Does kriging always use only distance?

No. It uses a dependence model and the arrangement of all included observations.

Must its weights be positive?

No. Negative weights can occur under valid configurations.

Is its uncertainty complete?

Not necessarily. Reported variance often conditions on fitted model parameters and a particular prediction target.

AI Glossary / Statistics and probability / Advanced

Kriging

Definition

Kriging is a family of prediction methods that use a modeled covariance or variogram to combine observed values when estimating an unobserved value, often at an unsampled spatial location. Under the specified mean and dependence assumptions, kriging chooses a linear unbiased predictor with minimum prediction-error variance within the relevant class. Its weights account for relationships among observations as well as their relationships with the prediction target. It is therefore more than distance-based averaging. The meaning of best is conditional on the model and does not guarantee accurate predictions when the assumptions are wrong.

What it means

Nearby measurements often resemble one another, but two nearby observations may also provide overlapping information. Kriging represents that dependence explicitly. A cluster of nearly redundant measurements need not receive the same combined influence as independent measurements providing distinct information. Different forms handle the mean differently: simple kriging treats it as known, ordinary kriging assumes an unknown constant mean, and universal kriging accommodates a specified trend structure. Kriging is closely related to Gaussian-process prediction under matched assumptions, but the best linear unbiased prediction argument itself does not require a full Gaussian distribution.

How it works

Define the prediction target and model the mean and spatial dependence, using exploratory analysis and a valid covariance or variogram. For ordinary kriging, choose weights w that sum to one and minimize Var[wᵀZ − Z₀]. If C is the observation covariance matrix, c the observation-target covariance vector and C₀₀ the target variance, this error variance is wᵀCw − 2wᵀc + C₀₀. Solve the resulting constrained linear system. Validate predictions at held-out locations using a design that reflects intended use. The reported kriging variance usually conditions on the specified or fitted dependence model and may omit uncertainty from estimating it.

Three practical examples

Environmental mapping: estimate a measured quantity between monitoring locations while assessing spatial prediction uncertainty.
Resource characterization: combine sampled values using a defensible model of spatial dependence and a clearly defined spatial support.
Computer experiments: predict simulator responses across input settings using a kriging surrogate. The input space need not be a geographic map, but its covariance assumptions still need validation.

Think of it this way

Think of consulting a group of witnesses whose observations overlap. You should consider not only how relevant each witness is to the question, but also how much they repeat one another's information. Kriging assigns weights using that modeled information structure. It can produce negative weights in some settings, so the result is not always a conventional weighted average bounded by the observed minimum and maximum.

Worked example

Consider two observations Z₁ = 10 and Z₂ = 14 with covariance matrix C = [[1, 0.2], [0.2, 1]]. Let their covariances with a target be c = (0.6, 0.3), and let the target variance be 1. Under an unknown constant mean, ordinary kriging imposes w₁ + w₂ = 1. Minimizing the error variance gives weights 11/16 and 5/16. The prediction is (11/16)10 + (5/16)14 = 11.25. Here Cw = (0.75, 0.45), wᵀc = 0.50625 and wᵀCw = 0.65625. The prediction-error variance is therefore 1 − 2(0.50625) + 0.65625 = 0.64375. Equal weights would predict 12 with modeled error variance 0.7. The kriging weights reduce that variance under the supplied model; they do not prove that 11.25 is closer to the unknown realized target in every instance.

Illustrative case study

Fictional case study: Aspen Mapping estimates a sensor measurement across an industrial site. An initial isotropic model treats dependence as the same in every direction, but held-out predictions reveal errors near a physical barrier. The team investigates directional dependence and whether separate regions or a revised covariance model are appropriate. It also distinguishes uncertainty in the latent spatial field from noise in a future sensor reading. The final map documents those assumptions instead of presenting a visually smooth surface as a complete statement of certainty.

Common misconceptions and mistakes

Common mistakes include treating kriging as inverse-distance weighting, interpreting best linear unbiased as universally best, and confusing an error variance with guaranteed accuracy. Trend misspecification, anisotropy, barriers and unmodeled measurement error can undermine predictions. The nugget component may represent microscale variation, measurement error or a modeling choice, with different implications for the prediction target. Exact interpolation depends on the observation and noise assumptions. Spatially adjacent training and test locations can make random cross-validation overly optimistic for prediction in genuinely unsampled regions.

Business use

Businesses can use kriging to guide sampling, map operating conditions and investigate spatial uncertainty. Define whether the target is a point value, an area average or a noisy future observation, because the appropriate covariance calculations differ. Compare predictive error and uncertainty calibration with simpler alternatives. Keep records of sampling locations, support, trend assumptions and fitted dependence parameters. Where decisions depend on poorly sampled regions, targeted additional measurements may be more useful than finer display resolution; a denser map does not create new observational information.

Check your understanding

Why can the kriging prediction differ from the simple average?

Answer: The supplied target covariances favor the first observation, while the observation covariance accounts for shared information. The weights minimize the modeled error variance subject to unbiasedness, rather than giving equal influence by default.

Related terms

Gaussian process
variogram
spatial covariance
ordinary kriging
universal kriging
nugget effect
best linear unbiased prediction

Learn next

derive the ordinary-kriging constraint and compare predictions under alternative valid covariance models. Then examine spatial validation and the distinction between predicting a latent field, a noisy observation and an area average.

Sources

Sources: U.S. Geological Survey, geostatistical methods report, section 3.3, https://pubs.usgs.gov/wri/1998/4145/report.pdf. Esri, How Kriging Works, https://pro.arcgis.com/en/pro-app/3.5/tool-reference/spatial-analyst/how-kriging-works.htm. NIST statistical glossary, https://www.itl.nist.gov/div898/handbook/glossary.htm. Calculation and fictional case are original illustrations.

Previous: Krichevsky - Trofimov estimator Next: Kronecker product