Regression and Classification by Zonal Kriging

Abstract

Consider a family $Z=\{{\mathbf{x}}_{\mathbf{i}},y_i$,$1\le i\le N\}$ of $N$ pairs of vectors ${\mathbf{x}}_{\mathbf{i}}\in {\mathbb{R}}^d$ and scalars $y_i$ that we aim to predict for a new sample vector ${\mathbf{x}}_0$. Kriging models $y$ as a sum of a deterministic function $m$, a drift which depends on the point $\mathbf{x}$, and a random function $z$ with zero mean. The zonality hypothesis interprets $y$ as a weighted sum of $d$ random functions of a single independent variables, each of which is a kriging, with a quadratic form for the variograms drift. We can therefore construct an unbiased estimator $y^{\ast }({\mathbf{x}}_0)={\sum }_i{\lambda }^{i}z({\mathbf{x}}_{\mathbf{i}})$ de $y({\mathbf{x}}_0)$ with minimal variance $E[y^{\ast }({\mathbf{x}}_0)-y({\mathbf{x}}_0){]}^2$, with the help of the known training set points. We give the explicitly closed form for ${\lambda }^i$ without having calculated the inverse of the matrices.

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