##  [Kriging](/kriging-1) 

 Definition

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

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

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

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

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

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

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

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

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.