Kriging, RBFs and machine learning: the mathematics they share
Kriging, dual kriging, radial basis function interpolation and kernel-based machine learning all compute an estimate as a sum of weights multiplied by values.
Ordinary kriging finds weights with ω = K⁻¹k(s₀), where K holds the covariances between all samples and k(s₀) those between the estimation point and each sample. Dual kriging reverses the order: it first fits weights to the known values with ω = K⁻¹z, then estimates with e(s₀) = ωk₀. RBF interpolation has the same form plus an optional polynomial, and equals dual kriging when the polynomial is zero. Support vector machines and RBF neural networks have close variants. This is part 3 of 4.
A short recap
All interpolation methods estimate the value at an unknown location as the sum of weights multiplied by the known sample values. The methods differ in how they obtain the weights.
Inverse distance weighting
IDW weights depend on distance as 1/dp, where the power p is typically 2 for 2D data and 3 for 3D data. It is simple, but it has known limitations: it uses no model of how the data are related, and it cannot give an estimation variance.
Kriging
Kriging calculates the weights as ω = K⁻¹k(s₀) and the estimate as the vector product of the weights and the sample values. K is the covariance matrix between all samples, and k(s₀) is the vector of covariances between the unknown location and the samples.
Dual kriging and RBFs
Dual kriging changes the order of the calculation. It first fits weights to the known values, ω = K⁻¹z(s), and then estimates using those pre-calculated weights. It gives the same estimates and is much faster for repeated evaluation.
RBF interpolation follows almost identical mathematics, e(s₀) = ωk₀ + P(s), where P(s) is a polynomial. When P(s) is zero, RBF interpolation is exactly dual kriging.
Machine learning
For a support vector machine, the estimate is e(s₀) = ωk₀ + b, the RBF form with a bias constant b. A neural network built from RBF nodes calculates its predictions exactly as dual kriging does, e(s₀) = ωk₀, with weights multiplying local covariances.
| Method | Estimate | How the weights are found |
|---|---|---|
| IDW | e = Σ ω v | From 1/dp |
| Ordinary kriging | e = ω · z | ω = K⁻¹k(s₀), recalculated for each estimate |
| Dual kriging | e = ωk₀ | ω = K⁻¹z, fitted once |
| RBF interpolation | e = ωk₀ + P(s) | Fitted once, with a polynomial term |
| Support vector machine | e = ωk₀ + b | Fitted with a bias constant |
| RBF neural network | e = ωk₀ | Trained |
What this means in practice
Because the underlying maths is shared, you can switch between these methods on the same data and compare the results directly. In GEOREKA, dual ordinary kriging, RBF and SVR all use all the sample points to produce estimates, so there is no search neighbourhood to set up or tune.
Despite different terminology, these methods share fundamental mathematics. The main difference is how the weights are calculated.
Apply it
Compare estimators on one model
Use the same samples and domains to compare RBF, SVR and dual ordinary kriging, and see how much the choice of weights changes the result. Resource estimation in GEOREKA →
Common questions
Kriging, RBFs and ML mathematics FAQ
Is RBF interpolation the same as kriging?
They are mathematically very close. RBF interpolation with a zero polynomial term is the same as dual kriging, which produces the same estimates as ordinary kriging.
What is dual kriging?
Dual kriging first fits weights to the known sample values, then uses those fixed weights to estimate at any location. It is faster than standard kriging when many estimates are needed.
Do I need a search neighbourhood in GEOREKA?
No. Dual ordinary kriging, RBF and SVR in GEOREKA use all points to produce estimates, so there is no neighbourhood analysis to set up.