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A soil map can look precise while its uncertainty is only calibrated for places that resemble the sampled locations. This guide separates marginal coverage, local coverage and extrapolation risk.
Open the chapter contents · Soil Science root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Build the interval
Route 2 · Work the calibration
Route 3 · Stress the map
Route 4 · Check limits and evidence
Route 5 · Transfer and continue
CHAPTER 1 OF 12 · BUILD THE INTERVAL
1. A prediction interval is a coverage promise, not a decoration
A digital soil map often predicts a property such as organic carbon, clay fraction or bulk density at every grid cell. A point prediction does not show how uncertain the map is. Conformal prediction can wrap a fitted model with data-driven intervals, but its clean finite-sample guarantee depends on an exchangeability condition that spatial data may not satisfy.
The practical question is therefore two-part: how was the interval calibrated, and where should its coverage promise apply? A national average coverage rate can coexist with severe undercoverage in remote soil–landscape combinations.
This article extends the Soil Science root, spatial cross-validation and buffer distance and spatial block bootstrap coverage calibration.
CHAPTER 2 OF 12 · BUILD THE INTERVAL
2. Split conformal turns calibration residuals into an interval
Fit the soil model on a training set. On a separate calibration set, compute nonconformity scores such as absolute residuals R_j = |Y_j – f(X_j)|. For a target miscoverage level alpha and n calibration points, choose the empirical score at rank ceiling[(n+1)(1-alpha)]. A new point prediction f(X_new) receives the interval f(X_new) plus or minus that score.
Under exchangeability of calibration and test examples, this construction gives marginal finite-sample coverage at least 1-alpha, subject to the usual tie convention. It does not require the regression model to be correct. It also does not promise that every region, soil class or individual grid cell achieves the same coverage.
Spatial dependence and uneven monitoring challenge exchangeability. Nearby calibration points can be easier than distant deployment cells; road-accessible farms can be overrepresented; and covariates in an unmonitored landscape may lie outside the training support.
CHAPTER 3 OF 12 · WORK THE CALIBRATION
3. Worked example: the finite-sample rank is visible
Suppose a spatially separated calibration design yields twelve absolute residuals, already sorted:
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute residual | 0.10 | 0.12 | 0.14 | 0.15 | 0.17 | 0.18 | 0.20 | 0.22 | 0.25 | 0.28 | 0.35 | 0.48 |
For alpha = 0.10 and n = 12, the rank is ceiling(13 × 0.90) = 12. The conformal radius is therefore 0.48. A point prediction of 1.25 receives [0.77, 1.73]. The small calibration set makes the 90-percent interval use the maximum score.
That apparent conservatism is finite-sample arithmetic. Replacing the twelfth score with the eleventh because it looks less extreme would change the coverage contract.
CHAPTER 4 OF 12 · WORK THE CALIBRATION
4. A random split and a spatial split answer different deployment questions
If calibration points are randomly interleaved with nearby training points, residuals may be small because the model has local support. A spatially blocked or buffered calibration set tests greater separation and may produce larger scores.
| Calibration design | 90% radius in teaching ledger | Near-sampled audit coverage | Remote-zone audit coverage |
|---|---|---|---|
| Randomly interleaved | 0.18 | 0.92 | 0.68 |
| Spatially separated | 0.48 | 0.91 | 0.87 |
| Weighted for declared shift | 0.44 | 0.90 | 0.89 |
The wider spatial interval is not automatically better. It is better aligned only if the deployment task resembles the spatial separation used in calibration. The split must match the prediction question.
CHAPTER 5 OF 12 · WORK THE CALIBRATION
5. Coverage maps reveal where the promise is weak
A coverage map compares held-out observations with their prediction intervals and aggregates the hit rate by region, environmental stratum or distance-to-sample band. It should also show the number of audit observations and interval width; a rate from three points is not comparable with a rate from three hundred.
Map at least three quantities together: empirical coverage, median interval width and support or dissimilarity. A method can reach nominal coverage by making remote intervals extremely wide. Another can produce narrow intervals that fail systematically outside sampled landscapes.
Do not tune the method repeatedly on the same final audit map. That turns the audit into training. Keep a final spatial holdout or use nested resampling when method selection is extensive.
CHAPTER 6 OF 12 · STRESS THE MAP
6. Covariate shift changes which calibration scores matter
Covariate shift means the distribution of predictors differs between calibration and deployment while the conditional outcome mechanism is assumed stable. Weighted conformal prediction can reweight calibration scores by a test-to-training density ratio. Tibshirani, Barber, Candès and Ramdas showed how weighted exchangeability extends conformal prediction under such shift.
The ratio must be known or estimated accurately enough. In soil mapping, remote predictor combinations may have almost no calibration support, causing very large or unstable weights. A formal interval can then become extremely wide or effectively uninformative.
Group-weighted conformal prediction can improve guarantees when a finite set of known groups explains the sampling shift. But a group label is not magic: the grouping must correspond to the selection mechanism and retain enough calibration data.
CHAPTER 7 OF 12 · STRESS THE MAP
7. Changed context: fixed monitoring locations
If soil observations come from fixed stations rather than locations sampled at random, global exchangeability usually does not hold. Mao, Martin and Reich developed model-free spatial prediction methods based on exact or approximate local exchangeability. Their global method targets marginal validity when locations are random; their local method provides approximate validity under fixed locations and broad spatial processes.
The word approximate matters. Local neighbourhood size creates a bias–variance trade-off. A tiny neighbourhood may contain too few calibration values; a large one may mix different soil–landscape regimes.
Record the sampling design, not only coordinates. Two maps with identical points can have different inferential meanings if one design was probability-based and the other followed roads or voluntary farms.
CHAPTER 8 OF 12 · STRESS THE MAP
8. Changed context: extrapolation beyond environmental support
Consider a peat landscape when the calibration set contains mainly mineral soils. The model may output a number, and split conformal may attach a familiar-width interval, yet neither action creates support. Coverage under ordinary exchangeability does not guarantee local validity in an unseen covariate region.
Add an area-of-applicability or dissimilarity layer. Flag cells where predictor combinations are far from training support. Treat unbounded or very wide weighted intervals as honest warnings rather than software failures.
The transfer check is to ask whether the target cell could plausibly have appeared in the calibration sample. If not, the map needs an extrapolation label, new sampling or a narrower claim.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Soil-model uncertainty is more than residual calibration
Conformal intervals can absorb many sources of predictive error if the calibration sample represents them, but they do not identify the cause. Laboratory error, depth mismatch, temporal change, geolocation error and covariate uncertainty can all enter the residuals.
Coverage of a scalar property also differs from simultaneous coverage of an entire map. A 90-percent marginal interval at each cell does not imply a 90-percent chance that every cell is covered at once. Multiple-property and functional targets need their own score and guarantee.
Keep unit conversions and physical bounds visible. Truncating a conformal interval at zero may be scientifically sensible for some properties, but it changes finite-sample coverage unless the transformation and score are constructed accordingly.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Limits and misconceptions
Distribution-free does not mean assumption-free. Exchangeability or a justified weighted alternative is the key assumption. Spatial blocking does not automatically restore it; blocking is a design choice that can make the audit more deployment-like.
Nominal marginal coverage is not conditional coverage at every location. A coverage map is a diagnostic, not a new theorem. Estimated covariate-shift weights can be noisy, especially in areas with weak overlap.
Finally, a wider interval is not automatically more calibrated, and a narrow interval is not automatically more useful. Report coverage and width together, stratified by meaningful spatial and environmental groups.
CHAPTER 11 OF 12 · TRANSFER AND CONTINUE
11. Evidence and verification route
Mao, Martin and Reich present Valid model-free spatial prediction, including global and local spatial conformal methods. Tibshirani, Barber, Candès and Ramdas provide Conformal Prediction Under Covariate Shift. Roberts and colleagues explain why spatial structure changes resampling design in Cross-validation strategies for data with temporal, spatial, hierarchical or phylogenetic structure.
Verification should reproduce the calibration rank, retain fold and buffer definitions, map coverage and width by support class, and keep a final untouched spatial audit. Save the score definition, alpha, tie rule, random seed and all density-ratio or group-weight calculations.
CHAPTER 12 OF 12 · TRANSFER AND CONTINUE
12. Frequently asked questions
Does conformal prediction make any soil model valid?
It can calibrate predictive intervals under its sampling assumptions; it does not repair a meaningless target, biased measurements or unsupported extrapolation.
Why can overall coverage be good while one region is poor?
Marginal coverage averages across the target distribution. Overcoverage in common regions can hide undercoverage in rare ones.
Should every map use spatial blocking?
Use a design that matches deployment. Blocking and buffers are tools for creating relevant separation, not universal rituals.
What if covariate-shift weights are extreme?
Treat that as weak overlap. Stabilisation changes the guarantee, so report it; new sampling or a narrower mapping domain may be more honest.
Final transfer check and connected reading
Before trusting a coverage map, answer: What is the target population of grid cells? How were training, calibration and audit locations sampled? Is the guarantee marginal, local, weighted or approximate? Where is covariate support weak? Do coverage and interval width remain acceptable in those regions?
Continue through the Soil Science root, spatial cross-validation and spatial block bootstrap calibration. The transferable habit is to connect every interval to a sampling design, a target region and an auditable coverage claim.
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