A geostatistical interpolation and regression technique that models a spatial or functional field as a Gaussian process, producing the best linear unbiased predictor and an associated estimate of prediction uncertainty.

Principle

Assume the unknown field is a realization of a Gaussian process with a specified mean and covariance (kernel); condition that process on observed data to compute predictive mean and variance at unobserved locations, often using covariance parameters estimated from data.

Demonstration

Given observations of a spatial variable at sample locations, choose a parametric covariance function (e.g., squared-exponential), estimate its parameters by maximum likelihood, then compute the kriging mean and kriging variance at new locations to interpolate and quantify uncertainty.

Misapplication

Using kriging with an inappropriate covariance model or without validating stationarity assumptions can yield misleading predictions and overconfident uncertainty estimates; blindly applying kriging to non-Gaussian heavy-tailed data without transformation can be inappropriate.

Consequence

Applied correctly, kriging yields smooth interpolants with principled uncertainty quantification, enables optimal linear prediction under Gaussian assumptions, and supports spatial design decisions based on predicted variance.

Reversal

The reversal is purely deterministic interpolation (e.g., inverse distance weighting or spline interpolation) that provides point estimates without a probabilistic uncertainty model and lacks formal optimality guarantees under stochastic assumptions.

Boundary

Applies when modeling tasks are compatible with Gaussian process assumptions or when linear unbiased predictors are desired; excludes settings where data are strongly nonstationary without appropriate modelling, or where computational cost of full Gaussian process inference is prohibitive without approximation.

Semantic Tension

Tension between model complexity and interpretability: flexible covariance models can fit complex dependence but risk overfitting and heavy computation; simpler kernels are interpretable and stable but may miss structure.

Synthesis

Kriging treats interpolation as probabilistic prediction under a Gaussian process prior: fit a covariance structure to observed data, compute conditional predictive distributions at targets, and report both point predictions and associated uncertainties to guide decision-making.