · p. 16
  • In stationary nonrandom designs, all methods retain near-nominal coverage, while whitening is roughly 40% narrower than naive conformal and 6–39% narrower than GSCP and LSCP.Table 3 reports coverage and mean width for the designs.Source: §5.2 · p. 14, Table 3 · p. 17

6 Fine particulate matter monitoring network

In the PM2.5 application, whitening produced the narrowest intervals in random folds while remaining broadly near nominal coverage; in whole-state holdout, it was comparable to LSCP as all methods undercovered. A regional diagnostic tracked realized coverage across risk groups.

6.1 Data and evaluation

  • The PM2.5 network motivates spatial conformal prediction because most fine-grid cells lack monitors, while naive conformal prediction ignores correlation among nearby monitors.Reliable predictions at unmonitored locations support air-quality exposure assessment.Source: §6.1 · p. 17
  • The application analyzed 2022 annual mean PM2.5 at 723 EPA monitors; spatial dependence accounted for 67% of residual variance and had a practical range around 460 kilometres.A typical monitor had around 69 neighbours within that range.Source: §6.1 · p. 17
  • Evaluation used random monitor folds and whole-state holdouts, with monitors split into fitting and calibration sets for prediction at held-out sites.Random-fold tests held out each of 20 folds in turn; whole-state holdout predicted each of 49 regions using observations outside it.Source: §6.1 · p. 17

6.2 Random folds

  • Across random folds, whitening was broadly near nominal coverage and had intervals approximately 6% narrower than GSCP, 4% narrower than LSCP, and 18% narrower than naive split conformal.Its intervals were narrowest in every distance quintile except the nearest, where GSCP and LSCP were narrower but undercovered by seven to eight percentage points.Source: §6.2 · p. 18, Figure 5 · p. 18

6.3 Whole-state holdout

  • In whole-state holdout, all four methods fell short of nominal coverage; LSCP and whitening performed best and were practically indistinguishable in coverage and width.Calibration scores came from regions that did not always mirror the predicted region.Source: §6.3 · pp. 18–19
  • Across 19 regions, the diagnostic’s relative-scale measure had Spearman correlation −0.90 with realized coverage; estimated coverage ranged from 0.769 in the highest-risk tercile to 0.987 in the lowest.Realized coverage was within one percentage point of estimates in the lower two terciles; the highest-tercile estimate overstated it by about 2.5 points.Source: §6.3 · p. 19, Table 5 · p. 19

7 Discussion

Sequential conditioning uses calibration residuals to capture spatial information left unused by existing spatial conformal methods. Its coverage and efficiency guarantees depend on covariance and residual-law assumptions, while a diagnostic flags regions at risk under misspecification.

7 Discussion

  • Sequential conditioning on calibration residuals addresses shared spatial information left in both calibration and target residuals by existing spatial conformal methods.Under an elliptical residual law and correct working covariance, it restores score exchangeability; under additional conditions, it stabilizes the threshold and yields an interval asymptotically as short as the oracle.Source: §7 · p. 19
  • When the working covariance captures residual dependence sufficiently well, whitening has nominal coverage and shorter, more stable intervals than alternatives; in whole-state holdout it was comparable to LSCP.In the air-quality application, its random-fold width advantage over GSCP and LSCP was largest at monitors far from the fitting set.Source: §7 · p. 19
  • Covariance misspecification may cause undercoverage for whitening, GSCP, and LSCP, while the proposed diagnostic identifies regions at risk without knowing the true data-generating process.The diagnostic captured coverage loss in simulations and identified high-risk regions in the application.Source: §7 · pp. 19–20
  • Exact coverage requires an elliptical residual law and correct working covariance; efficiency and misspecification results treat the estimated covariance as fixed.Further conditions are needed for efficiency and local-coverage outcomes.Source: §7 · p. 20

Supplementary Material for

The supplement collects proofs, implementation details, and supporting numerical results. Its sequential-prediction appendix links the implemented predictors to a matrix transformation.

Supplementary Material for

  • The supplement contains technical proofs, implementation details, and supporting numerical results for the main text.It covers exact-whitening theory, imperfect whitening, the prospective relative-scale diagnostic, simulation designs, and PM2.5 data processing.Source: Supplementary Material for · p. 22

S1 Sequential prediction is triangular whitening

S2 Exact coverage after whitening

Under the theorem’s assumptions, whitening makes transformed calibration and target errors exchangeable. Their exchangeable ranks yield the finite-sample coverage guarantee, with an unbounded interval in the extreme rank case.

S2 Exact coverage after whitening

S3 Interval width and asymptotic efficiency

Whitening uses the full labelled sample to set interval width and, under the stated assumptions, is asymptotically as short as the shortest conditionally valid interval.

S3.2 Proof of asymptotic oracle efficiency

S3 Interval width and asymptotic efficiency

  • Whitening replaces GSCP’s fitting-set predictive variance with variance given the full labelled sample, and the predictive variances determine their asymptotic width ratio.It also replaces the GSCP calibration quantile with a quantile of whitened scores.Source: S3 Interval width and asymptotic… · p. 23

S3.1 Proof of the width ratio

  • Under the width-ratio proof’s assumptions, the two sets of absolute calibration scores converge to a common limit, so their ratio is 1 + oP(1).The proof uses this convergence to establish the stated width ratio.Source: S3.1 Proof of the width ratio · p. 23

S4 Coverage when whitening is imperfect

The coverage error under imperfect whitening is bounded by combining a rank-based exchangeability result with the distance between the reference and actual error laws.

S4 Coverage when whitening is imperfect

  • Under the exchangeable reference law, coverage is within 1/(m + 1) of nominal; replacing it with the actual law changes the probability by at most total variation distance.The total variation bound applies to the coverage event.Source: S4 Coverage when whitening is imperfect · p. 25

S5 Relative scale and the prospective coverage diagnostic

The relative predictive scale characterizes expected coverage, and the paper turns it into a residual-based diagnostic for regions at risk of undercoverage.

S5 Relative scale and the prospective coverage diagnostic

  • The coverage-error bound requires the true covariance to evaluate, so Theorem 4 instead identifies regions at risk of undercoverage using residuals alone.The diagnostic estimates regional scale and compares it with the conformal threshold actually used.Source: S5 Relative scale and the prospective… · p. 25
  • The relative predictive scale characterizes expected coverage: values above one indicate possible undercoverage, while values below one predict conservative coverage.A value of one corresponds to approximately nominal coverage.Source: S5 Relative scale and the prospective… · p. 27

S5.1 Prospective implementation of the diagnostic

  • The prospective diagnostic estimates relative scale from held-out observations in the target region, transforming each as a target given the fitting and calibration data.These diagnostic observations are not used for fitting, calibration, or evaluation.Source: S5.1 Prospective implementation of the… · pp. 27–28
  • When diagnostic and target errors share a scale and the diagnostic errors meet the stated correlation condition, the estimated coverage is consistent; varying scales yield a regional average approximation.The regional estimate uses the root-mean-square of the diagnostic transformed errors.Source: S5.1 Prospective implementation of the… · p. 28

S6 Simulation designs and implementation choices

The simulations vary sampling geometry and implementation choices; whitening has the narrowest intervals in the reported spatial t settings, with factorization choices producing nearly identical results.

S6.1 Data-generating mechanisms

S6.1.1 Regime B: predictive information in calibration

S6.1.2 Regime C: calibration–target comparability

  • Regime C varies calibration–target comparability through designs including quadrant, clustered calibration, block holdout, varying range, disjoint halves, and distant patch.In clustered calibration, the calibration locations are nearest a point from which targets are drawn among the farthest locations.Source: S6.1.2 Regime C: calibration–target… · p. 29

S6.2.1 Software and covariance estimation

  • All methods fit the mean with ranger using 500 trees; whitening estimates exponential working-covariance parameters with BRISC_estimation and 15 nearest neighbours.GSCP and LSCP use the covariance-estimation routine supplied with scp.Source: S6.2.1 Software and covariance… · p. 29

S6.2.2 Choice of LSCP bandwidth

S6.2.3 Factorization and ordering

  • Across the nine Regime A designs, five whitening configurations using different orderings or factorization give nearly identical coverage and width.The alternatives include a fresh random permutation, two structured orderings, and an NNGP factor with K = 25.Source: S6.2.3 Factorization and ordering · p. 30, Table S2 · p. 31

S6.2.4 Departure from Gaussianity

S7 Fine particulate matter monitoring network data processing

The section describes the PM2.5 monitor data, covariates, and coordinate processing used in the analysis.

S7 Fine particulate matter monitoring network data processing

  • The dataset uses 2022 annual mean PM2.5 measurements from monitors with at least 250 daily observations, retaining one monitor per site in the contiguous United States and three agency-listed Mexico monitors.Monitors were selected from the Environmental Protection Agency’s Air Quality System.Source: S7 Fine particulate matter monitoring… · p. 31
  • Monitor covariates include elevation, categorical land use, urban, suburban, or rural setting, annual-mean wind speed, and 2020 population density.Land use and location setting come from the agency’s site listing; wind speed and population density come from TerraClimate and WorldPop, respectively.Source: S7 Fine particulate matter monitoring… · p. 31
  • Coordinates are projected to Albers equal-area planar kilometres, divided by 4547 km to fit the unit square, and LSCP uses bandwidth h = 0.40.The projection addresses distortion in raw latitude-longitude distances; scaling matches the simulations.Source: S7 Fine particulate matter monitoring… · p. 31
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