Closely Related Alternatives to Kriging Interpolation

There are close connections between several seemingly different topics; Kriging; radial basis function (RBF) interpolations, spline interpolations and the solutions of Partial Differential Equations (PDEs). Consequently, these techniques can produce families of identical interpolations. More interestingly, the connection with PDE solution methods allows the calculation of some Kriging interpolations that satisfy boundary conditions, and apply to finite domains. Periodic boundary conditions aside, finite domain interpolation is at best awkward in the Kriging or RBF views. Finally, the PDE view allows the connection to be made with such methods as signal processing and chaotic systems theory. The relationship between stochastic PDEs and Markov random field theory allows one to complete the circle, landing back in the domain of random fields and statistical estimation theory. Traversing this circle of relationships and the various branches that emanate from each of the “nodes” opens up a variety of alternative approaches to nonstationary, non-linear statistical estimation theories that generalise Kriging methods into these domains.